diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/FINAL_REPORT.md b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/FINAL_REPORT.md
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@@ -0,0 +1,38 @@
+# Information-management RL v1 — final decision
+
+## Decision
+
+**Rejected. `verified-math-a025` remains the leader.** The public root was not modified and model size stayed at 487,800,064 parameters.
+
+## Work completed
+
+- 105 manually specified long multi-turn cases: 63 train, 21 dev, 21 held-out test.
+- Seven skills: current-state updates, stale-step avoidance, selective retrieval, delta summaries, conflict clarification, source boundaries, and accumulated constraints.
+- 252 stochastic Budgie rollouts and 63 manual ideal demonstrations.
+- Qwen3.8-27B group rewards with explicit stale-fact, missing-state, format, honesty, and repetition penalties.
+- Offline clipped token-level GRPO/PPO with immutable-leader KL.
+- A longer three-epoch GRPO run plus manual-demonstration behavior cloning.
+- An eight-epoch high-quality ideal-response consolidation after RL.
+
+## Held-out information-management results
+
+| Model | Dev pass | Dev loops | Test pass | Test loops |
+|---|---:|---:|---:|---:|
+| leader | 3/21 | 0 | 1/21 | 1 |
+| hybrid alpha .025 | 4/21 | 0 | 2/21 | 1 |
+| full hybrid source | 7/21 | 2 | 5/21 | 2 |
+
+The full source learns conflict clarification and some selective/source-boundary behavior, but increases repetition and still fails all held-out current-state-update cases. The small alpha gives a modest no-loop transfer gain.
+
+## Protected reasoning gate
+
+| Model | GSM /30 | MATH /15 | ARC /30 | FOLIO /30 |
+|---|---:|---:|---:|---:|
+| leader | 5 | 3 | 11 | 13 |
+| hybrid alpha .025 | 3 | 1 | 11 | 13 |
+
+The only low-drift information candidate regresses GSM and MATH sharply, so it is rejected before HelpSteer/expanded gates. Smaller alphas do not improve the information holdout.
+
+## Conclusion
+
+Manual long-form RL produced measurable specialist behavior, but not a Pareto-safe general assistant. Do not promote or continue from this branch. The durable curriculum, rewards, rollouts, code, reports, and one reconstructable hybrid source are retained for research; future work must restart from `verified-math-a025`.
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/INFORMATION_RL_POLICY.md b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/INFORMATION_RL_POLICY.md
new file mode 100644
index 0000000000000000000000000000000000000000..bf30470fd30d730317e5e29b1d11c8b2ac914ff1
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/INFORMATION_RL_POLICY.md
@@ -0,0 +1,65 @@
+# Budgie information-management RL v1
+
+## Objective
+
+Improve long English dialogue without increasing Budgie's 487,800,064 parameters. The target behavior is to maintain a compact internal conversation state: use the newest correction, retain active constraints, answer only the requested slice of information, distinguish supplied facts from unknown facts, ask one focused clarification on unresolved conflicts, avoid repeating attempted actions, and stop after a complete answer.
+
+Immutable base: `verified-math-a025` / `ba2_vmath_a025`, SHA-256 `7404de2c08df968f41800d1362704a0ad4106ea0116cd0cd4b1709612a2a66be`.
+
+## Manual curriculum
+
+Seven independently scored skills:
+
+1. state updates and stale-fact suppression;
+2. avoiding previously attempted troubleshooting steps;
+3. selective retrieval from distracting context;
+4. delta-only summaries after updates;
+5. focused clarification under conflicting records;
+6. source boundaries and explicit uncertainty;
+7. accumulated exact constraints.
+
+There are 105 manually specified multi-turn cases: 63 train, 21 development, and 21 held-out test. Topic instances are disjoint across splits. Every case contains an ideal response, explicit required/forbidden state anchors, format checks, and repetition checks.
+
+## Reward
+
+Each rollout receives:
+
+- deterministic state/constraint reward;
+- stale-information penalties;
+- missing-current-fact penalties;
+- repetition and duplicate-sentence penalties;
+- canned-closing penalty;
+- Qwen3.8-27B semantic scores for state accuracy, instruction following, relevance, honesty, and non-repetition;
+- a hard cap when an explicit current fact or format constraint is violated.
+
+The semantic judge sees the full conversation, ideal reference, checks, and all candidate answers at once. Qwen thinking is disabled. Candidate order is rotated.
+
+## RL algorithm
+
+Use offline one-epoch clipped group-relative policy optimization:
+
+- four stochastic Budgie rollouts per train case plus one verified teacher demonstration;
+- rewards normalized within each case;
+- token-level PPO clipping against stored leader log-probabilities;
+- sampled-action KL penalty to the immutable leader;
+- entropy regularization;
+- FP32 master weights and BF16 autocast;
+- gradient clipping;
+- protected random/synthetic reasoning replay;
+- conservative updates and task-vector interpolation.
+
+This is not DPO. Strong and weak DPO previously failed; the new run uses explicit per-rollout scalar rewards, group advantages, PPO clipping, and leader KL.
+
+## Stop rules
+
+A candidate is rejected immediately if any of these occur:
+
+- held-out information-management reward or pass rate does not improve;
+- repetition-loop rate rises;
+- honesty/source-boundary pass rate falls;
+- fixed GSM8K, MATH-500, ARC-Challenge, FOLIO, or HelpSteer2 falls below `5/30`, `3/15`, `11/30`, `13/30`, or `97/200`;
+- the 36-prompt dialogue gate loses honesty or increases loops;
+- parameters, tokenizer, or architecture change;
+- NaN/Inf or malformed weights appear.
+
+Only a candidate passing the fixed gate may run the expanded/full gate. Rejected branches never become the next base, and the public root is not overwritten automatically.
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/MATH_AUGMENTED_RL_POLICY.md b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/MATH_AUGMENTED_RL_POLICY.md
new file mode 100644
index 0000000000000000000000000000000000000000..abcc8ae192effd2bc11951b046979dbd5e7892e8
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/MATH_AUGMENTED_RL_POLICY.md
@@ -0,0 +1,13 @@
+# Budgie information + mathematics RL v2
+
+Start from `verified-math-a025`, not from the rejected information specialist.
+
+Joint groups:
+
+- 63 manual long-dialogue information-management groups retained from v1;
+- 96 random, decontaminated, symbolically verified mathematics problems from `open-r1/OpenR1-Math-220k`;
+- no GSM8K, MATH-500, Hendrycks MATH, ARC, or FOLIO training rows.
+
+Each math group contains stochastic Budgie solutions plus a verified reference solution. Rewards are +1 for symbolic equivalence, negative for incorrect/malformed/repetitive solutions, and group-normalized. The policy objective is clipped token-level GRPO/PPO with KL to the immutable leader. Information and math groups are mixed at no less than 40:60, so dialogue gradients cannot dominate protected reasoning. Manual ideal responses and verified math references may receive a small behavior-cloning anchor.
+
+Candidates must improve held-out information-management tests without increasing loops, and must preserve at least GSM 5/30, MATH 3/15, ARC 11/30, FOLIO 13/30, and HelpSteer 97/200 before expanded evaluation. Model size and tokenizer remain unchanged.
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/STAGE1_REPORT.md b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/STAGE1_REPORT.md
new file mode 100644
index 0000000000000000000000000000000000000000..32933431a330da2b1424bc477dc3101a50164413
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/STAGE1_REPORT.md
@@ -0,0 +1,13 @@
+# Information-management RL v1 — preliminary decision
+
+A manual long-dialogue curriculum of 105 cases was built across state updates, stale-step avoidance, selective retrieval, delta summaries, conflict clarification, source boundaries, and accumulated constraints. Sixty-three train cases produced 252 stochastic Budgie rollouts plus 63 manual ideal demonstrations. Qwen3.8-27B assigned group rewards. Training used clipped token-level GRPO/PPO with leader KL, followed by a longer GRPO run and manual high-quality consolidation. Model size stayed unchanged.
+
+## Held-out deterministic gate
+
+| Model | Dev pass | Dev loops | Test pass | Test loops |
+|---|---:|---:|---:|---:|
+| leader | 3/21 | 0 | 1/21 | 1 |
+| hybrid alpha 0.025 | 4/21 | 0 | 2/21 | 1 |
+| full hybrid source | 7/21 | 2 | 5/21 | 2 |
+
+The small alpha has a modest transfer gain without extra loops. The full source improves conflict handling and selective retrieval but increases repetitions. Neither is promoted yet. `hybrid-a0p025` must pass fixed GSM/MATH/ARC/FOLIO/HelpSteer and Qwen pairwise dialogue gates before retention.
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/cases.jsonl b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/cases.jsonl
new file mode 100644
index 0000000000000000000000000000000000000000..e5879359ad951a60fd80177d07d568650584563f
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/cases.jsonl
@@ -0,0 +1,105 @@
+{"id": "state_update_00", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this community workshop. The initial record is Saturday at 10:00; Room Cedar."}, {"role": "assistant", "content": "Recorded the initial community workshop details."}, {"role": "user", "content": "Correction: replace Saturday at 10:00 with Sunday at 14:00. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Give the final time and room in one sentence."}], "ideal": "The community workshop is Sunday at 14:00 in Room Cedar.", "required": ["Sunday", "14:00", "Room Cedar"], "forbidden": ["Saturday at 10:00"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_01", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this software release. The initial record is version 3.4 on May 8; staged rollout."}, {"role": "assistant", "content": "Recorded the initial software release details."}, {"role": "user", "content": "Correction: replace version 3.4 on May 8 with version 3.5 on May 10. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "State the current version, date, and rollout method in one sentence."}], "ideal": "Version 3.5 will launch on May 10 through a staged rollout.", "required": ["3.5", "May 10", "staged rollout"], "forbidden": ["version 3.4 on May 8"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_02", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this supplier order. The initial record is 240 blue folders; deliver to Dock 2."}, {"role": "assistant", "content": "Recorded the initial supplier order details."}, {"role": "user", "content": "Correction: replace 240 blue folders with 180 green folders. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Write the final order instruction in one sentence."}], "ideal": "Order 180 green folders and deliver them to Dock 2.", "required": ["180", "green folders", "Dock 2"], "forbidden": ["240 blue folders"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_03", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this interview schedule. The initial record is Mira at 09:30; video call."}, {"role": "assistant", "content": "Recorded the initial interview schedule details."}, {"role": "user", "content": "Correction: replace Mira at 09:30 with Jon at 11:15. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Give only the current candidate, time, and format."}], "ideal": "Jon at 11:15 by video call.", "required": ["Jon", "11:15", "video call"], "forbidden": ["Mira at 09:30"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_04", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this museum visit. The initial record is 12 adult tickets; entry at 16:20."}, {"role": "assistant", "content": "Recorded the initial museum visit details."}, {"role": "user", "content": "Correction: replace 12 adult tickets with 9 adult tickets and 3 child tickets. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Summarize the final booking in one sentence."}], "ideal": "Book 9 adult tickets and 3 child tickets for entry at 16:20.", "required": ["9 adult", "3 child", "16:20"], "forbidden": ["12 adult tickets"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_05", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this newsletter campaign. The initial record is send Tuesday to 8,000 readers; subject line: Spring Notes."}, {"role": "assistant", "content": "Recorded the initial newsletter campaign details."}, {"role": "user", "content": "Correction: replace send Tuesday to 8,000 readers with send Thursday to 6,500 readers. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Give the current send plan in one sentence."}], "ideal": "Send “Spring Notes” on Thursday to 6,500 readers.", "required": ["Spring Notes", "Thursday", "6,500"], "forbidden": ["send Tuesday to 8,000 readers"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_06", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this lab sample. The initial record is label K-17 at 4°C; amber container."}, {"role": "assistant", "content": "Recorded the initial lab sample details."}, {"role": "user", "content": "Correction: replace label K-17 at 4°C with label K-19 at -20°C. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "State the final storage record in one sentence."}], "ideal": "Store sample K-19 at -20°C in an amber container.", "required": ["K-19", "-20", "amber"], "forbidden": ["label K-17 at 4°C"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_07", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this training session. The initial record is 90 minutes for 25 people; beginner level."}, {"role": "assistant", "content": "Recorded the initial training session details."}, {"role": "user", "content": "Correction: replace 90 minutes for 25 people with 60 minutes for 18 people. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Give the final duration, audience size, and level."}], "ideal": "Run a 60-minute beginner session for 18 people.", "required": ["60-minute", "18 people", "beginner"], "forbidden": ["90 minutes for 25 people"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_08", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this catering request. The initial record is vegetarian lunch for 40; no peanuts."}, {"role": "assistant", "content": "Recorded the initial catering request details."}, {"role": "user", "content": "Correction: replace vegetarian lunch for 40 with vegan lunch for 32. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "State the current catering requirement in one sentence."}], "ideal": "Provide a peanut-free vegan lunch for 32 people.", "required": ["vegan", "32", "peanut-free"], "forbidden": ["vegetarian lunch for 40"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_09", "split": "dev", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this archive transfer. The initial record is Box A7 to Shelf 4; scan the index first."}, {"role": "assistant", "content": "Recorded the initial archive transfer details."}, {"role": "user", "content": "Correction: replace Box A7 to Shelf 4 with Box C2 to Shelf 9. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Give the current transfer instruction only."}], "ideal": "Scan the index, then move Box C2 to Shelf 9.", "required": ["scan the index", "Box C2", "Shelf 9"], "forbidden": ["Box A7 to Shelf 4"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_10", "split": "dev", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this podcast recording. The initial record is episode 18 on Wednesday; guest: Dr. Vale."}, {"role": "assistant", "content": "Recorded the initial podcast recording details."}, {"role": "user", "content": "Correction: replace episode 18 on Wednesday with episode 21 on Friday. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "State the latest recording plan in one sentence."}], "ideal": "Record episode 21 with Dr. Vale on Friday.", "required": ["episode 21", "Dr. Vale", "Friday"], "forbidden": ["episode 18 on Wednesday"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_11", "split": "dev", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this bike rental. The initial record is two city bikes for 3 hours; pickup at North Gate."}, {"role": "assistant", "content": "Recorded the initial bike rental details."}, {"role": "user", "content": "Correction: replace two city bikes for 3 hours with three cargo bikes for 2 hours. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Give the final rental details in one sentence."}], "ideal": "Rent three cargo bikes for two hours, with pickup at North Gate.", "required": ["three cargo bikes", "two hours", "North Gate"], "forbidden": ["two city bikes for 3 hours"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_12", "split": "test", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this library display. The initial record is history books in June; second-floor lobby."}, {"role": "assistant", "content": "Recorded the initial library display details."}, {"role": "user", "content": "Correction: replace history books in June with science magazines in July. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "State the final display plan without mentioning the old one."}], "ideal": "Display science magazines in the second-floor lobby in July.", "required": ["science magazines", "July", "second-floor lobby"], "forbidden": ["history books in June"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_13", "split": "test", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this garden delivery. The initial record is six rose bushes on Monday; use the east entrance."}, {"role": "assistant", "content": "Recorded the initial garden delivery details."}, {"role": "user", "content": "Correction: replace six rose bushes on Monday with four maple saplings on Wednesday. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Give the updated delivery instruction in one sentence."}], "ideal": "Deliver four maple saplings on Wednesday through the east entrance.", "required": ["four maple saplings", "Wednesday", "east entrance"], "forbidden": ["six rose bushes on Monday"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "state_update_14", "split": "test", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this translation job. The initial record is French, 1,200 words, formal; deadline 18:00 UTC."}, {"role": "assistant", "content": "Recorded the initial translation job details."}, {"role": "user", "content": "Correction: replace French, 1,200 words, formal with Spanish, 900 words, conversational. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Give only the current language, length, tone, and deadline."}], "ideal": "Translate 900 words into conversational Spanish by 18:00 UTC.", "required": ["900 words", "Spanish", "conversational", "18:00 UTC"], "forbidden": ["French, 1,200 words, formal"], "checks": {"max_words": 30, "no_canned_closing": true}}
+{"id": "already_tried_00", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a laptop that connects to Wi-Fi but cannot reach websites. So far I have already restarted the laptop; reconnected to the network. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Renew the dhcp lease.\n2. Test with a different dns resolver.", "required": ["renew the DHCP lease", "test with a different DNS resolver"], "forbidden": ["restarted the laptop", "reconnected to the network"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_01", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a printer producing faint pages. So far I have already replaced the paper; restarted the printer. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Check the ink or toner level.\n2. Run the print-head cleaning utility.", "required": ["check the ink or toner level", "run the print-head cleaning utility"], "forbidden": ["replaced the paper", "restarted the printer"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_02", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a Python service returning a port-in-use error. So far I have already restarted the service; changed the browser. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Identify the process holding the port.\n2. Stop it or configure a different port.", "required": ["identify the process holding the port", "stop it or configure a different port"], "forbidden": ["restarted the service", "changed the browser"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_03", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a phone that will not charge. So far I have already rebooted the phone; tried another wall outlet. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Inspect and gently clean the charging port.\n2. Test a known-good cable and adapter.", "required": ["inspect and gently clean the charging port", "test a known-good cable and adapter"], "forbidden": ["rebooted the phone", "tried another wall outlet"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_04", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a spreadsheet with formulas showing as text. So far I have already reopened the file; changed the font. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Set the cells to general format.\n2. Re-enter each formula beginning with an equals sign.", "required": ["set the cells to General format", "re-enter each formula beginning with an equals sign"], "forbidden": ["reopened the file", "changed the font"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_05", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a web page whose new CSS is not visible. So far I have already refreshed the page; restarted the editor. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Perform a hard refresh or clear the browser cache.\n2. Verify that the edited stylesheet is the one loaded by the page.", "required": ["perform a hard refresh or clear the browser cache", "verify that the edited stylesheet is the one loaded by the page"], "forbidden": ["refreshed the page", "restarted the editor"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_06", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a Bluetooth keyboard that pairs but sends no keystrokes. So far I have already replaced the batteries; paired it again. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Remove conflicting keyboard profiles.\n2. Test the keyboard on another device.", "required": ["remove conflicting keyboard profiles", "test the keyboard on another device"], "forbidden": ["replaced the batteries", "paired it again"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_07", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a Git push rejected as non-fast-forward. So far I have already retried the push; checked the internet connection. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Fetch and inspect the remote changes.\n2. Rebase or merge before pushing again.", "required": ["fetch and inspect the remote changes", "rebase or merge before pushing again"], "forbidden": ["retried the push", "checked the internet connection"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_08", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a microwave that powers on but does not heat. So far I have already used a different outlet; changed the timer. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Stop using it and check the door latch for visible damage.\n2. Contact a qualified repair technician.", "required": ["stop using it and check the door latch for visible damage", "contact a qualified repair technician"], "forbidden": ["used a different outlet", "changed the timer"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_09", "split": "dev", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with headphones playing sound on only one side. So far I have already restarted the music app; raised the volume. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Check the device balance setting.\n2. Test the headphones with another source.", "required": ["check the device balance setting", "test the headphones with another source"], "forbidden": ["restarted the music app", "raised the volume"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_10", "split": "dev", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a calendar invitation showing the wrong time zone. So far I have already reopened the invitation; changed the event color. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Verify the calendar account time zone.\n2. Recreate the event with an explicit zone.", "required": ["verify the calendar account time zone", "recreate the event with an explicit zone"], "forbidden": ["reopened the invitation", "changed the event color"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_11", "split": "dev", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a Docker container exiting immediately. So far I have already restarted Docker; pulled the image again. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Inspect the container logs.\n2. Run the image with its entrypoint overridden for diagnosis.", "required": ["inspect the container logs", "run the image with its entrypoint overridden for diagnosis"], "forbidden": ["restarted Docker", "pulled the image again"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_12", "split": "test", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a camera whose photos look unexpectedly dark. So far I have already charged the battery; cleaned the lens. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Check exposure compensation.\n2. Test in aperture-priority mode with a wider aperture.", "required": ["check exposure compensation", "test in aperture-priority mode with a wider aperture"], "forbidden": ["charged the battery", "cleaned the lens"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_13", "split": "test", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with an e-reader that cannot open one book. So far I have already restarted the e-reader; increased screen brightness. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Remove and download the book again.\n2. Verify that its format is supported and not corrupted.", "required": ["remove and download the book again", "verify that its format is supported and not corrupted"], "forbidden": ["restarted the e-reader", "increased screen brightness"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "already_tried_14", "split": "test", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a smart speaker that hears commands but plays no audio. So far I have already restarted the speaker; raised the spoken volume. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Check whether audio is routed to another device.\n2. Unlink and relink the music service.", "required": ["check whether audio is routed to another device", "unlink and relink the music service"], "forbidden": ["restarted the speaker", "raised the spoken volume"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}
+{"id": "selective_00", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The Cedar project owner is Nila. Its budget is €18,400, its review is on 12 June, and the prototype color is orange."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "What is the review date? Answer with the date only."}], "ideal": "12 June", "required": ["12 June"], "forbidden": ["Nila", "18,400", "orange"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_01", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Order RX-4 contains 36 valves, ships by rail, arrives at Terminal B, and is insured for $9,000."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "Which terminal receives the order? Answer with the terminal only."}], "ideal": "Terminal B", "required": ["Terminal B"], "forbidden": ["36", "rail", "9,000"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_02", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Leah chose the elm desk, Omar chose the glass table, Priya chose the oak shelf, and Zane chose the steel cabinet."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "What did Priya choose? Give only the item."}], "ideal": "oak shelf", "required": ["oak shelf"], "forbidden": ["elm", "glass", "steel"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_03", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The rehearsal starts at 17:40 in Studio 3. Doors open at 17:10, the conductor is Ivo, and parking closes at 22:00."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "Where is the rehearsal? Answer with the location only."}], "ideal": "Studio 3", "required": ["Studio 3"], "forbidden": ["17:40", "Ivo", "22:00"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_04", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Dataset Lumen has 8,200 rows, uses a CC BY 4.0 license, was updated in March, and has 14 columns."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "What license does Dataset Lumen use? Give only the license."}], "ideal": "CC BY 4.0", "required": ["CC BY 4.0"], "forbidden": ["8,200", "March", "14"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_05", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The blue key opens the west cabinet, the brass key opens the archive, and the small black key opens the equipment locker."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "Which key opens the archive? Answer with the key only."}], "ideal": "brass key", "required": ["brass key"], "forbidden": ["blue", "black", "west"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_06", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Flight 608 departs at 06:25 from Gate 11, boards at 05:50, and allows one 8 kg cabin bag."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "What is the departure gate? Give only the gate."}], "ideal": "Gate 11", "required": ["Gate 11"], "forbidden": ["06:25", "05:50", "8 kg"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_07", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The Orion room seats 18, the Maple room seats 10, and the Harbor room seats 26. Only Harbor has a projector."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "Which room has a projector? Give only its name."}], "ideal": "Harbor", "required": ["Harbor"], "forbidden": ["18", "10", "26"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_08", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Invoice 771 is due on 4 August, totals £620, belongs to Kestrel Ltd, and uses purchase order PO-19."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "What is the purchase-order code? Return only the code."}], "ideal": "PO-19", "required": ["PO-19"], "forbidden": ["August", "620", "Kestrel"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_09", "split": "dev", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The north sensor reports humidity, the east sensor reports pressure, and the south sensor reports temperature."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "Which sensor reports pressure? Answer with the direction only."}], "ideal": "east", "required": ["east"], "forbidden": ["north", "south", "temperature"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_10", "split": "dev", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Package M2 weighs 4.3 kg, uses priority shipping, has tracking code TR-882, and contains textiles."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "Give the tracking code only."}], "ideal": "TR-882", "required": ["TR-882"], "forbidden": ["4.3", "priority", "textiles"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_11", "split": "dev", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Aster Café opens at 07:30, stops serving breakfast at 11:00, closes at 19:00, and is closed on Mondays."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "On which day is the café closed? One word only."}], "ideal": "Monday", "required": ["Monday"], "forbidden": ["07:30", "11:00", "19:00"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_12", "split": "test", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The field notebook is in Drawer 6, the spare battery is in Drawer 2, and the lens cloth is in Drawer 9."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "Where is the spare battery? Give only the drawer."}], "ideal": "Drawer 2", "required": ["Drawer 2"], "forbidden": ["Drawer 6", "Drawer 9", "notebook"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_13", "split": "test", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Team Quartz uses Matrix for chat, Forgejo for code, MinIO for files, and Jitsi for calls."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "What does Team Quartz use for code? Answer with the product only."}], "ideal": "Forgejo", "required": ["Forgejo"], "forbidden": ["Matrix", "MinIO", "Jitsi"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "selective_14", "split": "test", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The safety drill coordinator is Hana, the assembly point is Courtyard C, the alarm test is at 15:05, and attendance is 44."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "Give the assembly point only."}], "ideal": "Courtyard C", "required": ["Courtyard C"], "forbidden": ["Hana", "15:05", "44"], "checks": {"max_words": 8, "no_canned_closing": true}}
+{"id": "delta_00", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The launch plan used a $12,000 budget, a Monday release, and email promotion."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Budget is now $9,500."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "Release moves to Wednesday."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Budget: $9,500.\n- Release: Wednesday.", "required": ["Budget: $9,500", "Release: Wednesday"], "forbidden": ["email promotion."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_01", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The course had 10 lessons, weekly quizzes, and a final presentation."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "The course now has 8 lessons."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "The final presentation is replaced by a written reflection."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Length: 8 lessons.\n- Final task: written reflection.", "required": ["Length: 8 lessons", "Final task: written reflection"], "forbidden": ["a final presentation."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_02", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The trip used Hotel Nova, train travel, and a four-night stay."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Accommodation changes to Harbor Hostel."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "The stay is shortened to three nights."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Accommodation: Harbor Hostel.\n- Stay: three nights.", "required": ["Accommodation: Harbor Hostel", "Stay: three nights"], "forbidden": ["a four-night stay."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_03", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The app supported English and German and required Android 12."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Spanish support is added."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "Minimum Android version changes to 13."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Language added: Spanish.\n- Minimum Android: 13.", "required": ["Language added: Spanish", "Minimum Android: 13"], "forbidden": ["required Android 12."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_04", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The event expected 75 guests, used Hall A, and started at 18:00."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Attendance estimate rises to 90."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "The venue changes to Hall C."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Expected guests: 90.\n- Venue: Hall C.", "required": ["Expected guests: 90", "Venue: Hall C"], "forbidden": ["started at 18:00."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_05", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The report covered Q1, used PDF format, and was due Friday."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Coverage expands to Q1 and Q2."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "Delivery format changes to HTML."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Coverage: Q1 and Q2.\n- Format: HTML.", "required": ["Coverage: Q1 and Q2", "Format: HTML"], "forbidden": ["was due Friday."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_06", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The device used a 2,000 mAh battery, USB-C charging, and a black enclosure."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Battery capacity increases to 2,600 mAh."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "The enclosure changes to silver."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Battery: 2,600 mAh.\n- Enclosure: silver.", "required": ["Battery: 2,600 mAh", "Enclosure: silver"], "forbidden": ["a black enclosure."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_07", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The study sampled 120 adults, used phone interviews, and ran for six weeks."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Sample size increases to 150 adults."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "Interviews move online."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Sample: 150 adults.\n- Interview mode: online.", "required": ["Sample: 150 adults", "Interview mode: online"], "forbidden": ["ran for six weeks."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_08", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The menu offered soup, pasta, and cake, with service at noon."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Cake is replaced by fruit."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "Service moves to 13:00."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Dessert: fruit.\n- Service time: 13:00.", "required": ["Dessert: fruit", "Service time: 13:00"], "forbidden": ["with service at noon."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_09", "split": "dev", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The backup ran daily at 01:00 and retained files for 14 days."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "The schedule changes to every six hours."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "Retention increases to 30 days."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Schedule: every six hours.\n- Retention: 30 days.", "required": ["Schedule: every six hours", "Retention: 30 days"], "forbidden": ["retained files for 14 days."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_10", "split": "dev", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The exhibit featured ceramics, opened in April, and occupied Gallery 2."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Opening moves to May."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "The exhibit moves to Gallery 5."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Opening: May.\n- Location: Gallery 5.", "required": ["Opening: May", "Location: Gallery 5"], "forbidden": ["occupied Gallery 2."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_11", "split": "dev", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The subscription cost $18 monthly and included five team seats."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Price changes to $20 monthly."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "The plan now includes eight seats."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Monthly price: $20.\n- Seats: eight.", "required": ["Monthly price: $20", "Seats: eight"], "forbidden": ["included five team seats."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_12", "split": "test", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The greenhouse target was 22°C, 55% humidity, and morning watering."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Temperature target drops to 20°C."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "Watering moves to evening."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Temperature: 20°C.\n- Watering: evening.", "required": ["Temperature: 20°C", "Watering: evening"], "forbidden": ["morning watering."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_13", "split": "test", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The reading group met biweekly, discussed two chapters, and used Room 8."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Meetings become weekly."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "The group will discuss one chapter per meeting."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Frequency: weekly.\n- Reading: one chapter per meeting.", "required": ["Frequency: weekly", "Reading: one chapter per meeting"], "forbidden": ["used Room 8."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "delta_14", "split": "test", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The API allowed 100 requests per minute and returned XML by default."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "The limit rises to 160 requests per minute."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "The default response changes to JSON."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Rate limit: 160 requests per minute.\n- Default format: JSON.", "required": ["Rate limit: 160 requests per minute", "Default format: JSON"], "forbidden": ["returned XML by default."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}
+{"id": "conflict_00", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The venue manager says the room holds 40 people, while the signed booking says 28."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Which capacity should I put on the invitation? Ask one focused clarification instead of choosing a value."}], "ideal": "Should I use the venue manager’s figure of 40 or the signed booking’s figure of 28?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_01", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The project brief names Lina as owner, but the approval email names Marcus."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Who should I list as owner? Ask one focused clarification instead of choosing a value."}], "ideal": "Should Lina or Marcus be treated as the authoritative project owner?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_02", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "One itinerary says the train leaves at 08:10; the ticket says 08:45."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Tell me when to arrive. Ask one focused clarification instead of choosing a value."}], "ideal": "Which departure time is current: 08:10 on the itinerary or 08:45 on the ticket?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_03", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The spreadsheet total is €4,720, but the signed quote is €4,270."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "What amount should I approve? Ask one focused clarification instead of choosing a value."}], "ideal": "Which amount is authoritative: €4,720 in the spreadsheet or €4,270 in the signed quote?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_04", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The label says refrigerate the sample, while the protocol says store it at room temperature."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Where should I put it? Ask one focused clarification instead of choosing a value."}], "ideal": "Should the sample follow the label’s refrigeration instruction or the protocol’s room-temperature instruction?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_05", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The calendar lists Tuesday, but the client’s last message says Thursday."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Schedule the call. Ask one focused clarification instead of choosing a value."}], "ideal": "Should I schedule the call for Tuesday or for Thursday from the client’s latest message?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_06", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The inventory system shows 16 units, while the physical count found 11."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "How many can I promise? Ask one focused clarification instead of choosing a value."}], "ideal": "Which quantity should govern the promise: 16 in the system or 11 physically counted?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_07", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The style guide requires sentence case, but the campaign brief asks for title case."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Which capitalization should I use? Ask one focused clarification instead of choosing a value."}], "ideal": "Should the style guide’s sentence case or the campaign brief’s title case take priority?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_08", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The map marks the path open, but a ranger’s notice marks it closed."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Can I plan the route through it? Ask one focused clarification instead of choosing a value."}], "ideal": "Is the ranger’s closure notice current and authoritative over the map?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_09", "split": "dev", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The database record gives Ana’s region as West, while her submitted form says North."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Which region belongs in the report? Ask one focused clarification instead of choosing a value."}], "ideal": "Should the report use West from the database or North from Ana’s submitted form?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_10", "split": "dev", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The product page promises a two-year warranty, but the box insert says one year."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "What warranty should support quote? Ask one focused clarification instead of choosing a value."}], "ideal": "Which warranty term is authoritative: two years on the product page or one year in the box insert?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_11", "split": "dev", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The teacher’s rubric allows 1,000 words, but the assignment portal shows 800."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "How long should my essay be? Ask one focused clarification instead of choosing a value."}], "ideal": "Should the essay follow the 1,000-word rubric or the 800-word portal limit?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_12", "split": "test", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The maintenance log says filter type F7, while the installed-unit plate specifies F9."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Which replacement should I order? Ask one focused clarification instead of choosing a value."}], "ideal": "Should the replacement match F7 in the log or F9 on the installed unit’s plate?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_13", "split": "test", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The grant call says applications close at 17:00 local time, but the portal countdown implies 16:00."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Give me the deadline. Ask one focused clarification instead of choosing a value."}], "ideal": "Which deadline is authoritative: 17:00 in the grant call or 16:00 implied by the portal?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "conflict_14", "split": "test", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The recipe card calls for 4 grams of yeast, while the chef’s correction says 7 grams."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "How much yeast should I use? Ask one focused clarification instead of choosing a value."}], "ideal": "Should I follow the original 4 grams or the chef’s corrected 7 grams?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}
+{"id": "source_boundary_00", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The supplied memo says the pilot had 48 participants and lasted six weeks. It does not report the dropout count."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "How many participants dropped out? Answer directly without guessing."}], "ideal": "The dropout count is not provided in the memo, so it cannot be determined from the supplied information.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_01", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The inspection note confirms a cracked seal but gives no cause."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "What caused the seal to crack? Answer directly without guessing."}], "ideal": "The note confirms the crack but does not identify its cause.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_02", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The article excerpt says the town planted 2,000 trees. It gives no survival survey."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "How many trees survived? Answer directly without guessing."}], "ideal": "The excerpt does not provide survival data, so the number cannot be determined.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_03", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The invoice lists labor and materials but no payment date."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "On what date was it paid? Answer directly without guessing."}], "ideal": "No payment date is provided, so the payment date cannot be verified.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_04", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The experiment log records a temperature rise after minute 12 but does not identify the mechanism."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "Which mechanism caused the rise? Answer directly without guessing."}], "ideal": "The log shows when the rise occurred but does not establish its mechanism.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_05", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The meeting minutes approve the proposal without recording individual votes."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "Who voted against it? Answer directly without guessing."}], "ideal": "Individual votes were not recorded, so the minutes cannot identify who voted against it.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_06", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The product sheet gives a 500-cycle rating under lab conditions but no field-life estimate."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "How many years will it last in normal use? Answer directly without guessing."}], "ideal": "The sheet does not provide enough usage data to convert the lab cycle rating into years of normal use.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_07", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The survey summary reports an average score of 7.2 but omits the response count."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "How many people answered? Answer directly without guessing."}], "ideal": "The response count is missing, so it cannot be determined from the summary.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_08", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The shipping notice says the parcel left the depot yesterday but has no live tracking update."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "Where is it right now? Answer directly without guessing."}], "ideal": "The notice does not contain live location data, so the parcel’s current location cannot be verified.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_09", "split": "dev", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The biography states that Noor studied chemistry and later founded a company. It does not name the university."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "Which university did Noor attend? Answer directly without guessing."}], "ideal": "The university is not named in the supplied biography.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_10", "split": "dev", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The budget shows projected revenue and costs but no actual results."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "What profit did the project make? Answer directly without guessing."}], "ideal": "Actual profit cannot be calculated because only projections are provided.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_11", "split": "dev", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The case summary says the server failed after an update but contains no logs."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "Which component failed first? Answer directly without guessing."}], "ideal": "The summary does not include logs or a failure sequence, so the first failed component cannot be identified.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_12", "split": "test", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The catalog dates the vase to the 1880s but does not name its maker."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "Who made the vase? Answer directly without guessing."}], "ideal": "The maker is not identified in the catalog entry.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_13", "split": "test", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The clinical abstract reports improvement but omits the control group’s numbers."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "How much better was treatment than control? Answer directly without guessing."}], "ideal": "The difference cannot be calculated without the control group’s results.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "source_boundary_14", "split": "test", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The energy report gives annual consumption but no building floor area."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "What is consumption per square meter? Answer directly without guessing."}], "ideal": "Consumption per square meter cannot be calculated without the floor area.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}
+{"id": "constraint_00", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Draft a reminder about tomorrow’s 09:00 safety briefing."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: exactly two sentences; include 'Room 4'; do not use 'urgent'. Return only the requested content."}], "ideal": "Tomorrow’s safety briefing begins at 09:00 in Room 4. Please arrive five minutes early.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "sentence_count": 2}}
+{"id": "constraint_01", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "List ways to protect a laptop while traveling."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: exactly three bullet points; each under eight words; do not mention insurance. Return only the requested content."}], "ideal": "- Use a padded, water-resistant sleeve.\n- Keep it in your carry-on.\n- Enable encryption and a screen lock.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "bullet_count": 3}}
+{"id": "constraint_02", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Write a status line for a completed data import."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: exactly eight words; start with 'Import complete:'. Return only the requested content."}], "ideal": "Import complete: all records passed validation without errors.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "exact_words": 8}}
+{"id": "constraint_03", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Give a compact JSON object for a green task due Friday."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: valid JSON only; keys exactly title, color, due. Return only the requested content."}], "ideal": "{\"title\":\"task\",\"color\":\"green\",\"due\":\"Friday\"}", "required": [], "forbidden": [], "checks": {"no_canned_closing": true}}
+{"id": "constraint_04", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Decline a lunch invitation politely."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: one sentence; include 'thank you'; do not give an excuse. Return only the requested content."}], "ideal": "Thank you for inviting me, but I’ll have to decline.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true}}
+{"id": "constraint_05", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Summarize the action: Eli will send the draft by noon."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: start with 'Owner:'; one line; include the deadline. Return only the requested content."}], "ideal": "Owner: Eli — send the draft by noon.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true}}
+{"id": "constraint_06", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Describe a calm workspace."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: exactly six words; include 'light'. Return only the requested content."}], "ideal": "Soft light, clear desk, quiet focus.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "exact_words": 6}}
+{"id": "constraint_07", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Answer whether backups reduce data-loss risk."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: begin with Yes or No; then exactly one explanatory sentence. Return only the requested content."}], "ideal": "Yes. Backups provide a separate copy when original data is lost or damaged.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true}}
+{"id": "constraint_08", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Give two reasons to label cables."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: exactly two numbered items; no introduction. Return only the requested content."}], "ideal": "1. Faster troubleshooting.\n2. Safer equipment changes.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "numbered_items": 2}}
+{"id": "constraint_09", "split": "dev", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Write a meeting subject about reviewing the Atlas budget."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: under seven words; include 'Atlas'; do not use punctuation. Return only the requested content."}], "ideal": "Atlas budget review", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "max_words": 7}}
+{"id": "constraint_10", "split": "dev", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Explain version control to a beginner."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: one sentence; use the word history; under 20 words. Return only the requested content."}], "ideal": "Version control keeps a history of changes so work can be compared, shared, or restored.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "max_words": 20}}
+{"id": "constraint_11", "split": "dev", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Provide a CSV header for name, role, and start date."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: one line only; exactly three comma-separated fields; lowercase. Return only the requested content."}], "ideal": "name,role,start_date", "required": [], "forbidden": [], "checks": {"no_canned_closing": true}}
+{"id": "constraint_12", "split": "test", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Write a friendly confirmation for a 15:30 call."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: exactly two sentences; include Tuesday; do not start with Sure. Return only the requested content."}], "ideal": "Tuesday at 15:30 works for me. I’ll be ready for the call.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "sentence_count": 2}}
+{"id": "constraint_13", "split": "test", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Give a four-word warning about a wet floor."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: exactly four words; end with 'carefully'. Return only the requested content."}], "ideal": "Wet floor; walk carefully.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "exact_words": 4}}
+{"id": "constraint_14", "split": "test", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Return the enabled state and retry count."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: valid JSON only; keys exactly enabled and retries; enabled true; retries 3. Return only the requested content."}], "ideal": "{\"enabled\":true,\"retries\":3}", "required": [], "forbidden": [], "checks": {"no_canned_closing": true}}
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/cheap_dev_gate.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/cheap_dev_gate.json
new file mode 100644
index 0000000000000000000000000000000000000000..db439bfbb3a8f1a304f9a411e818e85479416c7c
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/cheap_dev_gate.json
@@ -0,0 +1,121 @@
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+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/cheap_dev_more.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/cheap_dev_more.json
new file mode 100644
index 0000000000000000000000000000000000000000..3d8381956726589a412efb911532b399dcc1196b
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/cheap_dev_more.json
@@ -0,0 +1,104 @@
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+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/constrained_training_report.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/constrained_training_report.json
new file mode 100644
index 0000000000000000000000000000000000000000..bdc2ba48cd2bcadc852d69640d3a387c88c1cb0f
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/constrained_training_report.json
@@ -0,0 +1,1409 @@
+{
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+  "math_to_info_per_update": "2:1",
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+  ],
+  "source_path": "/marimo/budgie_information_rl_v1/info_math_constrained_source.pt",
+  "seconds": 29.05026912689209,
+  "parameter_count_change": 0,
+  "benchmark_family_rows": 0
+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/curriculum_manifest.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/curriculum_manifest.json
new file mode 100644
index 0000000000000000000000000000000000000000..bacea0fe8e71c299c60a29b7203ce3be5e300b1d
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/curriculum_manifest.json
@@ -0,0 +1,20 @@
+{
+  "rows": 105,
+  "splits": {
+    "train": 63,
+    "dev": 21,
+    "test": 21
+  },
+  "categories": {
+    "state_update": 15,
+    "already_tried": 15,
+    "selective_answer": 15,
+    "delta_summary": 15,
+    "conflict_clarification": 15,
+    "source_boundary": 15,
+    "constraint_stack": 15
+  },
+  "ideal_reference_pass": 105,
+  "benchmark_family_rows": 0,
+  "parameter_count_change": 0
+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/encoded_rollouts.jsonl b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/encoded_rollouts.jsonl
new file mode 100644
index 0000000000000000000000000000000000000000..419d6c6a089e52aa7976ea2d80ea31e5f976f018
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/encoded_rollouts.jsonl
@@ -0,0 +1,315 @@
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+{"case_id": "state_update_00", "category": "state_update", "origin": "manual_ideal", "answer": "The community workshop is Sunday at 14:00 in Room Cedar.", "reward": 1, "advantage": 1.9785576908013356, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 1773, 11992, 30, 378, 3791, 2004, 314, 14986, 418, 216, 49160, 49159, 42, 49159, 49159, 43, 19690, 40074, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 1773, 11992, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 14986, 418, 216, 49160, 49159, 42, 49159, 49159, 351, 10528, 418, 216, 49160, 49163, 42, 49159, 49159, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 260, 3403, 655, 284, 3589, 281, 582, 6330, 30, 49154, 49158, 198, 504, 1773, 11992, 314, 10528, 418, 216, 49160, 49163, 42, 49159, 49159, 281, 19690, 40074, 30, 49154], "prefix_len": 182, "old_logprobs": [-3.5383315086364746, -4.427703857421875, -0.04108788073062897, -2.577716588973999, -10.673654556274414, -0.21951349079608917, -0.03779958561062813, -0.019542768597602844, -0.02538992464542389, -0.0007671750499866903, -0.0025566292461007833, -0.0005283151986077428, -3.266495943069458, -5.5789642333984375, -9.353328704833984, -0.15448707342147827, -0.010090411640703678]}
+{"case_id": "state_update_01", "category": "state_update", "origin": "budgie", "answer": "I have the current software release, date, and rollout method.", "reward": -0.42999999999999994, "advantage": -0.4076500646438513, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 3197, 4706, 30, 378, 3791, 2004, 314, 3586, 216, 49162, 30, 49163, 335, 2405, 216, 49167, 43, 32797, 5089, 447, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 3197, 4706, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 3586, 216, 49162, 30, 49163, 335, 2405, 216, 49167, 351, 3586, 216, 49162, 30, 49164, 335, 2405, 216, 49160, 49159, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 1574, 3586, 28, 3688, 28, 284, 5089, 447, 1341, 281, 582, 6330, 30, 49154, 49158, 198, 57, 457, 260, 1574, 3197, 4706, 28, 3688, 28, 284, 5089, 447, 1341, 30, 49154], "prefix_len": 192, "old_logprobs": [-2.182314872741699, -2.39223575592041, -0.5647780299186707, -0.5094540119171143, -1.748270034790039, -0.40003639459609985, -1.2524850368499756, -0.3873175084590912, -0.19827283918857574, -0.013853820040822029, -0.05804019421339035, -0.023208757862448692, -0.28303906321525574, -0.961670458316803, -0.20979776978492737]}
+{"case_id": "state_update_01", "category": "state_update", "origin": "budgie", "answer": "I've been using the current software for quite some time, and it's been a implant of changes. Please note that this is a general state of mind; the actual software will change over time and may not always be the same.", "reward": -0.6399999999999999, "advantage": -0.7584964317553625, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 3197, 4706, 30, 378, 3791, 2004, 314, 3586, 216, 49162, 30, 49163, 335, 2405, 216, 49167, 43, 32797, 5089, 447, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 3197, 4706, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 3586, 216, 49162, 30, 49163, 335, 2405, 216, 49167, 351, 3586, 216, 49162, 30, 49164, 335, 2405, 216, 49160, 49159, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 1574, 3586, 28, 3688, 28, 284, 5089, 447, 1341, 281, 582, 6330, 30, 49154, 49158, 198, 57, 3543, 719, 1015, 260, 1574, 3197, 327, 3100, 634, 655, 28, 284, 357, 506, 719, 253, 15286, 282, 1971, 30, 8765, 4109, 338, 451, 314, 253, 2210, 1215, 282, 1945, 43, 260, 4277, 3197, 523, 1363, 690, 655, 284, 654, 441, 1811, 325, 260, 1142, 30, 49154], "prefix_len": 192, "old_logprobs": [-2.182314872741699, -2.01723575592041, -1.729001760482788, -2.1185314655303955, -0.5257917046546936, -1.3288952112197876, -1.0836007595062256, -2.441427230834961, -2.3608412742614746, -0.14491309225559235, -0.007985570468008518, -0.9283064007759094, -0.7330425381660461, -1.9417873620986938, -0.7937408685684204, -1.1829017400741577, -1.9474642276763916, -11.066364288330078, -0.7858725190162659, -2.5755717754364014, -0.7184830904006958, -4.212847709655762, -0.9547610282897949, -0.27527037262916565, -2.2488765716552734, -0.7317131161689758, -0.701717734336853, -1.145960807800293, -5.877935409545898, -0.7216809391975403, -1.484148383140564, -2.1449546813964844, -2.0332305431365967, -1.2244635820388794, -1.6854379177093506, -3.716559648513794, -1.3538484573364258, -1.1305971145629883, -0.005727428011596203, -2.435258388519287, -2.9338929653167725, -1.7755340337753296, -0.7023258805274963, -0.5942155718803406, -3.0526890754699707, -0.3054719865322113, -0.9218202829360962, -0.188208669424057]}
+{"case_id": "state_update_01", "category": "state_update", "origin": "budgie", "answer": "The current software release has now been released.", "reward": -0.42999999999999994, "advantage": -0.4076500646438513, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 3197, 4706, 30, 378, 3791, 2004, 314, 3586, 216, 49162, 30, 49163, 335, 2405, 216, 49167, 43, 32797, 5089, 447, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 3197, 4706, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 3586, 216, 49162, 30, 49163, 335, 2405, 216, 49167, 351, 3586, 216, 49162, 30, 49164, 335, 2405, 216, 49160, 49159, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 1574, 3586, 28, 3688, 28, 284, 5089, 447, 1341, 281, 582, 6330, 30, 49154, 49158, 198, 504, 1574, 3197, 4706, 553, 1209, 719, 5118, 30, 49154], "prefix_len": 192, "old_logprobs": [-1.9323148727416992, -0.39758652448654175, -1.2457704544067383, -0.4551083445549011, -2.248504638671875, -2.8999903202056885, -0.2673014998435974, -2.1644890308380127, -1.0399905443191528, -0.21345637738704681]}
+{"case_id": "state_update_01", "category": "state_update", "origin": "budgie", "answer": "Current version of the software has.\"The existing software has been released\" on May 8; the software for the next ~8-hour period has been rolled out.\"", "reward": -0.42999999999999994, "advantage": -0.4076500646438513, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 3197, 4706, 30, 378, 3791, 2004, 314, 3586, 216, 49162, 30, 49163, 335, 2405, 216, 49167, 43, 32797, 5089, 447, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 3197, 4706, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 3586, 216, 49162, 30, 49163, 335, 2405, 216, 49167, 351, 3586, 216, 49162, 30, 49164, 335, 2405, 216, 49160, 49159, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 1574, 3586, 28, 3688, 28, 284, 5089, 447, 1341, 281, 582, 6330, 30, 49154, 49158, 198, 12183, 3586, 282, 260, 3197, 553, 1270, 504, 3832, 3197, 553, 719, 5118, 18, 335, 2405, 216, 49167, 43, 260, 3197, 327, 260, 1867, 11417, 49167, 29, 12137, 1657, 553, 719, 18221, 578, 1270, 49154], "prefix_len": 192, "old_logprobs": [-3.682314872741699, -1.58769953250885, -3.171445608139038, -1.5763535499572754, -0.026024868711829185, -4.401530742645264, -9.872038841247559, -1.784026026725769, -6.195160865783691, -0.8585687279701233, -0.8301941752433777, -0.5408469438552856, -1.8432621955871582, -3.4771511554718018, -1.2085751295089722, -0.10100419074296951, -0.0054865493439137936, -1.137941598892212, -1.9565064907073975, -0.7309503555297852, -1.9613854885101318, -4.311127185821533, -0.9002496600151062, -2.530369281768799, -10.186836242675781, -2.696715831756592, -1.140228033065796, -0.7184978127479553, -1.7487094402313232, -0.39507707953453064, -0.5033702850341797, -2.6057348251342773, -0.09109486639499664, -0.3765963315963745, -0.060345325618982315]}
+{"case_id": "state_update_01", "category": "state_update", "origin": "manual_ideal", "answer": "Version 3.5 will launch on May 10 through a staged rollout.", "reward": 1, "advantage": 1.9814466256869163, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 3197, 4706, 30, 378, 3791, 2004, 314, 3586, 216, 49162, 30, 49163, 335, 2405, 216, 49167, 43, 32797, 5089, 447, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 3197, 4706, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 3586, 216, 49162, 30, 49163, 335, 2405, 216, 49167, 351, 3586, 216, 49162, 30, 49164, 335, 2405, 216, 49160, 49159, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 1574, 3586, 28, 3688, 28, 284, 5089, 447, 1341, 281, 582, 6330, 30, 49154, 49158, 198, 22348, 216, 49162, 30, 49164, 523, 7198, 335, 2405, 216, 49160, 49159, 738, 253, 32797, 5089, 447, 30, 49154], "prefix_len": 192, "old_logprobs": [-11.9323148727417, -3.4971835613250732, -0.04634242504835129, -0.0607236884534359, -0.9866122007369995, -4.195736408233643, -5.95051383972168, -0.7424324154853821, -0.0545981265604496, -0.00043418517452664673, -0.015037745237350464, -0.012626232579350471, -9.417236328125, -4.423313617706299, -12.553691864013672, -0.25751763582229614, -0.0843658521771431, -0.7633506059646606, -0.03928326070308685]}
+{"case_id": "state_update_02", "category": "state_update", "origin": "budgie", "answer": "The final order instructions are:", "reward": -0.42999999999999994, "advantage": -0.8682338346767601, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 21671, 1686, 30, 378, 3791, 2004, 314, 216, 49161, 49163, 49159, 4461, 28291, 43, 5486, 288, 47491, 216, 49161, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 21671, 1686, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49161, 49163, 49159, 4461, 28291, 351, 216, 49160, 49167, 49159, 2654, 28291, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 19161, 260, 3403, 1686, 5785, 281, 582, 6330, 30, 49154, 49158, 198, 504, 3403, 1686, 6388, 359, 42, 49154], "prefix_len": 178, "old_logprobs": [-3.003546714782715, -0.33808356523513794, -0.05979806184768677, -2.4054512977600098, -1.0500785112380981, -2.1435728073120117, -0.7297799587249756]}
+{"case_id": "state_update_02", "category": "state_update", "origin": "budgie", "answer": "To scanner the supplier order, I will follow this instruction:", "reward": -0.42999999999999994, "advantage": -0.8682338346767601, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 21671, 1686, 30, 378, 3791, 2004, 314, 216, 49161, 49163, 49159, 4461, 28291, 43, 5486, 288, 47491, 216, 49161, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 21671, 1686, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49161, 49163, 49159, 4461, 28291, 351, 216, 49160, 49167, 49159, 2654, 28291, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 19161, 260, 3403, 1686, 5785, 281, 582, 6330, 30, 49154, 49158, 198, 2068, 28361, 260, 21671, 1686, 28, 339, 523, 1066, 451, 5785, 42, 49154], "prefix_len": 178, "old_logprobs": [-5.003546714782715, -12.306171417236328, -1.2147693634033203, -2.455315113067627, -0.35932812094688416, -0.15901438891887665, -0.4706861972808838, -1.3697532415390015, -1.1453897953033447, -1.7420918941497803, -1.982351541519165, -0.26221179962158203, -0.24606849253177643]}
+{"case_id": "state_update_02", "category": "state_update", "origin": "budgie", "answer": "The final order instruction was logged, with a focus on the green folders, 180 on', and a no-no, 240 blue on'.", "reward": -0.010000000000000009, "advantage": -0.06502191713438298, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 21671, 1686, 30, 378, 3791, 2004, 314, 216, 49161, 49163, 49159, 4461, 28291, 43, 5486, 288, 47491, 216, 49161, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 21671, 1686, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49161, 49163, 49159, 4461, 28291, 351, 216, 49160, 49167, 49159, 2654, 28291, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 19161, 260, 3403, 1686, 5785, 281, 582, 6330, 30, 49154, 49158, 198, 504, 3403, 1686, 5785, 436, 25649, 28, 351, 253, 1593, 335, 260, 2654, 28291, 28, 216, 49160, 49167, 49159, 335, 923, 284, 253, 787, 29, 4607, 28, 216, 49161, 49163, 49159, 4461, 335, 4289, 49154], "prefix_len": 178, "old_logprobs": [-3.003546714782715, -0.33808356523513794, -0.05979806184768677, -0.3429514169692993, -0.9636045098304749, -12.123274803161621, -3.099088668823242, -1.7900879383087158, -2.0157723426818848, -4.05908727645874, -0.014208228327333927, -1.466055989265442, -3.81107497215271, -1.7424765825271606, -2.3640501499176025, -2.1958231925964355, -0.08992916345596313, -0.010233079083263874, -0.00365672679618001, -4.539299964904785, -10.613958358764648, -0.5975590944290161, -1.3776049613952637, -4.549203872680664, -0.6797870993614197, -4.320714473724365, -4.849735736846924, -1.4240440130233765, -0.09555119276046753, -0.14493413269519806, -0.0015689932042732835, -0.49514687061309814, -1.1618576049804688, -1.3103933334350586, -0.03406720608472824]}
+{"case_id": "state_update_02", "category": "state_update", "origin": "budgie", "answer": "E Regulations: The holding company ahead of the delivery line to cautioned the order details for 240 blue and 180 green folders.", "reward": -0.010000000000000009, "advantage": -0.06502191713438298, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 21671, 1686, 30, 378, 3791, 2004, 314, 216, 49161, 49163, 49159, 4461, 28291, 43, 5486, 288, 47491, 216, 49161, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 21671, 1686, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49161, 49163, 49159, 4461, 28291, 351, 216, 49160, 49167, 49159, 2654, 28291, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 19161, 260, 3403, 1686, 5785, 281, 582, 6330, 30, 49154, 49158, 198, 53, 23904, 42, 378, 6961, 2727, 5976, 282, 260, 7037, 1761, 288, 44020, 260, 1686, 3841, 327, 216, 49161, 49163, 49159, 4461, 284, 216, 49160, 49167, 49159, 2654, 28291, 30, 49154], "prefix_len": 178, "old_logprobs": [-6.378546714782715, -11.103530883789062, -2.116166591644287, -3.155580997467041, -10.698890686035156, -1.7386643886566162, -6.331302165985107, -0.2076580971479416, -0.7511017918586731, -0.6182068586349487, -3.684450149536133, -2.3268136978149414, -11.057600021362305, -1.9556164741516113, -1.3767269849777222, -0.5032850503921509, -2.4963533878326416, -1.155957818031311, -0.7571561932563782, -0.3845262825489044, -0.009706325829029083, -0.19945494830608368, -2.9250028133392334, -0.0763554722070694, -0.006228203885257244, -0.01336037740111351, -0.0008323303773067892, -0.012377472594380379, -0.12370631843805313, -0.46911007165908813, -0.02123723365366459]}
+{"case_id": "state_update_02", "category": "state_update", "origin": "manual_ideal", "answer": "Order 180 green folders and deliver them to Dock 2.", "reward": 1, "advantage": 1.866511503622286, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 21671, 1686, 30, 378, 3791, 2004, 314, 216, 49161, 49163, 49159, 4461, 28291, 43, 5486, 288, 47491, 216, 49161, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 21671, 1686, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49161, 49163, 49159, 4461, 28291, 351, 216, 49160, 49167, 49159, 2654, 28291, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 19161, 260, 3403, 1686, 5785, 281, 582, 6330, 30, 49154, 49158, 198, 22977, 216, 49160, 49167, 49159, 2654, 28291, 284, 5486, 601, 288, 47491, 216, 49161, 30, 49154], "prefix_len": 178, "old_logprobs": [-9.566046714782715, -4.2038679122924805, -1.1436903476715088, -2.5676956176757812, -0.03286396712064743, -0.41533493995666504, -0.22635702788829803, -4.731751441955566, -6.975926399230957, -1.8430001735687256, -0.19318139553070068, -10.354192733764648, -1.2525413036346436, -0.020943094044923782, -0.3498145639896393, -0.01330462098121643]}
+{"case_id": "state_update_03", "category": "state_update", "origin": "budgie", "answer": "My approach would be slightly different, but I can help with any questions you have.", "reward": -0.42999999999999994, "advantage": -0.6673862687557579, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 5473, 8034, 30, 378, 3791, 2004, 314, 26918, 418, 216, 49159, 49168, 42, 49162, 49159, 43, 3341, 946, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 5473, 8034, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 26918, 418, 216, 49159, 49168, 42, 49162, 49159, 351, 12711, 418, 216, 49160, 49160, 42, 49160, 49164, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 805, 260, 1574, 8991, 28, 655, 28, 284, 4624, 30, 49154, 49158, 198, 5965, 2424, 736, 325, 5359, 896, 28, 564, 339, 416, 724, 351, 750, 2029, 346, 457, 30, 49154], "prefix_len": 183, "old_logprobs": [-4.326745986938477, -4.76550817489624, -4.969233512878418, -0.48068881034851074, -4.410386562347412, -0.13317370414733887, -0.8049441576004028, -0.8912198543548584, -0.19238382577896118, -1.6420888900756836, -1.100159764289856, -1.0243210792541504, -2.590599775314331, -0.5677677392959595, -1.5265631675720215, -0.6220433115959167, -0.8176565170288086, -0.08922603726387024]}
+{"case_id": "state_update_03", "category": "state_update", "origin": "budgie", "answer": "Maura battle opened on Friday, was at 11:15, and was scheduled to have a video call at 11:30.", "reward": -0.010000000000000009, "advantage": 0.09018733361564288, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 5473, 8034, 30, 378, 3791, 2004, 314, 26918, 418, 216, 49159, 49168, 42, 49162, 49159, 43, 3341, 946, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 5473, 8034, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 26918, 418, 216, 49159, 49168, 42, 49162, 49159, 351, 12711, 418, 216, 49160, 49160, 42, 49160, 49164, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 805, 260, 1574, 8991, 28, 655, 28, 284, 4624, 30, 49154, 49158, 198, 61, 47522, 5182, 6572, 335, 11655, 28, 436, 418, 216, 49160, 49160, 42, 49160, 49164, 28, 284, 436, 13831, 288, 457, 253, 3341, 946, 418, 216, 49160, 49160, 42, 49162, 49159, 30, 49154], "prefix_len": 183, "old_logprobs": [-5.326745986938477, -8.626222610473633, -6.5864715576171875, -8.161759376525879, -2.883364200592041, -3.7708051204681396, -1.023082971572876, -4.936332702636719, -3.370548725128174, -0.37009796500205994, -0.9596756100654602, -1.4749424457550049, -0.03481169044971466, -0.7544945478439331, -0.0845256969332695, -0.6643867492675781, -0.29505640268325806, -2.900233268737793, -4.465874671936035, -0.15680471062660217, -3.133070945739746, -0.5619394779205322, -3.586489677429199, -0.00844219047576189, -0.6714667677879333, -0.06445977836847305, -0.10835328698158264, -0.5844367742538452, -0.0004659997357521206, -1.304310917854309, -0.0854954868555069, -0.41875532269477844, -0.05074358358979225]}
+{"case_id": "state_update_03", "category": "state_update", "origin": "budgie", "answer": "The current candidate is pesky at 10:15, and the date is July 20th.", "reward": -0.42999999999999994, "advantage": -0.6673862687557579, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 5473, 8034, 30, 378, 3791, 2004, 314, 26918, 418, 216, 49159, 49168, 42, 49162, 49159, 43, 3341, 946, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 5473, 8034, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 26918, 418, 216, 49159, 49168, 42, 49162, 49159, 351, 12711, 418, 216, 49160, 49160, 42, 49160, 49164, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 805, 260, 1574, 8991, 28, 655, 28, 284, 4624, 30, 49154, 49158, 198, 504, 1574, 8991, 314, 40556, 418, 216, 49160, 49159, 42, 49160, 49164, 28, 284, 260, 3688, 314, 4185, 216, 49161, 49159, 373, 30, 49154], "prefix_len": 183, "old_logprobs": [-3.0142459869384766, -0.2978593707084656, -0.09090189635753632, -0.2986128628253937, -11.135499954223633, -0.7699233293533325, -0.2450428009033203, -0.32588687539100647, -2.583019733428955, -0.005536105018109083, -1.311431646347046, -0.43491601943969727, -2.5261662006378174, -1.1826359033584595, -0.2878291606903076, -2.9117462635040283, -0.2918950021266937, -5.273179531097412, -0.002693003974854946, -1.2923455238342285, -1.6037988662719727, -0.6436251997947693, -0.4241991639137268, -0.0873442068696022]}
+{"case_id": "state_update_03", "category": "state_update", "origin": "budgie", "answer": "I will provide the current candidate, time, and format as it is currently live.", "reward": -0.42999999999999994, "advantage": -0.6673862687557579, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 5473, 8034, 30, 378, 3791, 2004, 314, 26918, 418, 216, 49159, 49168, 42, 49162, 49159, 43, 3341, 946, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 5473, 8034, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 26918, 418, 216, 49159, 49168, 42, 49162, 49159, 351, 12711, 418, 216, 49160, 49160, 42, 49160, 49164, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 805, 260, 1574, 8991, 28, 655, 28, 284, 4624, 30, 49154, 49158, 198, 57, 523, 1538, 260, 1574, 8991, 28, 655, 28, 284, 4624, 347, 357, 314, 4112, 2330, 30, 49154], "prefix_len": 183, "old_logprobs": [-1.764245867729187, -1.8303418159484863, -1.457067608833313, -0.17918415367603302, -0.1615101546049118, -0.10692660510540009, -0.4582066535949707, -0.075696662068367, -0.02799289859831333, -0.05585115775465965, -0.00846133939921856, -1.554681658744812, -5.136227607727051, -0.5404329299926758, -1.3316599130630493, -6.141698837280273, -0.22190310060977936, -0.10541688650846481]}
+{"case_id": "state_update_03", "category": "state_update", "origin": "manual_ideal", "answer": "Jon at 11:15 by video call.", "reward": 1, "advantage": 1.9119714726516308, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 5473, 8034, 30, 378, 3791, 2004, 314, 26918, 418, 216, 49159, 49168, 42, 49162, 49159, 43, 3341, 946, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 5473, 8034, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 26918, 418, 216, 49159, 49168, 42, 49162, 49159, 351, 12711, 418, 216, 49160, 49160, 42, 49160, 49164, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 805, 260, 1574, 8991, 28, 655, 28, 284, 4624, 30, 49154, 49158, 198, 38976, 418, 216, 49160, 49160, 42, 49160, 49164, 411, 3341, 946, 30, 49154], "prefix_len": 183, "old_logprobs": [-7.764245986938477, -3.9574272632598877, -0.013010022230446339, -0.04316810145974159, -0.04397064074873924, -0.0017591729993000627, -0.0349603109061718, -0.004594721365720034, -7.372682571411133, -7.998364448547363, -0.01016593724489212, -0.4122934937477112, -0.03801616653800011]}
+{"case_id": "state_update_04", "category": "state_update", "origin": "budgie", "answer": "UnderHeader Earth's temperature increases to 12°C due to the passing of the Honolulu series, leading to asq ration of new life.", "reward": -0.42999999999999994, "advantage": -0.5888992037956788, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 6508, 3048, 30, 378, 3791, 2004, 314, 216, 49160, 49161, 2490, 22830, 43, 6082, 418, 216, 49160, 49165, 42, 49161, 49159, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 6508, 3048, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49160, 49161, 2490, 22830, 351, 216, 49168, 2490, 22830, 284, 216, 49162, 811, 22830, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 13764, 3298, 836, 260, 3403, 40199, 281, 582, 6330, 30, 49154, 49158, 198, 7134, 29223, 2591, 506, 2779, 4495, 288, 216, 49160, 49161, 4742, 51, 1568, 288, 260, 7685, 282, 260, 47502, 3086, 28, 2899, 288, 347, 97, 30884, 282, 725, 1029, 30, 49154], "prefix_len": 183, "old_logprobs": [-5.0442399978637695, -10.122404098510742, -11.051313400268555, -2.1852970123291016, -4.514230251312256, -2.7721176147460938, -4.173905372619629, -0.42726826667785645, -0.9462746381759644, -1.3974008560180664, -1.4567666053771973, -0.04879852011799812, -3.8683383464813232, -0.0037748999893665314, -1.6944665908813477, -5.694677352905273, -0.07981526851654053, -1.9472204446792603, -11.999358177185059, -5.746438026428223, -1.4898607730865479, -1.5066139698028564, -0.027094705030322075, -9.10806655883789, -8.953259468078613, -11.57337760925293, -5.749485492706299, -2.462455987930298, -2.616421937942505, -1.5990586280822754, -0.00903517846018076]}
+{"case_id": "state_update_04", "category": "state_update", "origin": "budgie", "answer": "Track the museum visit details.", "reward": -0.42999999999999994, "advantage": -0.5888992037956788, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 6508, 3048, 30, 378, 3791, 2004, 314, 216, 49160, 49161, 2490, 22830, 43, 6082, 418, 216, 49160, 49165, 42, 49161, 49159, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 6508, 3048, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49160, 49161, 2490, 22830, 351, 216, 49168, 2490, 22830, 284, 216, 49162, 811, 22830, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 13764, 3298, 836, 260, 3403, 40199, 281, 582, 6330, 30, 49154, 49158, 198, 35435, 260, 6508, 3048, 3841, 30, 49154], "prefix_len": 183, "old_logprobs": [-9.23173999786377, -0.6403732895851135, -2.3605475425720215, -0.08378312736749649, -1.4426425695419312, -1.4070402383804321, -0.033505599945783615]}
+{"case_id": "state_update_04", "category": "state_update", "origin": "budgie", "answer": "I was able to book my tour in 12 adult and 3 child tickets, as you mentioned.", "reward": -0.22, "advantage": -0.2118600794142992, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 6508, 3048, 30, 378, 3791, 2004, 314, 216, 49160, 49161, 2490, 22830, 43, 6082, 418, 216, 49160, 49165, 42, 49161, 49159, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 6508, 3048, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49160, 49161, 2490, 22830, 351, 216, 49168, 2490, 22830, 284, 216, 49162, 811, 22830, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 13764, 3298, 836, 260, 3403, 40199, 281, 582, 6330, 30, 49154, 49158, 198, 57, 436, 1730, 288, 1863, 957, 4770, 281, 216, 49160, 49161, 2490, 284, 216, 49162, 811, 22830, 28, 347, 346, 4415, 30, 49154], "prefix_len": 183, "old_logprobs": [-2.0442397594451904, -1.95679771900177, -1.6277629137039185, -0.0003700763627421111, -0.687701940536499, -2.2691149711608887, -3.4628584384918213, -3.329988718032837, -1.5678906440734863, -0.23791392147541046, -0.17203575372695923, -0.14156284928321838, -1.3912184238433838, -0.013639586046338081, -0.028798207640647888, -0.01609804667532444, -0.09907423704862595, -1.1666998863220215, -3.4433422088623047, -4.206454753875732, -0.6633468866348267, -0.19042924046516418, -0.07345316559076309]}
+{"case_id": "state_update_04", "category": "state_update", "origin": "budgie", "answer": "Finalorted; I’ll resend it in one sentence.", "reward": -0.42999999999999994, "advantage": -0.5888992037956788, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 6508, 3048, 30, 378, 3791, 2004, 314, 216, 49160, 49161, 2490, 22830, 43, 6082, 418, 216, 49160, 49165, 42, 49161, 49159, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 6508, 3048, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49160, 49161, 2490, 22830, 351, 216, 49168, 2490, 22830, 284, 216, 49162, 811, 22830, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 13764, 3298, 836, 260, 3403, 40199, 281, 582, 6330, 30, 49154, 49158, 198, 29288, 8146, 43, 339, 417, 764, 499, 486, 357, 281, 582, 6330, 30, 49154], "prefix_len": 183, "old_logprobs": [-4.2942399978637695, -8.787094116210938, -5.781290531158447, -1.4003405570983887, -1.4937820434570312, -0.6389973759651184, -7.299647808074951, -0.777152955532074, -1.9243346452713013, -1.9918861389160156, -0.07626555860042572, -0.24808678030967712, -0.059657689183950424, -0.028145436197519302]}
+{"case_id": "state_update_04", "category": "state_update", "origin": "manual_ideal", "answer": "Book 9 adult tickets and 3 child tickets for entry at 16:20.", "reward": 1, "advantage": 1.9785576908013356, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 6508, 3048, 30, 378, 3791, 2004, 314, 216, 49160, 49161, 2490, 22830, 43, 6082, 418, 216, 49160, 49165, 42, 49161, 49159, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 6508, 3048, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49160, 49161, 2490, 22830, 351, 216, 49168, 2490, 22830, 284, 216, 49162, 811, 22830, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 13764, 3298, 836, 260, 3403, 40199, 281, 582, 6330, 30, 49154, 49158, 198, 15654, 216, 49168, 2490, 22830, 284, 216, 49162, 811, 22830, 327, 6082, 418, 216, 49160, 49165, 42, 49161, 49159, 30, 49154], "prefix_len": 183, "old_logprobs": [-9.85673999786377, -9.20533561706543, -2.354046106338501, -0.16022135317325592, -0.2826578915119171, -1.797003984451294, -0.04427941143512726, -0.011824870482087135, -0.034839436411857605, -0.013158626854419708, -0.8698741793632507, -7.3142805099487305, -0.6724840998649597, -0.0960933193564415, -0.0169005636125803, -0.1170908510684967, -0.00987926498055458, -0.01625123620033264, -0.009833347983658314, -0.27301594614982605, -0.010545482859015465]}
+{"case_id": "state_update_05", "category": "state_update", "origin": "budgie", "answer": "To send the issue to 8,000 readers on Tuesday, first plan to send the issue to 8,000xi + 6,500yi for 6,500 years or 6,500x + 6,500 = 7,500x + 6,500.", "reward": -0.42999999999999994, "advantage": -0.5888992037956788, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 21300, 4878, 30, 378, 3791, 2004, 314, 5091, 14801, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 43, 2138, 1761, 42, 11652, 17304, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 21300, 4878, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 5091, 14801, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 351, 5091, 14963, 288, 216, 49165, 28, 49164, 49159, 49159, 4419, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 260, 1574, 5091, 1252, 281, 582, 6330, 30, 49154, 49158, 198, 2068, 5091, 260, 2843, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 335, 14801, 28, 808, 1252, 288, 5091, 260, 2843, 288, 216, 49167, 28, 49159, 49159, 49159, 3951, 1232, 216, 49165, 28, 49164, 49159, 49159, 40196, 327, 216, 49165, 28, 49164, 49159, 49159, 929, 355, 216, 49165, 28, 49164, 49159, 49159, 104, 1232, 216, 49165, 28, 49164, 49159, 49159, 446, 216, 49166, 28, 49164, 49159, 49159, 104, 1232, 216, 49165, 28, 49164, 49159, 49159, 30, 49154], "prefix_len": 190, "old_logprobs": [-4.142342567443848, -0.4285723865032196, -0.8195868730545044, -5.92687463760376, -1.523445963859558, -0.5879451632499695, -0.24158625304698944, -0.24571163952350616, -0.012594802305102348, -0.0005535738891921937, -0.002789774676784873, -0.24449141323566437, -0.679291307926178, -1.8418140411376953, -0.22558711469173431, -5.198014736175537, -2.5863406658172607, -0.2972193956375122, -1.0318186283111572, -1.3145506381988525, -2.760875701904297, -0.30556535720825195, -0.8186837434768677, -1.073420524597168, -0.015300637111067772, -0.18602274358272552, -0.0022695516236126423, -0.023225879296660423, -10.131132125854492, -2.514826774597168, -0.092682845890522, -0.4241476058959961, -0.017452243715524673, -0.01563364267349243, -0.0037214565090835094, -0.008785877376794815, -1.9489936828613281, -3.2768795490264893, -1.1719837188720703, -0.6951864361763, -0.1805761456489563, -0.041393883526325226, -0.003614679677411914, -0.005909945350140333, -4.384097576141357, -4.3663411140441895, -1.140722632408142, -0.749198853969574, -0.7303641438484192, -0.07386989146471024, -0.004994060844182968, -0.009335437789559364, -1.4623682498931885, -0.5325133800506592, -0.034092437475919724, -0.21209536492824554, -0.16809850931167603, -0.010380680672824383, -0.0007372760446742177, -0.0020728071685880423, -1.4281175136566162, -0.07347752898931503, -0.8353811502456665, -0.18221211433410645, -1.015766978263855, -0.34029465913772583, -0.030237112194299698, -0.3517102897167206, -0.5056867003440857, -0.04553108289837837, -0.21789135038852692, -0.17754141986370087, -0.1042933315038681, -0.002185577293857932, -0.001570302527397871, -2.355341911315918, -0.12835323810577393]}
+{"case_id": "state_update_05", "category": "state_update", "origin": "budgie", "answer": "I will send the final update on the critical summer issue by 6,500 readers tomorrow, and I will make the other detail unchanged.", "reward": -0.22, "advantage": -0.2118600794142992, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 21300, 4878, 30, 378, 3791, 2004, 314, 5091, 14801, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 43, 2138, 1761, 42, 11652, 17304, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 21300, 4878, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 5091, 14801, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 351, 5091, 14963, 288, 216, 49165, 28, 49164, 49159, 49159, 4419, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 260, 1574, 5091, 1252, 281, 582, 6330, 30, 49154, 49158, 198, 57, 523, 5091, 260, 3403, 6850, 335, 260, 2609, 4018, 2843, 411, 216, 49165, 28, 49164, 49159, 49159, 4419, 13759, 28, 284, 339, 523, 919, 260, 550, 2202, 23814, 30, 49154], "prefix_len": 190, "old_logprobs": [-3.0060229301452637, -1.2424448728561401, -0.5515133142471313, -0.5130346417427063, -4.7831950187683105, -1.796970009803772, -1.532900094985962, -0.47287267446517944, -9.011159896850586, -5.338616371154785, -2.408080816268921, -3.1197924613952637, -2.695772647857666, -0.3421853184700012, -0.06882225722074509, -0.014625833369791508, -0.0024595982395112514, -0.006586035247892141, -0.12955838441848755, -3.5535101890563965, -3.252488851547241, -3.7707691192626953, -0.9342402815818787, -0.41890770196914673, -3.4290575981140137, -1.6713664531707764, -0.5288388133049011, -0.2688244581222534, -1.577697515487671, -0.5435155630111694, -0.01670362986624241]}
+{"case_id": "state_update_05", "category": "state_update", "origin": "budgie", "answer": "I will send the update to 8,000 readers on the 6th to 7th of the following week.", "reward": -0.42999999999999994, "advantage": -0.5888992037956788, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 21300, 4878, 30, 378, 3791, 2004, 314, 5091, 14801, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 43, 2138, 1761, 42, 11652, 17304, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 21300, 4878, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 5091, 14801, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 351, 5091, 14963, 288, 216, 49165, 28, 49164, 49159, 49159, 4419, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 260, 1574, 5091, 1252, 281, 582, 6330, 30, 49154, 49158, 198, 57, 523, 5091, 260, 6850, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 335, 260, 216, 49165, 373, 288, 216, 49166, 373, 282, 260, 1695, 2605, 30, 49154], "prefix_len": 190, "old_logprobs": [-3.0060229301452637, -1.2424448728561401, -0.5515133142471313, -0.5130346417427063, -3.1581950187683105, -1.4380321502685547, -0.5557759404182434, -0.31851017475128174, -0.17199499905109406, -0.03282024711370468, -0.0006173135479912162, -0.002619171282276511, -0.26521164178848267, -0.27626997232437134, -2.4549174308776855, -2.681670665740967, -1.5346542596817017, -0.2873789668083191, -4.052506923675537, -0.4985194802284241, -1.0019021034240723, -0.004868676420301199, -0.586889922618866, -1.5764961242675781, -1.0974599123001099, -0.39694005250930786, -0.18614566326141357, -0.021511100232601166]}
+{"case_id": "state_update_05", "category": "state_update", "origin": "budgie", "answer": "Write a concise message to send to 8,000 readers on a Friday to get to know the story behind the story.", "reward": -0.42999999999999994, "advantage": -0.5888992037956788, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 21300, 4878, 30, 378, 3791, 2004, 314, 5091, 14801, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 43, 2138, 1761, 42, 11652, 17304, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 21300, 4878, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 5091, 14801, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 351, 5091, 14963, 288, 216, 49165, 28, 49164, 49159, 49159, 4419, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 260, 1574, 5091, 1252, 281, 582, 6330, 30, 49154, 49158, 198, 19161, 253, 19484, 3714, 288, 5091, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 335, 253, 11655, 288, 820, 288, 699, 260, 1977, 2893, 260, 1977, 30, 49154], "prefix_len": 190, "old_logprobs": [-4.631022930145264, -1.0301132202148438, -2.741150140762329, -3.116569757461548, -0.6857938766479492, -0.8834426403045654, -0.7747357487678528, -0.5956671833992004, -0.04777552932500839, -0.20744043588638306, -0.003901965217664838, -0.00014256415306590497, -0.0007918321643956006, -0.19599242508411407, -0.9128596186637878, -2.202897071838379, -2.8578052520751953, -3.0254077911376953, -3.5086278915405273, -4.147344589233398, -0.9087522029876709, -1.2935587167739868, -3.2551355361938477, -3.1779041290283203, -0.36537444591522217, -2.6803812980651855, -0.7012664079666138, -0.020527886226773262]}
+{"case_id": "state_update_05", "category": "state_update", "origin": "manual_ideal", "answer": "Send “Spring Notes” on Thursday to 6,500 readers.", "reward": 1, "advantage": 1.9785576908013356, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 21300, 4878, 30, 378, 3791, 2004, 314, 5091, 14801, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 43, 2138, 1761, 42, 11652, 17304, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 21300, 4878, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 5091, 14801, 288, 216, 49167, 28, 49159, 49159, 49159, 4419, 351, 5091, 14963, 288, 216, 49165, 28, 49164, 49159, 49159, 4419, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 260, 1574, 5091, 1252, 281, 582, 6330, 30, 49154, 49158, 198, 29752, 619, 32155, 17304, 573, 335, 14963, 288, 216, 49165, 28, 49164, 49159, 49159, 4419, 30, 49154], "prefix_len": 190, "old_logprobs": [-11.568523406982422, -10.042241096496582, -6.641674518585205, -7.720275402069092, -0.2262086272239685, -1.4692745208740234, -3.018886089324951, -0.9353724122047424, -1.024758219718933, -0.033710088580846786, -0.05431262403726578, -0.004838899709284306, -0.0023975218646228313, -0.00045885046711191535, -0.07742434740066528, -0.4486429691314697, -0.028442908078432083]}
+{"case_id": "state_update_06", "category": "state_update", "origin": "budgie", "answer": "I will keep the previous details in mind, keeping the rule in mind and [\"K-17 at 4°C\", \"K-19 at -20°C\"].", "reward": -0.010000000000000009, "advantage": -0.06502191713438298, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 1587, 4315, 30, 378, 3791, 2004, 314, 4368, 659, 29, 49160, 49166, 418, 216, 49163, 4742, 51, 43, 35115, 7934, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 1587, 4315, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 4368, 659, 29, 49160, 49166, 418, 216, 49163, 4742, 51, 351, 4368, 659, 29, 49160, 49168, 418, 731, 49161, 49159, 4742, 51, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 3403, 4759, 2004, 281, 582, 6330, 30, 49154, 49158, 198, 57, 523, 1446, 260, 2672, 3841, 281, 1945, 28, 4825, 260, 3938, 281, 1945, 284, 9523, 59, 29, 49160, 49166, 418, 216, 49163, 4742, 51, 1002, 476, 59, 29, 49160, 49168, 418, 731, 49161, 49159, 4742, 51, 16777, 49154], "prefix_len": 188, "old_logprobs": [-1.2065651416778564, -2.1260573863983154, -1.013885498046875, -0.095981664955616, -3.779801845550537, -1.227989673614502, -1.7078089714050293, -1.3219845294952393, -2.1555633544921875, -3.49072003364563, -0.54564368724823, -7.427108287811279, -0.6096417307853699, -0.04531646519899368, -1.3605835437774658, -12.719605445861816, -0.32264775037765503, -0.028116807341575623, -0.008502001874148846, -0.5331705212593079, -1.0328587293624878, -0.29241836071014404, -0.002146204235032201, -0.011557881720364094, -0.00035446559195406735, -1.2491272687911987, -0.016203850507736206, -0.061645206063985825, -0.0010548033751547337, -0.002367552602663636, -0.0071356212720274925, -0.0011478032683953643, -0.0019051750423386693, -0.0001445904199499637, -4.494089080253616e-05, -0.001498410478234291, -4.076874756719917e-05, -2.4213907718658447, -0.11879850178956985]}
+{"case_id": "state_update_06", "category": "state_update", "origin": "budgie", "answer": "I am moving to the new storage record from the new plane at -20°C, now labeled K-19 in the leanings of the new plane.", "reward": -0.010000000000000009, "advantage": -0.06502191713438298, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 1587, 4315, 30, 378, 3791, 2004, 314, 4368, 659, 29, 49160, 49166, 418, 216, 49163, 4742, 51, 43, 35115, 7934, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 1587, 4315, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 4368, 659, 29, 49160, 49166, 418, 216, 49163, 4742, 51, 351, 4368, 659, 29, 49160, 49168, 418, 731, 49161, 49159, 4742, 51, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 3403, 4759, 2004, 281, 582, 6330, 30, 49154, 49158, 198, 57, 744, 4138, 288, 260, 725, 4759, 2004, 429, 260, 725, 8303, 418, 731, 49161, 49159, 4742, 51, 28, 1209, 12380, 659, 29, 49160, 49168, 281, 260, 10681, 616, 282, 260, 725, 8303, 30, 49154], "prefix_len": 188, "old_logprobs": [-1.2065651416778564, -2.9385573863983154, -6.3906145095825195, -0.8969383835792542, -2.299490451812744, -2.033522605895996, -0.7091928124427795, -1.5068151950836182, -4.915513038635254, -1.2539650201797485, -2.5666773319244385, -9.033952713012695, -2.5159926414489746, -0.8134377002716064, -0.12497220933437347, -0.02185429073870182, -0.2371794730424881, -0.001207337831147015, -2.4941487312316895, -3.917492389678955, -1.9382885694503784, -0.17243534326553345, -0.008973272517323494, -0.009349845349788666, -0.15208397805690765, -4.238330364227295, -0.6028160452842712, -10.857549667358398, -4.370842933654785, -0.5582073330879211, -0.4653944671154022, -1.820735216140747, -1.5674768686294556, -0.33150702714920044, -0.029016977176070213]}
+{"case_id": "state_update_06", "category": "state_update", "origin": "budgie", "answer": "snee posss a new storage record.", "reward": -0.42999999999999994, "advantage": -0.8682338346767601, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 1587, 4315, 30, 378, 3791, 2004, 314, 4368, 659, 29, 49160, 49166, 418, 216, 49163, 4742, 51, 43, 35115, 7934, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 1587, 4315, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 4368, 659, 29, 49160, 49166, 418, 216, 49163, 4742, 51, 351, 4368, 659, 29, 49160, 49168, 418, 731, 49161, 49159, 4742, 51, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 3403, 4759, 2004, 281, 582, 6330, 30, 49154, 49158, 198, 99, 23236, 1178, 99, 253, 725, 4759, 2004, 30, 49154], "prefix_len": 188, "old_logprobs": [-14.269064903259277, -11.291692733764648, -12.39181137084961, -2.6987996101379395, -2.285006523132324, -1.8529386520385742, -3.2221665382385254, -0.40150588750839233, -2.0529940128326416, -0.04237131401896477]}
+{"case_id": "state_update_06", "category": "state_update", "origin": "budgie", "answer": "I'll keep the other detail as is.", "reward": -0.42999999999999994, "advantage": -0.8682338346767601, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 1587, 4315, 30, 378, 3791, 2004, 314, 4368, 659, 29, 49160, 49166, 418, 216, 49163, 4742, 51, 43, 35115, 7934, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 1587, 4315, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 4368, 659, 29, 49160, 49166, 418, 216, 49163, 4742, 51, 351, 4368, 659, 29, 49160, 49168, 418, 731, 49161, 49159, 4742, 51, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 3403, 4759, 2004, 281, 582, 6330, 30, 49154, 49158, 198, 57, 3060, 1446, 260, 550, 2202, 347, 314, 30, 49154], "prefix_len": 188, "old_logprobs": [-1.2065651416778564, -3.6260573863983154, -0.9265859127044678, -0.222093865275383, -0.5666009783744812, -0.05939957872033119, -1.0292214155197144, -0.213322713971138, -0.2889602482318878, -0.028217755258083344]}
+{"case_id": "state_update_06", "category": "state_update", "origin": "manual_ideal", "answer": "Store sample K-19 at -20°C in an amber container.", "reward": 1, "advantage": 1.866511503622286, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 1587, 4315, 30, 378, 3791, 2004, 314, 4368, 659, 29, 49160, 49166, 418, 216, 49163, 4742, 51, 43, 35115, 7934, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 1587, 4315, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 4368, 659, 29, 49160, 49166, 418, 216, 49163, 4742, 51, 351, 4368, 659, 29, 49160, 49168, 418, 731, 49161, 49159, 4742, 51, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 3403, 4759, 2004, 281, 582, 6330, 30, 49154, 49158, 198, 29405, 4315, 659, 29, 49160, 49168, 418, 731, 49161, 49159, 4742, 51, 281, 354, 35115, 7934, 30, 49154], "prefix_len": 188, "old_logprobs": [-9.269064903259277, -5.845562934875488, -4.996070861816406, -0.008692994713783264, -0.005134609993547201, -0.10507577657699585, -0.7274929285049438, -0.2556789517402649, -0.016765058040618896, -0.003941981587558985, -0.06779260188341141, -0.0011767374817281961, -2.274580478668213, -3.8804540634155273, -10.2813081741333, -0.5819387435913086, -0.22399911284446716, -0.014042736031115055]}
+{"case_id": "state_update_07", "category": "state_update", "origin": "budgie", "answer": "80 minutes.", "reward": -0.42999999999999994, "advantage": -0.6707581132756656, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 2534, 6094, 30, 378, 3791, 2004, 314, 216, 49168, 49159, 3487, 327, 216, 49161, 49164, 701, 43, 25906, 1106, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 2534, 6094, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49168, 49159, 3487, 327, 216, 49161, 49164, 701, 351, 216, 49165, 49159, 3487, 327, 216, 49160, 49167, 701, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 260, 3403, 8687, 28, 5936, 2203, 28, 284, 1106, 30, 49154, 49158, 198, 49167, 49159, 3487, 30, 49154], "prefix_len": 186, "old_logprobs": [-3.6086792945861816, -0.7768667340278625, -2.3086578845977783, -1.7154515981674194, -0.20238806307315826]}
+{"case_id": "state_update_07", "category": "state_update", "origin": "budgie", "answer": "25+18+60=103.", "reward": -0.42999999999999994, "advantage": -0.6707581132756656, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 2534, 6094, 30, 378, 3791, 2004, 314, 216, 49168, 49159, 3487, 327, 216, 49161, 49164, 701, 43, 25906, 1106, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 2534, 6094, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49168, 49159, 3487, 327, 216, 49161, 49164, 701, 351, 216, 49165, 49159, 3487, 327, 216, 49160, 49167, 701, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 260, 3403, 8687, 28, 5936, 2203, 28, 284, 1106, 30, 49154, 49158, 198, 49161, 49164, 27, 49160, 49167, 27, 49165, 49159, 45, 49160, 49159, 49162, 30, 49154], "prefix_len": 186, "old_logprobs": [-3.3586792945861816, -0.22043584287166595, -3.7247095108032227, -0.8901780843734741, -0.04800314083695412, -1.1062546968460083, -0.5261842012405396, -0.038345709443092346, -0.4033920466899872, -0.06425756961107254, -0.13932521641254425, -0.0011104138102382421, -1.7382471561431885, -0.2013663500547409]}
+{"case_id": "state_update_07", "category": "state_update", "origin": "budgie", "answer": "18.5.", "reward": -0.42999999999999994, "advantage": -0.6707581132756656, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 2534, 6094, 30, 378, 3791, 2004, 314, 216, 49168, 49159, 3487, 327, 216, 49161, 49164, 701, 43, 25906, 1106, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 2534, 6094, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49168, 49159, 3487, 327, 216, 49161, 49164, 701, 351, 216, 49165, 49159, 3487, 327, 216, 49160, 49167, 701, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 260, 3403, 8687, 28, 5936, 2203, 28, 284, 1106, 30, 49154, 49158, 198, 49160, 49167, 30, 49164, 30, 49154], "prefix_len": 186, "old_logprobs": [-2.2336792945861816, -0.5642526745796204, -2.405764102935791, -1.953848958015442, -3.315310001373291, -1.9623727798461914]}
+{"case_id": "state_update_07", "category": "state_update", "origin": "budgie", "answer": "60 minutes for 18 people.", "reward": 0, "advantage": 0.1045805660483564, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 2534, 6094, 30, 378, 3791, 2004, 314, 216, 49168, 49159, 3487, 327, 216, 49161, 49164, 701, 43, 25906, 1106, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 2534, 6094, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49168, 49159, 3487, 327, 216, 49161, 49164, 701, 351, 216, 49165, 49159, 3487, 327, 216, 49160, 49167, 701, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 260, 3403, 8687, 28, 5936, 2203, 28, 284, 1106, 30, 49154, 49158, 198, 49165, 49159, 3487, 327, 216, 49160, 49167, 701, 30, 49154], "prefix_len": 186, "old_logprobs": [-4.108679294586182, -0.07651273161172867, -0.5226006507873535, -1.1168982982635498, -0.09121653437614441, -0.05781902000308037, -0.012628233060240746, -0.021688543260097504, -1.147094964981079, -0.05306094139814377]}
+{"case_id": "state_update_07", "category": "state_update", "origin": "manual_ideal", "answer": "Run a 60-minute beginner session for 18 people.", "reward": 1, "advantage": 1.9076937737786406, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 2534, 6094, 30, 378, 3791, 2004, 314, 216, 49168, 49159, 3487, 327, 216, 49161, 49164, 701, 43, 25906, 1106, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 2534, 6094, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 216, 49168, 49159, 3487, 327, 216, 49161, 49164, 701, 351, 216, 49165, 49159, 3487, 327, 216, 49160, 49167, 701, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 26533, 260, 3403, 8687, 28, 5936, 2203, 28, 284, 1106, 30, 49154, 49158, 198, 17249, 253, 216, 49165, 49159, 29, 15598, 25906, 6094, 327, 216, 49160, 49167, 701, 30, 49154], "prefix_len": 186, "old_logprobs": [-10.358678817749023, -2.3336992263793945, -2.5680956840515137, -0.8826658129692078, -0.058629002422094345, -0.06797950714826584, -0.0022018731106072664, -10.550015449523926, -1.9191888570785522, -3.4754137992858887, -0.21925224363803864, -0.6674404144287109, -0.030846331268548965, -0.13625475764274597, -0.594115138053894, -0.08746162801980972]}
+{"case_id": "state_update_08", "category": "state_update", "origin": "budgie", "answer": "I’m ready and waiting for you!", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 25446, 3116, 30, 378, 3791, 2004, 314, 20851, 11241, 327, 216, 49163, 49159, 43, 787, 28087, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 25446, 3116, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 20851, 11241, 327, 216, 49163, 49159, 351, 19913, 11241, 327, 216, 49162, 49161, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 1574, 25446, 9702, 281, 582, 6330, 30, 49154, 49158, 198, 57, 417, 93, 4421, 284, 7003, 327, 346, 17, 49154], "prefix_len": 175, "old_logprobs": [-1.3456062078475952, -2.5628437995910645, -0.38350096344947815, -2.384777784347534, -4.927680969238281, -1.4390218257904053, -0.8731493949890137, -3.1222002506256104, -4.570001125335693, -0.29057496786117554]}
+{"case_id": "state_update_08", "category": "state_update", "origin": "budgie", "answer": "I’ll be using the previous data.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 25446, 3116, 30, 378, 3791, 2004, 314, 20851, 11241, 327, 216, 49163, 49159, 43, 787, 28087, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 25446, 3116, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 20851, 11241, 327, 216, 49163, 49159, 351, 19913, 11241, 327, 216, 49162, 49161, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 1574, 25446, 9702, 281, 582, 6330, 30, 49154, 49158, 198, 57, 417, 764, 325, 1015, 260, 2672, 940, 30, 49154], "prefix_len": 175, "old_logprobs": [-1.3456062078475952, -2.5628437995910645, -2.2585010528564453, -4.468459606170654, -5.2908220291137695, -0.5310598015785217, -6.4495720863342285, -3.0083093643188477, -1.8824471235275269, -0.03498989716172218]}
+{"case_id": "state_update_08", "category": "state_update", "origin": "budgie", "answer": "I’ll put the new value on the gemstones in a single sentence.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 25446, 3116, 30, 378, 3791, 2004, 314, 20851, 11241, 327, 216, 49163, 49159, 43, 787, 28087, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 25446, 3116, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 20851, 11241, 327, 216, 49163, 49159, 351, 19913, 11241, 327, 216, 49162, 49161, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 1574, 25446, 9702, 281, 582, 6330, 30, 49154, 49158, 198, 57, 417, 764, 1830, 260, 725, 1685, 335, 260, 44948, 281, 253, 2244, 6330, 30, 49154], "prefix_len": 175, "old_logprobs": [-1.3456062078475952, -2.5628437995910645, -2.2585010528564453, -4.593459606170654, -0.7080067992210388, -1.9453485012054443, -2.2202134132385254, -3.4975662231445312, -0.5213855504989624, -10.809959411621094, -2.597259044647217, -2.6953859329223633, -1.6028194427490234, -0.09010271728038788, -0.25971555709838867, -0.05474214628338814]}
+{"case_id": "state_update_08", "category": "state_update", "origin": "budgie", "answer": "I will now state the current catering requirement in one sentence.\n\nFinal answer: 40", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 25446, 3116, 30, 378, 3791, 2004, 314, 20851, 11241, 327, 216, 49163, 49159, 43, 787, 28087, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 25446, 3116, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 20851, 11241, 327, 216, 49163, 49159, 351, 19913, 11241, 327, 216, 49162, 49161, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 1574, 25446, 9702, 281, 582, 6330, 30, 49154, 49158, 198, 57, 523, 1209, 1215, 260, 1574, 25446, 9702, 281, 582, 6330, 30, 198, 198, 29288, 2988, 42, 216, 49163, 49159, 49154], "prefix_len": 175, "old_logprobs": [-1.3456062078475952, -2.0628437995910645, -1.6740164756774902, -0.6133262515068054, -0.1693822145462036, -0.07913824915885925, -0.38156333565711975, -0.07772880047559738, -0.5929628014564514, -0.0779685527086258, -0.07235874235630035, -0.09358212351799011, -3.1231069564819336, -0.11207514256238937, -1.0632978677749634, -0.19515864551067352, -0.0008688965463079512, -2.6808865070343018, -0.18143737316131592, -0.002292506629601121, -0.2566838562488556]}
+{"case_id": "state_update_08", "category": "state_update", "origin": "manual_ideal", "answer": "Provide a peanut-free vegan lunch for 32 people.", "reward": 1, "advantage": 1.9999965035026162, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 35435, 451, 25446, 3116, 30, 378, 3791, 2004, 314, 20851, 11241, 327, 216, 49163, 49159, 43, 787, 28087, 30, 49154, 49158, 198, 24664, 277, 260, 3791, 25446, 3116, 3841, 30, 49154, 49157, 198, 16116, 8676, 42, 4819, 20851, 11241, 327, 216, 49163, 49159, 351, 19913, 11241, 327, 216, 49162, 49161, 30, 5545, 260, 550, 2202, 23814, 30, 49154, 49158, 198, 7134, 5623, 43, 339, 417, 764, 722, 260, 19611, 1685, 429, 1209, 335, 30, 49154, 49157, 198, 10024, 260, 1574, 25446, 9702, 281, 582, 6330, 30, 49154, 49158, 198, 37264, 253, 21856, 29, 5725, 19913, 11241, 327, 216, 49162, 49161, 701, 30, 49154], "prefix_len": 175, "old_logprobs": [-10.533105850219727, -0.5862517952919006, -11.68420124053955, -0.2806343138217926, -1.005324125289917, -8.487152099609375, -1.554023265838623, -0.5555850267410278, -0.26638659834861755, -0.16384416818618774, -0.008100276812911034, -2.250577926635742, -1.027247667312622, -0.02073364332318306]}
+{"case_id": "already_tried_00", "category": "already_tried", "origin": "budgie", "answer": "I will not repeat any of these next-step actions.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 17066, 338, 13766, 288, 15030, 29, 14366, 564, 2526, 3470, 7976, 30, 1644, 1869, 339, 457, 2221, 26389, 277, 260, 17066, 43, 298, 8222, 288, 260, 3082, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 523, 441, 9960, 750, 282, 623, 1867, 29, 5528, 3643, 30, 49154], "prefix_len": 164, "old_logprobs": [-0.7194951176643372, -2.1211442947387695, -1.4568545818328857, -1.91269052028656, -0.38842830061912537, -0.8333019018173218, -1.68100106716156, -1.2612073421478271, -2.1116442680358887, -1.8285363912582397, -0.33898022770881653, -0.35161903500556946, -0.18513654172420502]}
+{"case_id": "already_tried_00", "category": "already_tried", "origin": "budgie", "answer": "I will not send you any further information, and I have already tried the previous steps.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 17066, 338, 13766, 288, 15030, 29, 14366, 564, 2526, 3470, 7976, 30, 1644, 1869, 339, 457, 2221, 26389, 277, 260, 17066, 43, 298, 8222, 288, 260, 3082, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 523, 441, 5091, 346, 750, 2030, 1096, 28, 284, 339, 457, 2221, 5655, 260, 2672, 3301, 30, 49154], "prefix_len": 164, "old_logprobs": [-0.7194951176643372, -2.1211442947387695, -1.4568545818328857, -1.47519052028656, -0.9855451583862305, -1.9222025871276855, -0.9454700350761414, -3.6108195781707764, -2.903158664703369, -1.7718747854232788, -0.40648573637008667, -4.54929780960083, -0.8945069909095764, -3.291956901550293, -1.640981912612915, -2.9177796840667725, -1.570460557937622, -0.20784103870391846, -0.13063205778598785]}
+{"case_id": "already_tried_00", "category": "already_tried", "origin": "budgie", "answer": "I can’t be sure what to do next, but I can provide two next-step actions if it’s necessary.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 17066, 338, 13766, 288, 15030, 29, 14366, 564, 2526, 3470, 7976, 30, 1644, 1869, 339, 457, 2221, 26389, 277, 260, 17066, 43, 298, 8222, 288, 260, 3082, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 416, 417, 100, 325, 2090, 732, 288, 536, 1867, 28, 564, 339, 416, 1538, 827, 1867, 29, 5528, 3643, 585, 357, 417, 99, 2371, 30, 49154], "prefix_len": 164, "old_logprobs": [-0.7328633666038513, -3.1183958053588867, -3.4893603324890137, -0.0010413468116894364, -6.0204973220825195, -1.0543689727783203, -1.2151226997375488, -1.9304155111312866, -0.15296395123004913, -1.2505111694335938, -0.7279033660888672, -0.38434556126594543, -0.6443548798561096, -0.9756972789764404, -3.346477746963501, -2.861705780029297, -1.0412417650222778, -2.58689546585083, -1.4772539138793945, -1.6506705284118652, -3.823317766189575, -3.134157657623291, -1.4300200939178467, -0.011660868301987648, -2.417806386947632, -0.31221234798431396, -0.11062686890363693]}
+{"case_id": "already_tried_00", "category": "already_tried", "origin": "budgie", "answer": "I will not repeat any of them again.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 17066, 338, 13766, 288, 15030, 29, 14366, 564, 2526, 3470, 7976, 30, 1644, 1869, 339, 457, 2221, 26389, 277, 260, 17066, 43, 298, 8222, 288, 260, 3082, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 523, 441, 9960, 750, 282, 601, 1163, 30, 49154], "prefix_len": 164, "old_logprobs": [-0.7328633666038513, -2.1183958053588867, -1.4880273342132568, -1.9018419981002808, -0.3892260193824768, -0.8373094201087952, -1.9331645965576172, -1.746121883392334, -0.18261192739009857, -0.10391487926244736]}
+{"case_id": "already_tried_00", "category": "already_tried", "origin": "manual_ideal", "answer": "1. Renew the dhcp lease.\n2. Test with a different dns resolver.", "reward": 1, "advantage": 1.9999965035026162, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 17066, 338, 13766, 288, 15030, 29, 14366, 564, 2526, 3470, 7976, 30, 1644, 1869, 339, 457, 2221, 26389, 277, 260, 17066, 43, 298, 8222, 288, 260, 3082, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 22043, 260, 29818, 15766, 23745, 30, 198, 49161, 30, 4246, 351, 253, 896, 287, 5400, 5838, 331, 30, 49154], "prefix_len": 164, "old_logprobs": [-2.482863426208496, -0.06908641010522842, -13.755659103393555, -5.667713165283203, -12.284743309020996, -8.818910598754883, -12.686405181884766, -2.425290584564209, -0.38481467962265015, -0.007976227439939976, -0.0013190149329602718, -6.052673816680908, -6.6858906745910645, -1.893613576889038, -2.2719883918762207, -7.225600242614746, -3.9939334392547607, -10.051290512084961, -0.07764541357755661, -0.49846941232681274, -0.650230348110199]}
+{"case_id": "already_tried_01", "category": "already_tried", "origin": "budgie", "answer": "I need the printer to print two next-step fixations.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 11825, 6216, 22495, 5158, 30, 1644, 1869, 339, 457, 2221, 7233, 260, 1941, 43, 26389, 277, 260, 11825, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 737, 260, 11825, 288, 1628, 827, 1867, 29, 5528, 4107, 491, 30, 49154], "prefix_len": 155, "old_logprobs": [-0.5592483282089233, -2.220686912536621, -2.363193988800049, -1.0757242441177368, -0.21017511188983917, -2.167112350463867, -2.037212371826172, -2.2131776809692383, -1.9196587800979614, -2.4990034103393555, -6.216026306152344, -1.014387607574463, -0.7829963564872742, -0.24909886717796326]}
+{"case_id": "already_tried_01", "category": "already_tried", "origin": "budgie", "answer": "I need one nextFeb informative:\n\nI need one next Mar 1.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 11825, 6216, 22495, 5158, 30, 1644, 1869, 339, 457, 2221, 7233, 260, 1941, 43, 26389, 277, 260, 11825, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 737, 582, 1867, 16516, 17628, 42, 198, 198, 57, 737, 582, 1867, 2828, 216, 49160, 30, 49154], "prefix_len": 155, "old_logprobs": [-0.5592483282089233, -2.220686912536621, -4.144443988800049, -1.4297032356262207, -12.130006790161133, -10.467945098876953, -3.9653382301330566, -2.057284355163574, -0.9557596445083618, -0.6312481164932251, -1.49005925655365, -1.1127692461013794, -0.20610526204109192, -3.8287341594696045, -2.016425132751465, -1.261644721031189, -3.550677537918091, -0.39650923013687134]}
+{"case_id": "already_tried_01", "category": "already_tried", "origin": "budgie", "answer": "I need one next consumed: the printer stops working.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 11825, 6216, 22495, 5158, 30, 1644, 1869, 339, 457, 2221, 7233, 260, 1941, 43, 26389, 277, 260, 11825, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 737, 582, 1867, 9495, 42, 260, 11825, 12326, 1891, 30, 49154], "prefix_len": 155, "old_logprobs": [-0.5592483282089233, -2.220686912536621, -4.144443988800049, -1.4297032356262207, -9.010225296020508, -3.2725915908813477, -3.121455669403076, -0.479411780834198, -4.208871841430664, -0.992333710193634, -0.6869819164276123, -0.09533219784498215]}
+{"case_id": "already_tried_01", "category": "already_tried", "origin": "budgie", "answer": "I need to re-implement the printer production process for a more accurate and accurate result.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 11825, 6216, 22495, 5158, 30, 1644, 1869, 339, 457, 2221, 7233, 260, 1941, 43, 26389, 277, 260, 11825, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 737, 288, 298, 29, 306, 11954, 260, 11825, 2262, 980, 327, 253, 540, 5161, 284, 5161, 966, 30, 49154], "prefix_len": 155, "old_logprobs": [-0.5592483282089233, -2.220686912536621, -1.8319441080093384, -4.573211669921875, -0.5115247964859009, -4.322471618652344, -0.6139869689941406, -0.42282530665397644, -1.4981882572174072, -3.751455307006836, -0.6405293941497803, -3.821859836578369, -1.6617761850357056, -2.168039560317993, -0.21423731744289398, -1.6535143852233887, -1.1454740762710571, -1.6005107164382935, -0.033682309091091156, -0.22940167784690857]}
+{"case_id": "already_tried_01", "category": "already_tried", "origin": "manual_ideal", "answer": "1. Check the ink or toner level.\n2. Run the print-head cleaning utility.", "reward": 1, "advantage": 1.9999965035026162, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 11825, 6216, 22495, 5158, 30, 1644, 1869, 339, 457, 2221, 7233, 260, 1941, 43, 26389, 277, 260, 11825, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 7023, 260, 12923, 355, 9241, 259, 1106, 30, 198, 49161, 30, 13823, 260, 1628, 29, 2699, 7811, 9981, 30, 49154], "prefix_len": 155, "old_logprobs": [-2.809630870819092, -0.0623425655066967, -3.984689950942993, -0.41377007961273193, -8.475607872009277, -7.085564613342285, -6.309340000152588, -0.0790967270731926, -1.7886168956756592, -0.9768006801605225, -0.3927249014377594, -0.0077149523422122, -0.0006202919175848365, -7.081876754760742, -0.28889158368110657, -4.394766807556152, -4.46046257019043, -5.377265930175781, -6.867912769317627, -10.59903335571289, -0.14742140471935272, -2.7028284072875977]}
+{"case_id": "already_tried_02", "category": "already_tried", "origin": "budgie", "answer": "You can do this by using the following two next-use actions:", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 5905, 2754, 10900, 253, 2399, 29, 254, 29, 1726, 4514, 30, 1644, 1869, 339, 457, 2221, 26389, 277, 260, 2754, 43, 4247, 260, 11177, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 2683, 416, 536, 451, 411, 1015, 260, 1695, 827, 1867, 29, 1726, 3643, 42, 49154], "prefix_len": 161, "old_logprobs": [-3.7118186950683594, -1.0271368026733398, -1.7752652168273926, -1.733241319656372, -0.47820156812667847, -2.652477502822876, -0.5645446181297302, -2.057248115539551, -1.6917212009429932, -1.0014983415603638, -1.9772089719772339, -3.8095061779022217, -2.4984772205352783, -0.7805072069168091, -1.5580804347991943]}
+{"case_id": "already_tried_02", "category": "already_tried", "origin": "budgie", "answer": "First, try to change why the service is currently not working.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 5905, 2754, 10900, 253, 2399, 29, 254, 29, 1726, 4514, 30, 1644, 1869, 339, 457, 2221, 26389, 277, 260, 2754, 43, 4247, 260, 11177, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 5345, 28, 1576, 288, 1363, 1701, 260, 2754, 314, 4112, 441, 1891, 30, 49154], "prefix_len": 161, "old_logprobs": [-4.211818695068359, -0.2274508774280548, -2.7606935501098633, -0.7162795066833496, -5.488691329956055, -7.687767028808594, -0.2882246673107147, -2.5853111743927, -0.33569517731666565, -4.671785354614258, -1.2522302865982056, -0.15514476597309113, -0.22308994829654694, -1.1967992782592773]}
+{"case_id": "already_tried_02", "category": "already_tried", "origin": "budgie", "answer": "I will give you the two next maforim steps.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 5905, 2754, 10900, 253, 2399, 29, 254, 29, 1726, 4514, 30, 1644, 1869, 339, 457, 2221, 26389, 277, 260, 2754, 43, 4247, 260, 11177, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 523, 1928, 346, 260, 827, 1867, 12102, 1710, 306, 3301, 30, 49154], "prefix_len": 161, "old_logprobs": [-0.7118187546730042, -2.233222484588623, -1.8590795993804932, -0.4332693815231323, -2.0811285972595215, -1.6649999618530273, -0.06441239267587662, -9.627325057983398, -7.730014324188232, -6.876345634460449, -3.474062204360962, -0.747039258480072, -0.16602161526679993]}
+{"case_id": "already_tried_02", "category": "already_tried", "origin": "budgie", "answer": "I need your help with identifying the next exterminability issue.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 5905, 2754, 10900, 253, 2399, 29, 254, 29, 1726, 4514, 30, 1644, 1869, 339, 457, 2221, 26389, 277, 260, 2754, 43, 4247, 260, 11177, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 737, 469, 724, 351, 7401, 260, 1867, 41094, 1307, 2843, 30, 49154], "prefix_len": 161, "old_logprobs": [-0.7118187546730042, -2.045722484588623, -2.34932279586792, -0.7633939385414124, -0.24455377459526062, -4.7122392654418945, -0.9234203100204468, -1.2850109338760376, -10.754249572753906, -4.34330415725708, -0.7045241594314575, -0.7333616018295288, -0.6408247351646423]}
+{"case_id": "already_tried_02", "category": "already_tried", "origin": "manual_ideal", "answer": "1. Identify the process holding the port.\n2. Stop it or configure a different port.", "reward": 1, "advantage": 1.9999965035026162, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 5905, 2754, 10900, 253, 2399, 29, 254, 29, 1726, 4514, 30, 1644, 1869, 339, 457, 2221, 26389, 277, 260, 2754, 43, 4247, 260, 11177, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 12716, 260, 980, 6961, 260, 2399, 30, 198, 49161, 30, 17178, 357, 355, 24095, 253, 896, 2399, 30, 49154], "prefix_len": 161, "old_logprobs": [-2.4686391353607178, -0.062346819788217545, -7.235079765319824, -0.3474428653717041, -7.162546634674072, -7.368337154388428, -1.0699586868286133, -4.912797927856445, -1.5900094509124756, -0.347547709941864, -0.023412451148033142, -0.0011472079204395413, -6.7536211013793945, -4.899114608764648, -3.894364833831787, -10.822293281555176, -1.8073842525482178, -6.031850337982178, -1.6463570594787598, -0.42962563037872314, -2.0667316913604736]}
+{"case_id": "already_tried_03", "category": "already_tried", "origin": "budgie", "answer": "I’ll fix the phone and then tell you the next one.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 5460, 338, 523, 441, 5363, 30, 1644, 1869, 339, 457, 2221, 5201, 2053, 277, 260, 5460, 43, 5655, 1372, 4923, 19558, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 417, 764, 4107, 260, 5460, 284, 965, 2505, 346, 260, 1867, 582, 30, 49154], "prefix_len": 158, "old_logprobs": [-0.6716872453689575, -2.0992231369018555, -1.0387156009674072, -5.899169921875, -0.7959761619567871, -3.0144712924957275, -1.8277387619018555, -2.2094242572784424, -3.5041115283966064, -0.19307486712932587, -1.9456530809402466, -0.05143069103360176, -0.9977462291717529, -0.3777262568473816, -0.07041953504085541]}
+{"case_id": "already_tried_03", "category": "already_tried", "origin": "budgie", "answer": "I received an error in the phone, so I cannot proceed.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 5460, 338, 523, 441, 5363, 30, 1644, 1869, 339, 457, 2221, 5201, 2053, 277, 260, 5460, 43, 5655, 1372, 4923, 19558, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 3656, 354, 4514, 281, 260, 5460, 28, 588, 339, 2526, 7219, 30, 49154], "prefix_len": 158, "old_logprobs": [-0.6716872453689575, -6.7867231369018555, -1.5228455066680908, -0.6393566131591797, -2.448225736618042, -1.3416097164154053, -2.9956531524658203, -1.6182492971420288, -0.5996202230453491, -0.09552595019340515, -2.2415199279785156, -2.8164241313934326, -1.1471611261367798, -0.21686981618404388]}
+{"case_id": "already_tried_03", "category": "already_tried", "origin": "budgie", "answer": "I will not receive further information about this new call.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 5460, 338, 523, 441, 5363, 30, 1644, 1869, 339, 457, 2221, 5201, 2053, 277, 260, 5460, 43, 5655, 1372, 4923, 19558, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 523, 441, 3796, 2030, 1096, 563, 451, 725, 946, 30, 49154], "prefix_len": 158, "old_logprobs": [-0.6716872453689575, -2.0992231369018555, -0.9639589190483093, -5.198906421661377, -2.7267260551452637, -3.8825252056121826, -1.750442624092102, -2.127859115600586, -3.1703083515167236, -2.6896109580993652, -0.39430347084999084, -0.1110399141907692]}
+{"case_id": "already_tried_03", "category": "already_tried", "origin": "budgie", "answer": "I’m ready for the next call.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 5460, 338, 523, 441, 5363, 30, 1644, 1869, 339, 457, 2221, 5201, 2053, 277, 260, 5460, 43, 5655, 1372, 4923, 19558, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 417, 93, 4421, 327, 260, 1867, 946, 30, 49154], "prefix_len": 158, "old_logprobs": [-0.6716872453689575, -2.0992231369018555, -0.913715660572052, -2.4417502880096436, -2.0865678787231445, -1.8693488836288452, -0.05760004743933678, -3.269818067550659, -0.5519683957099915, -0.2287387251853943]}
+{"case_id": "already_tried_03", "category": "already_tried", "origin": "manual_ideal", "answer": "1. Inspect and gently clean the charging port.\n2. Test a known-good cable and adapter.", "reward": 1, "advantage": 1.9999965035026162, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 5460, 338, 523, 441, 5363, 30, 1644, 1869, 339, 457, 2221, 5201, 2053, 277, 260, 5460, 43, 5655, 1372, 4923, 19558, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 29161, 284, 9014, 2577, 260, 15478, 2399, 30, 198, 49161, 30, 4246, 253, 1343, 29, 12018, 11045, 284, 33303, 30, 49154], "prefix_len": 158, "old_logprobs": [-2.4214324951171875, -0.07590574026107788, -12.526939392089844, -4.405955791473389, -11.702948570251465, -5.401874542236328, -0.1255924105644226, -6.51625394821167, -4.155086040496826, -0.7675564885139465, -0.42107242345809937, -0.004277841653674841, -0.0009554826538078487, -4.444795608520508, -5.090394496917725, -7.864786148071289, -8.513676643371582, -6.527530670166016, -6.394984245300293, -3.488046169281006, -12.876324653625488, -1.0289692878723145, -1.905796766281128]}
+{"case_id": "already_tried_04", "category": "already_tried", "origin": "budgie", "answer": "I need it in a text-based format.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 26337, 351, 18306, 5369, 347, 1694, 30, 1644, 1869, 339, 457, 2221, 47848, 260, 2301, 43, 4247, 260, 11713, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 737, 357, 281, 253, 1694, 29, 2542, 4624, 30, 49154], "prefix_len": 156, "old_logprobs": [-0.5582013726234436, -1.9571317434310913, -4.28559684753418, -3.262272357940674, -1.5585328340530396, -3.266849994659424, -1.7064430713653564, -0.19106540083885193, -1.5690630674362183, -0.8603324294090271, -0.7606700658798218]}
+{"case_id": "already_tried_04", "category": "already_tried", "origin": "budgie", "answer": "I’m sure you’re working on the next step, but I wouldWays need to ensure there’s no confusion.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 26337, 351, 18306, 5369, 347, 1694, 30, 1644, 1869, 339, 457, 2221, 47848, 260, 2301, 43, 4247, 260, 11713, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 417, 93, 2090, 346, 417, 257, 1891, 335, 260, 1867, 1833, 28, 564, 339, 736, 71, 646, 737, 288, 2381, 665, 417, 99, 787, 9563, 30, 49154], "prefix_len": 156, "old_logprobs": [-0.5582013726234436, -1.7696317434310913, -1.0669193267822266, -2.8968019485473633, -1.6545603275299072, -1.0423330068588257, -0.23829743266105652, -4.196124076843262, -0.3730406165122986, -2.937309980392456, -2.350522041320801, -2.058976650238037, -1.5330169200897217, -1.4857605695724487, -1.1214826107025146, -4.360185623168945, -8.920464515686035, -4.335386753082275, -3.3347136974334717, -1.0524187088012695, -4.507737159729004, -3.4314115047454834, -0.8590435981750488, -0.001158519764430821, -0.2544155716896057, -3.0624680519104004, -1.0704331398010254, -0.10588763654232025]}
+{"case_id": "already_tried_04", "category": "already_tried", "origin": "budgie", "answer": "I repeat this, and I will ask you for a secondNextFirst – this is a new, much better way to simulate a big experiment.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 26337, 351, 18306, 5369, 347, 1694, 30, 1644, 1869, 339, 457, 2221, 47848, 260, 2301, 43, 4247, 260, 11713, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 9960, 451, 28, 284, 339, 523, 1998, 346, 327, 253, 1796, 8693, 5345, 816, 451, 314, 253, 725, 28, 1083, 1365, 970, 288, 18463, 253, 2066, 4308, 30, 49154], "prefix_len": 156, "old_logprobs": [-0.5582013726234436, -6.988381862640381, -2.677971363067627, -3.8074002265930176, -1.651473045349121, -1.2259149551391602, -1.233887791633606, -3.3357300758361816, -0.8440439701080322, -3.621135711669922, -2.3172621726989746, -1.6440041065216064, -7.582242012023926, -5.178114891052246, -6.1364946365356445, -3.493340253829956, -0.9767433404922485, -1.5921334028244019, -2.442875385284424, -3.5662450790405273, -4.539247035980225, -2.0804619789123535, -1.5666191577911377, -0.4536041021347046, -9.200784683227539, -2.065477132797241, -5.855262756347656, -4.383793354034424, -0.7100578546524048, -0.29981687664985657]}
+{"case_id": "already_tried_04", "category": "already_tried", "origin": "budgie", "answer": "I need two next deep areas to take a look at.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 26337, 351, 18306, 5369, 347, 1694, 30, 1644, 1869, 339, 457, 2221, 47848, 260, 2301, 43, 4247, 260, 11713, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 737, 827, 1867, 2276, 1721, 288, 1188, 253, 1492, 418, 30, 49154], "prefix_len": 156, "old_logprobs": [-0.5582013726234436, -1.9571317434310913, -3.0355966091156006, -0.49191734194755554, -8.962010383605957, -10.421004295349121, -1.6245478391647339, -4.141678810119629, -1.817957878112793, -2.1014761924743652, -0.10068561136722565, -0.41657811403274536, -0.27137860655784607]}
+{"case_id": "already_tried_04", "category": "already_tried", "origin": "manual_ideal", "answer": "1. Set the cells to general format.\n2. Re-enter each formula beginning with an equals sign.", "reward": 1, "advantage": 1.9999965035026162, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 26337, 351, 18306, 5369, 347, 1694, 30, 1644, 1869, 339, 457, 2221, 47848, 260, 2301, 43, 4247, 260, 11713, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 5427, 260, 1960, 288, 2210, 4624, 30, 198, 49161, 30, 942, 29, 9294, 971, 7961, 3616, 351, 354, 16473, 914, 30, 49154], "prefix_len": 156, "old_logprobs": [-2.5366811752319336, -0.05895092710852623, -5.379383563995361, -0.49988335371017456, -6.9888105392456055, -1.0980387926101685, -9.810358047485352, -2.906806468963623, -1.1560015678405762, -0.12997300922870636, -0.026597077026963234, -0.0015050761867314577, -6.812226295471191, -2.033811092376709, -4.588705062866211, -5.4619574546813965, -3.259209394454956, -13.009403228759766, -0.3968242406845093, -4.563922882080078, -3.6742186546325684, -0.028917616233229637, -0.9336638450622559, -3.6549174785614014]}
+{"case_id": "already_tried_05", "category": "already_tried", "origin": "budgie", "answer": "I’m not sure which two next features to look at, but these two are important:\n\n1.Fix the new C Saskatchewan position in the viewport.\n2.Acknowrome the properties of the viewport.\n3.Inter wx_uit(oleum.w betting_on_x).\n4.Acknowrome the properties of the viewport.\n5.Subtract 1 from the viewport width.\n6.Acknowrome the properties of the viewport.\n7.Acknowrome the properties of the viewport.\n\nThese next features help the viewport align with the viewport’s size, and it can be useful for a single view where the viewport itself is small.\n\nI should also note that these are called \"next features\" in the future, so I shouldd't Theorem those.", "reward": -0.85, "advantage": -0.97924556143768, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 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+{"case_id": "already_tried_05", "category": "already_tried", "origin": "budgie", "answer": "I cannot give more than one next management action.", "reward": -0.42999999999999994, "advantage": -0.31801865500066134, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 3877, 3013, 3449, 725, 25298, 314, 441, 6178, 30, 1644, 1869, 339, 457, 2221, 46691, 260, 3013, 43, 26389, 277, 260, 7056, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 2526, 1928, 540, 670, 582, 1867, 2343, 2353, 30, 49154], "prefix_len": 159, "old_logprobs": [-0.7100666165351868, -3.777735710144043, -2.6734251976013184, -3.22499680519104, -0.35873115062713623, -1.5463873147964478, -0.831476628780365, -5.496971607208252, -0.12125314772129059, -0.7606141567230225, -0.3267732262611389]}
+{"case_id": "already_tried_05", "category": "already_tried", "origin": "budgie", "answer": "I will provide these two next applicability actions now.", "reward": -0.42999999999999994, "advantage": -0.31801865500066134, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 3877, 3013, 3449, 725, 25298, 314, 441, 6178, 30, 1644, 1869, 339, 457, 2221, 46691, 260, 3013, 43, 26389, 277, 260, 7056, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 523, 1538, 623, 827, 1867, 32039, 3643, 1209, 30, 49154], "prefix_len": 159, "old_logprobs": [-0.7100666165351868, -2.152735710144043, -2.4513187408447266, -4.025435924530029, -0.5540184378623962, -0.132544606924057, -10.972318649291992, -1.7330927848815918, -4.720358848571777, -0.5072550773620605, -0.2585401237010956]}
+{"case_id": "already_tried_05", "category": "already_tried", "origin": "budgie", "answer": "I’ve already anger the page; I will citing the same steps I stopped to avoid repetition and the information I previously tried.", "reward": -0.42999999999999994, "advantage": -0.31801865500066134, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 3877, 3013, 3449, 725, 25298, 314, 441, 6178, 30, 1644, 1869, 339, 457, 2221, 46691, 260, 3013, 43, 26389, 277, 260, 7056, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 417, 307, 2221, 9846, 260, 3013, 43, 339, 523, 18617, 260, 1142, 3301, 339, 8976, 288, 2482, 18388, 284, 260, 1096, 339, 5094, 5655, 30, 49154], "prefix_len": 159, "old_logprobs": [-0.7100666165351868, -1.965235710144043, -2.089510917663574, -1.8620085716247559, -11.425339698791504, -0.8865642547607422, -2.8686649799346924, -1.5737154483795166, -1.6602377891540527, -2.072779417037964, -11.897180557250977, -1.1366617679595947, -4.28336238861084, -1.8001099824905396, -1.4219783544540405, -6.560004234313965, -1.8321201801300049, -1.9381917715072632, -1.549168348312378, -2.8814103603363037, -4.343197345733643, -4.6260576248168945, -0.9877489805221558, -5.888538360595703, -0.49820753931999207, -0.27870500087738037, -0.0586448535323143]}
+{"case_id": "already_tried_05", "category": "already_tried", "origin": "manual_ideal", "answer": "1. Perform a hard refresh or clear the browser cache.\n2. Verify that the edited stylesheet is the one loaded by the page.", "reward": 1, "advantage": 1.9333015264396638, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 3877, 3013, 3449, 725, 25298, 314, 441, 6178, 30, 1644, 1869, 339, 457, 2221, 46691, 260, 3013, 43, 26389, 277, 260, 7056, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 25859, 253, 1759, 24325, 355, 2437, 260, 11177, 13842, 30, 198, 49161, 30, 36830, 338, 260, 14220, 6825, 27628, 314, 260, 582, 13049, 411, 260, 3013, 30, 49154], "prefix_len": 159, "old_logprobs": [-2.460066556930542, -0.07619398087263107, -7.535190582275391, -0.5372382402420044, -5.619928359985352, -9.301209449768066, -5.258175849914551, -5.0599799156188965, -1.053034782409668, -7.220737934112549, -9.198378562927246, -0.7885396480560303, -0.6492361426353455, -0.005747577641159296, -0.0007782529573887587, -6.10923957824707, -0.9153022766113281, -0.3901046812534332, -11.807281494140625, -6.287891387939453, -10.117032051086426, -0.7732503414154053, -3.780475378036499, -5.039531707763672, -10.733394622802734, -2.873204231262207, -0.5729126334190369, -2.220417022705078, -0.6327050924301147, -2.1451621055603027]}
+{"case_id": "already_tried_06", "category": "already_tried", "origin": "budgie", "answer": "I will not give a new question.", "reward": -0.42999999999999994, "advantage": -0.5888992037956788, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 27573, 13473, 338, 9180, 564, 14137, 787, 1646, 45646, 30, 1644, 1869, 339, 457, 2221, 7233, 260, 9513, 43, 20054, 357, 1163, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 523, 441, 1928, 253, 725, 1962, 30, 49154], "prefix_len": 159, "old_logprobs": [-0.5932253003120422, -2.1474273204803467, -1.5038424730300903, -3.2077136039733887, -2.467026472091675, -1.264600157737732, -5.2047882080078125, -1.3122416734695435, -0.1412235051393509]}
+{"case_id": "already_tried_06", "category": "already_tried", "origin": "budgie", "answer": "1. online learning: access to a computer with 1-way digital connection.", "reward": -0.42999999999999994, "advantage": -0.5888992037956788, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 27573, 13473, 338, 9180, 564, 14137, 787, 1646, 45646, 30, 1644, 1869, 339, 457, 2221, 7233, 260, 9513, 43, 20054, 357, 1163, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 2329, 1380, 42, 1594, 288, 253, 2583, 351, 216, 49160, 29, 2384, 3182, 3890, 30, 49154], "prefix_len": 159, "old_logprobs": [-2.3620123863220215, -0.08046234399080276, -6.493180751800537, -6.226981163024902, -1.742205023765564, -5.44648551940918, -1.0049434900283813, -2.4179763793945312, -2.8933520317077637, -1.5215957164764404, -4.032413482666016, -1.475454568862915, -1.9983755350112915, -3.7989606857299805, -5.365976810455322, -2.6631217002868652, -0.8119489550590515, -1.1298434734344482]}
+{"case_id": "already_tried_06", "category": "already_tried", "origin": "budgie", "answer": "1. Back stabilize the battery.\n2. Stretch the keyboard.", "reward": -0.22, "advantage": -0.2118600794142992, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 27573, 13473, 338, 9180, 564, 14137, 787, 1646, 45646, 30, 1644, 1869, 339, 457, 2221, 7233, 260, 9513, 43, 20054, 357, 1163, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 8279, 23173, 260, 6979, 30, 198, 49161, 30, 625, 30250, 260, 13473, 30, 49154], "prefix_len": 159, "old_logprobs": [-2.3432252407073975, -0.0912376418709755, -5.931471824645996, -10.346261024475098, -0.4655523896217346, -0.3392833173274994, -0.7111531496047974, -0.346289724111557, -0.0022703842259943485, -0.0017294225981459022, -4.839558124542236, -2.91135573387146, -0.9010117650032043, -3.5636518001556396, -0.716224193572998, -1.1914210319519043]}
+{"case_id": "already_tried_06", "category": "already_tried", "origin": "budgie", "answer": "I’d like to know what you'd like to do next.", "reward": -0.42999999999999994, "advantage": -0.5888992037956788, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 27573, 13473, 338, 9180, 564, 14137, 787, 1646, 45646, 30, 1644, 1869, 339, 457, 2221, 7233, 260, 9513, 43, 20054, 357, 1163, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 417, 84, 702, 288, 699, 732, 346, 6737, 702, 288, 536, 1867, 30, 49154], "prefix_len": 159, "old_logprobs": [-0.5932253003120422, -1.8349273204803467, -2.403473377227783, -1.178954839706421, -0.9294155836105347, -3.8752009868621826, -1.1817294359207153, -1.3623218536376953, -5.639820098876953, -0.15691913664340973, -0.4270450174808502, -0.20045588910579681, -0.49252402782440186, -0.5578424334526062, -0.08925394713878632]}
+{"case_id": "already_tried_06", "category": "already_tried", "origin": "manual_ideal", "answer": "1. Remove conflicting keyboard profiles.\n2. Test the keyboard on another device.", "reward": 1, "advantage": 1.9785576908013356, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 27573, 13473, 338, 9180, 564, 14137, 787, 1646, 45646, 30, 1644, 1869, 339, 457, 2221, 7233, 260, 9513, 43, 20054, 357, 1163, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 15812, 19812, 13473, 12628, 30, 198, 49161, 30, 4246, 260, 13473, 335, 1372, 3476, 30, 49154], "prefix_len": 159, "old_logprobs": [-2.3620123863220215, -0.08046234399080276, -5.985368251800537, -11.805105209350586, -9.911874771118164, -12.907414436340332, -1.2601728439331055, -0.6819016337394714, -0.01107639167457819, -0.0014316319720819592, -5.968137264251709, -1.021337866783142, -3.5476348400115967, -3.6819891929626465, -3.466904640197754, -2.321570634841919, -0.4498688578605652, -1.3703248500823975]}
+{"case_id": "already_tried_07", "category": "already_tried", "origin": "budgie", "answer": "I will not repeat any steps.", "reward": -0.42999999999999994, "advantage": -0.5888992037956788, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 24156, 5220, 11238, 347, 1603, 29, 7987, 29, 10403, 30, 1644, 1869, 339, 457, 2221, 829, 2041, 260, 5220, 43, 12775, 260, 5594, 3890, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 523, 441, 9960, 750, 3301, 30, 49154], "prefix_len": 161, "old_logprobs": [-0.7213464975357056, -2.0324344635009766, -1.5640637874603271, -1.6283061504364014, -0.32369109988212585, -1.770092487335205, -1.380621314048767, -0.30591946840286255]}
+{"case_id": "already_tried_07", "category": "already_tried", "origin": "budgie", "answer": "1. ==> Check the internet connection\n2. ==> Change the problem to a new one", "reward": -0.22, "advantage": -0.2118600794142992, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 24156, 5220, 11238, 347, 1603, 29, 7987, 29, 10403, 30, 1644, 1869, 339, 457, 2221, 829, 2041, 260, 5220, 43, 12775, 260, 5594, 3890, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 1758, 46, 7023, 260, 5594, 3890, 198, 49161, 30, 1758, 46, 7662, 260, 1732, 288, 253, 725, 582, 49154], "prefix_len": 161, "old_logprobs": [-2.4503848552703857, -0.06520342826843262, -10.551506996154785, -3.2306439876556396, -2.348357915878296, -0.6904765367507935, -2.7869718074798584, -0.054701514542102814, -1.7382144927978516, -0.015895791351795197, -0.0017470349557697773, -0.9637435674667358, -0.1383247822523117, -4.3886213302612305, -0.23069033026695251, -0.5482220649719238, -1.9157862663269043, -1.0909942388534546, -0.6067140102386475, -0.9375890493392944, -4.182873725891113]}
+{"case_id": "already_tried_07", "category": "already_tried", "origin": "budgie", "answer": "I initially tried the options 1. Check the internet connection; 2. Check the problem; 3. Re-read the problem statement, and note down a few things.", "reward": -0.42999999999999994, "advantage": -0.5888992037956788, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 24156, 5220, 11238, 347, 1603, 29, 7987, 29, 10403, 30, 1644, 1869, 339, 457, 2221, 829, 2041, 260, 5220, 43, 12775, 260, 5594, 3890, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 7083, 5655, 260, 3416, 216, 49160, 30, 7023, 260, 5594, 3890, 43, 216, 49161, 30, 7023, 260, 1732, 43, 216, 49162, 30, 942, 29, 1040, 260, 1732, 4814, 28, 284, 4109, 1187, 253, 1443, 1495, 30, 49154], "prefix_len": 161, "old_logprobs": [-0.7213464975357056, -7.282434463500977, -1.1295170783996582, -1.7381129264831543, -7.155224800109863, -5.666495323181152, -0.6595124006271362, -2.593600034713745, -2.6172354221343994, -0.4366123080253601, -0.902073860168457, -0.0739838182926178, -1.8188742399215698, -0.06894710659980774, -0.010007445700466633, -0.007052054163068533, -0.8220622539520264, -0.2902086675167084, -2.1712310314178467, -1.894116997718811, -0.09199056774377823, -0.0012207920663058758, -0.00607240479439497, -1.9123021364212036, -1.4262425899505615, -2.3094334602355957, -0.06292963027954102, -0.6868686676025391, -1.4027739763259888, -3.4420833587646484, -1.0811314582824707, -4.910377025604248, -1.2673389911651611, -3.677945137023926, -2.022339344024658, -1.9443222284317017, -1.229845404624939, -0.28906604647636414]}
+{"case_id": "already_tried_07", "category": "already_tried", "origin": "budgie", "answer": "I’ll answer the new task directly.", "reward": -0.42999999999999994, "advantage": -0.5888992037956788, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 24156, 5220, 11238, 347, 1603, 29, 7987, 29, 10403, 30, 1644, 1869, 339, 457, 2221, 829, 2041, 260, 5220, 43, 12775, 260, 5594, 3890, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 417, 764, 2988, 260, 725, 3856, 3319, 30, 49154], "prefix_len": 161, "old_logprobs": [-0.7213464975357056, -1.9699345827102661, -1.292940616607666, -6.298867702484131, -0.5500489473342896, -1.3932528495788574, -2.878106117248535, -1.3881187438964844, -0.6508511304855347, -0.08684190362691879]}
+{"case_id": "already_tried_07", "category": "already_tried", "origin": "manual_ideal", "answer": "1. Fetch and inspect the remote changes.\n2. Rebase or merge before pushing again.", "reward": 1, "advantage": 1.9785576908013356, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 24156, 5220, 11238, 347, 1603, 29, 7987, 29, 10403, 30, 1644, 1869, 339, 457, 2221, 829, 2041, 260, 5220, 43, 12775, 260, 5594, 3890, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 34313, 284, 8121, 260, 6431, 1971, 30, 198, 49161, 30, 942, 6822, 355, 16905, 1092, 10388, 1163, 30, 49154], "prefix_len": 161, "old_logprobs": [-2.4503848552703857, -0.06520342826843262, -12.989006996154785, -5.464072227478027, -8.950897216796875, -0.24509289860725403, -7.896091938018799, -10.260087966918945, -1.2832419872283936, -0.44534480571746826, -0.008849567733705044, -0.0012890134239569306, -3.290130615234375, -9.780823707580566, -6.7295403480529785, -6.9217143058776855, -9.231553077697754, -6.348733901977539, -6.369965076446533, -0.1653491109609604, -1.2403899431228638]}
+{"case_id": "already_tried_08", "category": "already_tried", "origin": "budgie", "answer": "I will not send you through those two next Printing.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 19375, 338, 6599, 335, 564, 1072, 441, 2817, 30, 1644, 1869, 339, 457, 2221, 804, 253, 896, 19558, 43, 4247, 260, 23123, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 523, 441, 5091, 346, 738, 967, 827, 1867, 25613, 30, 49154], "prefix_len": 159, "old_logprobs": [-0.6645570993423462, -2.150458812713623, -1.6232755184173584, -1.2006123065948486, -0.7167249917984009, -0.2587294578552246, -0.28638628125190735, -1.5935059785842896, -0.6136084794998169, -10.01675796508789, -0.8564862012863159, -0.19056373834609985]}
+{"case_id": "already_tried_08", "category": "already_tried", "origin": "budgie", "answer": "1. I removed the power source and found it was down.\n2. I changed the positive compensation account.\n3. I changed the power source again.\n\nFinal answer: 3", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 19375, 338, 6599, 335, 564, 1072, 441, 2817, 30, 1644, 1869, 339, 457, 2221, 804, 253, 896, 19558, 43, 4247, 260, 23123, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 339, 5178, 260, 1149, 2257, 284, 983, 357, 436, 1187, 30, 198, 49161, 30, 339, 4247, 260, 2731, 11985, 2051, 30, 198, 49162, 30, 339, 4247, 260, 1149, 2257, 1163, 30, 198, 198, 29288, 2988, 42, 216, 49162, 49154], "prefix_len": 159, "old_logprobs": [-2.2895569801330566, -0.0594402439892292, -4.229940891265869, -6.133904457092285, -0.24539299309253693, -1.71654212474823, -1.0537196397781372, -1.5973553657531738, -4.539947509765625, -2.5318984985351562, -1.5408161878585815, -3.6658246517181396, -0.2982119023799896, -0.8860485553741455, -0.002937528071925044, -0.0012453190283849835, -0.2569902539253235, -2.591334581375122, -0.06537998467683792, -9.382467269897461, -9.618371963500977, -7.926178932189941, -1.0428203344345093, -0.2815646231174469, -0.09522610157728195, -0.001701218425296247, -0.07656849920749664, -1.6568830013275146, -0.08036642521619797, -2.589773416519165, -0.14317037165164948, -1.7943421602249146, -0.4487248659133911, -0.46204090118408203, -1.4103171825408936, -0.9666457176208496, -0.29395270347595215, -0.0020115396473556757, -0.43341097235679626, -1.4344583749771118, -0.06567786633968353]}
+{"case_id": "already_tried_08", "category": "already_tried", "origin": "budgie", "answer": "I will not ask you to repeat any of those steps again.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 19375, 338, 6599, 335, 564, 1072, 441, 2817, 30, 1644, 1869, 339, 457, 2221, 804, 253, 896, 19558, 43, 4247, 260, 23123, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 523, 441, 1998, 346, 288, 9960, 750, 282, 967, 3301, 1163, 30, 49154], "prefix_len": 159, "old_logprobs": [-0.6645570993423462, -2.150458812713623, -1.6232755184173584, -3.0756123065948486, -0.9527013301849365, -0.8356502652168274, -2.229659080505371, -0.9449266195297241, -0.9738049507141113, -1.336275339126587, -1.1260066032409668, -0.8224331140518188, -0.18900573253631592, -0.16266599297523499]}
+{"case_id": "already_tried_08", "category": "already_tried", "origin": "budgie", "answer": "I will ask you to change the power limit of the chain, then change the current power level of the chain, depending on your current power level.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 19375, 338, 6599, 335, 564, 1072, 441, 2817, 30, 1644, 1869, 339, 457, 2221, 804, 253, 896, 19558, 43, 4247, 260, 23123, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 57, 523, 1998, 346, 288, 1363, 260, 1149, 3673, 282, 260, 5891, 28, 965, 1363, 260, 1574, 1149, 1106, 282, 260, 5891, 28, 4509, 335, 469, 1574, 1149, 1106, 30, 49154], "prefix_len": 159, "old_logprobs": [-0.6645570993423462, -2.150458812713623, -3.2482755184173584, -1.0951493978500366, -0.975055992603302, -2.4905152320861816, -0.3288327157497406, -3.8516104221343994, -6.383537769317627, -0.8529859781265259, -0.09433654695749283, -7.976990699768066, -3.2467005252838135, -3.6422557830810547, -3.007019281387329, -0.11417610198259354, -4.47685432434082, -1.4658546447753906, -1.9810106754302979, -1.0232558250427246, -0.1401853859424591, -1.3338392972946167, -1.182328462600708, -8.133707046508789, -0.00963029358536005, -0.8908677101135254, -1.0017108917236328, -0.9219918251037598, -0.3504718840122223, -0.4214595556259155, -0.1099691167473793]}
+{"case_id": "already_tried_08", "category": "already_tried", "origin": "manual_ideal", "answer": "1. Stop using it and check the door latch for visible damage.\n2. Contact a qualified repair technician.", "reward": 1, "advantage": 1.9999965035026162, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 57, 737, 724, 351, 253, 19375, 338, 6599, 335, 564, 1072, 441, 2817, 30, 1644, 1869, 339, 457, 2221, 804, 253, 896, 19558, 43, 4247, 260, 23123, 30, 21691, 4247, 260, 1732, 30, 49154, 49158, 198, 57, 417, 764, 2482, 9613, 346, 738, 967, 1142, 3301, 1163, 30, 49154, 49157, 198, 26533, 3869, 827, 1867, 10444, 3643, 28, 347, 22266, 3301, 30, 3315, 441, 9960, 3534, 339, 2221, 5655, 30, 49154, 49158, 198, 49160, 30, 17178, 1015, 357, 284, 2545, 260, 6644, 303, 1503, 327, 6178, 3055, 30, 198, 49161, 30, 13225, 253, 11645, 7698, 28027, 30, 49154], "prefix_len": 159, "old_logprobs": [-2.2895569801330566, -0.0594402439892292, -7.306112766265869, -4.727433204650879, -5.371179103851318, -2.6986985206604004, -2.811258554458618, -1.0851712226867676, -8.30691909790039, -8.355438232421875, -2.89854097366333, -3.463578701019287, -6.741083145141602, -5.039363861083984, -0.23248234391212463, -0.19816754758358002, -0.005099978763610125, -0.0009152276325039566, -9.656671524047852, -1.2962075471878052, -6.835844039916992, -4.741815567016602, -7.637845993041992, -1.2681267261505127, -1.297078251838684]}
+{"case_id": "selective_00", "category": "selective_answer", "origin": "budgie", "answer": "I am here to provide a direct answer and acknowledge that I cannot verify the date of the review.", "reward": -0.22, "advantage": -0.7248268722033092, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 40074, 1391, 7615, 314, 442, 9194, 30, 4781, 6050, 314, 24927, 49160, 49167, 28, 49163, 49159, 49159, 28, 624, 2523, 314, 335, 216, 49160, 49161, 4019, 28, 284, 260, 15575, 2380, 314, 10245, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 2523, 3688, 47, 19842, 351, 260, 3688, 805, 30, 49154, 49158, 198, 57, 744, 1535, 288, 1538, 253, 1454, 2988, 284, 10406, 338, 339, 2526, 12198, 260, 3688, 282, 260, 2523, 30, 49154], "prefix_len": 147, "old_logprobs": [-0.8247277140617371, -1.7663670778274536, -4.576011657714844, -0.266767293214798, -1.1310213804244995, -2.917008876800537, -3.6941192150115967, -0.060949042439460754, -2.001164197921753, -6.057967662811279, -0.6808152794837952, -1.8048509359359741, -0.612459123134613, -0.8030996322631836, -1.005478858947754, -0.17865625023841858, -1.3015899658203125, -0.18004010617733002, -0.059215106070041656, -0.7924914956092834, -0.08311724662780762]}
+{"case_id": "selective_00", "category": "selective_answer", "origin": "budgie", "answer": "I am prefrontal to provide a date, so we can always ask time-out without having to re convolution previous ones.", "reward": -0.22, "advantage": -0.7248268722033092, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 40074, 1391, 7615, 314, 442, 9194, 30, 4781, 6050, 314, 24927, 49160, 49167, 28, 49163, 49159, 49159, 28, 624, 2523, 314, 335, 216, 49160, 49161, 4019, 28, 284, 260, 15575, 2380, 314, 10245, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 2523, 3688, 47, 19842, 351, 260, 3688, 805, 30, 49154, 49158, 198, 57, 744, 37044, 288, 1538, 253, 3688, 28, 588, 392, 416, 1811, 1998, 655, 29, 447, 1355, 1953, 288, 298, 29714, 2672, 2911, 30, 49154], "prefix_len": 147, "old_logprobs": [-0.8247277140617371, -1.7663670778274536, -10.724449157714844, -1.0512351989746094, -1.5912201404571533, -1.6410902738571167, -0.9872123003005981, -2.520721435546875, -1.6584988832473755, -3.5139620304107666, -0.44660764932632446, -5.892853736877441, -4.612531661987305, -9.254012107849121, -1.405447244644165, -1.2166941165924072, -3.2860641479492188, -2.894855499267578, -0.14335061609745026, -2.893913984298706, -10.06965160369873, -3.2355172634124756, -4.074301719665527, -0.13236103951931, -0.02413278818130493]}
+{"case_id": "selective_00", "category": "selective_answer", "origin": "budgie", "answer": "I should state the date.", "reward": -0.010000000000000009, "advantage": -0.26357340807393065, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 40074, 1391, 7615, 314, 442, 9194, 30, 4781, 6050, 314, 24927, 49160, 49167, 28, 49163, 49159, 49159, 28, 624, 2523, 314, 335, 216, 49160, 49161, 4019, 28, 284, 260, 15575, 2380, 314, 10245, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 2523, 3688, 47, 19842, 351, 260, 3688, 805, 30, 49154, 49158, 198, 57, 868, 1215, 260, 3688, 30, 49154], "prefix_len": 147, "old_logprobs": [-0.8130863904953003, -3.9641175270080566, -1.679347038269043, -0.23455414175987244, -0.06072290241718292, -1.6102912425994873, -0.04821809008717537]}
+{"case_id": "selective_00", "category": "selective_answer", "origin": "budgie", "answer": "The date is June 12th.", "reward": 0, "advantage": -0.24160895740110308, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 40074, 1391, 7615, 314, 442, 9194, 30, 4781, 6050, 314, 24927, 49160, 49167, 28, 49163, 49159, 49159, 28, 624, 2523, 314, 335, 216, 49160, 49161, 4019, 28, 284, 260, 15575, 2380, 314, 10245, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 2523, 3688, 47, 19842, 351, 260, 3688, 805, 30, 49154, 49158, 198, 504, 3688, 314, 4019, 216, 49160, 49161, 373, 30, 49154], "prefix_len": 147, "old_logprobs": [-0.8130863904953003, -0.7556851506233215, -0.677800714969635, -2.3033409118652344, -0.051719166338443756, -0.024732597172260284, -0.2932860553264618, -0.9559962749481201, -0.23776966333389282, -0.047639597207307816]}
+{"case_id": "selective_00", "category": "selective_answer", "origin": "manual_ideal", "answer": "12 June", "reward": 1, "advantage": 1.954836109881652, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 40074, 1391, 7615, 314, 442, 9194, 30, 4781, 6050, 314, 24927, 49160, 49167, 28, 49163, 49159, 49159, 28, 624, 2523, 314, 335, 216, 49160, 49161, 4019, 28, 284, 260, 15575, 2380, 314, 10245, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 2523, 3688, 47, 19842, 351, 260, 3688, 805, 30, 49154, 49158, 198, 49160, 49161, 4019, 49154], "prefix_len": 147, "old_logprobs": [-8.377707481384277, -1.240557074546814, -0.08422023057937622, -2.032186269760132]}
+{"case_id": "selective_01", "category": "selective_answer", "origin": "budgie", "answer": "The terminal receives the order is B.", "reward": -0.010000000000000009, "advantage": -0.3685925517242877, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 22977, 428, 72, 29, 49163, 3593, 216, 49162, 49165, 18586, 28, 7761, 411, 6358, 28, 20623, 418, 37154, 389, 28, 284, 314, 38684, 327, 1885, 49168, 28, 49159, 49159, 49159, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 12892, 11287, 260, 1686, 47, 19842, 351, 260, 12892, 805, 30, 49154, 49158, 198, 504, 12892, 11287, 260, 1686, 314, 389, 30, 49154], "prefix_len": 143, "old_logprobs": [-1.1740760803222656, -0.08822384476661682, -0.39480137825012207, -0.028767282143235207, -0.014352306723594666, -0.7074620127677917, -3.5495102405548096, -1.6351805925369263, -0.08583404123783112]}
+{"case_id": "selective_01", "category": "selective_answer", "origin": "budgie", "answer": "I am Mé with the details.", "reward": -0.010000000000000009, "advantage": -0.3685925517242877, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 22977, 428, 72, 29, 49163, 3593, 216, 49162, 49165, 18586, 28, 7761, 411, 6358, 28, 20623, 418, 37154, 389, 28, 284, 314, 38684, 327, 1885, 49168, 28, 49159, 49159, 49159, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 12892, 11287, 260, 1686, 47, 19842, 351, 260, 12892, 805, 30, 49154, 49158, 198, 57, 744, 32623, 351, 260, 3841, 30, 49154], "prefix_len": 143, "old_logprobs": [-0.9240760803222656, -1.3530209064483643, -9.534682273864746, -5.0445404052734375, -0.414177805185318, -0.19860637187957764, -0.4224373698234558, -0.08253444731235504]}
+{"case_id": "selective_01", "category": "selective_answer", "origin": "budgie", "answer": "I was asked to find the terminal receiving the order, but I am an AI assistant.", "reward": -0.22, "advantage": -0.8778322613433693, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 22977, 428, 72, 29, 49163, 3593, 216, 49162, 49165, 18586, 28, 7761, 411, 6358, 28, 20623, 418, 37154, 389, 28, 284, 314, 38684, 327, 1885, 49168, 28, 49159, 49159, 49159, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 12892, 11287, 260, 1686, 47, 19842, 351, 260, 12892, 805, 30, 49154, 49158, 198, 57, 436, 3413, 288, 1042, 260, 12892, 7418, 260, 1686, 28, 564, 339, 744, 354, 5646, 11173, 30, 49154], "prefix_len": 143, "old_logprobs": [-0.864103376865387, -3.4119760990142822, -1.8627101182937622, -0.5301386117935181, -2.1722373962402344, -0.766478955745697, -0.14041300117969513, -2.434999465942383, -0.10607881844043732, -0.02334560640156269, -0.8916440010070801, -0.4607009291648865, -0.7655344009399414, -2.091986656188965, -2.0084829330444336, -0.009128974750638008, -0.5894355773925781, -2.344130516052246, -0.3226543366909027]}
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+{"case_id": "selective_04", "category": "selective_answer", "origin": "budgie", "answer": "L owners can choose from a range of license options.", "reward": -0.22, "advantage": -0.4824499661806777, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 21931, 450, 17065, 553, 216, 49167, 28, 49161, 49159, 49159, 10055, 28, 2925, 253, 17280, 18172, 216, 49163, 30, 49159, 9768, 28, 436, 8044, 281, 3903, 28, 284, 553, 216, 49160, 49163, 8935, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 9768, 1072, 31399, 450, 17065, 722, 47, 13843, 805, 260, 9768, 30, 49154, 49158, 198, 60, 5139, 416, 3525, 429, 253, 1845, 282, 9768, 3416, 30, 49154], "prefix_len": 147, "old_logprobs": [-0.9729524254798889, -9.917146682739258, -2.010348320007324, -3.0050673484802246, -0.9435200691223145, -1.0245310068130493, -2.10697865486145, -0.007881850004196167, -1.3646087646484375, -1.5632104873657227, -1.5191311836242676, -0.045075707137584686]}
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+{"case_id": "selective_05", "category": "selective_answer", "origin": "budgie", "answer": "The blue key opens the aesthetic.", "reward": -0.22, "advantage": -0.7196428393330484, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4461, 1646, 10573, 260, 5846, 19565, 28, 260, 20933, 1646, 10573, 260, 17428, 28, 284, 260, 1165, 2632, 1646, 10573, 260, 3813, 8645, 259, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 1646, 10573, 260, 17428, 47, 19842, 351, 260, 1646, 805, 30, 49154, 49158, 198, 504, 4461, 1646, 10573, 260, 13495, 30, 49154], "prefix_len": 138, "old_logprobs": [-0.22965174913406372, -0.7791144847869873, -0.0027638589963316917, -0.10261783748865128, -0.0008485292200930417, -10.729280471801758, -1.6211206912994385, -0.03498920425772667]}
+{"case_id": "selective_05", "category": "selective_answer", "origin": "budgie", "answer": "I must be missing a key.", "reward": -0.010000000000000009, "advantage": -0.25889589951615766, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4461, 1646, 10573, 260, 5846, 19565, 28, 260, 20933, 1646, 10573, 260, 17428, 28, 284, 260, 1165, 2632, 1646, 10573, 260, 3813, 8645, 259, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 1646, 10573, 260, 17428, 47, 19842, 351, 260, 1646, 805, 30, 49154, 49158, 198, 57, 1251, 325, 6782, 253, 1646, 30, 49154], "prefix_len": 138, "old_logprobs": [-1.854651689529419, -5.02083158493042, -4.280564785003662, -1.4757877588272095, -1.1785601377487183, -0.04801620915532112, -0.5379562973976135, -0.3495781719684601]}
+{"case_id": "selective_05", "category": "selective_answer", "origin": "budgie", "answer": "I have the key only.", "reward": -0.010000000000000009, "advantage": -0.25889589951615766, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4461, 1646, 10573, 260, 5846, 19565, 28, 260, 20933, 1646, 10573, 260, 17428, 28, 284, 260, 1165, 2632, 1646, 10573, 260, 3813, 8645, 259, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 1646, 10573, 260, 17428, 47, 19842, 351, 260, 1646, 805, 30, 49154, 49158, 198, 57, 457, 260, 1646, 805, 30, 49154], "prefix_len": 138, "old_logprobs": [-1.854651689529419, -0.45833173394203186, -0.07211387157440186, -0.3705003261566162, -1.360901117324829, -0.06903722882270813, -0.06454326957464218]}
+{"case_id": "selective_05", "category": "selective_answer", "origin": "budgie", "answer": "The blue key opens theArtaman.", "reward": -0.22, "advantage": -0.7196428393330484, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4461, 1646, 10573, 260, 5846, 19565, 28, 260, 20933, 1646, 10573, 260, 17428, 28, 284, 260, 1165, 2632, 1646, 10573, 260, 3813, 8645, 259, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 1646, 10573, 260, 17428, 47, 19842, 351, 260, 1646, 805, 30, 49154, 49158, 198, 504, 4461, 1646, 10573, 260, 9393, 14971, 30, 49154], "prefix_len": 138, "old_logprobs": [-0.22965174913406372, -0.7791144847869873, -0.0027638589963316917, -0.10261783748865128, -0.0008485292200930417, -7.760530948638916, -8.973271369934082, -1.313027262687683, -0.03796027600765228]}
+{"case_id": "selective_05", "category": "selective_answer", "origin": "manual_ideal", "answer": "brass key", "reward": 1, "advantage": 1.957077477698412, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4461, 1646, 10573, 260, 5846, 19565, 28, 260, 20933, 1646, 10573, 260, 17428, 28, 284, 260, 1165, 2632, 1646, 10573, 260, 3813, 8645, 259, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 1646, 10573, 260, 17428, 47, 19842, 351, 260, 1646, 805, 30, 49154, 49158, 198, 1612, 520, 1646, 49154], "prefix_len": 138, "old_logprobs": [-14.41715145111084, -2.880685329437256, -0.9613747596740723, -1.8933430910110474]}
+{"case_id": "selective_06", "category": "selective_answer", "origin": "budgie", "answer": "The departure gate is the one from which the flight departs.", "reward": -0.22, "advantage": -0.368200729640723, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 54, 2501, 216, 49165, 49159, 49167, 7425, 99, 418, 216, 49159, 49165, 42, 49161, 49164, 429, 18447, 216, 49160, 49160, 28, 12087, 418, 216, 49159, 49164, 42, 49164, 49159, 28, 284, 2518, 582, 216, 49167, 11134, 18510, 8727, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 19740, 10166, 47, 13843, 805, 260, 10166, 30, 49154, 49158, 198, 504, 19740, 10166, 314, 260, 582, 429, 527, 260, 6688, 7425, 99, 30, 49154], "prefix_len": 150, "old_logprobs": [-0.5831629633903503, -1.412279486656189, -0.03215855360031128, -0.15639902651309967, -1.8872697353363037, -2.7893571853637695, -2.295363426208496, -0.22874781489372253, -0.20346826314926147, -1.2279529571533203, -1.0020747184753418, -0.00041321321623399854, -0.16913431882858276, -0.020952317863702774]}
+{"case_id": "selective_06", "category": "selective_answer", "origin": "budgie", "answer": "The departure gate is at 06:15.", "reward": -0.010000000000000009, "advantage": 0.014582207114484028, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 54, 2501, 216, 49165, 49159, 49167, 7425, 99, 418, 216, 49159, 49165, 42, 49161, 49164, 429, 18447, 216, 49160, 49160, 28, 12087, 418, 216, 49159, 49164, 42, 49164, 49159, 28, 284, 2518, 582, 216, 49167, 11134, 18510, 8727, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 19740, 10166, 47, 13843, 805, 260, 10166, 30, 49154, 49158, 198, 504, 19740, 10166, 314, 418, 216, 49159, 49165, 42, 49160, 49164, 30, 49154], "prefix_len": 150, "old_logprobs": [-0.5831629633903503, -1.412279486656189, -0.03215855360031128, -0.15639902651309967, -1.4810197353363037, -1.3596785068511963, -1.1211270093917847, -1.289710521697998, -0.001650877296924591, -3.2586238384246826, -0.26329317688941956, -0.8340132236480713, -0.05123138055205345]}
+{"case_id": "selective_06", "category": "selective_answer", "origin": "budgie", "answer": "The flight departs at 06:25 from recharge 11, which is just after 05:50.", "reward": -0.6399999999999999, "advantage": -1.1337666031511369, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 54, 2501, 216, 49165, 49159, 49167, 7425, 99, 418, 216, 49159, 49165, 42, 49161, 49164, 429, 18447, 216, 49160, 49160, 28, 12087, 418, 216, 49159, 49164, 42, 49164, 49159, 28, 284, 2518, 582, 216, 49167, 11134, 18510, 8727, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 19740, 10166, 47, 13843, 805, 260, 10166, 30, 49154, 49158, 198, 504, 6688, 7425, 99, 418, 216, 49159, 49165, 42, 49161, 49164, 429, 23424, 216, 49160, 49160, 28, 527, 314, 915, 990, 216, 49159, 49164, 42, 49164, 49159, 30, 49154], "prefix_len": 150, "old_logprobs": [-0.5753034353256226, -2.0472583770751953, -1.9236822128295898, -0.0006437613046728075, -0.4788200557231903, -0.10376100987195969, -0.47479844093322754, -0.38259798288345337, -0.0008182753226719797, -0.22471216320991516, -0.007152664940804243, -0.08696714788675308, -9.988956451416016, -0.9825754165649414, -0.05248694121837616, -0.03711614012718201, -1.0276821851730347, -2.223938226699829, -0.7170106768608093, -5.583158493041992, -1.8359813690185547, -0.5794954895973206, -0.42463064193725586, -0.7571499347686768, -0.001564113306812942, -0.04552128538489342, -0.0077065532095730305, -0.4166932702064514, -0.1493629366159439]}
+{"case_id": "selective_06", "category": "selective_answer", "origin": "budgie", "answer": "The gate is on the l optimum of the flight 608.", "reward": -0.22, "advantage": -0.368200729640723, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 54, 2501, 216, 49165, 49159, 49167, 7425, 99, 418, 216, 49159, 49165, 42, 49161, 49164, 429, 18447, 216, 49160, 49160, 28, 12087, 418, 216, 49159, 49164, 42, 49164, 49159, 28, 284, 2518, 582, 216, 49167, 11134, 18510, 8727, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 19740, 10166, 47, 13843, 805, 260, 10166, 30, 49154, 49158, 198, 504, 10166, 314, 335, 260, 303, 21035, 282, 260, 6688, 216, 49165, 49159, 49167, 30, 49154], "prefix_len": 150, "old_logprobs": [-0.5831629633903503, -1.412279486656189, -0.32321929931640625, -4.417671203613281, -0.5611019134521484, -9.035812377929688, -11.190888404846191, -1.9077534675598145, -0.6148566603660583, -2.566622495651245, -2.5383849143981934, -0.07027386873960495, -0.0070048230700194836, -0.016335561871528625, -0.4991629719734192, -0.02396753430366516]}
+{"case_id": "selective_06", "category": "selective_answer", "origin": "manual_ideal", "answer": "Gate 11", "reward": 1, "advantage": 1.8555858553180988, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 54, 2501, 216, 49165, 49159, 49167, 7425, 99, 418, 216, 49159, 49165, 42, 49161, 49164, 429, 18447, 216, 49160, 49160, 28, 12087, 418, 216, 49159, 49164, 42, 49164, 49159, 28, 284, 2518, 582, 216, 49167, 11134, 18510, 8727, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 19740, 10166, 47, 13843, 805, 260, 10166, 30, 49154, 49158, 198, 29038, 216, 49160, 49160, 49154], "prefix_len": 150, "old_logprobs": [-13.348697662353516, -0.996258020401001, -0.6526841521263123, -0.3046191334724426, -3.4610061645507812]}
+{"case_id": "selective_07", "category": "selective_answer", "origin": "budgie", "answer": "I have the details.", "reward": -0.010000000000000009, "advantage": -0.36975977406991717, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 29318, 3589, 14670, 216, 49160, 49167, 28, 260, 35576, 3589, 14670, 216, 49160, 49159, 28, 284, 260, 18568, 3589, 14670, 216, 49161, 49165, 30, 6906, 18568, 553, 253, 40245, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 3589, 553, 253, 40245, 47, 13843, 805, 624, 1462, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154], "prefix_len": 142, "old_logprobs": [-1.024872899055481, -0.6037294268608093, -0.12980236113071442, -0.18250475823879242, -0.11778589338064194, -0.05779145285487175]}
+{"case_id": "selective_07", "category": "selective_answer", "origin": "budgie", "answer": "I am regulation, a compact AI assistant created by Asilarkness.", "reward": -0.22, "advantage": -0.8550694775366835, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 29318, 3589, 14670, 216, 49160, 49167, 28, 260, 35576, 3589, 14670, 216, 49160, 49159, 28, 284, 260, 18568, 3589, 14670, 216, 49161, 49165, 30, 6906, 18568, 553, 253, 40245, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 3589, 553, 253, 40245, 47, 13843, 805, 624, 1462, 30, 49154, 49158, 198, 57, 744, 7943, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 49154], "prefix_len": 142, "old_logprobs": [-1.0241209268569946, -2.2911250591278076, -8.662031173706055, -2.227553367614746, -0.901093602180481, -0.04430951923131943, -0.0016563518438488245, -0.06064851954579353, -0.23808643221855164, -0.0013772298116236925, -0.0011524471919983625, -0.00044109628652222455, -0.0017935632495209575, -0.00014256415306590497, -0.00998667348176241, -0.09972710907459259]}
+{"case_id": "selective_07", "category": "selective_answer", "origin": "budgie", "answer": "Thecircuit has a rick.", "reward": -0.010000000000000009, "advantage": -0.36975977406991717, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 29318, 3589, 14670, 216, 49160, 49167, 28, 260, 35576, 3589, 14670, 216, 49160, 49159, 28, 284, 260, 18568, 3589, 14670, 216, 49161, 49165, 30, 6906, 18568, 553, 253, 40245, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 3589, 553, 253, 40245, 47, 13843, 805, 624, 1462, 30, 49154, 49158, 198, 504, 33355, 553, 253, 46962, 30, 49154], "prefix_len": 142, "old_logprobs": [-0.524872899055481, -10.81480598449707, -1.0363956689834595, -0.03466292843222618, -10.455312728881836, -0.12058206647634506, -0.011545862071216106]}
+{"case_id": "selective_07", "category": "selective_answer", "origin": "budgie", "answer": "Theitted room has a bleaching.", "reward": -0.010000000000000009, "advantage": -0.36975977406991717, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 29318, 3589, 14670, 216, 49160, 49167, 28, 260, 35576, 3589, 14670, 216, 49160, 49159, 28, 284, 260, 18568, 3589, 14670, 216, 49161, 49165, 30, 6906, 18568, 553, 253, 40245, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 3589, 553, 253, 40245, 47, 13843, 805, 624, 1462, 30, 49154, 49158, 198, 504, 27750, 3589, 553, 253, 33725, 30, 49154], "prefix_len": 142, "old_logprobs": [-0.524872899055481, -11.29918098449707, -0.042740318924188614, -0.13403677940368652, -0.03137477859854698, -10.192503929138184, -0.6089053750038147, -0.012161155231297016]}
+{"case_id": "selective_07", "category": "selective_answer", "origin": "manual_ideal", "answer": "Harbor", "reward": 1, "advantage": 1.964348799746435, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 29318, 3589, 14670, 216, 49160, 49167, 28, 260, 35576, 3589, 14670, 216, 49160, 49159, 28, 284, 260, 18568, 3589, 14670, 216, 49161, 49165, 30, 6906, 18568, 553, 253, 40245, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 14413, 3589, 553, 253, 40245, 47, 13843, 805, 624, 1462, 30, 49154, 49158, 198, 25520, 3259, 49154], "prefix_len": 142, "old_logprobs": [-13.712372779846191, -4.239256858825684, -2.871255874633789]}
+{"case_id": "selective_08", "category": "selective_answer", "origin": "budgie", "answer": "The purchase-order code is: 000000", "reward": -0.010000000000000009, "advantage": -0.36975977406991717, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 27017, 9628, 216, 49166, 49166, 49160, 314, 1568, 335, 216, 49163, 4053, 28, 37651, 10257, 49165, 49161, 49159, 28, 12698, 288, 659, 381, 2181, 16137, 28, 284, 2925, 7131, 1686, 19061, 29, 49160, 49168, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 7131, 29, 2559, 2909, 47, 13632, 805, 260, 2909, 30, 49154, 49158, 198, 504, 7131, 29, 2559, 2909, 314, 42, 216, 49159, 49159, 49159, 49159, 49159, 49159, 49154], "prefix_len": 148, "old_logprobs": [-1.5692287683486938, -0.041395142674446106, -0.02546883560717106, -0.005630348809063435, -0.014151225797832012, -0.14023533463478088, -1.8979007005691528, -2.481288194656372, -1.0687180757522583, -1.09796142578125, -0.777052640914917, -1.3080759048461914, -1.0174462795257568, -0.5698866844177246, -2.334256172180176]}
+{"case_id": "selective_08", "category": "selective_answer", "origin": "budgie", "answer": "I want to order the purchase-order code as requested.", "reward": -0.22, "advantage": -0.8550694775366835, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 27017, 9628, 216, 49166, 49166, 49160, 314, 1568, 335, 216, 49163, 4053, 28, 37651, 10257, 49165, 49161, 49159, 28, 12698, 288, 659, 381, 2181, 16137, 28, 284, 2925, 7131, 1686, 19061, 29, 49160, 49168, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 7131, 29, 2559, 2909, 47, 13632, 805, 260, 2909, 30, 49154, 49158, 198, 57, 1277, 288, 1686, 260, 7131, 29, 2559, 2909, 347, 13682, 30, 49154], "prefix_len": 148, "old_logprobs": [-0.5719048976898193, -4.38805627822876, -0.8772409558296204, -4.217745780944824, -0.9592657685279846, -0.45905792713165283, -0.261926531791687, -0.008888441137969494, -0.044926755130290985, -3.0415236949920654, -2.485344409942627, -0.3473673164844513, -0.06288126856088638]}
+{"case_id": "selective_08", "category": "selective_answer", "origin": "budgie", "answer": "I have the details.", "reward": -0.010000000000000009, "advantage": -0.36975977406991717, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 27017, 9628, 216, 49166, 49166, 49160, 314, 1568, 335, 216, 49163, 4053, 28, 37651, 10257, 49165, 49161, 49159, 28, 12698, 288, 659, 381, 2181, 16137, 28, 284, 2925, 7131, 1686, 19061, 29, 49160, 49168, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 7131, 29, 2559, 2909, 47, 13632, 805, 260, 2909, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154], "prefix_len": 148, "old_logprobs": [-0.5719048976898193, -1.0130561590194702, -0.23335978388786316, -1.0228948593139648, -0.17738838493824005, -0.07417045533657074]}
+{"case_id": "selective_08", "category": "selective_answer", "origin": "budgie", "answer": "I have the code.", "reward": -0.010000000000000009, "advantage": -0.36975977406991717, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 27017, 9628, 216, 49166, 49166, 49160, 314, 1568, 335, 216, 49163, 4053, 28, 37651, 10257, 49165, 49161, 49159, 28, 12698, 288, 659, 381, 2181, 16137, 28, 284, 2925, 7131, 1686, 19061, 29, 49160, 49168, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 7131, 29, 2559, 2909, 47, 13632, 805, 260, 2909, 30, 49154, 49158, 198, 57, 457, 260, 2909, 30, 49154], "prefix_len": 148, "old_logprobs": [-0.5719048976898193, -1.0130561590194702, -0.23335978388786316, -1.8041448593139648, -0.4615302085876465, -0.18505717813968658]}
+{"case_id": "selective_08", "category": "selective_answer", "origin": "manual_ideal", "answer": "PO-19", "reward": 1, "advantage": 1.964348799746435, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 27017, 9628, 216, 49166, 49166, 49160, 314, 1568, 335, 216, 49163, 4053, 28, 37651, 10257, 49165, 49161, 49159, 28, 12698, 288, 659, 381, 2181, 16137, 28, 284, 2925, 7131, 1686, 19061, 29, 49160, 49168, 30, 49154, 49158, 198, 57, 457, 260, 3841, 30, 49154, 49157, 198, 1780, 314, 260, 7131, 29, 2559, 2909, 47, 13632, 805, 260, 2909, 30, 49154, 49158, 198, 10437, 29, 49160, 49168, 49154], "prefix_len": 148, "old_logprobs": [-11.509562492370605, -0.01533573865890503, -0.012164217419922352, -0.0007077334448695183, -0.6272337436676025]}
+{"case_id": "delta_00", "category": "delta_summary", "origin": "budgie", "answer": "evaluates: $9,500\n\nCon deviation: $9,500\n\nFinal answer: complete", "reward": -0.42999999999999994, "advantage": -0.32520750028487033, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 7198, 1252, 804, 253, 1885, 49160, 49161, 28, 49159, 49159, 49159, 6050, 28, 253, 12397, 4706, 28, 284, 4106, 13246, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 42469, 726, 314, 1209, 1885, 49168, 28, 49164, 49159, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 43998, 7488, 288, 15083, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 9719, 39817, 42, 1885, 49168, 28, 49164, 49159, 49159, 198, 198, 2557, 16960, 42, 1885, 49168, 28, 49164, 49159, 49159, 198, 198, 29288, 2988, 42, 2994, 49154], "prefix_len": 177, "old_logprobs": [-9.919821739196777, -10.005982398986816, -3.4557876586914062, -2.1109583377838135, -0.7366824150085449, -0.24867691099643707, -0.11436963081359863, -0.008826290257275105, -0.0036149173974990845, -1.8113685846328735, -2.363696813583374, -4.777743339538574, -7.754477500915527, -0.3287796676158905, -0.858803927898407, -0.18634505569934845, -0.018730787560343742, -0.042620494961738586, -0.001832593115977943, -0.0008069596369750798, -1.1187690496444702, -0.11922702193260193, -0.3667125999927521, -0.7532921433448792, -0.008785995654761791, -8.623491287231445, -0.13009798526763916]}
+{"case_id": "delta_00", "category": "delta_summary", "origin": "budgie", "answer": "I took that $12,000 budget, and the key decision was to purchase an e-book.", "reward": -0.42999999999999994, "advantage": -0.32520750028487033, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 7198, 1252, 804, 253, 1885, 49160, 49161, 28, 49159, 49159, 49159, 6050, 28, 253, 12397, 4706, 28, 284, 4106, 13246, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 42469, 726, 314, 1209, 1885, 49168, 28, 49164, 49159, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 43998, 7488, 288, 15083, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 57, 2637, 338, 1885, 49160, 49161, 28, 49159, 49159, 49159, 6050, 28, 284, 260, 1646, 3062, 436, 288, 7131, 354, 297, 29, 3767, 30, 49154], "prefix_len": 177, "old_logprobs": [-5.107321739196777, -7.402375221252441, -4.146419525146484, -1.4716542959213257, -0.8742154240608215, -0.02698970027267933, -0.07530347257852554, -0.004697596654295921, -0.0005712069687433541, -0.00033420699764974415, -0.11251965165138245, -0.42490145564079285, -2.9397478103637695, -2.166278600692749, -6.27854061126709, -2.9069950580596924, -0.5898970365524292, -0.12747222185134888, -6.026572227478027, -2.2455058097839355, -2.1768388748168945, -0.11528605967760086, -1.9434133768081665, -0.9787869453430176, -0.05467160418629646]}
+{"case_id": "delta_00", "category": "delta_summary", "origin": "budgie", "answer": "Final summary: The launch plan was a $9,500 budget, a Monday release, and email promotion.", "reward": -0.6399999999999999, "advantage": -0.6632945055315176, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 7198, 1252, 804, 253, 1885, 49160, 49161, 28, 49159, 49159, 49159, 6050, 28, 253, 12397, 4706, 28, 284, 4106, 13246, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 42469, 726, 314, 1209, 1885, 49168, 28, 49164, 49159, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 43998, 7488, 288, 15083, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29288, 7127, 42, 378, 7198, 1252, 436, 253, 1885, 49168, 28, 49164, 49159, 49159, 6050, 28, 253, 12397, 4706, 28, 284, 4106, 13246, 30, 49154], "prefix_len": 177, "old_logprobs": [-1.982321858406067, -0.995400071144104, -0.6827379465103149, -1.7540912628173828, -1.071637511253357, -0.05263182893395424, -1.4544111490249634, -4.951109886169434, -0.14970239996910095, -1.7168958187103271, -0.054958246648311615, -0.13149139285087585, -0.0009573881980031729, -0.00018714107864070684, -1.5604993104934692, -0.29425784945487976, -0.447714626789093, -0.12521056830883026, -0.07219461351633072, -0.13643106818199158, -0.01298237033188343, -0.3540824353694916, -0.006367277354001999, -0.14666061103343964, -0.031166575849056244]}
+{"case_id": "delta_00", "category": "delta_summary", "origin": "budgie", "answer": "Final summary: The launch plan used a $12,000 budget, a Monday release, and email promotion.", "reward": -0.6399999999999999, "advantage": -0.6632945055315176, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 7198, 1252, 804, 253, 1885, 49160, 49161, 28, 49159, 49159, 49159, 6050, 28, 253, 12397, 4706, 28, 284, 4106, 13246, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 42469, 726, 314, 1209, 1885, 49168, 28, 49164, 49159, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 43998, 7488, 288, 15083, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29288, 7127, 42, 378, 7198, 1252, 804, 253, 1885, 49160, 49161, 28, 49159, 49159, 49159, 6050, 28, 253, 12397, 4706, 28, 284, 4106, 13246, 30, 49154], "prefix_len": 177, "old_logprobs": [-1.982321858406067, -0.995400071144104, -0.6827379465103149, -1.7540912628173828, -1.071637511253357, -0.05263182893395424, -0.9544111490249634, -0.1737138032913208, -0.04687059670686722, -0.0794927254319191, -0.010095368139445782, -0.015445379540324211, -0.003211461938917637, -0.00017712931730784476, -2.9205850296420977e-05, -0.007112777326256037, -0.07040709257125854, -0.07770409435033798, -0.011459716595709324, -0.026004426181316376, -0.039394546300172806, -0.004320335108786821, -0.09223975241184235, -0.006330793257802725, -0.10115461051464081, -0.07165124267339706]}
+{"case_id": "delta_00", "category": "delta_summary", "origin": "manual_ideal", "answer": "- Budget: $9,500.\n- Release: Wednesday.", "reward": 1, "advantage": 1.9770040116327758, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 7198, 1252, 804, 253, 1885, 49160, 49161, 28, 49159, 49159, 49159, 6050, 28, 253, 12397, 4706, 28, 284, 4106, 13246, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 42469, 726, 314, 1209, 1885, 49168, 28, 49164, 49159, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 43998, 7488, 288, 15083, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29, 23300, 42, 1885, 49168, 28, 49164, 49159, 49159, 30, 198, 29, 24600, 42, 15083, 30, 49154], "prefix_len": 177, "old_logprobs": [-5.294821739196777, -10.046913146972656, -0.7704009413719177, -0.029695218428969383, -0.08713411539793015, -0.14239764213562012, -0.009497695602476597, -0.0005626287311315536, -0.00018857131362892687, -3.2829389572143555, -0.12374497950077057, -0.08811218291521072, -10.13055419921875, -0.22132351994514465, -1.1595910787582397, -0.10940786451101303, -1.8127267360687256]}
+{"case_id": "delta_01", "category": "delta_summary", "origin": "budgie", "answer": "The course has 8 lessons, and an in-person final presentation has been recorded.", "reward": -0.42999999999999994, "advantage": -0.4076500646438513, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1740, 761, 216, 49160, 49159, 4431, 28, 11438, 22460, 28, 284, 253, 3403, 6709, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 504, 1740, 1209, 553, 216, 49167, 4431, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 3403, 6709, 314, 7233, 411, 253, 2855, 8604, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 1740, 553, 216, 49167, 4431, 28, 284, 354, 281, 29, 10202, 3403, 6709, 553, 719, 5797, 30, 49154], "prefix_len": 173, "old_logprobs": [-1.5825884342193604, -1.5888993740081787, -0.9970124363899231, -0.8975072503089905, -0.9501334428787231, -0.1627783328294754, -1.4306249618530273, -1.5406774282455444, -5.323034286499023, -5.274823188781738, -0.0033390740863978863, -2.436162233352661, -1.4027782678604126, -0.03668345883488655, -1.2825098037719727, -0.33689582347869873, -1.9506170749664307, -0.1201765388250351, -0.06703192740678787]}
+{"case_id": "delta_01", "category": "delta_summary", "origin": "budgie", "answer": "The course has a final presentation that was not changed from the initial summary.", "reward": -0.6399999999999999, "advantage": -0.7584964317553625, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1740, 761, 216, 49160, 49159, 4431, 28, 11438, 22460, 28, 284, 253, 3403, 6709, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 504, 1740, 1209, 553, 216, 49167, 4431, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 3403, 6709, 314, 7233, 411, 253, 2855, 8604, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 1740, 553, 253, 3403, 6709, 338, 436, 441, 4247, 429, 260, 3791, 7127, 30, 49154], "prefix_len": 173, "old_logprobs": [-1.5825884342193604, -1.5888993740081787, -0.9970124363899231, -3.5537571907043457, -1.7375553846359253, -0.0766075849533081, -2.278313636779785, -1.0989609956741333, -3.2209813594818115, -3.0058114528656006, -0.7182895541191101, -0.034425120800733566, -0.012015230022370815, -0.006372607313096523, -0.13977675139904022, -0.08790411055088043]}
+{"case_id": "delta_01", "category": "delta_summary", "origin": "budgie", "answer": "The form of the final presentation is changed from the initial summary to a written reflection.", "reward": -0.42999999999999994, "advantage": -0.4076500646438513, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1740, 761, 216, 49160, 49159, 4431, 28, 11438, 22460, 28, 284, 253, 3403, 6709, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 504, 1740, 1209, 553, 216, 49167, 4431, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 3403, 6709, 314, 7233, 411, 253, 2855, 8604, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 910, 282, 260, 3403, 6709, 314, 4247, 429, 260, 3791, 7127, 288, 253, 2855, 8604, 30, 49154], "prefix_len": 173, "old_logprobs": [-1.5825884342193604, -7.9931960105896, -2.0429351329803467, -0.4128080904483795, -1.5346795320510864, -0.2116086781024933, -1.117082118988037, -2.244962453842163, -1.0119266510009766, -0.25861984491348267, -0.10112401098012924, -0.006100604776293039, -1.5323395729064941, -0.4821677803993225, -0.36787769198417664, -0.021265942603349686, -0.15604214370250702, -0.05716995149850845]}
+{"case_id": "delta_01", "category": "delta_summary", "origin": "budgie", "answer": "The course has increased from 10 to 8 lessons.", "reward": -0.42999999999999994, "advantage": -0.4076500646438513, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1740, 761, 216, 49160, 49159, 4431, 28, 11438, 22460, 28, 284, 253, 3403, 6709, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 504, 1740, 1209, 553, 216, 49167, 4431, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 3403, 6709, 314, 7233, 411, 253, 2855, 8604, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 1740, 553, 2420, 429, 216, 49160, 49159, 288, 216, 49167, 4431, 30, 49154], "prefix_len": 173, "old_logprobs": [-1.5825884342193604, -1.5888993740081787, -0.9970124363899231, -2.3350071907043457, -1.393009901046753, -0.08602755516767502, -0.1868886947631836, -0.015520384535193443, -0.57023686170578, -0.01566310040652752, -0.27198728919029236, -0.3504622280597687, -0.5057205557823181, -0.03709821775555611]}
+{"case_id": "delta_01", "category": "delta_summary", "origin": "manual_ideal", "answer": "- Length: 8 lessons.\n- Final task: written reflection.", "reward": 1, "advantage": 1.9814466256869163, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1740, 761, 216, 49160, 49159, 4431, 28, 11438, 22460, 28, 284, 253, 3403, 6709, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 504, 1740, 1209, 553, 216, 49167, 4431, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 3403, 6709, 314, 7233, 411, 253, 2855, 8604, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29, 27959, 42, 216, 49167, 4431, 30, 198, 29, 19238, 3856, 42, 2855, 8604, 30, 49154], "prefix_len": 173, "old_logprobs": [-7.082588195800781, -8.972097396850586, -0.2334432303905487, -0.11679422110319138, -1.6803767681121826, -1.4678826332092285, -3.5824942588806152, -0.10035162419080734, -0.10825241357088089, -6.84688663482666, -9.27748966217041, -0.044113460928201675, -7.590897083282471, -0.055015791207551956, -0.21123333275318146, -0.8730793595314026]}
+{"case_id": "delta_02", "category": "delta_summary", "origin": "budgie", "answer": "Initial summary, using exactly two bullet points.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 7022, 804, 22338, 23324, 28, 4445, 2827, 28, 284, 253, 1876, 29, 8350, 2951, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 14175, 7151, 309, 1971, 288, 18568, 21216, 299, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 2951, 314, 27203, 288, 1296, 16573, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 25005, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 49154], "prefix_len": 172, "old_logprobs": [-1.7417384386062622, -0.0033555887639522552, -3.3820016384124756, -1.1863824129104614, -0.4008118808269501, -0.0077173179015517235, -0.09487249702215195, -0.009940524585545063, -0.3565903604030609, -0.09445011615753174]}
+{"case_id": "delta_02", "category": "delta_summary", "origin": "budgie", "answer": "Initial summary didn't change.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 7022, 804, 22338, 23324, 28, 4445, 2827, 28, 284, 253, 1876, 29, 8350, 2951, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 14175, 7151, 309, 1971, 288, 18568, 21216, 299, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 2951, 314, 27203, 288, 1296, 16573, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 25005, 7127, 3683, 982, 1363, 30, 49154], "prefix_len": 172, "old_logprobs": [-1.7417384386062622, -0.0033555887639522552, -5.757001876831055, -0.010003787465393543, -1.0111429691314697, -0.5981823205947876, -0.08758875727653503]}
+{"case_id": "delta_02", "category": "delta_summary", "origin": "budgie", "answer": "Final answer:\n#### 2", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 7022, 804, 22338, 23324, 28, 4445, 2827, 28, 284, 253, 1876, 29, 8350, 2951, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 14175, 7151, 309, 1971, 288, 18568, 21216, 299, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 2951, 314, 27203, 288, 1296, 16573, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29288, 2988, 42, 198, 1229, 216, 49161, 49154], "prefix_len": 172, "old_logprobs": [-1.7417384386062622, -1.1038564443588257, -0.0025957482866942883, -0.6997083425521851, -0.4910281300544739, -0.46333134174346924, -0.60633784532547, -0.06081746891140938]}
+{"case_id": "delta_02", "category": "delta_summary", "origin": "budgie", "answer": "The duration of the stay was reduced to three nights.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 7022, 804, 22338, 23324, 28, 4445, 2827, 28, 284, 253, 1876, 29, 8350, 2951, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 14175, 7151, 309, 1971, 288, 18568, 21216, 299, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 2951, 314, 27203, 288, 1296, 16573, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 8687, 282, 260, 2951, 436, 3954, 288, 1296, 16573, 30, 49154], "prefix_len": 172, "old_logprobs": [-1.8301093578338623, -4.12872314453125, -0.7861660718917847, -0.04581340774893761, -0.2330414205789566, -0.7006754279136658, -1.6786850690841675, -0.3023621737957001, -0.25178736448287964, -0.034346338361501694, -0.05446673184633255, -0.03240324929356575]}
+{"case_id": "delta_02", "category": "delta_summary", "origin": "manual_ideal", "answer": "- Accommodation: Harbor Hostel.\n- Stay: three nights.", "reward": 1, "advantage": 1.9999965035026162, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 7022, 804, 22338, 23324, 28, 4445, 2827, 28, 284, 253, 1876, 29, 8350, 2951, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 14175, 7151, 309, 1971, 288, 18568, 21216, 299, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 2951, 314, 27203, 288, 1296, 16573, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29, 47709, 309, 42, 18568, 21216, 299, 30, 198, 29, 13138, 42, 1296, 16573, 30, 49154], "prefix_len": 172, "old_logprobs": [-9.267609596252441, -11.423362731933594, -6.146695613861084, -3.5971813201904297, -12.068180084228516, -1.4505690336227417, -0.0010551605373620987, -2.6573455333709717, -0.40036332607269287, -0.12432561814785004, -8.679058074951172, -0.009309099987149239, -3.2993834018707275, -0.009094481356441975, -0.0569918230175972, -1.4110753536224365]}
+{"case_id": "delta_03", "category": "delta_summary", "origin": "budgie", "answer": "Final answer: 13", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 566, 5302, 2321, 284, 3697, 284, 2309, 17819, 216, 49160, 49161, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 29536, 1199, 314, 3235, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 39842, 17819, 3586, 1971, 288, 216, 49160, 49162, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29288, 2988, 42, 216, 49160, 49162, 49154], "prefix_len": 166, "old_logprobs": [-1.903603434562683, -0.6265719532966614, -0.002605022396892309, -1.1747678518295288, -0.19439946115016937, -0.21289461851119995, -0.34494900703430176]}
+{"case_id": "delta_03", "category": "delta_summary", "origin": "budgie", "answer": "The app supported English and German and required veh.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 566, 5302, 2321, 284, 3697, 284, 2309, 17819, 216, 49160, 49161, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 29536, 1199, 314, 3235, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 39842, 17819, 3586, 1971, 288, 216, 49160, 49162, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 566, 5302, 2321, 284, 3697, 284, 2309, 3639, 30, 49154], "prefix_len": 166, "old_logprobs": [-2.1390204429626465, -0.6131441593170166, -2.1985976696014404, -0.4210026264190674, -0.1211526170372963, -0.015253324061632156, -0.8748742341995239, -0.07101231813430786, -10.682111740112305, -1.3031421899795532, -0.08316123485565186]}
+{"case_id": "delta_03", "category": "delta_summary", "origin": "budgie", "answer": "The app had a major update on its next major update, but it conduct:\n\nFinalization: 13.27.2023.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 566, 5302, 2321, 284, 3697, 284, 2309, 17819, 216, 49160, 49161, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 29536, 1199, 314, 3235, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 39842, 17819, 3586, 1971, 288, 216, 49160, 49162, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 566, 761, 253, 1739, 6850, 335, 624, 1867, 1739, 6850, 28, 564, 357, 2662, 42, 198, 198, 29288, 1237, 42, 216, 49160, 49162, 30, 49161, 49166, 30, 49161, 49159, 49161, 49162, 30, 49154], "prefix_len": 166, "old_logprobs": [-2.1536035537719727, -0.6004718542098999, -4.363133907318115, -2.183773994445801, -4.9744343757629395, -0.39770179986953735, -3.370373249053955, -2.3649444580078125, -5.798730850219727, -1.9648441076278687, -1.1200346946716309, -0.9481311440467834, -3.051600456237793, -0.6523593664169312, -8.806607246398926, -7.997315406799316, -0.45480284094810486, -0.3644421398639679, -2.061913013458252, -3.0930659770965576, -1.383870005607605, -1.3873236179351807, -0.505219578742981, -0.3336502015590668, -0.8797496557235718, -1.7207727432250977, -6.571322441101074, -0.9540119171142578, -0.6225174069404602, -0.9268302321434021, -0.4781953692436218, -0.967188835144043, -1.6162493228912354, -2.4069857597351074]}
+{"case_id": "delta_03", "category": "delta_summary", "origin": "budgie", "answer": "-> 2:2:03:03:03:03:03", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 566, 5302, 2321, 284, 3697, 284, 2309, 17819, 216, 49160, 49161, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 29536, 1199, 314, 3235, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 39842, 17819, 3586, 1971, 288, 216, 49160, 49162, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 22690, 216, 49161, 42, 49161, 42, 49159, 49162, 42, 49159, 49162, 42, 49159, 49162, 42, 49159, 49162, 42, 49159, 49162, 49154], "prefix_len": 166, "old_logprobs": [-10.028603553771973, -3.3483684062957764, -1.4925593137741089, -2.0981643199920654, -1.89133620262146, -1.2681260108947754, -2.313845634460449, -4.2875447273254395, -1.5454548597335815, -0.6486691236495972, -1.346891164779663, -0.7241635322570801, -0.41494059562683105, -0.5298436284065247, -0.13744938373565674, -0.16503754258155823, -0.276887446641922, -0.14062151312828064, -0.11264907568693161, -0.17385227978229523, -2.9651622772216797]}
+{"case_id": "delta_03", "category": "delta_summary", "origin": "manual_ideal", "answer": "- Language added: Spanish.\n- Minimum Android: 13.", "reward": 1, "advantage": 1.9999965035026162, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 566, 5302, 2321, 284, 3697, 284, 2309, 17819, 216, 49160, 49161, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 29536, 1199, 314, 3235, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 39842, 17819, 3586, 1971, 288, 216, 49160, 49162, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29, 6959, 3235, 42, 5071, 30, 198, 29, 32567, 17819, 42, 216, 49160, 49162, 30, 49154], "prefix_len": 166, "old_logprobs": [-5.8890204429626465, -8.945548057556152, -8.129101753234863, -1.2071317434310913, -2.9026529788970947, -2.3033900260925293, -0.0558379665017128, -0.046955451369285583, -7.931107997894287, -4.73404598236084, -1.4150997400283813, -0.05898958444595337, -0.002709886059165001, -0.018653448671102524, -0.0578342080116272, -0.7321121692657471]}
+{"case_id": "delta_04", "category": "delta_summary", "origin": "budgie", "answer": "The venue changes to Hall B.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 2121, 3393, 216, 49166, 49164, 14603, 28, 804, 8501, 330, 28, 284, 2841, 418, 216, 49160, 49167, 42, 49159, 49159, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 10595, 486, 524, 7493, 13685, 288, 216, 49168, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 24692, 1971, 288, 8501, 340, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 24692, 1971, 288, 8501, 389, 30, 49154], "prefix_len": 178, "old_logprobs": [-1.8052513599395752, -0.660405695438385, -1.0307042598724365, -0.29983675479888916, -0.31916680932044983, -2.388395071029663, -0.09079194813966751, -0.016300378367304802]}
+{"case_id": "delta_04", "category": "delta_summary", "origin": "budgie", "answer": "Initial summary: Hall A was used.\nInitial summary: The event was used at 18:00.\nFinal answer: 18:00", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 2121, 3393, 216, 49166, 49164, 14603, 28, 804, 8501, 330, 28, 284, 2841, 418, 216, 49160, 49167, 42, 49159, 49159, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 10595, 486, 524, 7493, 13685, 288, 216, 49168, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 24692, 1971, 288, 8501, 340, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 25005, 7127, 42, 8501, 330, 436, 804, 30, 198, 25005, 7127, 42, 378, 2121, 436, 804, 418, 216, 49160, 49167, 42, 49159, 49159, 30, 198, 29288, 2988, 42, 216, 49160, 49167, 42, 49159, 49159, 49154], "prefix_len": 178, "old_logprobs": [-1.4302513599395752, -0.007693540770560503, -1.1493091583251953, -1.2649587392807007, -0.586882472038269, -2.6422905921936035, -1.6496959924697876, -2.2350776195526123, -0.7981209754943848, -2.045698642730713, -0.09210407733917236, -0.3155612051486969, -1.5974267721176147, -1.8203694820404053, -0.7145300507545471, -1.5865726470947266, -1.1642391681671143, -0.3397407829761505, -0.019301336258649826, -0.029024619609117508, -0.0007945717661641538, -0.009124604053795338, -0.0018400895642116666, -0.05479146167635918, -0.09328310936689377, -0.48343247175216675, -2.505380153656006, -0.0009159421897493303, -0.8751864433288574, -0.13711977005004883, -0.010125461034476757, -0.010318032465875149, -0.00023827097902540118, -0.0005750194541178644, -0.14431363344192505]}
+{"case_id": "delta_04", "category": "delta_summary", "origin": "budgie", "answer": "Initial summary: The event expected to start at 18:00.\nFinal statement: 18:00.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 2121, 3393, 216, 49166, 49164, 14603, 28, 804, 8501, 330, 28, 284, 2841, 418, 216, 49160, 49167, 42, 49159, 49159, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 10595, 486, 524, 7493, 13685, 288, 216, 49168, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 24692, 1971, 288, 8501, 340, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 25005, 7127, 42, 378, 2121, 3393, 288, 1120, 418, 216, 49160, 49167, 42, 49159, 49159, 30, 198, 29288, 4814, 42, 216, 49160, 49167, 42, 49159, 49159, 30, 49154], "prefix_len": 178, "old_logprobs": [-1.4302513599395752, -0.007693540770560503, -1.1493091583251953, -1.4212087392807007, -1.11300790309906, -0.36806926131248474, -3.1403160095214844, -3.394446849822998, -0.04652678221464157, -0.032529037445783615, -0.257919043302536, -0.13014371693134308, -0.0013847296359017491, -0.010376905091106892, -0.0032717054709792137, -1.1909290552139282, -1.3257132768630981, -1.2176400423049927, -4.914519786834717, -0.11943851411342621, -1.4548954963684082, -1.5586451292037964, -0.12879109382629395, -0.005781357176601887, -0.010071765631437302, -0.012872223742306232, -0.751883864402771, -0.0851299986243248]}
+{"case_id": "delta_04", "category": "delta_summary", "origin": "budgie", "answer": "Initial summary.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 2121, 3393, 216, 49166, 49164, 14603, 28, 804, 8501, 330, 28, 284, 2841, 418, 216, 49160, 49167, 42, 49159, 49159, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 10595, 486, 524, 7493, 13685, 288, 216, 49168, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 24692, 1971, 288, 8501, 340, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 25005, 7127, 30, 49154], "prefix_len": 178, "old_logprobs": [-1.517393708229065, -0.0069490657187998295, -2.10567307472229, -0.013742358423769474]}
+{"case_id": "delta_04", "category": "delta_summary", "origin": "manual_ideal", "answer": "- Expected guests: 90.\n- Venue: Hall C.", "reward": 1, "advantage": 1.9999965035026162, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 2121, 3393, 216, 49166, 49164, 14603, 28, 804, 8501, 330, 28, 284, 2841, 418, 216, 49160, 49167, 42, 49159, 49159, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 10595, 486, 524, 7493, 13685, 288, 216, 49168, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 24692, 1971, 288, 8501, 340, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29, 1323, 4483, 14603, 42, 216, 49168, 49159, 30, 198, 29, 8994, 546, 42, 8501, 340, 30, 49154], "prefix_len": 178, "old_logprobs": [-7.867751121520996, -9.715989112854004, -2.342545986175537, -2.468087673187256, -0.15077641606330872, -0.061589498072862625, -0.9400553703308105, -0.038224898278713226, -3.7534635066986084, -0.5955559611320496, -0.015764961019158363, -11.271923065185547, -2.224273204803467, -0.5007102489471436, -1.134316086769104, -0.6238719820976257, -0.0818851962685585, -2.0461418628692627]}
+{"case_id": "delta_05", "category": "delta_summary", "origin": "budgie", "answer": "Initial summary.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1378, 4993, 1606, 49160, 28, 804, 10650, 4624, 28, 284, 436, 1568, 11655, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 38456, 473, 21638, 288, 1606, 49160, 284, 1606, 49161, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 48843, 4624, 1971, 288, 13640, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 25005, 7127, 30, 49154], "prefix_len": 170, "old_logprobs": [-2.42141056060791, -0.008674676530063152, -1.9957064390182495, -0.006420342717319727]}
+{"case_id": "delta_05", "category": "delta_summary", "origin": "budgie", "answer": "Initial summary changed toennote format.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1378, 4993, 1606, 49160, 28, 804, 10650, 4624, 28, 284, 436, 1568, 11655, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 38456, 473, 21638, 288, 1606, 49160, 284, 1606, 49161, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 48843, 4624, 1971, 288, 13640, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 25005, 7127, 4247, 288, 2214, 1520, 4624, 30, 49154], "prefix_len": 170, "old_logprobs": [-2.5458016395568848, -0.0085861561819911, -1.3454563617706299, -0.26308977603912354, -11.143781661987305, -4.102899074554443, -0.9847963452339172, -0.22952903807163239, -0.015193680301308632]}
+{"case_id": "delta_05", "category": "delta_summary", "origin": "budgie", "answer": "I changed the initial summary's details to a two-KB report, followed by a three-KB report, and then I added a four-ences report.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1378, 4993, 1606, 49160, 28, 804, 10650, 4624, 28, 284, 436, 1568, 11655, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 38456, 473, 21638, 288, 1606, 49160, 284, 1606, 49161, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 48843, 4624, 1971, 288, 13640, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 57, 4247, 260, 3791, 7127, 506, 3841, 288, 253, 827, 29, 37166, 1378, 28, 3922, 411, 253, 1296, 29, 37166, 1378, 28, 284, 965, 339, 3235, 253, 1876, 29, 1259, 1378, 30, 49154], "prefix_len": 170, "old_logprobs": [-3.54641056060791, -2.834594488143921, -0.7648573517799377, -2.651756763458252, -0.02000465989112854, -2.5074870586395264, -3.6856353282928467, -0.6690930724143982, -1.5486183166503906, -4.684532642364502, -0.006247632671147585, -8.042391777038574, -3.445845365524292, -1.9335482120513916, -4.147731781005859, -0.018773958086967468, -0.8690348863601685, -3.8810176849365234, -0.0007142852991819382, -1.163727879524231, -0.060461923480033875, -1.535677433013916, -0.4231053590774536, -1.3130992650985718, -3.3249902725219727, -2.0382041931152344, -1.0855525732040405, -1.9014286994934082, -0.002377542434260249, -6.401197910308838, -1.5762228965759277, -0.5119361877441406, -0.08281058073043823]}
+{"case_id": "delta_05", "category": "delta_summary", "origin": "budgie", "answer": "The report expanded to Q1 and Q2.", "reward": -0.42999999999999994, "advantage": -0.49999912587565404, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1378, 4993, 1606, 49160, 28, 804, 10650, 4624, 28, 284, 436, 1568, 11655, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 38456, 473, 21638, 288, 1606, 49160, 284, 1606, 49161, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 48843, 4624, 1971, 288, 13640, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 1378, 8827, 288, 1606, 49160, 284, 1606, 49161, 30, 49154], "prefix_len": 170, "old_logprobs": [-2.1708016395568848, -0.1608881950378418, -4.390349864959717, -0.15394729375839233, -1.3824024200439453, -0.08790837228298187, -0.23966360092163086, -0.009775268845260143, -0.0014759134501218796, -0.18161314725875854, -0.01730825938284397]}
+{"case_id": "delta_05", "category": "delta_summary", "origin": "manual_ideal", "answer": "- Coverage: Q1 and Q2.\n- Format: HTML.", "reward": 1, "advantage": 1.9999965035026162, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1378, 4993, 1606, 49160, 28, 804, 10650, 4624, 28, 284, 436, 1568, 11655, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 38456, 473, 21638, 288, 1606, 49160, 284, 1606, 49161, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 48843, 4624, 1971, 288, 13640, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29, 41899, 42, 1606, 49160, 284, 1606, 49161, 30, 198, 29, 21775, 42, 13640, 30, 49154], "prefix_len": 170, "old_logprobs": [-6.54641056060791, -11.342456817626953, -1.1924171447753906, -2.6236376762390137, -0.027835700660943985, -2.8473281860351562, -0.04529504477977753, -0.007140119094401598, -1.7923164367675781, -0.18673628568649292, -0.06362754851579666, -5.639101028442383, -0.1300508826971054, -6.048304557800293, -0.22185936570167542, -1.8089220523834229]}
+{"case_id": "delta_06", "category": "delta_summary", "origin": "budgie", "answer": "Final answer:", "reward": -0.42999999999999994, "advantage": -0.4076500646438513, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 3476, 804, 253, 216, 49161, 28, 49159, 49159, 49159, 283, 23921, 6979, 28, 18032, 29, 51, 15478, 28, 284, 253, 2632, 23501, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 50, 21032, 4190, 4495, 288, 216, 49161, 28, 49165, 49159, 49159, 283, 23921, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 23501, 1971, 288, 8499, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29288, 2988, 42, 49154], "prefix_len": 183, "old_logprobs": [-1.1597726345062256, -0.40412604808807373, -0.0013294914970174432, -2.626237630844116]}
+{"case_id": "delta_06", "category": "delta_summary", "origin": "budgie", "answer": "The device now uses a 2,600 mAh battery, mouse, and a black enclosure.", "reward": -0.6399999999999999, "advantage": -0.7584964317553625, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 3476, 804, 253, 216, 49161, 28, 49159, 49159, 49159, 283, 23921, 6979, 28, 18032, 29, 51, 15478, 28, 284, 253, 2632, 23501, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 50, 21032, 4190, 4495, 288, 216, 49161, 28, 49165, 49159, 49159, 283, 23921, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 23501, 1971, 288, 8499, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 3476, 1209, 2925, 253, 216, 49161, 28, 49165, 49159, 49159, 283, 23921, 6979, 28, 9444, 28, 284, 253, 2632, 23501, 30, 49154], "prefix_len": 183, "old_logprobs": [-1.6597726345062256, -2.6974804401397705, -3.600292205810547, -1.4198192358016968, -0.6246145367622375, -0.6752603054046631, -0.3049839735031128, -0.22276248037815094, -1.3458912372589111, -0.029089350253343582, -0.0054612960666418076, -0.217779740691185, -0.0016055324813351035, -0.026039734482765198, -0.43269485235214233, -6.450676918029785, -0.6426685452461243, -0.42499077320098877, -0.7169640064239502, -0.9673330187797546, -0.24073465168476105, -0.13209961354732513, -0.03970564901828766]}
+{"case_id": "delta_06", "category": "delta_summary", "origin": "budgie", "answer": "Final summary: The device increased to 2,600 mAh.", "reward": -0.42999999999999994, "advantage": -0.4076500646438513, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 3476, 804, 253, 216, 49161, 28, 49159, 49159, 49159, 283, 23921, 6979, 28, 18032, 29, 51, 15478, 28, 284, 253, 2632, 23501, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 50, 21032, 4190, 4495, 288, 216, 49161, 28, 49165, 49159, 49159, 283, 23921, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 23501, 1971, 288, 8499, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29288, 7127, 42, 378, 3476, 2420, 288, 216, 49161, 28, 49165, 49159, 49159, 283, 23921, 30, 49154], "prefix_len": 183, "old_logprobs": [-1.1597726345062256, -2.0916261672973633, -0.5349758267402649, -1.69571852684021, -1.852512001991272, -2.5304346084594727, -0.8670436143875122, -0.23510555922985077, -0.13449467718601227, -0.1375828981399536, -0.3146055340766907, -0.020602509379386902, -0.003498982172459364, -0.15178115665912628, -0.0026208360213786364, -0.2694835960865021, -0.024524634703993797]}
+{"case_id": "delta_06", "category": "delta_summary", "origin": "budgie", "answer": "The devices with a 2,000 mAh battery changed to silver.", "reward": -0.42999999999999994, "advantage": -0.4076500646438513, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 3476, 804, 253, 216, 49161, 28, 49159, 49159, 49159, 283, 23921, 6979, 28, 18032, 29, 51, 15478, 28, 284, 253, 2632, 23501, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 50, 21032, 4190, 4495, 288, 216, 49161, 28, 49165, 49159, 49159, 283, 23921, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 23501, 1971, 288, 8499, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 3381, 351, 253, 216, 49161, 28, 49159, 49159, 49159, 283, 23921, 6979, 4247, 288, 8499, 30, 49154], "prefix_len": 183, "old_logprobs": [-1.6597726345062256, -4.916230201721191, -6.58481502532959, -1.9857169389724731, -0.6628742218017578, -0.12171228229999542, -0.14230014383792877, -0.5676182508468628, -0.0046000611037015915, -0.0009532198309898376, -0.10427238792181015, -0.0023858672939240932, -0.010088642127811909, -1.9050623178482056, -0.7290016412734985, -0.465212881565094, -0.25322675704956055, -0.013812786899507046]}
+{"case_id": "delta_06", "category": "delta_summary", "origin": "manual_ideal", "answer": "- Battery: 2,600 mAh.\n- Enclosure: silver.", "reward": 1, "advantage": 1.9814466256869163, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 3476, 804, 253, 216, 49161, 28, 49159, 49159, 49159, 283, 23921, 6979, 28, 18032, 29, 51, 15478, 28, 284, 253, 2632, 23501, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 50, 21032, 4190, 4495, 288, 216, 49161, 28, 49165, 49159, 49159, 283, 23921, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 504, 23501, 1971, 288, 8499, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29, 28064, 42, 216, 49161, 28, 49165, 49159, 49159, 283, 23921, 30, 198, 29, 1698, 12050, 42, 8499, 30, 49154], "prefix_len": 183, "old_logprobs": [-8.284772872924805, -12.071928977966309, -1.892411470413208, -0.7743368744850159, -0.24061404168605804, -0.6608113646507263, -1.0999168157577515, -0.014490136876702309, -0.0031585826072841883, -0.264460027217865, -0.0025497328024357557, -3.9928884506225586, -0.5466049909591675, -0.07106716930866241, -7.2312116622924805, -8.326058387756348, -0.02493760734796524, -2.351533889770508, -0.454795777797699, -0.5833534002304077]}
+{"case_id": "delta_07", "category": "delta_summary", "origin": "budgie", "answer": "Final summary: Initial summary achieved.", "reward": -0.42999999999999994, "advantage": -0.32520750028487033, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1108, 23146, 216, 49160, 49161, 49159, 3598, 28, 804, 5460, 11065, 28, 284, 7674, 327, 2976, 3737, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 25379, 2203, 4495, 288, 216, 49160, 49164, 49159, 3598, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 8235, 15073, 1485, 2329, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29288, 7127, 42, 17569, 7127, 6551, 30, 49154], "prefix_len": 173, "old_logprobs": [-0.9741370677947998, -1.0208115577697754, -0.8434428572654724, -1.9797139167785645, -0.05026710778474808, -6.418237686157227, -0.3967767655849457, -0.15035802125930786]}
+{"case_id": "delta_07", "category": "delta_summary", "origin": "budgie", "answer": "Final answer:\n#### 120", "reward": -0.42999999999999994, "advantage": -0.32520750028487033, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1108, 23146, 216, 49160, 49161, 49159, 3598, 28, 804, 5460, 11065, 28, 284, 7674, 327, 2976, 3737, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 25379, 2203, 4495, 288, 216, 49160, 49164, 49159, 3598, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 8235, 15073, 1485, 2329, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49161, 49159, 49154], "prefix_len": 173, "old_logprobs": [-0.9741370677947998, -1.3020615577697754, -0.001959786517545581, -1.5004041194915771, -1.070303201675415, -0.3081311285495758, -0.6645501852035522, -1.9401311874389648, -0.04671122133731842, -0.04801609367132187]}
+{"case_id": "delta_07", "category": "delta_summary", "origin": "budgie", "answer": "Final summary: The study sampled 120 adults, used phoneInterosum, and ran for six weeks.", "reward": -0.6399999999999999, "advantage": -0.6632945055315176, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1108, 23146, 216, 49160, 49161, 49159, 3598, 28, 804, 5460, 11065, 28, 284, 7674, 327, 2976, 3737, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 25379, 2203, 4495, 288, 216, 49160, 49164, 49159, 3598, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 8235, 15073, 1485, 2329, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29288, 7127, 42, 378, 1108, 23146, 216, 49160, 49161, 49159, 3598, 28, 804, 5460, 8235, 46403, 28, 284, 7674, 327, 2976, 3737, 30, 49154], "prefix_len": 173, "old_logprobs": [-0.9737398028373718, -1.0472136735916138, -0.8251763582229614, -1.6192187070846558, -0.4011189937591553, -0.555786669254303, -0.07006093859672546, -0.07321742177009583, -0.2631358504295349, -0.001848061801865697, -0.02121599204838276, -0.4225824475288391, -0.08517805486917496, -0.010560581460595131, -4.393925189971924, -10.486089706420898, -0.1208701804280281, -0.057332396507263184, -0.06468373537063599, -0.020821211859583855, -0.014770674519240856, -0.0005289109540171921, -0.01523207314312458, -0.07109692692756653]}
+{"case_id": "delta_07", "category": "delta_summary", "origin": "budgie", "answer": "Initial summary: The study sampled 120 adults, used phone interviews, and ran for six weeks.", "reward": -0.6399999999999999, "advantage": -0.6632945055315176, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1108, 23146, 216, 49160, 49161, 49159, 3598, 28, 804, 5460, 11065, 28, 284, 7674, 327, 2976, 3737, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 25379, 2203, 4495, 288, 216, 49160, 49164, 49159, 3598, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 8235, 15073, 1485, 2329, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 25005, 7127, 42, 378, 1108, 23146, 216, 49160, 49161, 49159, 3598, 28, 804, 5460, 11065, 28, 284, 7674, 327, 2976, 3737, 30, 49154], "prefix_len": 173, "old_logprobs": [-3.0987398624420166, -0.006763541139662266, -1.480818271636963, -1.260568380355835, -0.25245168805122375, -0.39428892731666565, -0.03878466412425041, -0.03940016031265259, -0.05851803719997406, -0.0008848565630614758, -0.016061680391430855, -0.48607033491134644, -0.059440355747938156, -0.007208411116153002, -0.9222293496131897, -0.029949890449643135, -0.009147047996520996, -0.03553686663508415, -0.007542110979557037, -0.005048384889960289, -0.00017271934484597296, -0.009102867916226387, -0.3169609010219574]}
+{"case_id": "delta_07", "category": "delta_summary", "origin": "manual_ideal", "answer": "- Sample: 150 adults.\n- Interview mode: online.", "reward": 1, "advantage": 1.9770040116327758, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 1108, 23146, 216, 49160, 49161, 49159, 3598, 28, 804, 5460, 11065, 28, 284, 7674, 327, 2976, 3737, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 25379, 2203, 4495, 288, 216, 49160, 49164, 49159, 3598, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 8235, 15073, 1485, 2329, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29, 21530, 42, 216, 49160, 49164, 49159, 3598, 30, 198, 29, 29789, 5723, 42, 2329, 30, 49154], "prefix_len": 173, "old_logprobs": [-7.6612396240234375, -8.754792213439941, -3.3618321418762207, -0.23205207288265228, -0.2006494253873825, -2.2188897132873535, -0.017714032903313637, -1.530894160270691, -2.326871395111084, -0.4696493148803711, -0.07919100672006607, -11.094978332519531, -8.75245475769043, -0.011655919253826141, -5.533929347991943, -0.27953487634658813, -1.410570502281189]}
+{"case_id": "delta_08", "category": "delta_summary", "origin": "budgie", "answer": "Final answer: The menu offered soup, pasta, and cake, with service at noon.", "reward": -0.6399999999999999, "advantage": -0.5779547990464586, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 10592, 5340, 14562, 28, 19904, 28, 284, 16253, 28, 351, 2754, 418, 27616, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 51, 575, 314, 7233, 411, 4354, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 18043, 7488, 288, 216, 49160, 49162, 42, 49159, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29288, 2988, 42, 378, 10592, 5340, 14562, 28, 19904, 28, 284, 16253, 28, 351, 2754, 418, 27616, 30, 49154], "prefix_len": 171, "old_logprobs": [-1.9456720352172852, -0.5305362939834595, -0.0014285369543358684, -3.5238428115844727, -0.957151472568512, -0.7833350896835327, -0.645799994468689, -0.06316109001636505, -0.18364384770393372, -0.013873923569917679, -0.005430114455521107, -0.008105243556201458, -0.1594451367855072, -0.02175048366189003, -0.00915130041539669, -0.008748418651521206, -0.006308048963546753, -0.01878097839653492, -0.036505326628685]}
+{"case_id": "delta_08", "category": "delta_summary", "origin": "budgie", "answer": "Final answer:\n1. Pizza is now served at 13:00.\n2. Carrots and chicken are served at midnight.\n3. We have the same type of soup and pasta, but the chicken is served on a tiramisu.\n4. The cake is served at 13:00.\n\nFinal answer: 13:00", "reward": -0.6399999999999999, "advantage": -0.5779547990464586, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 10592, 5340, 14562, 28, 19904, 28, 284, 16253, 28, 351, 2754, 418, 27616, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 51, 575, 314, 7233, 411, 4354, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 18043, 7488, 288, 216, 49160, 49162, 42, 49159, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29288, 2988, 42, 198, 49160, 30, 377, 16017, 314, 1209, 5019, 418, 216, 49160, 49162, 42, 49159, 49159, 30, 198, 49161, 30, 2974, 13073, 284, 9610, 359, 5019, 418, 25232, 30, 198, 49162, 30, 1046, 457, 260, 1142, 1502, 282, 14562, 284, 19904, 28, 564, 260, 9610, 314, 5019, 335, 253, 31659, 814, 271, 101, 30, 198, 49163, 30, 378, 16253, 314, 5019, 418, 216, 49160, 49162, 42, 49159, 49159, 30, 198, 198, 29288, 2988, 42, 216, 49160, 49162, 42, 49159, 49159, 49154], "prefix_len": 171, "old_logprobs": [-1.9456720352172852, -0.5305362939834595, -0.0014285369543358684, -1.6332178115844727, -1.130880355834961, -1.9881620407104492, -5.095293045043945, -1.4144632816314697, -2.1142237186431885, -3.1680550575256348, -0.42460063099861145, -0.5865340828895569, -0.5228161811828613, -0.37520632147789, -1.067118763923645, -0.006371067836880684, -0.027981190010905266, -0.013138038106262684, -0.23989948630332947, -0.2247539609670639, -0.05355766788125038, -0.0012011463986709714, -4.350783348083496, -1.5420911312103271, -1.3090956211090088, -3.5247035026550293, -1.0810126066207886, -1.7105368375778198, -0.18492263555526733, -5.625565052032471, -0.10945145785808563, -0.7025584578514099, -0.8048457503318787, -0.0049583567306399345, -6.514702320098877, -1.8148982524871826, -3.9025888442993164, -2.9575836658477783, -3.370227813720703, -0.022359881550073624, -3.955308437347412, -1.511765718460083, -0.7822349667549133, -1.6771883964538574, -1.5575063228607178, -1.5871354341506958, -3.2670464515686035, -0.6599374413490295, -1.6770472526550293, -3.57531476020813, -1.3048770427703857, -8.083662033081055, -1.6875295639038086, -0.1685594916343689, -0.06437382847070694, -0.8514163494110107, -0.23298437893390656, -0.19925923645496368, -0.0015669699059799314, -1.0275626182556152, -1.1632078886032104, -0.3862214684486389, -0.7886272072792053, -0.20375563204288483, -0.5824205875396729, -0.0804777443408966, -0.3652079105377197, -0.0019135037437081337, -0.00765769649296999, -0.0010836211731657386, -0.06484440714120865, -0.12966331839561462, -1.2004003524780273, -0.008523395285010338, -0.0006068295333534479, -0.0003302744007669389, -1.8533742427825928, -0.11792353540658951, -0.7431961297988892, -0.061046842485666275, -0.0008487674640491605, -0.000949885172303766, -0.2214537411928177]}
+{"case_id": "delta_08", "category": "delta_summary", "origin": "budgie", "answer": "The menu offered soup, pasta, and cake, with service at noon.", "reward": -0.6399999999999999, "advantage": -0.5779547990464586, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 10592, 5340, 14562, 28, 19904, 28, 284, 16253, 28, 351, 2754, 418, 27616, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 51, 575, 314, 7233, 411, 4354, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 18043, 7488, 288, 216, 49160, 49162, 42, 49159, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 10592, 5340, 14562, 28, 19904, 28, 284, 16253, 28, 351, 2754, 418, 27616, 30, 49154], "prefix_len": 171, "old_logprobs": [-1.6956720352172852, -0.6204361319541931, -1.4920213222503662, -0.557081401348114, -0.06051769107580185, -0.17043691873550415, -0.007304509170353413, -0.003774306271225214, -0.01545406598597765, -0.13094061613082886, -0.07292987406253815, -0.011625166051089764, -0.020243702456355095, -0.0196004007011652, -0.025043409317731857, -0.04427006095647812]}
+{"case_id": "delta_08", "category": "delta_summary", "origin": "budgie", "answer": "The menu changed from 12:00 to 13:00.", "reward": -0.42999999999999994, "advantage": -0.24992639958765783, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 10592, 5340, 14562, 28, 19904, 28, 284, 16253, 28, 351, 2754, 418, 27616, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 51, 575, 314, 7233, 411, 4354, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 18043, 7488, 288, 216, 49160, 49162, 42, 49159, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 504, 10592, 4247, 429, 216, 49160, 49161, 42, 49159, 49159, 288, 216, 49160, 49162, 42, 49159, 49159, 30, 49154], "prefix_len": 171, "old_logprobs": [-1.6956720352172852, -0.6204361319541931, -3.273271322250366, -0.28989237546920776, -2.2719979286193848, -0.16884730756282806, -0.5201476812362671, -0.0048093595542013645, -0.18242749571800232, -0.005739280954003334, -0.7492838501930237, -0.017825404182076454, -0.09956114739179611, -0.6859952807426453, -0.0006475735572166741, -0.0012209111591801047, -0.00027092601521871984, -0.20982985198497772, -0.03020588308572769]}
+{"case_id": "delta_08", "category": "delta_summary", "origin": "manual_ideal", "answer": "- Dessert: fruit.\n- Service time: 13:00.", "reward": 1, "advantage": 1.9837907967270338, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 25005, 7127, 42, 378, 10592, 5340, 14562, 28, 19904, 28, 284, 16253, 28, 351, 2754, 418, 27616, 30, 49154, 49158, 198, 25005, 7127, 5797, 30, 49154, 49157, 198, 51, 575, 314, 7233, 411, 4354, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 18043, 7488, 288, 216, 49160, 49162, 42, 49159, 49159, 30, 49154, 49158, 198, 31113, 30, 49154, 49157, 198, 6106, 805, 732, 4247, 429, 260, 3791, 7127, 28, 1015, 3869, 827, 16425, 2876, 30, 3315, 441, 9960, 23814, 3841, 30, 49154, 49158, 198, 29, 422, 19029, 42, 4354, 30, 198, 29, 5538, 655, 42, 216, 49160, 49162, 42, 49159, 49159, 30, 49154], "prefix_len": 171, "old_logprobs": [-6.195672035217285, -4.537581443786621, -3.736365795135498, -0.6342588067054749, -7.503355503082275, -2.7645161151885986, -0.25939813256263733, -0.08348006755113602, -7.358335494995117, -10.515188217163086, -0.03584226965904236, -1.2384413480758667, -0.09508898854255676, -0.4183218777179718, -0.0161435604095459, -0.005151332356035709, -0.001572206849232316, -0.6918658018112183, -2.134291648864746]}
+{"case_id": "conflict_00", "category": "conflict_clarification", "origin": "budgie", "answer": "You should put 90 people on the resiliency, not 90 people on theinee.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 24692, 8864, 1921, 260, 3589, 6650, 216, 49163, 49159, 701, 28, 979, 260, 8239, 40199, 1921, 216, 49161, 49167, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 14413, 4190, 868, 339, 1830, 335, 260, 23362, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 2683, 868, 1830, 216, 49168, 49159, 701, 335, 260, 29529, 28, 441, 216, 49168, 49159, 701, 335, 260, 33243, 30, 49154], "prefix_len": 140, "old_logprobs": [-2.475294589996338, -0.1964922547340393, -0.15153101086616516, -0.48730725049972534, -4.648344039916992, -0.16383253037929535, -0.5240986347198486, -0.4254584312438965, -0.09275825321674347, -11.615055084228516, -2.2338767051696777, -0.6729718446731567, -0.3390410244464874, -2.5144546031951904, -0.4268534183502197, -1.2336899042129517, -1.066962480545044, -0.1417037695646286, -11.756094932556152, -0.206974059343338, -0.02176494710147381]}
+{"case_id": "conflict_00", "category": "conflict_clarification", "origin": "budgie", "answer": "The venue manager should not put yourpx on the invite, instead give you a number.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 24692, 8864, 1921, 260, 3589, 6650, 216, 49163, 49159, 701, 28, 979, 260, 8239, 40199, 1921, 216, 49161, 49167, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 14413, 4190, 868, 339, 1830, 335, 260, 23362, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 504, 24692, 8864, 868, 441, 1830, 469, 20653, 335, 260, 14725, 28, 2489, 1928, 346, 253, 1230, 30, 49154], "prefix_len": 140, "old_logprobs": [-0.9752947092056274, -2.623786687850952, -0.0710202008485794, -0.2776562571525574, -4.25377893447876, -1.127421498298645, -4.824211597442627, -8.462652206420898, -2.205298900604248, -0.09936440736055374, -2.2602226734161377, -2.466294288635254, -2.1967015266418457, -2.4876914024353027, -1.3367606401443481, -1.4449876546859741, -1.8270996809005737, -0.7422977089881897, -0.020780697464942932]}
+{"case_id": "conflict_00", "category": "conflict_clarification", "origin": "budgie", "answer": "You should put 50 people on the invitation to make it the largest capacity.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 24692, 8864, 1921, 260, 3589, 6650, 216, 49163, 49159, 701, 28, 979, 260, 8239, 40199, 1921, 216, 49161, 49167, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 14413, 4190, 868, 339, 1830, 335, 260, 23362, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 2683, 868, 1830, 216, 49164, 49159, 701, 335, 260, 23362, 288, 919, 357, 260, 3995, 4190, 30, 49154], "prefix_len": 140, "old_logprobs": [-2.475294589996338, -0.1964922547340393, -0.15153101086616516, -0.48730725049972534, -1.7733442783355713, -0.1814476102590561, -0.538396954536438, -0.4020655155181885, -0.09085214883089066, -2.6896109580993652, -2.6426661014556885, -3.0964303016662598, -1.0530303716659546, -3.0876193046569824, -2.26945161819458, -3.2690672874450684, -1.1735589504241943, -0.030809573829174042]}
+{"case_id": "conflict_00", "category": "conflict_clarification", "origin": "budgie", "answer": "The room should hold 20 people, not 40.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 24692, 8864, 1921, 260, 3589, 6650, 216, 49163, 49159, 701, 28, 979, 260, 8239, 40199, 1921, 216, 49161, 49167, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 14413, 4190, 868, 339, 1830, 335, 260, 23362, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 504, 3589, 868, 2205, 216, 49161, 49159, 701, 28, 441, 216, 49163, 49159, 30, 49154], "prefix_len": 140, "old_logprobs": [-0.9752947092056274, -1.4987866878509521, -0.6496959328651428, -0.9551618099212646, -0.3462375998497009, -2.0387377738952637, -1.5391179323196411, -0.3616401255130768, -1.6089060306549072, -1.0480722188949585, -0.16451020538806915, -0.0689692497253418, -0.007887172512710094, -0.0796210914850235, -0.02617596462368965]}
+{"case_id": "conflict_00", "category": "conflict_clarification", "origin": "manual_ideal", "answer": "Should I use the venue manager’s figure of 40 or the signed booking’s figure of 28?", "reward": 1, "advantage": 1.9999959016477427, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 24692, 8864, 1921, 260, 3589, 6650, 216, 49163, 49159, 701, 28, 979, 260, 8239, 40199, 1921, 216, 49161, 49167, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 14413, 4190, 868, 339, 1830, 335, 260, 23362, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 20646, 339, 722, 260, 24692, 8864, 417, 99, 4110, 282, 216, 49163, 49159, 355, 260, 8239, 40199, 417, 99, 4110, 282, 216, 49161, 49167, 47, 49154], "prefix_len": 140, "old_logprobs": [-12.051738739013672, -0.3997626006603241, -4.640422344207764, -1.1943730115890503, -4.364699363708496, -0.4345000684261322, -3.048759698867798, -6.4490144723095e-05, -7.728677749633789, -2.2468483448028564, -0.1443074494600296, -0.2886996567249298, -0.02731361612677574, -3.8062171936035156, -0.5026355385780334, -0.7201125025749207, -1.0344184637069702, -0.2131602168083191, -0.00038675934774801135, -0.32164034247398376, -0.07198087126016617, -0.004192373715341091, -0.008175479248166084, -0.013062618672847748, -0.4152412712574005, -0.056457482278347015]}
+{"case_id": "conflict_01", "category": "conflict_clarification", "origin": "budgie", "answer": "You should choose the value associated with the fact that Marcus is the owner.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 1391, 5453, 3834, 450, 1787, 347, 7615, 28, 564, 260, 11175, 4106, 3834, 30685, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 10576, 868, 339, 1398, 347, 7615, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 2683, 868, 3525, 260, 1685, 2188, 351, 260, 1027, 338, 30685, 314, 260, 7615, 30, 49154], "prefix_len": 133, "old_logprobs": [-1.4688369035720825, -0.1715576946735382, -1.5531755685806274, -1.7302703857421875, -1.1967015266418457, -3.505126714706421, -0.00183723377995193, -0.9062537550926208, -6.260148525238037, -0.41741982102394104, -1.3455901145935059, -0.9332642555236816, -0.30759382247924805, -0.04541512206196785, -0.5531184077262878, -0.07900916039943695]}
+{"case_id": "conflict_01", "category": "conflict_clarification", "origin": "budgie", "answer": "I should choose the one with the focus of providing a reasonable value.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 1391, 5453, 3834, 450, 1787, 347, 7615, 28, 564, 260, 11175, 4106, 3834, 30685, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 10576, 868, 339, 1398, 347, 7615, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 57, 868, 3525, 260, 582, 351, 260, 1593, 282, 3258, 253, 8541, 1685, 30, 49154], "prefix_len": 133, "old_logprobs": [-1.0938369035720825, -0.2963385581970215, -1.8827966451644897, -1.428468108177185, -2.211941957473755, -2.1323299407958984, -0.2865491211414337, -2.5992236137390137, -1.0592626333236694, -2.0314040184020996, -0.9679545164108276, -7.131972789764404, -1.926835298538208, -0.7485162019729614, -0.1651826798915863]}
+{"case_id": "conflict_01", "category": "conflict_clarification", "origin": "budgie", "answer": "You should list it as the owner, not the other way around.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 1391, 5453, 3834, 450, 1787, 347, 7615, 28, 564, 260, 11175, 4106, 3834, 30685, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 10576, 868, 339, 1398, 347, 7615, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 2683, 868, 1398, 357, 347, 260, 7615, 28, 441, 260, 550, 970, 1130, 30, 49154], "prefix_len": 133, "old_logprobs": [-1.4688369035720825, -0.1715576946735382, -0.9906755685806274, -3.8364243507385254, -0.1961655616760254, -1.9568467140197754, -0.04169098660349846, -0.6692357063293457, -1.1032168865203857, -0.5129051208496094, -1.1678766012191772, -0.5940930247306824, -0.002830429933965206, -0.021766463294625282, -0.012830797582864761]}
+{"case_id": "conflict_01", "category": "conflict_clarification", "origin": "budgie", "answer": "I should mention you as the owner instead of listing a value.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 1391, 5453, 3834, 450, 1787, 347, 7615, 28, 564, 260, 11175, 4106, 3834, 30685, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 10576, 868, 339, 1398, 347, 7615, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 57, 868, 3311, 346, 347, 260, 7615, 2489, 282, 15656, 253, 1685, 30, 49154], "prefix_len": 133, "old_logprobs": [-1.0938369035720825, -0.2963385581970215, -3.5702967643737793, -4.304631233215332, -0.7828856706619263, -0.6345930099487305, -0.049422748386859894, -2.566774845123291, -0.03570551052689552, -4.580392837524414, -1.4858460426330566, -0.14863206446170807, -0.10413049906492233, -0.021250654011964798]}
+{"case_id": "conflict_01", "category": "conflict_clarification", "origin": "manual_ideal", "answer": "Should Lina or Marcus be treated as the authoritative project owner?", "reward": 1, "advantage": 1.9999959016477427, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 1391, 5453, 3834, 450, 1787, 347, 7615, 28, 564, 260, 11175, 4106, 3834, 30685, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 10576, 868, 339, 1398, 347, 7615, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 20646, 450, 1787, 355, 30685, 325, 4838, 347, 260, 24730, 1391, 7615, 47, 49154], "prefix_len": 133, "old_logprobs": [-12.031336784362793, -9.23848819732666, -0.0011781663633883, -5.475086212158203, -0.24228818714618683, -0.22926026582717896, -8.338482856750488, -0.03183204308152199, -1.0906391143798828, -14.671916007995605, -8.575486183166504, -1.7769193649291992, -0.15892216563224792, -0.06987164914608002]}
+{"case_id": "conflict_02", "category": "conflict_clarification", "origin": "budgie", "answer": "The train leaves at 8:10; we'll arrive at 8:10.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 2705, 38900, 529, 1921, 260, 4445, 3711, 418, 216, 49159, 49167, 42, 49160, 49159, 43, 260, 19574, 1921, 216, 49159, 49167, 42, 49163, 49164, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 31530, 549, 645, 288, 11296, 30, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 504, 4445, 3711, 418, 216, 49167, 42, 49160, 49159, 43, 392, 3060, 11296, 418, 216, 49167, 42, 49160, 49159, 30, 49154], "prefix_len": 141, "old_logprobs": [-2.659721612930298, -1.5166741609573364, -0.1380867213010788, -0.07343333214521408, -0.03693440556526184, -1.3664121627807617, -0.0033522620797157288, -0.7805978059768677, -0.0863894373178482, -2.141788959503174, -3.7764344215393066, -2.7063918113708496, -1.2508981227874756, -0.42133957147598267, -0.37309858202934265, -0.20183107256889343, -0.0004807746736332774, -0.43989893794059753, -0.0520753487944603, -0.829798698425293, -0.022349273785948753]}
+{"case_id": "conflict_02", "category": "conflict_clarification", "origin": "budgie", "answer": "You should arrive at 9:10.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 2705, 38900, 529, 1921, 260, 4445, 3711, 418, 216, 49159, 49167, 42, 49160, 49159, 43, 260, 19574, 1921, 216, 49159, 49167, 42, 49163, 49164, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 31530, 549, 645, 288, 11296, 30, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 2683, 868, 11296, 418, 216, 49168, 42, 49160, 49159, 30, 49154], "prefix_len": 141, "old_logprobs": [-2.935267448425293, -0.6559098362922668, -0.051885105669498444, -1.3752511739730835, -0.2682430148124695, -3.500746488571167, -0.01695670746266842, -1.2876060009002686, -0.3106829822063446, -1.241313099861145, -0.041538212448358536]}
+{"case_id": "conflict_02", "category": "conflict_clarification", "origin": "budgie", "answer": "I ask one focus feature to check the proposed time.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 2705, 38900, 529, 1921, 260, 4445, 3711, 418, 216, 49159, 49167, 42, 49160, 49159, 43, 260, 19574, 1921, 216, 49159, 49167, 42, 49163, 49164, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 31530, 549, 645, 288, 11296, 30, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 57, 1998, 582, 1593, 3865, 288, 2545, 260, 5433, 655, 30, 49154], "prefix_len": 141, "old_logprobs": [-1.5602673292160034, -2.5557620525360107, -0.9665454626083374, -2.9376394748687744, -11.051849365234375, -2.4307541847229004, -4.36527156829834, -1.0864862203598022, -9.696033477783203, -1.0214247703552246, -0.44363313913345337, -0.010735614225268364]}
+{"case_id": "conflict_02", "category": "conflict_clarification", "origin": "budgie", "answer": "The train leaves at 08:10.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 2705, 38900, 529, 1921, 260, 4445, 3711, 418, 216, 49159, 49167, 42, 49160, 49159, 43, 260, 19574, 1921, 216, 49159, 49167, 42, 49163, 49164, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 31530, 549, 645, 288, 11296, 30, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 504, 4445, 3711, 418, 216, 49159, 49167, 42, 49160, 49159, 30, 49154], "prefix_len": 141, "old_logprobs": [-2.685267448425293, -1.454133152961731, -0.13919267058372498, -0.08013000339269638, -0.03577509894967079, -0.4918695092201233, -0.07678909599781036, -0.00042083943844772875, -0.27411970496177673, -0.014837272465229034, -1.110548734664917, -0.06508604437112808]}
+{"case_id": "conflict_02", "category": "conflict_clarification", "origin": "manual_ideal", "answer": "Which departure time is current: 08:10 on the itinerary or 08:45 on the ticket?", "reward": 1, "advantage": 1.9999959016477427, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 2705, 38900, 529, 1921, 260, 4445, 3711, 418, 216, 49159, 49167, 42, 49160, 49159, 43, 260, 19574, 1921, 216, 49159, 49167, 42, 49163, 49164, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 31530, 549, 645, 288, 11296, 30, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 14413, 19740, 655, 314, 1574, 42, 216, 49159, 49167, 42, 49160, 49159, 335, 260, 38900, 529, 355, 216, 49159, 49167, 42, 49163, 49164, 335, 260, 19574, 47, 49154], "prefix_len": 141, "old_logprobs": [-8.097221374511719, -4.983704566955566, -0.26120200753211975, -0.5340360403060913, -8.92785930633545, -6.345309734344482, -1.0913670063018799, -0.39120858907699585, -0.1464894860982895, -0.003408220363780856, -0.48113763332366943, -0.056464578956365585, -5.9498491287231445, -0.5869342088699341, -11.643343925476074, -0.024371428415179253, -2.6789541244506836, -0.05187729746103287, -0.15644164383411407, -0.054963212460279465, -0.0002812943421304226, -0.10466627031564713, -0.01524322759360075, -0.17332996428012848, -0.10416519641876221, -0.06623709201812744, -0.07755132019519806, -0.0657724067568779]}
+{"case_id": "conflict_03", "category": "conflict_clarification", "origin": "budgie", "answer": "I should probably review the total to see what's going on.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 26337, 2719, 314, 24927, 49163, 28, 49166, 49161, 49159, 28, 564, 260, 8239, 12219, 314, 24927, 49163, 28, 49161, 49166, 49159, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 1780, 1902, 868, 339, 25312, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 57, 868, 3105, 2523, 260, 2719, 288, 963, 732, 506, 2045, 335, 30, 49154], "prefix_len": 139, "old_logprobs": [-1.6565988063812256, -0.45539936423301697, -7.558897972106934, -4.665889263153076, -0.4383234977722168, -4.236650466918945, -3.1766769886016846, -1.6522088050842285, -2.1230640411376953, -2.253546714782715, -1.4023829698562622, -0.015277626924216747, -0.7527669072151184, -0.053262826055288315]}
+{"case_id": "conflict_03", "category": "conflict_clarification", "origin": "budgie", "answer": "1", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 26337, 2719, 314, 24927, 49163, 28, 49166, 49161, 49159, 28, 564, 260, 8239, 12219, 314, 24927, 49163, 28, 49161, 49166, 49159, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 1780, 1902, 868, 339, 25312, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 49160, 49154], "prefix_len": 139, "old_logprobs": [-3.5315988063812256, -3.884100914001465]}
+{"case_id": "conflict_03", "category": "conflict_clarification", "origin": "budgie", "answer": "I should wait before asking one focused clarification instead of choosing a value.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 26337, 2719, 314, 24927, 49163, 28, 49166, 49161, 49159, 28, 564, 260, 8239, 12219, 314, 24927, 49163, 28, 49161, 49166, 49159, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 1780, 1902, 868, 339, 25312, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 57, 868, 5847, 1092, 5617, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154], "prefix_len": 139, "old_logprobs": [-1.654300332069397, -0.47389286756515503, -10.198896408081055, -2.6906299591064453, -4.4827117919921875, -2.008209466934204, -0.28169092535972595, -0.4887729287147522, -0.8448553085327148, -0.01760007254779339, -0.69370037317276, -0.09259407222270966, -0.04086922109127045, -0.08932623267173767, -0.00996861606836319]}
+{"case_id": "conflict_03", "category": "conflict_clarification", "origin": "budgie", "answer": "The value should be 4,720 euros.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 26337, 2719, 314, 24927, 49163, 28, 49166, 49161, 49159, 28, 564, 260, 8239, 12219, 314, 24927, 49163, 28, 49161, 49166, 49159, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 1780, 1902, 868, 339, 25312, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 504, 1685, 868, 325, 216, 49163, 28, 49166, 49161, 49159, 39236, 30, 49154], "prefix_len": 139, "old_logprobs": [-2.2815988063812256, -2.7075300216674805, -0.5203148126602173, -0.061545222997665405, -2.6563732624053955, -1.1204602718353271, -0.5016376972198486, -0.8507553339004517, -0.12294063717126846, -0.016812769696116447, -1.815079927444458, -0.841162383556366, -0.007000679615885019]}
+{"case_id": "conflict_03", "category": "conflict_clarification", "origin": "manual_ideal", "answer": "Which amount is authoritative: €4,720 in the spreadsheet or €4,270 in the signed quote?", "reward": 1, "advantage": 1.9999959016477427, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 26337, 2719, 314, 24927, 49163, 28, 49166, 49161, 49159, 28, 564, 260, 8239, 12219, 314, 24927, 49163, 28, 49161, 49166, 49159, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 1780, 1902, 868, 339, 25312, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 14413, 1902, 314, 24730, 42, 24927, 49163, 28, 49166, 49161, 49159, 281, 260, 26337, 355, 24927, 49163, 28, 49161, 49166, 49159, 281, 260, 8239, 12219, 47, 49154], "prefix_len": 139, "old_logprobs": [-8.279794692993164, -2.73695707321167, -1.0533592700958252, -13.846280097961426, -5.821052551269531, -0.289279967546463, -0.16062426567077637, -0.09335010498762131, -0.42312270402908325, -0.014710884541273117, -0.0032106302678585052, -7.459678649902344, -2.8647358417510986, -10.167951583862305, -0.5115951895713806, -0.05500822886824608, -0.010647747665643692, -0.013666162267327309, -0.02762548439204693, -0.0021845067385584116, -0.0014509160537272692, -0.07457935810089111, -0.2541474997997284, -0.33059456944465637, -6.354251384735107, -0.33629339933395386, -0.07967954874038696]}
+{"case_id": "conflict_04", "category": "conflict_clarification", "origin": "budgie", "answer": "I should put the label inside the room.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4368, 1921, 11769, 3194, 260, 4315, 28, 979, 260, 6529, 1921, 4249, 357, 418, 3589, 2779, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 9576, 868, 339, 1830, 357, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 57, 868, 1830, 260, 4368, 2972, 260, 3589, 30, 49154], "prefix_len": 134, "old_logprobs": [-1.1017132997512817, -0.34638068079948425, -0.7105637192726135, -0.6841046810150146, -1.54689359664917, -2.5754001140594482, -0.3129989504814148, -0.6624277234077454, -2.1054277420043945, -0.07145193219184875]}
+{"case_id": "conflict_04", "category": "conflict_clarification", "origin": "budgie", "answer": "You should put the data in a cool, dark room.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4368, 1921, 11769, 3194, 260, 4315, 28, 979, 260, 6529, 1921, 4249, 357, 418, 3589, 2779, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 9576, 868, 339, 1830, 357, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 2683, 868, 1830, 260, 940, 281, 253, 3122, 28, 3605, 3589, 30, 49154], "prefix_len": 134, "old_logprobs": [-2.601713180541992, -0.2010265290737152, -0.3879639208316803, -0.4457787275314331, -3.317659378051758, -0.41430777311325073, -0.3400959074497223, -5.184813499450684, -1.0809130668640137, -1.5611975193023682, -0.6470497846603394, -0.5447888970375061, -0.07717276364564896]}
+{"case_id": "conflict_04", "category": "conflict_clarification", "origin": "budgie", "answer": "The CIP should give you clear instructions on how to save the data.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4368, 1921, 11769, 3194, 260, 4315, 28, 979, 260, 6529, 1921, 4249, 357, 418, 3589, 2779, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 9576, 868, 339, 1830, 357, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 504, 340, 6188, 868, 1928, 346, 2437, 6388, 335, 638, 288, 3898, 260, 940, 30, 49154], "prefix_len": 134, "old_logprobs": [-2.101713180541992, -9.388270378112793, -6.482463836669922, -1.025554895401001, -4.411762237548828, -1.5140340328216553, -2.764869451522827, -1.7212200164794922, -0.9739630818367004, -1.7049579620361328, -0.09908805787563324, -4.404426574707031, -0.32627078890800476, -1.0639839172363281, -0.7289502620697021, -0.027868973091244698]}
+{"case_id": "conflict_04", "category": "conflict_clarification", "origin": "budgie", "answer": "I should put a message on the record.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4368, 1921, 11769, 3194, 260, 4315, 28, 979, 260, 6529, 1921, 4249, 357, 418, 3589, 2779, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 9576, 868, 339, 1830, 357, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 57, 868, 1830, 253, 3714, 335, 260, 2004, 30, 49154], "prefix_len": 134, "old_logprobs": [-1.1017132997512817, -0.34638068079948425, -0.7105637192726135, -2.7466046810150146, -5.835944175720215, -2.34356951713562, -0.33130118250846863, -3.341024875640869, -1.834682583808899, -0.1350238025188446]}
+{"case_id": "conflict_04", "category": "conflict_clarification", "origin": "manual_ideal", "answer": "Should the sample follow the label’s refrigeration instruction or the protocol’s room-temperature instruction?", "reward": 1, "advantage": 1.9999959016477427, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4368, 1921, 11769, 3194, 260, 4315, 28, 979, 260, 6529, 1921, 4249, 357, 418, 3589, 2779, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 9576, 868, 339, 1830, 357, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 20646, 260, 4315, 1066, 260, 4368, 417, 99, 33668, 5785, 355, 260, 6529, 417, 99, 3589, 29, 19864, 5785, 47, 49154], "prefix_len": 134, "old_logprobs": [-11.038459777832031, -4.931400775909424, -3.7369296550750732, -6.832645893096924, -1.0810706615447998, -4.900527000427246, -4.579483985900879, -0.0006324436399154365, -12.435416221618652, -6.84245491027832, -3.261159896850586, -1.4853811264038086, -7.1187849044799805, -0.35364025831222534, -0.000277123210253194, -8.369056701660156, -5.5177226066589355, -7.887805938720703, -1.9889954328536987, -0.21420517563819885, -0.036409925669431686]}
+{"case_id": "conflict_05", "category": "conflict_clarification", "origin": "budgie", "answer": "The call was a mistake.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 11401, 8639, 14801, 28, 564, 260, 5688, 417, 99, 1596, 3714, 1921, 14963, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 38385, 260, 946, 30, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 504, 946, 436, 253, 11081, 30, 49154], "prefix_len": 129, "old_logprobs": [-1.369273066520691, -0.6221527457237244, -2.1829833984375, -2.4587643146514893, -2.5613667964935303, -0.27877163887023926, -0.4427655041217804]}
+{"case_id": "conflict_05", "category": "conflict_clarification", "origin": "budgie", "answer": "The call conflicts with a focus on Tuesday.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 11401, 8639, 14801, 28, 564, 260, 5688, 417, 99, 1596, 3714, 1921, 14963, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 38385, 260, 946, 30, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 504, 946, 7301, 351, 253, 1593, 335, 14801, 30, 49154], "prefix_len": 129, "old_logprobs": [-1.369273066520691, -0.6221527457237244, -4.2142333984375, -0.05104109272360802, -1.9059677124023438, -1.78994882106781, -1.9570647478103638, -1.8221893310546875, -0.597078800201416, -0.03428183123469353]}
+{"case_id": "conflict_05", "category": "conflict_clarification", "origin": "budgie", "answer": "I should now ask one focused clarification.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 11401, 8639, 14801, 28, 564, 260, 5688, 417, 99, 1596, 3714, 1921, 14963, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 38385, 260, 946, 30, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 57, 868, 1209, 1998, 582, 4782, 28879, 30, 49154], "prefix_len": 129, "old_logprobs": [-2.1192731857299805, -1.6966569423675537, -5.888864517211914, -1.3976011276245117, -0.49385255575180054, -0.2813304364681244, -0.8416077494621277, -2.176196575164795, -0.03296755254268646]}
+{"case_id": "conflict_05", "category": "conflict_clarification", "origin": "budgie", "answer": "You should not choose the value.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 11401, 8639, 14801, 28, 564, 260, 5688, 417, 99, 1596, 3714, 1921, 14963, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 38385, 260, 946, 30, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 2683, 868, 441, 3525, 260, 1685, 30, 49154], "prefix_len": 129, "old_logprobs": [-2.3692731857299805, -1.9035804271697998, -2.3958816528320312, -1.2141324281692505, -1.904921293258667, -0.41299423575401306, -1.4180408716201782, -0.10990867018699646]}
+{"case_id": "conflict_05", "category": "conflict_clarification", "origin": "manual_ideal", "answer": "Should I schedule the call for Tuesday or for Thursday from the client’s latest message?", "reward": 1, "advantage": 1.9999959016477427, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 11401, 8639, 14801, 28, 564, 260, 5688, 417, 99, 1596, 3714, 1921, 14963, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 38385, 260, 946, 30, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 20646, 339, 8034, 260, 946, 327, 14801, 355, 327, 14963, 429, 260, 5688, 417, 99, 6008, 3714, 47, 49154], "prefix_len": 129, "old_logprobs": [-10.74427318572998, -1.694443941116333, -8.82499885559082, -1.7283978462219238, -0.13911662995815277, -5.192424774169922, -2.3333544731140137, -1.5698754787445068, -4.81373929977417, -0.37786751985549927, -8.557109832763672, -1.5182855129241943, -0.586854100227356, -2.1323037147521973, -5.9960475482512265e-05, -6.59716796875, -1.530098557472229, -0.11563166975975037, -0.029767194762825966]}
+{"case_id": "conflict_06", "category": "conflict_clarification", "origin": "budgie", "answer": "I'm helpable, but it's important to do thorough research before making a decision.", "reward": -0.22, "advantage": -0.6079455199011327, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 13513, 817, 2744, 216, 49160, 49165, 4784, 28, 979, 260, 2099, 985, 983, 216, 49160, 49160, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 2020, 800, 416, 339, 8135, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 57, 5248, 724, 515, 28, 564, 357, 506, 1070, 288, 536, 9817, 1151, 1092, 1625, 253, 3062, 30, 49154], "prefix_len": 134, "old_logprobs": [-1.346938133239746, -4.109841346740723, -7.877164840698242, -1.7831639051437378, -0.5835847854614258, -0.5315516591072083, -2.7115821838378906, -0.38306915760040283, -1.2351709604263306, -0.09059172868728638, -4.699505805969238, -1.9751859903335571, -0.4338233470916748, -0.6789677143096924, -0.6966403722763062, -0.535118818283081, -2.3953866958618164, -0.08850910514593124, -0.03003920055925846]}
+{"case_id": "conflict_06", "category": "conflict_clarification", "origin": "budgie", "answer": "16?", "reward": 0, "advantage": -0.14354269219887855, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 13513, 817, 2744, 216, 49160, 49165, 4784, 28, 979, 260, 2099, 985, 983, 216, 49160, 49160, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 2020, 800, 416, 339, 8135, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 49160, 49165, 47, 49154], "prefix_len": 134, "old_logprobs": [-1.596938133239746, -0.5253070592880249, -5.155054092407227, -0.3715294599533081]}
+{"case_id": "conflict_06", "category": "conflict_clarification", "origin": "budgie", "answer": "16 can I promising, but only 11 are known.", "reward": -0.22, "advantage": -0.6079455199011327, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 13513, 817, 2744, 216, 49160, 49165, 4784, 28, 979, 260, 2099, 985, 983, 216, 49160, 49160, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 2020, 800, 416, 339, 8135, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 49160, 49165, 416, 339, 9410, 28, 564, 805, 216, 49160, 49160, 359, 1343, 30, 49154], "prefix_len": 134, "old_logprobs": [-1.596938133239746, -0.5253070592880249, -1.5300538539886475, -3.3650882244110107, -2.751739978790283, -2.185649871826172, -0.5832724571228027, -2.326430320739746, -0.3763207197189331, -0.09170271456241608, -0.3703038692474365, -1.854870080947876, -4.139369010925293, -0.4838813543319702, -0.008919160813093185]}
+{"case_id": "conflict_06", "category": "conflict_clarification", "origin": "budgie", "answer": "2. 10. 11.", "reward": -0.22, "advantage": -0.6079455199011327, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 13513, 817, 2744, 216, 49160, 49165, 4784, 28, 979, 260, 2099, 985, 983, 216, 49160, 49160, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 2020, 800, 416, 339, 8135, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 49161, 30, 216, 49160, 49159, 30, 216, 49160, 49160, 30, 49154], "prefix_len": 134, "old_logprobs": [-3.471938133239746, -2.6424202919006348, -2.1059486865997314, -1.1122362613677979, -1.5852688550949097, -1.594788670539856, -1.9645496606826782, -0.9060718417167664, -1.4454377889633179, -0.31912627816200256, -1.1225600242614746]}
+{"case_id": "conflict_06", "category": "conflict_clarification", "origin": "manual_ideal", "answer": "Which quantity should govern the promise: 16 in the system or 11 physically counted?", "reward": 1, "advantage": 1.9673792519022768, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 13513, 817, 2744, 216, 49160, 49165, 4784, 28, 979, 260, 2099, 985, 983, 216, 49160, 49160, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 2020, 800, 416, 339, 8135, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 14413, 9044, 868, 1330, 260, 8135, 42, 216, 49160, 49165, 281, 260, 817, 355, 216, 49160, 49160, 8582, 17288, 47, 49154], "prefix_len": 134, "old_logprobs": [-7.6122589111328125, -7.787771224975586, -4.3479766845703125, -15.985737800598145, -4.350830078125, -9.942694664001465, -6.818366050720215, -0.5981113910675049, -0.6171489357948303, -1.1821680068969727, -8.29248046875, -1.5944453477859497, -6.937166213989258, -0.6410129070281982, -0.1201367899775505, -0.0829826295375824, -0.19165539741516113, -9.96264362335205, -7.733682632446289, -4.039692401885986, -0.07216699421405792]}
+{"case_id": "conflict_07", "category": "conflict_clarification", "origin": "budgie", "answer": "The first step is to determine what is the focus of the campaign brief. This can be an established fact, a piece of information from a larger source, or any detail about a particular topic. This is the first step on your side.\n\nThe second step is to think about what makes your campaign short, what makes your campaign good. This can be a specific aspect of your issue, a specific local issue, or any information that might help you get a better answer. The second step is to decide which of the options you want to use.\n\nLet's say the goal is to get as much information as possible from the campaign short to give the full story, which requires a more precise and direct answer. However, if the campaign short doesn't provide enough context, you might want to consider using the following options:\n\n- The campaign short provides a specific, which is a detail from the campaign.\n- The campaign short provides", "reward": -0.42999999999999994, "advantage": -0.7992890537773156, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 3997, 3948, 3073, 6330, 1671, 28, 564, 260, 4878, 5453, 11421, 327, 5038, 1671, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 14413, 47570, 868, 339, 722, 47, 9430, 582, 4782, 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+{"case_id": "conflict_07", "category": "conflict_clarification", "origin": "budgie", "answer": "You can't ask which one is more recognizable or commonly used.", "reward": -0.22, "advantage": -0.39188443898790726, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 3997, 3948, 3073, 6330, 1671, 28, 564, 260, 4878, 5453, 11421, 327, 5038, 1671, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 14413, 47570, 868, 339, 722, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 2683, 416, 982, 1998, 527, 582, 314, 540, 26628, 355, 4278, 804, 30, 49154], "prefix_len": 132, "old_logprobs": [-2.2895379066467285, -2.2179219722747803, -2.6415669918060303, -2.7629661560058594, -2.8899288177490234, -2.8214008808135986, -1.2943341732025146, -1.3939194679260254, -9.779023170471191, -2.9023306369781494, -4.479959964752197, -2.4252750873565674, -0.2904644310474396, -0.06337930262088776]}
+{"case_id": "conflict_07", "category": "conflict_clarification", "origin": "budgie", "answer": "I should choose the start of a sentence, not the end.", "reward": -0.22, "advantage": -0.39188443898790726, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 3997, 3948, 3073, 6330, 1671, 28, 564, 260, 4878, 5453, 11421, 327, 5038, 1671, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 14413, 47570, 868, 339, 722, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 57, 868, 3525, 260, 1120, 282, 253, 6330, 28, 441, 260, 1112, 30, 49154], "prefix_len": 132, "old_logprobs": [-1.789537787437439, -0.5895469784736633, -0.9906036853790283, -1.2889318466186523, -6.49107551574707, -0.6696266531944275, -0.5478332042694092, -1.255197525024414, -3.094270706176758, -0.49617135524749756, -0.36838775873184204, -0.35723647475242615, -0.28936707973480225, -0.024634087458252907]}
+{"case_id": "conflict_07", "category": "conflict_clarification", "origin": "budgie", "answer": "The first thought is to choose a clarity to keep the focus on the fact.", "reward": -0.22, "advantage": -0.39188443898790726, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 3997, 3948, 3073, 6330, 1671, 28, 564, 260, 4878, 5453, 11421, 327, 5038, 1671, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 14413, 47570, 868, 339, 722, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 504, 808, 2275, 314, 288, 3525, 253, 13150, 288, 1446, 260, 1593, 335, 260, 1027, 30, 49154], "prefix_len": 132, "old_logprobs": [-1.664537787437439, -1.9165682792663574, -1.6570571660995483, -0.29933738708496094, -0.8673373460769653, -2.7291486263275146, -1.2160418033599854, -8.288867950439453, -3.514507293701172, -3.242486000061035, -0.7851459980010986, -1.7945195436477661, -0.10677769780158997, -0.37015774846076965, -3.9376168251037598, -1.0589396953582764, -0.3903599977493286]}
+{"case_id": "conflict_07", "category": "conflict_clarification", "origin": "manual_ideal", "answer": "Should the style guide’s sentence case or the campaign brief’s title case take priority?", "reward": 1, "advantage": 1.9749423707410376, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 3997, 3948, 3073, 6330, 1671, 28, 564, 260, 4878, 5453, 11421, 327, 5038, 1671, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 14413, 47570, 868, 339, 722, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 20646, 260, 3997, 3948, 417, 99, 6330, 1671, 355, 260, 4878, 5453, 417, 99, 5038, 1671, 1188, 9282, 47, 49154], "prefix_len": 132, "old_logprobs": [-11.53953742980957, -4.500533580780029, -4.980959892272949, -0.05604921281337738, -6.621548652648926, -0.0010306290350854397, -4.372220993041992, -0.08090530335903168, -5.6045331954956055, -1.0240572690963745, -0.2313915342092514, -0.8621571063995361, -0.7589856386184692, -0.0005416594794951379, -0.19829845428466797, -0.022333301603794098, -7.889586448669434, -2.503934144973755, -0.3438413441181183, -0.028050972148776054]}
+{"case_id": "conflict_08", "category": "conflict_clarification", "origin": "budgie", "answer": "No. The relationship between the open map and the closed ranger’s notice is not a one-to-one relationship.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4141, 8029, 260, 2050, 1440, 28, 564, 253, 412, 8876, 417, 99, 4887, 8029, 357, 6370, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 7306, 339, 1252, 260, 7410, 738, 357, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 5230, 30, 378, 1986, 826, 260, 1440, 4141, 284, 260, 6370, 412, 8876, 417, 99, 4887, 314, 441, 253, 582, 29, 1141, 29, 537, 1986, 30, 49154], "prefix_len": 136, "old_logprobs": [-0.5216156840324402, -0.2926916182041168, -1.3926414251327515, -6.1217193603515625, -0.5609779357910156, -0.7082793116569519, -2.005425214767456, -1.5702548027038574, -0.14481677114963531, -1.1404423713684082, -0.10687432438135147, -0.4533182382583618, -0.1923578530550003, -1.5375916957855225, -5.3165931603871286e-05, -0.7880738973617554, -0.35995128750801086, -1.351725697517395, -2.6763193607330322, -1.9365054368972778, -0.007060813717544079, -0.13715167343616486, -0.000782183778937906, -0.00134222989436239, -0.6056671142578125, -0.36110615730285645, -0.2728237807750702]}
+{"case_id": "conflict_08", "category": "conflict_clarification", "origin": "budgie", "answer": "I should not choose a value, I should choose one focused grows.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4141, 8029, 260, 2050, 1440, 28, 564, 253, 412, 8876, 417, 99, 4887, 8029, 357, 6370, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 7306, 339, 1252, 260, 7410, 738, 357, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 57, 868, 441, 3525, 253, 1685, 28, 339, 868, 3525, 582, 4782, 8759, 30, 49154], "prefix_len": 136, "old_logprobs": [-1.8107601404190063, -2.7848961353302, -1.1927145719528198, -2.2466139793395996, -0.33633673191070557, -0.04999510943889618, -1.7922065258026123, -1.5653870105743408, -0.12611113488674164, -1.080278754234314, -0.896695613861084, -3.232316493988037, -11.71069049835205, -0.47865432500839233, -0.01426452212035656]}
+{"case_id": "conflict_08", "category": "conflict_clarification", "origin": "budgie", "answer": "No. I cannot plan the route through it.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4141, 8029, 260, 2050, 1440, 28, 564, 253, 412, 8876, 417, 99, 4887, 8029, 357, 6370, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 7306, 339, 1252, 260, 7410, 738, 357, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 5230, 30, 339, 2526, 1252, 260, 7410, 738, 357, 30, 49154], "prefix_len": 136, "old_logprobs": [-0.43576017022132874, -0.2896139323711395, -1.0392526388168335, -1.1873936653137207, -0.3160950243473053, -0.33578577637672424, -0.022218234837055206, -0.20977041125297546, -0.09202928096055984, -0.6610497236251831, -0.3393379747867584]}
+{"case_id": "conflict_08", "category": "conflict_clarification", "origin": "budgie", "answer": "No. The goal is to plan the route, not the route itself.", "reward": -0.22, "advantage": -0.49999897541193566, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4141, 8029, 260, 2050, 1440, 28, 564, 253, 412, 8876, 417, 99, 4887, 8029, 357, 6370, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 7306, 339, 1252, 260, 7410, 738, 357, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 5230, 30, 378, 3491, 314, 288, 1252, 260, 7410, 28, 441, 260, 7410, 2581, 30, 49154], "prefix_len": 136, "old_logprobs": [-0.43576017022132874, -0.2896139323711395, -1.3830026388168335, -3.816788911819458, -0.3356768488883972, -0.27359652519226074, -1.5075846910476685, -0.3661591112613678, -0.06053385138511658, -1.4081964492797852, -0.145760640501976, -0.7430629134178162, -0.7450029850006104, -0.7481091618537903, -0.038357872515916824, -0.02741197682917118]}
+{"case_id": "conflict_08", "category": "conflict_clarification", "origin": "manual_ideal", "answer": "Is the ranger’s closure notice current and authoritative over the map?", "reward": 1, "advantage": 1.9999959016477427, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4141, 8029, 260, 2050, 1440, 28, 564, 253, 412, 8876, 417, 99, 4887, 8029, 357, 6370, 30, 49154, 49158, 198, 14359, 827, 4585, 4678, 30, 49154, 49157, 198, 7306, 339, 1252, 260, 7410, 738, 357, 47, 9430, 582, 4782, 28879, 2489, 282, 8362, 253, 1685, 30, 49154, 49158, 198, 4182, 260, 412, 8876, 417, 99, 20291, 4887, 1574, 284, 24730, 690, 260, 4141, 47, 49154], "prefix_len": 136, "old_logprobs": [-7.810760021209717, -1.6582965850830078, -7.101383686065674, -0.18859750032424927, -2.0599541664123535, -9.262132516596466e-05, -11.816156387329102, -6.1939215660095215, -6.240877151489258, -3.5102477073669434, -12.644166946411133, -5.938477993011475, -0.4289207458496094, -2.3950295448303223, -0.19434872269630432, -0.03085211105644703]}
+{"case_id": "source_boundary_00", "category": "source_boundary", "origin": "budgie", "answer": "I’m not sure how you got this, but you can find the number of participants who dropped out using the information you provided.", "reward": -0.010000000000000009, "advantage": -0.5061788432676426, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 12919, 37211, 1921, 260, 10657, 761, 216, 49163, 49167, 4697, 284, 15298, 2976, 3737, 30, 657, 1072, 441, 1378, 260, 28180, 985, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 4697, 10370, 578, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 417, 93, 441, 2090, 638, 346, 3363, 451, 28, 564, 346, 416, 1042, 260, 1230, 282, 4697, 617, 10370, 578, 1015, 260, 1096, 346, 2711, 30, 49154], "prefix_len": 141, "old_logprobs": [-1.0233293771743774, -2.624603748321533, -0.2437460869550705, -1.7138862609863281, -0.2165520191192627, -2.490556240081787, -1.92274808883667, -0.5805894136428833, -2.9428036212921143, -1.2461912631988525, -0.06966564059257507, -3.1577250957489014, -0.680553674697876, -2.189528703689575, -0.7803711891174316, -1.248146891593933, -0.26419398188591003, -0.12217089533805847, -0.24270041286945343, -0.11456642299890518, -0.006315156817436218, -3.941749334335327, -0.3705695569515228, -2.4777069091796875, -1.8234386444091797, -0.8842157125473022, -0.19391685724258423, -0.03837955370545387]}
+{"case_id": "source_boundary_00", "category": "source_boundary", "origin": "budgie", "answer": "archeological started by setting up a study group of 48, so they had 48 – 30 = 18 participants left.", "reward": -0.010000000000000009, "advantage": -0.5061788432676426, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 12919, 37211, 1921, 260, 10657, 761, 216, 49163, 49167, 4697, 284, 15298, 2976, 3737, 30, 657, 1072, 441, 1378, 260, 28180, 985, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 4697, 10370, 578, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 1487, 255, 1430, 2841, 411, 4054, 614, 253, 1108, 1528, 282, 216, 49163, 49167, 28, 588, 502, 761, 216, 49163, 49167, 816, 216, 49162, 49159, 446, 216, 49160, 49167, 4697, 2049, 30, 49154], "prefix_len": 141, "old_logprobs": [-13.335829734802246, -5.924513339996338, -8.539730072021484, -4.9321441650390625, -4.607439041137695, -5.969359397888184, -0.686614990234375, -0.985232949256897, -3.514556407928467, -1.9671478271484375, -1.478393793106079, -0.2543852627277374, -1.0600919723510742, -0.5095565915107727, -2.0016002655029297, -3.352902412414551, -3.1203770637512207, -1.5353769063949585, -0.3237977623939514, -0.05826941505074501, -0.0009153467253781855, -3.0132546424865723, -0.01641448773443699, -2.9143471717834473, -1.3565292358398438, -0.036626920104026794, -0.18876591324806213, -2.396077979938127e-05, -0.003198272082954645, -0.517913281917572, -0.8644662499427795, -0.27664220333099365, -0.22723625600337982]}
+{"case_id": "source_boundary_00", "category": "source_boundary", "origin": "budgie", "answer": "I’m Descriptive, so I don’t have real-time data or a current question.", "reward": -0.010000000000000009, "advantage": -0.5061788432676426, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 12919, 37211, 1921, 260, 10657, 761, 216, 49163, 49167, 4697, 284, 15298, 2976, 3737, 30, 657, 1072, 441, 1378, 260, 28180, 985, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 4697, 10370, 578, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 417, 93, 46145, 28, 588, 339, 1326, 417, 100, 457, 1345, 29, 2256, 940, 355, 253, 1574, 1962, 30, 49154], "prefix_len": 141, "old_logprobs": [-1.0233293771743774, -2.624603748321533, -0.2437460869550705, -12.129901885986328, -1.6385278701782227, -1.6066455841064453, -0.6665956974029541, -2.6551175117492676, -0.2018556147813797, -7.545663538621739e-05, -0.8950077891349792, -2.1543726921081543, -0.26828721165657043, -0.01651228405535221, -0.2526856064796448, -2.2335867881774902, -2.3762929439544678, -1.4163541793823242, -2.641524314880371, -0.6964870691299438, -0.545688807964325]}
+{"case_id": "source_boundary_00", "category": "source_boundary", "origin": "budgie", "answer": "It is not possible to know the number of participants who dropped out without more information.", "reward": 0, "advantage": -0.48136615487216994, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 12919, 37211, 1921, 260, 10657, 761, 216, 49163, 49167, 4697, 284, 15298, 2976, 3737, 30, 657, 1072, 441, 1378, 260, 28180, 985, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 4697, 10370, 578, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 1589, 314, 441, 1636, 288, 699, 260, 1230, 282, 4697, 617, 10370, 578, 1355, 540, 1096, 30, 49154], "prefix_len": 141, "old_logprobs": [-3.750535726547241, -1.267572283744812, -1.903840184211731, -0.1739702820777893, -0.01364969927817583, -0.9243790507316589, -0.9828577041625977, -0.3342346251010895, -0.007520222570747137, -0.01110751647502184, -0.1928575485944748, -0.02213766798377037, -0.0003623305819928646, -0.0910855084657669, -2.196976661682129, -1.002777338027954, -0.09125418215990067, -0.08114934712648392]}
+{"case_id": "source_boundary_00", "category": "source_boundary", "origin": "manual_ideal", "answer": "The dropout count is not provided in the memo, so it cannot be determined from the supplied information.", "reward": 1, "advantage": 1.9999026846750978, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 12919, 37211, 1921, 260, 10657, 761, 216, 49163, 49167, 4697, 284, 15298, 2976, 3737, 30, 657, 1072, 441, 1378, 260, 28180, 985, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 4697, 10370, 578, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 28180, 985, 314, 441, 2711, 281, 260, 37211, 28, 588, 357, 2526, 325, 5250, 429, 260, 12919, 1096, 30, 49154], "prefix_len": 141, "old_logprobs": [-2.148329257965088, -12.617807388305664, -0.4551210403442383, -1.6304835081100464, -2.076472282409668, -1.5419807434082031, -1.545581579208374, -0.045882754027843475, -4.56721305847168, -1.8398149013519287, -0.7181539535522461, -1.8189767599105835, -2.7410197257995605, -0.008395615965127945, -1.6209787130355835, -5.755026817321777, -0.30805402994155884, -11.26594352722168, -1.817848801612854, -0.1898621916770935, -0.14817535877227783]}
+{"case_id": "source_boundary_01", "category": "source_boundary", "origin": "budgie", "answer": "The seal caused the Selksі to crack.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 14789, 4109, 23351, 253, 27132, 11654, 564, 3894, 787, 1728, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 1780, 2952, 260, 11654, 288, 10540, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 11654, 2952, 260, 15945, 538, 40275, 288, 10540, 30, 49154], "prefix_len": 130, "old_logprobs": [-0.7561925649642944, -1.3691054582595825, -4.676443099975586, -0.03182730823755264, -10.507658004760742, -5.681236743927002, -10.108863830566406, -1.283211350440979, -1.9507622718811035, -0.593299150466919, -0.06050333008170128]}
+{"case_id": "source_boundary_01", "category": "source_boundary", "origin": "budgie", "answer": "The fellow folder was E Tables, not asteps.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 14789, 4109, 23351, 253, 27132, 11654, 564, 3894, 787, 1728, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 1780, 2952, 260, 11654, 288, 10540, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 6955, 12938, 436, 414, 32796, 28, 441, 3356, 11726, 30, 49154], "prefix_len": 130, "old_logprobs": [-0.7561925649642944, -9.915980339050293, -8.222617149353027, -2.4035887718200684, -10.462464332580566, -10.963391304016113, -2.2565016746520996, -2.7634403705596924, -11.401233673095703, -12.02122688293457, -0.3610314726829529, -0.03372841328382492]}
+{"case_id": "source_boundary_01", "category": "source_boundary", "origin": "budgie", "answer": "The seal was fragile without a cause.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 14789, 4109, 23351, 253, 27132, 11654, 564, 3894, 787, 1728, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 1780, 2952, 260, 11654, 288, 10540, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 11654, 436, 15725, 1355, 253, 1728, 30, 49154], "prefix_len": 130, "old_logprobs": [-0.7561925649642944, -1.3691054582595825, -1.8951928615570068, -9.16087532043457, -7.359156608581543, -0.9482406377792358, -1.2625982761383057, -0.34676826000213623, -0.027580954134464264]}
+{"case_id": "source_boundary_01", "category": "source_boundary", "origin": "budgie", "answer": "The seal was severed by a scientific process callediox.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 14789, 4109, 23351, 253, 27132, 11654, 564, 3894, 787, 1728, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 1780, 2952, 260, 11654, 288, 10540, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 11654, 436, 45109, 411, 253, 3097, 980, 1217, 8491, 30, 49154], "prefix_len": 130, "old_logprobs": [-0.7561925649642944, -1.3691054582595825, -1.8951928615570068, -10.59056282043457, -3.354879379272461, -1.098113775253296, -7.062286376953125, -2.091247081756592, -0.6125179529190063, -11.058281898498535, -1.074739933013916, -0.06798296421766281]}
+{"case_id": "source_boundary_01", "category": "source_boundary", "origin": "manual_ideal", "answer": "The note confirms the crack but does not identify its cause.", "reward": 1, "advantage": 1.9999950495172043, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 14789, 4109, 23351, 253, 27132, 11654, 564, 3894, 787, 1728, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 1780, 2952, 260, 11654, 288, 10540, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 4109, 23351, 260, 10540, 564, 1072, 441, 2669, 624, 1728, 30, 49154], "prefix_len": 130, "old_logprobs": [-0.7561925649642944, -8.212855339050293, -3.4384288787841797, -0.7771905660629272, -8.71940803527832, -1.5185878276824951, -2.036421775817871, -0.024983301758766174, -8.468749046325684, -4.518121719360352, -0.1682695895433426, -0.06363123655319214, -0.030904239043593407]}
+{"case_id": "source_boundary_02", "category": "source_boundary", "origin": "budgie", "answer": "I’ll use the information from the source material.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 2524, 20578, 1921, 260, 3102, 10641, 216, 49161, 28, 49159, 49159, 49159, 3417, 30, 657, 3894, 787, 6092, 4867, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 3417, 11663, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 417, 764, 722, 260, 1096, 429, 260, 2257, 1376, 30, 49154], "prefix_len": 137, "old_logprobs": [-1.0136322975158691, -2.217834949493408, -1.6904900074005127, -1.1564619541168213, -0.8677839040756226, -2.5131890773773193, -0.9608733654022217, -0.12801766395568848, -3.8184762001037598, -0.513185977935791, -0.7230939269065857, -0.029898405075073242]}
+{"case_id": "source_boundary_02", "category": "source_boundary", "origin": "budgie", "answer": "I’m goin’ to count the trees that survived.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 2524, 20578, 1921, 260, 3102, 10641, 216, 49161, 28, 49159, 49159, 49159, 3417, 30, 657, 3894, 787, 6092, 4867, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 3417, 11663, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 417, 93, 685, 254, 417, 288, 985, 260, 3417, 338, 11663, 30, 49154], "prefix_len": 137, "old_logprobs": [-1.0136322975158691, -2.217834949493408, -0.3154900074005127, -5.664246559143066, -0.7442248463630676, -0.09699071943759918, -0.3924863934516907, -4.097664833068848, -1.0305126905441284, -0.19940006732940674, -1.2071847915649414, -0.4822172224521637, -0.8141486644744873, -0.1524825096130371]}
+{"case_id": "source_boundary_02", "category": "source_boundary", "origin": "budgie", "answer": "Asilarkness planted 2000 trees.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 2524, 20578, 1921, 260, 3102, 10641, 216, 49161, 28, 49159, 49159, 49159, 3417, 30, 657, 3894, 787, 6092, 4867, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 3417, 11663, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 1653, 301, 792, 924, 10641, 216, 49161, 49159, 49159, 49159, 3417, 30, 49154], "prefix_len": 137, "old_logprobs": [-3.763632297515869, -0.026333075016736984, -0.002585760550573468, -6.615896563744172e-05, -0.544455885887146, -0.12729085981845856, -0.027315009385347366, -1.0683034658432007, -0.42101672291755676, -0.05368070676922798, -0.3126908242702484, -0.7051265835762024, -0.04749114438891411]}
+{"case_id": "source_boundary_02", "category": "source_boundary", "origin": "budgie", "answer": "I know 2,000 trees.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 2524, 20578, 1921, 260, 3102, 10641, 216, 49161, 28, 49159, 49159, 49159, 3417, 30, 657, 3894, 787, 6092, 4867, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 3417, 11663, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 699, 216, 49161, 28, 49159, 49159, 49159, 3417, 30, 49154], "prefix_len": 137, "old_logprobs": [-1.0136322975158691, -2.342834949493408, -2.038478374481201, -0.04070819541811943, -0.4861011207103729, -0.004657610785216093, -0.00028153270250186324, -0.0004085659747943282, -0.15039710700511932, -1.0517692565917969, -0.11522572487592697]}
+{"case_id": "source_boundary_02", "category": "source_boundary", "origin": "manual_ideal", "answer": "The excerpt does not provide survival data, so the number cannot be determined.", "reward": 1, "advantage": 1.9999950495172043, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 2524, 20578, 1921, 260, 3102, 10641, 216, 49161, 28, 49159, 49159, 49159, 3417, 30, 657, 3894, 787, 6092, 4867, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 3417, 11663, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 20578, 1072, 441, 1538, 6092, 940, 28, 588, 260, 1230, 2526, 325, 5250, 30, 49154], "prefix_len": 137, "old_logprobs": [-1.7636322975158691, -13.038537979125977, -10.125852584838867, -0.3476831018924713, -2.086818218231201, -8.932546615600586, -1.6256682872772217, -1.4945999383926392, -1.1216809749603271, -1.9662264585494995, -2.353785514831543, -5.982611179351807, -0.0033088955096900463, -0.4065680503845215, -0.31656575202941895, -0.09837427735328674]}
+{"case_id": "source_boundary_03", "category": "source_boundary", "origin": "budgie", "answer": "I found the information through an external source on the annual payment schedule.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 48686, 8639, 3496, 284, 2254, 564, 787, 9599, 3688, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 3280, 732, 3688, 436, 357, 5940, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 983, 260, 1096, 738, 354, 5004, 2257, 335, 260, 5316, 9599, 8034, 30, 49154], "prefix_len": 129, "old_logprobs": [-0.8927786946296692, -3.551790237426758, -1.021336317062378, -1.5577123165130615, -3.917097568511963, -2.629957675933838, -3.825624465942383, -0.09908093512058258, -5.677481174468994, -1.0713862180709839, -7.534938335418701, -2.8775618076324463, -2.269768476486206, -0.2813540995121002, -0.04328957572579384]}
+{"case_id": "source_boundary_03", "category": "source_boundary", "origin": "budgie", "answer": "I-I didn't know the date, it was paid on Friday.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 48686, 8639, 3496, 284, 2254, 564, 787, 9599, 3688, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 3280, 732, 3688, 436, 357, 5940, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 29, 57, 3683, 982, 699, 260, 3688, 28, 357, 436, 5940, 335, 11655, 30, 49154], "prefix_len": 129, "old_logprobs": [-0.8927786946296692, -5.239290237426758, -0.8997269868850708, -3.6406760215759277, -1.2243356704711914, -0.8237937688827515, -1.5804572105407715, -1.1461443901062012, -1.5249121189117432, -4.819476127624512, -0.2822822034358978, -0.197811558842659, -1.0224703550338745, -4.170210361480713, -0.5613791942596436, -0.02670525759458542]}
+{"case_id": "source_boundary_03", "category": "source_boundary", "origin": "budgie", "answer": "I know it was paid on February 17th.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 48686, 8639, 3496, 284, 2254, 564, 787, 9599, 3688, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 3280, 732, 3688, 436, 357, 5940, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 699, 357, 436, 5940, 335, 5096, 216, 49160, 49166, 373, 30, 49154], "prefix_len": 129, "old_logprobs": [-0.8927786946296692, -3.801790237426758, -0.8267160654067993, -0.25480085611343384, -0.09890277683734894, -0.4377340078353882, -2.6365814208984375, -0.007856657728552818, -0.9397374391555786, -3.795461654663086, -0.5298703908920288, -0.7561032176017761, -0.07450146228075027]}
+{"case_id": "source_boundary_03", "category": "source_boundary", "origin": "budgie", "answer": "It was paid on January 1, 2023.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 48686, 8639, 3496, 284, 2254, 564, 787, 9599, 3688, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 3280, 732, 3688, 436, 357, 5940, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 1589, 436, 5940, 335, 4450, 216, 49160, 28, 216, 49161, 49159, 49161, 49162, 30, 49154], "prefix_len": 129, "old_logprobs": [-1.3927786350250244, -0.058559074997901917, -0.06574584543704987, -0.3482254445552826, -3.4147560596466064, -0.0028709645848721266, -0.6839197874069214, -1.1863933801651, -0.03894026577472687, -0.009644461795687675, -0.0007124984404072165, -0.34716513752937317, -1.1725759506225586, -0.07268281280994415, -0.02647355943918228]}
+{"case_id": "source_boundary_03", "category": "source_boundary", "origin": "manual_ideal", "answer": "No payment date is provided, so the payment date cannot be verified.", "reward": 1, "advantage": 1.9999950495172043, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 48686, 8639, 3496, 284, 2254, 564, 787, 9599, 3688, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 3280, 732, 3688, 436, 357, 5940, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 5230, 9599, 3688, 314, 2711, 28, 588, 260, 9599, 3688, 2526, 325, 21548, 30, 49154], "prefix_len": 129, "old_logprobs": [-6.2677788734436035, -3.3014392852783203, -0.6650804281234741, -1.4609215259552002, -3.1495907306671143, -1.473803997039795, -0.632594108581543, -1.854618787765503, -5.551197528839111, -2.4405579566955566, -2.9595601558685303, -0.0017689309315755963, -6.715890407562256, -0.2065969705581665, -0.10466251522302628]}
+{"case_id": "source_boundary_04", "category": "source_boundary", "origin": "budgie", "answer": "The mechanism caused the rise.", "reward": -0.010000000000000009, "advantage": -0.5061788432676426, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4308, 1993, 4585, 253, 2779, 4095, 990, 8427, 216, 49160, 49161, 564, 1072, 441, 2669, 260, 6958, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 14413, 6958, 2952, 260, 4095, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 6958, 2952, 260, 4095, 30, 49154], "prefix_len": 136, "old_logprobs": [-0.25858476758003235, -2.030606746673584, -1.459545612335205, -0.008432142436504364, -0.05865924432873726, -0.3368956744670868, -0.4048720598220825]}
+{"case_id": "source_boundary_04", "category": "source_boundary", "origin": "budgie", "answer": "The mechanism caused the rise.", "reward": -0.010000000000000009, "advantage": -0.5061788432676426, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4308, 1993, 4585, 253, 2779, 4095, 990, 8427, 216, 49160, 49161, 564, 1072, 441, 2669, 260, 6958, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 14413, 6958, 2952, 260, 4095, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 6958, 2952, 260, 4095, 30, 49154], "prefix_len": 136, "old_logprobs": [-0.25858476758003235, -2.030606746673584, -1.459545612335205, -0.008432142436504364, -0.05865924432873726, -0.3368956744670868, -0.4048720598220825]}
+{"case_id": "source_boundary_04", "category": "source_boundary", "origin": "budgie", "answer": "The temperature rise occurred after minute 12, but we don’t have a specific mechanism.", "reward": 0, "advantage": -0.48136615487216994, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4308, 1993, 4585, 253, 2779, 4095, 990, 8427, 216, 49160, 49161, 564, 1072, 441, 2669, 260, 6958, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 14413, 6958, 2952, 260, 4095, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 2779, 4095, 5691, 990, 8427, 216, 49160, 49161, 28, 564, 392, 1326, 417, 100, 457, 253, 1678, 6958, 30, 49154], "prefix_len": 136, "old_logprobs": [-0.25781312584877014, -4.6232404708862305, -0.1604495346546173, -1.712378740310669, -1.215083122253418, -0.5519878268241882, -0.005810987204313278, -0.05730808153748512, -0.06138065457344055, -0.8818901777267456, -0.8989930152893066, -2.589543104171753, -1.486733078956604, -0.3488359749317169, -8.821098163025454e-05, -1.3151979446411133, -1.2997697591781616, -1.5769407749176025, -0.7952007055282593, -1.7025004625320435, -0.28587067127227783]}
+{"case_id": "source_boundary_04", "category": "source_boundary", "origin": "budgie", "answer": "The primary mechanism causing the rise is the non-radiation process.", "reward": -0.010000000000000009, "advantage": -0.5061788432676426, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4308, 1993, 4585, 253, 2779, 4095, 990, 8427, 216, 49160, 49161, 564, 1072, 441, 2669, 260, 6958, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 14413, 6958, 2952, 260, 4095, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 3003, 6958, 4439, 260, 4095, 314, 260, 1603, 29, 2270, 2809, 980, 30, 49154], "prefix_len": 136, "old_logprobs": [-0.25858476758003235, -4.249356746673584, -0.6461946368217468, -1.3591874837875366, -0.013797149993479252, -0.11486860364675522, -0.5778840780258179, -0.3759785592556, -6.955013275146484, -0.06691999733448029, -5.064012050628662, -0.8901267051696777, -3.2739458084106445, -0.8097789883613586, -0.11731212586164474]}
+{"case_id": "source_boundary_04", "category": "source_boundary", "origin": "manual_ideal", "answer": "The log shows when the rise occurred but does not establish its mechanism.", "reward": 1, "advantage": 1.9999026846750978, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4308, 1993, 4585, 253, 2779, 4095, 990, 8427, 216, 49160, 49161, 564, 1072, 441, 2669, 260, 6958, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 14413, 6958, 2952, 260, 4095, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 1993, 2744, 645, 260, 4095, 5691, 564, 1072, 441, 3348, 624, 6958, 30, 49154], "prefix_len": 136, "old_logprobs": [-0.25858476758003235, -8.702482223510742, -3.4908363819122314, -8.28211498260498, -0.19671495258808136, -3.677886962890625, -0.9317288398742676, -3.3673791885375977, -0.5914135575294495, -0.0035323624033480883, -7.017106056213379, -4.253974437713623, -1.252788782119751, -0.09740154445171356, -0.042375314980745316]}
+{"case_id": "source_boundary_05", "category": "source_boundary", "origin": "budgie", "answer": "I did not vote for it. I am torch Throughout the conversation and can only provide the facts and facts I know in the given format.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4751, 3487, 25312, 260, 10317, 1355, 10016, 1212, 14879, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 10576, 17845, 1523, 357, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 1250, 441, 7507, 327, 357, 30, 339, 744, 5984, 11067, 260, 6634, 284, 416, 805, 1538, 260, 5307, 284, 5307, 339, 699, 281, 260, 1836, 4624, 30, 49154], "prefix_len": 127, "old_logprobs": [-0.2299249768257141, -3.907691240310669, -0.1545768529176712, -1.7543163299560547, -1.3190752267837524, -0.27146920561790466, -0.34352898597717285, -1.2295315265655518, -1.1667908430099487, -9.136371612548828, -9.658509254455566, -0.8375434875488281, -1.421568512916565, -2.513751745223999, -4.780343055725098, -1.5837522745132446, -1.4307434558868408, -1.687981367111206, -0.5095171928405762, -1.0249944925308228, -0.5966634750366211, -2.18880033493042, -1.3298614025115967, -3.778589963912964, -1.8038544654846191, -4.33795690536499, -2.4200775623321533, -0.17740845680236816, -0.04242239147424698]}
+{"case_id": "source_boundary_05", "category": "source_boundary", "origin": "budgie", "answer": "I’m not sure, but based on what we know, [I] said did vote against it.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4751, 3487, 25312, 260, 10317, 1355, 10016, 1212, 14879, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 10576, 17845, 1523, 357, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 417, 93, 441, 2090, 28, 564, 1552, 335, 732, 392, 699, 28, 933, 57, 77, 1137, 1250, 7507, 1523, 357, 30, 49154], "prefix_len": 127, "old_logprobs": [-0.22673891484737396, -2.0963704586029053, -0.4396344721317291, -2.158859968185425, -0.21363422274589539, -1.3677514791488647, -0.12494075298309326, -1.6314345598220825, -0.006340980529785156, -3.621044397354126, -1.5764110088348389, -0.4707578122615814, -0.4387311637401581, -8.48768424987793, -1.4027493000030518, -2.7257168292999268, -2.9675180912017822, -5.46711540222168, -3.307905673980713, -0.3496250510215759, -0.19646140933036804, -0.17581050097942352, -0.050546515733003616]}
+{"case_id": "source_boundary_05", "category": "source_boundary", "origin": "budgie", "answer": "I did not vote against it.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4751, 3487, 25312, 260, 10317, 1355, 10016, 1212, 14879, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 10576, 17845, 1523, 357, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 1250, 441, 7507, 1523, 357, 30, 49154], "prefix_len": 127, "old_logprobs": [-0.22673891484737396, -3.9088704586029053, -0.15472279489040375, -1.754488229751587, -0.5691065788269043, -0.19405344128608704, -0.3242032527923584, -0.4536595940589905]}
+{"case_id": "source_boundary_05", "category": "source_boundary", "origin": "budgie", "answer": "I amHist ledger.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4751, 3487, 25312, 260, 10317, 1355, 10016, 1212, 14879, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 10576, 17845, 1523, 357, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 744, 8523, 30045, 30, 49154], "prefix_len": 127, "old_logprobs": [-0.22673891484737396, -2.4088704586029053, -9.482624053955078, -12.55118465423584, -1.4488798379898071, -0.435181200504303]}
+{"case_id": "source_boundary_05", "category": "source_boundary", "origin": "manual_ideal", "answer": "Individual votes were not recorded, so the minutes cannot identify who voted against it.", "reward": 1, "advantage": 1.9999950495172043, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4751, 3487, 25312, 260, 10317, 1355, 10016, 1212, 14879, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 10576, 17845, 1523, 357, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 23920, 14879, 592, 441, 5797, 28, 588, 260, 3487, 2526, 2669, 617, 17845, 1523, 357, 30, 49154], "prefix_len": 127, "old_logprobs": [-13.414238929748535, -0.9900743961334229, -1.6069750785827637, -4.002044677734375, -6.839709758758545, -1.2046489715576172, -1.102369785308838, -1.9373648166656494, -9.387782096862793, -5.859726905822754, -6.1569695472717285, -1.2878540754318237, -0.15706810355186462, -0.5329169034957886, -0.2353011816740036, -0.055106837302446365, -0.07350222766399384]}
+{"case_id": "source_boundary_06", "category": "source_boundary", "origin": "budgie", "answer": "I would expect 5 years to last in use.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 1406, 7381, 3894, 253, 216, 49164, 49159, 49159, 29, 12022, 14404, 656, 1587, 1920, 564, 787, 1955, 29, 3460, 7493, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 929, 523, 357, 1596, 281, 2955, 722, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 736, 1959, 216, 49164, 929, 288, 1596, 281, 722, 30, 49154], "prefix_len": 143, "old_logprobs": [-0.7148588299751282, -3.0916128158569336, -2.162916660308838, -1.1033669710159302, -1.7069523334503174, -0.8313695192337036, -0.8390942811965942, -0.13916714489459991, -0.7924445867538452, -1.5204598903656006, -0.45559802651405334, -0.035636261105537415]}
+{"case_id": "source_boundary_06", "category": "source_boundary", "origin": "budgie", "answer": "I’m off to introduce this product, but it will last for six years.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 1406, 7381, 3894, 253, 216, 49164, 49159, 49159, 29, 12022, 14404, 656, 1587, 1920, 564, 787, 1955, 29, 3460, 7493, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 929, 523, 357, 1596, 281, 2955, 722, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 417, 93, 767, 288, 7328, 451, 1406, 28, 564, 357, 523, 1596, 327, 2976, 929, 30, 49154], "prefix_len": 143, "old_logprobs": [-0.7148588299751282, -1.6541128158569336, -0.4631243944168091, -7.168235778808594, -1.2130964994430542, -6.742710113525391, -3.976715087890625, -1.6650168895721436, -2.0510666370391846, -1.109499454498291, -1.745910882949829, -1.5389845371246338, -0.5804595351219177, -1.2436708211898804, -4.200091361999512, -0.17688921093940735, -0.6693381667137146, -0.04185951128602028]}
+{"case_id": "source_boundary_06", "category": "source_boundary", "origin": "budgie", "answer": "I would guess around 20 years.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 1406, 7381, 3894, 253, 216, 49164, 49159, 49159, 29, 12022, 14404, 656, 1587, 1920, 564, 787, 1955, 29, 3460, 7493, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 929, 523, 357, 1596, 281, 2955, 722, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 736, 6820, 1130, 216, 49161, 49159, 929, 30, 49154], "prefix_len": 143, "old_logprobs": [-0.7142096757888794, -3.0945801734924316, -2.3556113243103027, -3.658329963684082, -0.13882315158843994, -1.728796124458313, -1.0925129652023315, -0.6342602968215942, -0.8838879466056824, -0.04171876981854439]}
+{"case_id": "source_boundary_06", "category": "source_boundary", "origin": "budgie", "answer": "I know it will last around 3 years.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 1406, 7381, 3894, 253, 216, 49164, 49159, 49159, 29, 12022, 14404, 656, 1587, 1920, 564, 787, 1955, 29, 3460, 7493, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 929, 523, 357, 1596, 281, 2955, 722, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 699, 357, 523, 1596, 1130, 216, 49162, 929, 30, 49154], "prefix_len": 143, "old_logprobs": [-0.7148588299751282, -3.0291128158569336, -0.716000497341156, -0.7415803074836731, -0.08427534997463226, -2.527463436126709, -0.18448400497436523, -2.190394401550293, -1.0439186096191406, -1.5537292957305908, -0.0635334700345993]}
+{"case_id": "source_boundary_06", "category": "source_boundary", "origin": "manual_ideal", "answer": "The sheet does not provide enough usage data to convert the lab cycle rating into years of normal use.", "reward": 1, "advantage": 1.9999950495172043, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 1406, 7381, 3894, 253, 216, 49164, 49159, 49159, 29, 12022, 14404, 656, 1587, 1920, 564, 787, 1955, 29, 3460, 7493, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 929, 523, 357, 1596, 281, 2955, 722, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 7381, 1072, 441, 1538, 2001, 6486, 940, 288, 7728, 260, 1587, 4834, 14404, 618, 929, 282, 2955, 722, 30, 49154], "prefix_len": 143, "old_logprobs": [-2.440791368484497, -9.828192710876465, -6.277126789093018, -0.0726001188158989, -3.395650863647461, -2.7141709327697754, -12.043331146240234, -0.3630906641483307, -0.2645212411880493, -8.186516761779785, -2.150437831878662, -7.6912760734558105, -6.038015842437744, -0.6878067255020142, -2.181535243988037, -0.6343536972999573, -2.919243335723877, -0.9182887077331543, -0.03010108694434166, -0.1366402953863144, -0.11711768060922623]}
+{"case_id": "source_boundary_07", "category": "source_boundary", "origin": "budgie", "answer": "I answered 10.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4867, 7127, 4631, 354, 3049, 6581, 282, 216, 49166, 30, 49161, 564, 9313, 842, 260, 2426, 985, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 701, 11739, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 11739, 216, 49160, 49159, 30, 49154], "prefix_len": 135, "old_logprobs": [-0.8943935632705688, -3.2838294506073, -0.3940379023551941, -1.4794182777404785, -0.796021580696106, -2.0124034881591797, -0.017362387850880623]}
+{"case_id": "source_boundary_07", "category": "source_boundary", "origin": "budgie", "answer": "I will find out the number of people who answered that question.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4867, 7127, 4631, 354, 3049, 6581, 282, 216, 49166, 30, 49161, 564, 9313, 842, 260, 2426, 985, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 701, 11739, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 523, 1042, 578, 260, 1230, 282, 701, 617, 11739, 338, 1962, 30, 49154], "prefix_len": 135, "old_logprobs": [-0.8943935632705688, -4.408829689025879, -3.5172653198242188, -1.1400806903839111, -1.910564661026001, -0.9204086661338806, -0.06556365638971329, -0.176049143075943, -0.14626510441303253, -0.5040399432182312, -4.136527061462402, -0.506546676158905, -1.2740817070007324, -0.24336214363574982]}
+{"case_id": "source_boundary_07", "category": "source_boundary", "origin": "budgie", "answer": "I answered 18.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4867, 7127, 4631, 354, 3049, 6581, 282, 216, 49166, 30, 49161, 564, 9313, 842, 260, 2426, 985, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 701, 11739, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 11739, 216, 49160, 49167, 30, 49154], "prefix_len": 135, "old_logprobs": [-0.8943935632705688, -3.2838294506073, -0.3940379023551941, -1.4794182777404785, -3.2960214614868164, -1.351190209388733, -0.00946805626153946]}
+{"case_id": "source_boundary_07", "category": "source_boundary", "origin": "budgie", "answer": "I will answer the question directly without guessing.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4867, 7127, 4631, 354, 3049, 6581, 282, 216, 49166, 30, 49161, 564, 9313, 842, 260, 2426, 985, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 701, 11739, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 523, 2988, 260, 1962, 3319, 1355, 29916, 30, 49154], "prefix_len": 135, "old_logprobs": [-0.8943935632705688, -4.408829689025879, -1.4547652006149292, -0.9154694080352783, -0.4889863431453705, -0.8143066763877869, -0.4077695608139038, -0.10419505089521408, -0.1633344292640686, -0.0887933000922203]}
+{"case_id": "source_boundary_07", "category": "source_boundary", "origin": "manual_ideal", "answer": "The response count is missing, so it cannot be determined from the summary.", "reward": 1, "advantage": 1.9999950495172043, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 4867, 7127, 4631, 354, 3049, 6581, 282, 216, 49166, 30, 49161, 564, 9313, 842, 260, 2426, 985, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 2020, 800, 701, 11739, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 2426, 985, 314, 6782, 28, 588, 357, 2526, 325, 5250, 429, 260, 7127, 30, 49154], "prefix_len": 135, "old_logprobs": [-3.0193934440612793, -5.610597610473633, -1.9539449214935303, -1.0264482498168945, -3.9566056728363037, -0.5857589244842529, -1.0915167331695557, -2.5574569702148438, -3.939601421356201, -0.026543445885181427, -1.1567084789276123, -5.744723796844482, -0.09449458867311478, -4.220023155212402, -0.389077365398407, -0.13133271038532257]}
+{"case_id": "source_boundary_08", "category": "source_boundary", "origin": "budgie", "answer": "I should not guess.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 12497, 4887, 1921, 260, 35598, 2049, 260, 41219, 18400, 564, 553, 787, 2330, 9524, 6850, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 9576, 314, 357, 1048, 1209, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 868, 441, 6820, 30, 49154], "prefix_len": 134, "old_logprobs": [-0.47267624735832214, -3.0236830711364746, -1.4353514909744263, -0.9981943368911743, -1.2561540603637695, -1.2499573230743408]}
+{"case_id": "source_boundary_08", "category": "source_boundary", "origin": "budgie", "answer": "I am on a budget.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 12497, 4887, 1921, 260, 35598, 2049, 260, 41219, 18400, 564, 553, 787, 2330, 9524, 6850, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 9576, 314, 357, 1048, 1209, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 744, 335, 253, 6050, 30, 49154], "prefix_len": 134, "old_logprobs": [-0.47267624735832214, -1.2111830711364746, -4.51975154876709, -1.1565322875976562, -6.480802536010742, -1.6327539682388306, -0.14234046638011932]}
+{"case_id": "source_boundary_08", "category": "source_boundary", "origin": "budgie", "answer": "I am here.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 12497, 4887, 1921, 260, 35598, 2049, 260, 41219, 18400, 564, 553, 787, 2330, 9524, 6850, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 9576, 314, 357, 1048, 1209, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 744, 1535, 30, 49154], "prefix_len": 134, "old_logprobs": [-0.47267624735832214, -1.2111830711364746, -4.39475154876709, -1.3693357706069946, -0.1533661186695099]}
+{"case_id": "source_boundary_08", "category": "source_boundary", "origin": "budgie", "answer": "I need a location information from an external source, so I can contact you there.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 12497, 4887, 1921, 260, 35598, 2049, 260, 41219, 18400, 564, 553, 787, 2330, 9524, 6850, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 9576, 314, 357, 1048, 1209, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 57, 737, 253, 3695, 1096, 429, 354, 5004, 2257, 28, 588, 339, 416, 2770, 346, 665, 30, 49154], "prefix_len": 134, "old_logprobs": [-0.47267624735832214, -2.5861830711364746, -2.5800158977508545, -4.473158359527588, -3.2028517723083496, -3.0792837142944336, -3.957773447036743, -0.9895243644714355, -0.00509487884119153, -1.7210756540298462, -1.8820284605026245, -0.29031506180763245, -1.141558289527893, -5.77787971496582, -1.1517517566680908, -3.259270191192627, -0.24400371313095093, -0.026080384850502014]}
+{"case_id": "source_boundary_08", "category": "source_boundary", "origin": "manual_ideal", "answer": "The notice does not contain live location data, so the parcel’s current location cannot be verified.", "reward": 1, "advantage": 1.9999950495172043, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 504, 12497, 4887, 1921, 260, 35598, 2049, 260, 41219, 18400, 564, 553, 787, 2330, 9524, 6850, 30, 49154, 49158, 198, 57, 417, 764, 722, 805, 260, 1096, 281, 338, 1376, 30, 49154, 49157, 198, 9576, 314, 357, 1048, 1209, 47, 19842, 3319, 1355, 29916, 30, 49154, 49158, 198, 504, 4887, 1072, 441, 2127, 2330, 3695, 940, 28, 588, 260, 35598, 417, 99, 1574, 3695, 2526, 325, 21548, 30, 49154], "prefix_len": 134, "old_logprobs": [-2.2355668544769287, -10.227718353271484, -6.633266448974609, -0.04050398990511894, -4.1435465812683105, -5.106439590454102, -7.12007999420166, -1.428864598274231, -2.1115200519561768, -0.6691135764122009, -2.8283302783966064, -11.816269874572754, -5.797732830047607, -0.0010071687866002321, -1.2578431367874146, -0.10044553130865097, -3.96830415725708, -0.010243226774036884, -4.362934589385986, -0.1901792287826538, -0.09733762592077255]}
+{"case_id": "constraint_00", "category": "constraint_stack", "origin": "budgie", "answer": "We need to find the intersection of the two sentences. The first sentence is \"I am a little bit late for a safety designing meeting.\" This means that I am late for an event and not part of a motion that calls for the meeting or that I am late for a meeting.\n\nThe second sentence is \"I am a bit late for a safety designing meeting.\" This means that I am late for a safety-related event (i.e., an event the other person is part of) but I am not part of a motion that calls for the event itself.\n\nHowever, the first sentence does not contain any explicit facts about the current time, so we cannot conclude that my time must be exactly two hours before 09:00.\n\nWe can say that the second sentence does not contain any facts about the current time, so we cannot conclude that I am late before or after 09:00.", "reward": -0.22, "advantage": -0.8550694775366835, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 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+{"case_id": "constraint_00", "category": "constraint_stack", "origin": "budgie", "answer": "I will only provide the requested content.", "reward": -0.010000000000000009, "advantage": -0.36975977406991717, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 52, 2663, 253, 13151, 563, 13759, 417, 99, 216, 49159, 49168, 42, 49159, 49159, 2950, 40268, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 827, 8545, 43, 1453, 637, 42122, 216, 49163, 36651, 536, 441, 722, 637, 3552, 293, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 523, 805, 1538, 260, 13682, 2627, 30, 49154], "prefix_len": 144, "old_logprobs": [-0.7129100561141968, -2.661337375640869, -2.787292003631592, -0.5528796315193176, -0.1953621357679367, -0.5086996555328369, -0.08798228204250336, -0.7691274881362915, -0.0732649490237236]}
+{"case_id": "constraint_00", "category": "constraint_stack", "origin": "budgie", "answer": "I will provide the requested content without repetition.", "reward": -0.010000000000000009, "advantage": -0.36975977406991717, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 52, 2663, 253, 13151, 563, 13759, 417, 99, 216, 49159, 49168, 42, 49159, 49159, 2950, 40268, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 827, 8545, 43, 1453, 637, 42122, 216, 49163, 36651, 536, 441, 722, 637, 3552, 293, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 523, 1538, 260, 13682, 2627, 1355, 18388, 30, 49154], "prefix_len": 144, "old_logprobs": [-0.7099709510803223, -2.697906732559204, -1.4621602296829224, -0.4386492967605591, -0.6000160574913025, -0.06474396586418152, -2.8472232818603516, -3.0164144039154053, -0.40706735849380493, -0.08940646797418594]}
+{"case_id": "constraint_00", "category": "constraint_stack", "origin": "budgie", "answer": "I am not sure, but I should consider the constraints from my perspective.", "reward": -0.010000000000000009, "advantage": -0.36975977406991717, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 52, 2663, 253, 13151, 563, 13759, 417, 99, 216, 49159, 49168, 42, 49159, 49159, 2950, 40268, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 827, 8545, 43, 1453, 637, 42122, 216, 49163, 36651, 536, 441, 722, 637, 3552, 293, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 744, 441, 2090, 28, 564, 339, 868, 1771, 260, 10195, 429, 957, 4939, 30, 49154], "prefix_len": 144, "old_logprobs": [-0.7099709510803223, -3.010406732559204, -4.391406059265137, -1.0811680555343628, -2.8648524284362793, -0.3615267872810364, -0.5851467847824097, -1.2171640396118164, -3.4713988304138184, -1.0525093078613281, -2.564345598220825, -5.012619972229004, -2.3617782592773438, -1.791538953781128, -0.24908623099327087, -1.106200933456421]}
+{"case_id": "constraint_00", "category": "constraint_stack", "origin": "manual_ideal", "answer": "Tomorrow’s safety briefing begins at 09:00 in Room 4. Please arrive five minutes early.", "reward": 1, "advantage": 1.964348799746435, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 52, 2663, 253, 13151, 563, 13759, 417, 99, 216, 49159, 49168, 42, 49159, 49159, 2950, 40268, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 827, 8545, 43, 1453, 637, 42122, 216, 49163, 36651, 536, 441, 722, 637, 3552, 293, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 24960, 6189, 417, 99, 2950, 40268, 5634, 418, 216, 49159, 49168, 42, 49159, 49159, 281, 19690, 216, 49163, 30, 8765, 11296, 2531, 3487, 1584, 30, 49154], "prefix_len": 144, "old_logprobs": [-11.074481964111328, -1.3861663341522217, -1.8290746212005615, -0.00019476900342851877, -1.6648316383361816, -9.546262741088867, -8.02365493774414, -0.7961843609809875, -0.09239345043897629, -0.5138339996337891, -0.07090394198894501, -0.0007830176036804914, -0.012101209722459316, -0.005451336968690157, -6.286070823669434, -5.043767929077148, -0.00887012667953968, -0.015498317778110504, -0.34242507815361023, -5.384272575378418, -5.361272811889648, -8.223636627197266, -0.08455166220664978, -2.8968489170074463, -0.8415803909301758, -0.06042758375406265]}
+{"case_id": "constraint_01", "category": "constraint_stack", "origin": "budgie", "answer": "I need this to be checked.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 6106, 1853, 288, 2037, 253, 17066, 979, 10323, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 1296, 16425, 2876, 43, 971, 656, 4475, 1924, 43, 536, 441, 3311, 6697, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 737, 451, 288, 325, 12775, 30, 49154], "prefix_len": 134, "old_logprobs": [-0.525261640548706, -1.9580793380737305, -3.6033129692077637, -2.658548355102539, -0.2788134217262268, -3.323679208755493, -0.7813342213630676, -0.2657419443130493]}
+{"case_id": "constraint_01", "category": "constraint_stack", "origin": "budgie", "answer": "I should not share the entire conversation, and I should not give any information about insurance.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 6106, 1853, 288, 2037, 253, 17066, 979, 10323, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 1296, 16425, 2876, 43, 971, 656, 4475, 1924, 43, 536, 441, 3311, 6697, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 868, 441, 2419, 260, 2415, 6634, 28, 284, 339, 868, 441, 1928, 750, 1096, 563, 6697, 30, 49154], "prefix_len": 134, "old_logprobs": [-0.525261640548706, -1.8330793380737305, -1.3717334270477295, -4.775572299957275, -1.6626595258712769, -1.0939499139785767, -0.4143465459346771, -1.358136773109436, -1.5900232791900635, -0.3809037506580353, -0.6151437163352966, -0.43777164816856384, -4.509643077850342, -0.9043041467666626, -1.725138783454895, -0.7760886549949646, -0.10487550497055054, -0.34080350399017334, -0.24965031445026398]}
+{"case_id": "constraint_01", "category": "constraint_stack", "origin": "budgie", "answer": "I should only include the information needed to answer the question, and not include any unnecessary information.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 6106, 1853, 288, 2037, 253, 17066, 979, 10323, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 1296, 16425, 2876, 43, 971, 656, 4475, 1924, 43, 536, 441, 3311, 6697, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 868, 805, 1453, 260, 1096, 2350, 288, 2988, 260, 1962, 28, 284, 441, 1453, 750, 11125, 1096, 30, 49154], "prefix_len": 134, "old_logprobs": [-0.5263993740081787, -1.8376455307006836, -3.182011604309082, -2.1516318321228027, -0.42771226167678833, -3.5471904277801514, -2.8158469200134277, -0.3030444085597992, -0.6553285121917725, -0.1823209971189499, -0.6931965351104736, -1.7768359184265137, -1.1750319004058838, -1.3744029998779297, -1.18544340133667, -0.355386883020401, -4.098664283752441, -2.46376895904541, -0.19240477681159973, -0.10298021882772446]}
+{"case_id": "constraint_01", "category": "constraint_stack", "origin": "budgie", "answer": "I need more information or context about the cost of the laptop.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 6106, 1853, 288, 2037, 253, 17066, 979, 10323, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 1296, 16425, 2876, 43, 971, 656, 4475, 1924, 43, 536, 441, 3311, 6697, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 737, 540, 1096, 355, 2468, 563, 260, 1708, 282, 260, 17066, 30, 49154], "prefix_len": 134, "old_logprobs": [-0.525261640548706, -1.9580793380737305, -2.8220629692077637, -1.025710940361023, -2.596644401550293, -1.4723175764083862, -2.07365083694458, -0.34170228242874146, -6.873394966125488, -0.7435566782951355, -0.6849386692047119, -0.14121480286121368, -1.3469551801681519, -0.25633397698402405]}
+{"case_id": "constraint_01", "category": "constraint_stack", "origin": "manual_ideal", "answer": "- Use a padded, water-resistant sleeve.\n- Keep it in your carry-on.\n- Enable encryption and a screen lock.", "reward": 1, "advantage": 1.9999950495172043, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 6106, 1853, 288, 2037, 253, 17066, 979, 10323, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 1296, 16425, 2876, 43, 971, 656, 4475, 1924, 43, 536, 441, 3311, 6697, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 29, 4076, 253, 43916, 28, 913, 29, 14943, 39279, 30, 198, 29, 5545, 357, 281, 469, 3631, 29, 258, 30, 198, 29, 45151, 15066, 284, 253, 5078, 8645, 30, 49154], "prefix_len": 134, "old_logprobs": [-8.88388729095459, -4.248902797698975, -1.6617984771728516, -12.385416984558105, -5.583153247833252, -5.942584991455078, -0.482768177986145, -8.298378944396973, -13.182074546813965, -2.580744504928589, -0.21426242589950562, -0.016726020723581314, -3.2843260765075684, -2.328516960144043, -2.983255624771118, -3.8620553016662598, -5.72519063949585, -1.555426836013794, -1.6673595905303955, -2.294215679168701, -0.036596234887838364, -0.015040211379528046, -11.780933380126953, -11.650826454162598, -3.3067545890808105, -5.23604154586792, -9.567856788635254, -6.94837760925293, -0.3373831510543823, -1.1055653095245361]}
+{"case_id": "constraint_02", "category": "constraint_stack", "origin": "budgie", "answer": "The task requires an import complete statement, which is not allowed for a simple task.", "reward": -0.010000000000000009, "advantage": -0.5061788432676426, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 19161, 253, 3559, 1761, 327, 253, 5371, 940, 752, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 4475, 1924, 43, 1120, 351, 637, 18754, 2994, 42, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 504, 3856, 3073, 354, 752, 2994, 4814, 28, 527, 314, 441, 3917, 327, 253, 2232, 3856, 30, 49154], "prefix_len": 131, "old_logprobs": [-1.0196832418441772, -2.690727710723877, -1.3602795600891113, -2.6904525756835938, -2.1863691806793213, -0.3480495512485504, -2.1157960891723633, -1.6024971008300781, -0.963029146194458, -1.4031496047973633, -1.98630952835083, -2.4040377140045166, -5.727386474609375, -1.2910021543502808, -3.3301658630371094, -3.280876398086548, -1.0060652494430542, -1.7147133350372314]}
+{"case_id": "constraint_02", "category": "constraint_stack", "origin": "budgie", "answer": "The format requires a status line that includes a conclusion, an answer, and a domain name if applicable.", "reward": -0.010000000000000009, "advantage": -0.5061788432676426, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 19161, 253, 3559, 1761, 327, 253, 5371, 940, 752, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 4475, 1924, 43, 1120, 351, 637, 18754, 2994, 42, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 504, 4624, 3073, 253, 3559, 1761, 338, 2978, 253, 4446, 28, 354, 2988, 28, 284, 253, 6902, 1462, 585, 9609, 30, 49154], "prefix_len": 131, "old_logprobs": [-1.0196832418441772, -5.018852710723877, -1.1724889278411865, -0.7215226292610168, -2.4084889888763428, -0.05682312697172165, -0.7834422588348389, -1.9664145708084106, -2.3769819736480713, -5.453387260437012, -1.3373843431472778, -2.728994131088257, -3.070509910583496, -0.6493176221847534, -0.19446679949760437, -0.7502038478851318, -8.72674560546875, -0.41697943210601807, -1.9953382015228271, -1.332265853881836, -0.11900447309017181, -0.551579475402832]}
+{"case_id": "constraint_02", "category": "constraint_stack", "origin": "budgie", "answer": "I shouldoc_print_text(text) to print the word count.", "reward": 0, "advantage": -0.48136615487216994, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 19161, 253, 3559, 1761, 327, 253, 5371, 940, 752, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 4475, 1924, 43, 1120, 351, 637, 18754, 2994, 42, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 868, 406, 79, 3272, 79, 2692, 24, 2692, 25, 288, 1628, 260, 2229, 985, 30, 49154], "prefix_len": 131, "old_logprobs": [-1.5196832418441772, -2.156658887863159, -10.379541397094727, -6.343707084655762, -2.987013101577759, -1.101137399673462, -1.737851858139038, -0.560512900352478, -1.3842074871063232, -0.6922652721405029, -2.825160503387451, -1.7284109592437744, -0.23835857212543488, -6.217965602874756, -1.0278104543685913, -0.9836747050285339, -0.08152958750724792]}
+{"case_id": "constraint_02", "category": "constraint_stack", "origin": "budgie", "answer": "The content must be exactly eight words, no more than nine words, and contain the following information:", "reward": -0.010000000000000009, "advantage": -0.5061788432676426, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 19161, 253, 3559, 1761, 327, 253, 5371, 940, 752, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 4475, 1924, 43, 1120, 351, 637, 18754, 2994, 42, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 504, 2627, 1251, 325, 3869, 4475, 1924, 28, 787, 540, 670, 7462, 1924, 28, 284, 2127, 260, 1695, 1096, 42, 49154], "prefix_len": 131, "old_logprobs": [-1.0196832418441772, -2.378227710723877, -2.277657985687256, -0.325332373380661, -0.9160503149032593, -0.06681150943040848, -0.01199449971318245, -1.5008879899978638, -2.4247822761535645, -1.236025094985962, -0.22541700303554535, -2.7804548740386963, -0.4315250515937805, -0.9370529055595398, -0.8777263164520264, -3.579904556274414, -2.1237454414367676, -1.2895104885101318, -2.283752679824829, -0.04113936424255371, -1.874595046043396]}
+{"case_id": "constraint_02", "category": "constraint_stack", "origin": "manual_ideal", "answer": "Import complete: all records passed validation without errors.", "reward": 1, "advantage": 1.9999026846750978, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 19161, 253, 3559, 1761, 327, 253, 5371, 940, 752, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 4475, 1924, 43, 1120, 351, 637, 18754, 2994, 42, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 18754, 2994, 42, 511, 4585, 4180, 13132, 1355, 6573, 30, 49154], "prefix_len": 131, "old_logprobs": [-7.644683361053467, -0.07974691689014435, -0.028593987226486206, -10.214856147766113, -10.243629455566406, -9.840221405029297, -13.813945770263672, -6.923023223876953, -3.035823106765747, -0.19852739572525024, -0.04104726389050484]}
+{"case_id": "constraint_03", "category": "constraint_stack", "origin": "budgie", "answer": "What is the name of the task that you are looking for?", "reward": 0, "advantage": -0.4937413264602444, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 26533, 253, 12633, 20110, 1569, 327, 253, 2654, 3856, 1568, 11655, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3993, 20110, 805, 43, 8889, 3869, 5038, 28, 2380, 28, 1568, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 1780, 314, 260, 1462, 282, 260, 3856, 338, 346, 359, 3012, 327, 47, 49154], "prefix_len": 134, "old_logprobs": [-1.381201982498169, -2.069222927093506, -0.18571597337722778, -3.7268433570861816, -0.13386505842208862, -0.07516802102327347, -0.7516379952430725, -4.018148422241211, -0.5654358863830566, -1.5885239839553833, -2.481372356414795, -0.14566583931446075, -0.04855995997786522, -0.06009255349636078]}
+{"case_id": "constraint_03", "category": "constraint_stack", "origin": "budgie", "answer": "To handle the requirements, I can use a ` AVC` object that contains the required properties.", "reward": 0, "advantage": -0.4937413264602444, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 26533, 253, 12633, 20110, 1569, 327, 253, 2654, 3856, 1568, 11655, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3993, 20110, 805, 43, 8889, 3869, 5038, 28, 2380, 28, 1568, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 2068, 5605, 260, 4292, 28, 339, 416, 722, 253, 2179, 21922, 51, 80, 1569, 338, 3593, 260, 2309, 3849, 30, 49154], "prefix_len": 134, "old_logprobs": [-3.507506847381592, -5.288956165313721, -1.0240898132324219, -4.012267112731934, -0.6331208944320679, -1.0498384237289429, -2.0682101249694824, -1.0372850894927979, -0.8655186891555786, -2.6071817874908447, -10.593762397766113, -4.088814735412598, -2.7042322158813477, -1.867988109588623, -1.853440523147583, -1.5563381910324097, -0.8492273688316345, -1.9506844282150269, -1.3142471313476562, -0.6763726472854614, -1.312433123588562]}
+{"case_id": "constraint_03", "category": "constraint_stack", "origin": "budgie", "answer": "Here is the desired content:\n```\nWhat are the constraints?\n\nWhat are the criteria?\n\nWhat are the keys?\n\nWhat are the values?\n\nWhat are the values?\n```", "reward": -0.010000000000000009, "advantage": -0.5186777570895497, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 26533, 253, 12633, 20110, 1569, 327, 253, 2654, 3856, 1568, 11655, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3993, 20110, 805, 43, 8889, 3869, 5038, 28, 2380, 28, 1568, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 4590, 314, 260, 6253, 2627, 42, 198, 3725, 198, 1780, 359, 260, 10195, 47, 198, 198, 1780, 359, 260, 7228, 47, 198, 198, 1780, 359, 260, 8889, 47, 198, 198, 1780, 359, 260, 2396, 47, 198, 198, 1780, 359, 260, 2396, 47, 198, 3725, 49154], "prefix_len": 134, "old_logprobs": [-2.9499425888061523, -1.285447120666504, -0.7383572459220886, -5.827037811279297, -3.928880214691162, -0.6023111343383789, -0.5094382762908936, -1.3311421871185303, -1.4323182106018066, -2.0938658714294434, -1.31199312210083, -0.23160451650619507, -0.6498602628707886, -1.1177043914794922, -0.2402665764093399, -2.589993953704834, -1.0777738094329834, -0.38491493463516235, -0.01780877448618412, -4.142121315002441, -0.5170422792434692, -0.09531724452972412, -0.3574489653110504, -0.9864910840988159, -0.8511396646499634, -0.017495587468147278, -2.7468533515930176, -0.25672709941864014, -0.03678193688392639, -0.0796477347612381, -0.6210684776306152, -0.37275251746177673, -0.00612560473382473, -1.4307208061218262, -0.010079790838062763, -0.02612951211631298, -0.19619710743427277, -1.089622974395752, -0.5845511555671692, -0.00803003553301096, -1.3140907287597656, -0.04020893573760986, -0.028626423329114914, -1.157066822052002, -0.4100048542022705]}
+{"case_id": "constraint_03", "category": "constraint_stack", "origin": "budgie", "answer": "I should follow the rules provided by Asilarkness.", "reward": 0, "advantage": -0.4937413264602444, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 26533, 253, 12633, 20110, 1569, 327, 253, 2654, 3856, 1568, 11655, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3993, 20110, 805, 43, 8889, 3869, 5038, 28, 2380, 28, 1568, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 868, 1066, 260, 3372, 2711, 411, 1032, 301, 792, 924, 30, 49154], "prefix_len": 134, "old_logprobs": [-2.381201982498169, -1.5555317401885986, -1.0172147750854492, -0.5895247459411621, -3.7197341918945312, -3.062234401702881, -1.2123996019363403, -0.4361439645290375, -0.0003815161471720785, -0.001505790394730866, -0.00016878610767889768, -0.497312068939209, -0.32370248436927795]}
+{"case_id": "constraint_03", "category": "constraint_stack", "origin": "manual_ideal", "answer": "{\"title\":\"task\",\"color\":\"green\",\"due\":\"Friday\"}", "reward": 1, "advantage": 1.9999017364702827, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 26533, 253, 12633, 20110, 1569, 327, 253, 2654, 3856, 1568, 11655, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3993, 20110, 805, 43, 8889, 3869, 5038, 28, 2380, 28, 1568, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 39428, 7260, 25535, 10738, 14460, 6713, 25535, 9298, 14460, 16395, 25535, 33709, 23597, 49154], "prefix_len": 134, "old_logprobs": [-11.84995174407959, -9.257038116455078, -13.800414085388184, -12.183327674865723, -11.157981872558594, -4.067418098449707, -11.028606414794922, -10.88536262512207, -5.011720180511475, -3.6803531646728516, -11.506437301635742, -9.164865493774414, -2.9428021907806396, -1.9935541152954102]}
+{"case_id": "constraint_04", "category": "constraint_stack", "origin": "budgie", "answer": "I should be able to answer your question, but please let me know what you would like me to know.", "reward": 0, "advantage": -0.49999875000312505, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 37751, 462, 253, 11241, 23362, 31348, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 582, 6330, 43, 1453, 637, 42094, 346, 36651, 536, 441, 1928, 354, 23031, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 868, 325, 1730, 288, 2988, 469, 1962, 28, 564, 5361, 1303, 549, 699, 732, 346, 736, 702, 549, 288, 699, 30, 49154], "prefix_len": 131, "old_logprobs": [-0.5158237218856812, -1.4167163372039795, -3.7000672817230225, -1.237973690032959, -0.0031190102454274893, -2.3810174465179443, -1.5227816104888916, -1.4204814434051514, -2.3081741333007812, -0.6979750394821167, -0.9701674580574036, -3.121659755706787, -0.0026430694852024317, -0.009454948827624321, -1.3447967767715454, -1.1286895275115967, -1.0195302963256836, -0.009904053062200546, -0.1897697001695633, -0.0006559127941727638, -1.3957220315933228, -0.7130571007728577, -0.01397373341023922]}
+{"case_id": "constraint_04", "category": "constraint_stack", "origin": "budgie", "answer": "I would like to make a lunch heartily.", "reward": 0, "advantage": -0.49999875000312505, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 37751, 462, 253, 11241, 23362, 31348, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 582, 6330, 43, 1453, 637, 42094, 346, 36651, 536, 441, 1928, 354, 23031, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 736, 702, 288, 919, 253, 11241, 1911, 878, 30, 49154], "prefix_len": 131, "old_logprobs": [-0.5130606293678284, -3.494447708129883, -1.1101619005203247, -0.42217472195625305, -4.194232940673828, -0.6185587644577026, -1.0246559381484985, -7.5288615226745605, -4.358006954193115, -0.7067521810531616, -0.14407265186309814]}
+{"case_id": "constraint_04", "category": "constraint_stack", "origin": "budgie", "answer": "I am Wren, Megan, the speaks, the wast steam. I will give one sentence, please follow the instructions, no other words.", "reward": 0, "advantage": -0.49999875000312505, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 37751, 462, 253, 11241, 23362, 31348, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 582, 6330, 43, 1453, 637, 42094, 346, 36651, 536, 441, 1928, 354, 23031, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 744, 47970, 28, 2867, 870, 28, 260, 13531, 28, 260, 436, 100, 9403, 30, 339, 523, 1928, 582, 6330, 28, 5361, 1066, 260, 6388, 28, 787, 550, 1924, 30, 49154], "prefix_len": 131, "old_logprobs": [-0.5625404119491577, -2.807755470275879, -10.669795989990234, -1.5742664337158203, -9.120248794555664, -0.8628376126289368, -0.9319315552711487, -3.55998158454895, -9.814140319824219, -2.5430901050567627, -2.8186562061309814, -8.54883098602295, -1.519439697265625, -10.470964431762695, -2.2092409133911133, -1.4887564182281494, -3.21681809425354, -3.13114595413208, -3.0051164627075195, -0.31345292925834656, -1.6603562831878662, -3.0497214794158936, -2.423258066177368, -1.2838257551193237, -1.9333118200302124, -2.4773833751678467, -2.8204357624053955, -6.17549991607666, -1.8197412490844727, -1.5618231296539307, -0.12317211180925369]}
+{"case_id": "constraint_04", "category": "constraint_stack", "origin": "budgie", "answer": "I ask you to follow the How to format, including necessary inputs and outputs.", "reward": 0, "advantage": -0.49999875000312505, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 37751, 462, 253, 11241, 23362, 31348, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 582, 6330, 43, 1453, 637, 42094, 346, 36651, 536, 441, 1928, 354, 23031, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 1998, 346, 288, 1066, 260, 1073, 288, 4624, 28, 1285, 2371, 9577, 284, 13594, 30, 49154], "prefix_len": 131, "old_logprobs": [-0.5130606293678284, -3.369447708129883, -0.45825475454330444, -0.31590139865875244, -2.160874605178833, -1.1968557834625244, -8.827113151550293, -0.8222372531890869, -6.334277629852295, -7.030549049377441, -3.379024028778076, -5.7179412841796875, -5.536815643310547, -1.203373908996582, -2.076366662979126, -0.47704076766967773, -0.08742164820432663]}
+{"case_id": "constraint_04", "category": "constraint_stack", "origin": "manual_ideal", "answer": "Thank you for inviting me, but I’ll have to decline.", "reward": 1, "advantage": 1.9999950000125002, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 37751, 462, 253, 11241, 23362, 31348, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 582, 6330, 43, 1453, 637, 42094, 346, 36651, 536, 441, 1928, 354, 23031, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 19503, 346, 327, 17746, 549, 28, 564, 339, 417, 764, 457, 288, 6683, 30, 49154], "prefix_len": 131, "old_logprobs": [-11.013060569763184, -0.05143623799085617, -1.3073797225952148, -7.197422981262207, -0.1759931594133377, -3.3067996501922607, -1.749855399131775, -0.599578320980072, -6.006180286407471, -3.955846071243286, -4.610511779785156, -0.2008388191461563, -8.413507461547852, -1.6498507261276245, -0.09166932106018066]}
+{"case_id": "constraint_05", "category": "constraint_stack", "origin": "budgie", "answer": "The task is to find and print the final content, including the deadline, that will be sent by the end of the conversation.", "reward": 0, "advantage": -0.49999875000312505, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 13764, 3298, 836, 260, 2353, 42, 15869, 523, 5091, 260, 10049, 411, 27616, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 1120, 351, 637, 35175, 259, 32266, 43, 582, 1761, 43, 1453, 260, 19652, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 504, 3856, 314, 288, 1042, 284, 1628, 260, 3403, 2627, 28, 1285, 260, 19652, 28, 338, 523, 325, 2362, 411, 260, 1112, 282, 260, 6634, 30, 49154], "prefix_len": 138, "old_logprobs": [-1.5098180770874023, -2.483572483062744, -0.7489947080612183, -0.2467542588710785, -2.7633984088897705, -2.0339128971099854, -1.9314895868301392, -0.4556346833705902, -4.788477897644043, -1.926527976989746, -2.6510024070739746, -0.7875831127166748, -0.565569281578064, -0.8136343359947205, -0.1986127346754074, -3.2060818672180176, -2.7716031074523926, -0.14203478395938873, -0.2655443549156189, -1.0023221969604492, -3.354571580886841, -1.7624999284744263, -0.015121704898774624, -0.20829366147518158, -2.401813507080078, -0.07646933197975159, -0.20912380516529083]}
+{"case_id": "constraint_05", "category": "constraint_stack", "origin": "budgie", "answer": "I should not include the end.", "reward": 0, "advantage": -0.49999875000312505, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 13764, 3298, 836, 260, 2353, 42, 15869, 523, 5091, 260, 10049, 411, 27616, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 1120, 351, 637, 35175, 259, 32266, 43, 582, 1761, 43, 1453, 260, 19652, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 868, 441, 1453, 260, 1112, 30, 49154], "prefix_len": 138, "old_logprobs": [-1.842599630355835, -1.3603878021240234, -1.619659662246704, -1.1073658466339111, -0.9006326794624329, -4.655689716339111, -4.9518351554870605, -0.20892930030822754]}
+{"case_id": "constraint_05", "category": "constraint_stack", "origin": "budgie", "answer": "The request goes to the `start_time` parameter.", "reward": 0, "advantage": -0.49999875000312505, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 13764, 3298, 836, 260, 2353, 42, 15869, 523, 5091, 260, 10049, 411, 27616, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 1120, 351, 637, 35175, 259, 32266, 43, 582, 1761, 43, 1453, 260, 19652, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 504, 3116, 3935, 288, 260, 2179, 3738, 79, 2256, 80, 8709, 30, 49154], "prefix_len": 138, "old_logprobs": [-1.467599630355835, -4.421277046203613, -2.585787773132324, -1.721732258796692, -1.262335181236267, -3.1761410236358643, -0.502252459526062, -1.160133957862854, -0.7914257049560547, -0.401998370885849, -1.3015695810317993, -1.318376064300537, -0.2276543527841568]}
+{"case_id": "constraint_05", "category": "constraint_stack", "origin": "budgie", "answer": "I'll be happy to help.", "reward": 0, "advantage": -0.49999875000312505, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 13764, 3298, 836, 260, 2353, 42, 15869, 523, 5091, 260, 10049, 411, 27616, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 1120, 351, 637, 35175, 259, 32266, 43, 582, 1761, 43, 1453, 260, 19652, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 3060, 325, 5587, 288, 724, 30, 49154], "prefix_len": 138, "old_logprobs": [-1.842599630355835, -3.5478878021240234, -2.8616397380828857, -0.4847692549228668, -0.0017533419886603951, -0.5475924015045166, -1.9651923179626465, -0.29389896988868713]}
+{"case_id": "constraint_05", "category": "constraint_stack", "origin": "manual_ideal", "answer": "Owner: Eli — send the draft by noon.", "reward": 1, "advantage": 1.9999950000125002, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 13764, 3298, 836, 260, 2353, 42, 15869, 523, 5091, 260, 10049, 411, 27616, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 1120, 351, 637, 35175, 259, 32266, 43, 582, 1761, 43, 1453, 260, 19652, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 35175, 259, 42, 15869, 1841, 5091, 260, 10049, 411, 27616, 30, 49154], "prefix_len": 138, "old_logprobs": [-12.592599868774414, -0.002428084844723344, -0.5492830276489258, -10.3245267868042, -11.35898494720459, -5.61904239654541, -0.5187625885009766, -7.105475902557373, -0.38979408144950867, -0.12241372466087341, -0.08907141536474228, -0.02847117930650711]}
+{"case_id": "constraint_06", "category": "constraint_stack", "origin": "budgie", "answer": "I should not provide the content unless it is specified.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 37964, 253, 9071, 26344, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 2976, 1924, 43, 1453, 637, 2501, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 868, 441, 1538, 260, 2627, 6365, 357, 314, 7663, 30, 49154], "prefix_len": 123, "old_logprobs": [-0.5935055017471313, -1.462934136390686, -2.3566231727600098, -3.4412736892700195, -1.2307287454605103, -1.407025694847107, -5.8180389404296875, -0.48974084854125977, -1.4558677673339844, -4.7359938621521, -1.520221471786499, -0.11191751062870026]}
+{"case_id": "constraint_06", "category": "constraint_stack", "origin": "budgie", "answer": "I should provide the content:\n\n\"I am an AI, so I cannot demonstrate or provide visual content.\"\n\nI must change the content to only the requested text.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 37964, 253, 9071, 26344, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 2976, 1924, 43, 1453, 637, 2501, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 868, 1538, 260, 2627, 42, 198, 198, 18, 57, 744, 354, 5646, 28, 588, 339, 2526, 6286, 355, 1538, 3145, 2627, 1270, 198, 198, 57, 1251, 1363, 260, 2627, 288, 805, 260, 13682, 1694, 30, 49154], "prefix_len": 123, "old_logprobs": [-0.5939081311225891, -1.5164073705673218, -2.1676077842712402, -0.7017182111740112, -2.0136470794677734, -3.7425405979156494, -1.509142279624939, -0.3844579756259918, -1.08560049533844, -0.8209452629089355, -1.2411600351333618, -4.560488700866699, -0.01772633008658886, -3.8277664184570312, -1.30368173122406, -0.2903395891189575, -1.3243464231491089, -8.481416702270508, -2.4328885078430176, -2.8766958713531494, -4.662540435791016, -1.0378141403198242, -1.0959529876708984, -1.204779863357544, -0.03538314998149872, -0.9605637788772583, -2.9392757415771484, -7.493187427520752, -0.4415707290172577, -3.0126943588256836, -1.0041160583496094, -2.0558605194091797, -1.0108522176742554, -0.29401296377182007, -2.8375563621520996, -0.768309473991394, -0.12752290070056915]}
+{"case_id": "constraint_06", "category": "constraint_stack", "origin": "budgie", "answer": "I should now include the word 'light' in the answer.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 37964, 253, 9071, 26344, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 2976, 1924, 43, 1453, 637, 2501, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 868, 1209, 1453, 260, 2229, 637, 2501, 23, 281, 260, 2988, 30, 49154], "prefix_len": 123, "old_logprobs": [-0.5935055017471313, -1.462934136390686, -6.73162317276001, -2.5157461166381836, -0.9250661730766296, -1.098178744316101, -0.0763116180896759, -0.06824732571840286, -0.7211688160896301, -0.27500635385513306, -0.5196376442909241, -2.8697714805603027, -0.23480349779129028, -0.13530601561069489]}
+{"case_id": "constraint_06", "category": "constraint_stack", "origin": "budgie", "answer": "I will provide the requested content, including the word 'light'.", "reward": -0.010000000000000009, "advantage": -0.4999987623793011, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 37964, 253, 9071, 26344, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 2976, 1924, 43, 1453, 637, 2501, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 523, 1538, 260, 13682, 2627, 28, 1285, 260, 2229, 637, 2501, 4289, 49154], "prefix_len": 123, "old_logprobs": [-0.5935055017471313, -2.9629340171813965, -0.953747034072876, -0.3714427351951599, -0.6302080154418945, -0.10277102887630463, -1.50216805934906, -1.9707825183868408, -1.132704496383667, -1.8108996152877808, -0.18670256435871124, -0.06129993125796318, -0.9208881855010986, -0.1821662187576294]}
+{"case_id": "constraint_06", "category": "constraint_stack", "origin": "manual_ideal", "answer": "Soft light, clear desk, quiet focus.", "reward": 1, "advantage": 1.9999950495172043, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 37964, 253, 9071, 26344, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 2976, 1924, 43, 1453, 637, 2501, 4289, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 22777, 1420, 28, 2437, 10917, 28, 8354, 1593, 30, 49154], "prefix_len": 123, "old_logprobs": [-12.531005859375, -6.209319591522217, -5.179167747497559, -6.130183696746826, -10.785547256469727, -0.5344614386558533, -7.011074542999268, -8.061287879943848, -1.3681730031967163, -0.024098459631204605]}
+{"case_id": "constraint_07", "category": "constraint_stack", "origin": "budgie", "answer": "No.", "reward": 0, "advantage": -0.49999875000312505, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 21350, 1991, 41936, 2369, 940, 29, 6915, 1621, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 1956, 351, 9230, 355, 2838, 43, 965, 3869, 582, 32384, 6330, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 5230, 30, 49154], "prefix_len": 131, "old_logprobs": [-1.2646522521972656, -1.9036030769348145, -0.6973457336425781]}
+{"case_id": "constraint_07", "category": "constraint_stack", "origin": "budgie", "answer": "No", "reward": 0, "advantage": -0.49999875000312505, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 21350, 1991, 41936, 2369, 940, 29, 6915, 1621, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 1956, 351, 9230, 355, 2838, 43, 965, 3869, 582, 32384, 6330, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 5230, 49154], "prefix_len": 131, "old_logprobs": [-1.2646522521972656, -0.46610313653945923]}
+{"case_id": "constraint_07", "category": "constraint_stack", "origin": "budgie", "answer": "No", "reward": 0, "advantage": -0.49999875000312505, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 21350, 1991, 41936, 2369, 940, 29, 6915, 1621, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 1956, 351, 9230, 355, 2838, 43, 965, 3869, 582, 32384, 6330, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 5230, 49154], "prefix_len": 131, "old_logprobs": [-1.2646522521972656, -0.46610313653945923]}
+{"case_id": "constraint_07", "category": "constraint_stack", "origin": "budgie", "answer": "No", "reward": 0, "advantage": -0.49999875000312505, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 21350, 1991, 41936, 2369, 940, 29, 6915, 1621, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 1956, 351, 9230, 355, 2838, 43, 965, 3869, 582, 32384, 6330, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 5230, 49154], "prefix_len": 131, "old_logprobs": [-1.2646522521972656, -0.46610313653945923]}
+{"case_id": "constraint_07", "category": "constraint_stack", "origin": "manual_ideal", "answer": "Yes. Backups provide a separate copy when original data is lost or damaged.", "reward": 1, "advantage": 1.9999950000125002, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 21350, 1991, 41936, 2369, 940, 29, 6915, 1621, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 1956, 351, 9230, 355, 2838, 43, 965, 3869, 582, 32384, 6330, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 10539, 30, 8279, 5835, 1538, 253, 4447, 3784, 645, 2798, 940, 314, 3590, 355, 7824, 30, 49154], "prefix_len": 131, "old_logprobs": [-2.3896522521972656, -4.498425006866455, -9.998640060424805, -5.2630743980407715, -7.806127548217773, -0.6963371634483337, -8.34837818145752, -6.017496585845947, -6.513453960418701, -9.680261611938477, -0.41340920329093933, -0.22787581384181976, -1.20621657371521, -2.5289173126220703, -3.5755138397216797, -0.5174980759620667, -0.28783005475997925]}
+{"case_id": "constraint_08", "category": "constraint_stack", "origin": "budgie", "answer": "The two-itemcolour must be a specific item or concept, not a general statement (or statement of fact).", "reward": -0.010000000000000009, "advantage": -0.36975977406991717, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 26533, 827, 3710, 288, 4368, 16494, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 827, 22266, 3723, 43, 787, 5887, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 504, 827, 29, 6692, 31991, 1251, 325, 253, 1678, 5033, 355, 1909, 28, 441, 253, 2210, 4814, 365, 270, 4814, 282, 1027, 595, 49154], "prefix_len": 125, "old_logprobs": [-1.7571604251861572, -2.844442367553711, -1.5966485738754272, -0.6842413544654846, -12.14094352722168, -1.6162850856781006, -0.573518693447113, -2.8522145748138428, -3.4976911544799805, -0.7296342253684998, -1.771132469177246, -0.7193829417228699, -1.2132930755615234, -0.7902111411094666, -1.2896676063537598, -1.7021211385726929, -1.821518898010254, -5.866916179656982, -0.8753083944320679, -2.5410468578338623, -1.5052292346954346, -0.4095531702041626, -0.4925262928009033, -0.4014826714992523]}
+{"case_id": "constraint_08", "category": "constraint_stack", "origin": "budgie", "answer": "I should not label a wire as an introduction. I should not call a wire an introduction. I should not call a wire an introduction. I should not call a wire an introduction. I should not call a wire an introduction. I should never call a wire an introduction.", "reward": -0.22, "advantage": -0.8550694775366835, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 26533, 827, 3710, 288, 4368, 16494, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 827, 22266, 3723, 43, 787, 5887, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 868, 441, 4368, 253, 6192, 347, 354, 5887, 30, 339, 868, 441, 946, 253, 6192, 354, 5887, 30, 339, 868, 441, 946, 253, 6192, 354, 5887, 30, 339, 868, 441, 946, 253, 6192, 354, 5887, 30, 339, 868, 441, 946, 253, 6192, 354, 5887, 30, 339, 868, 2093, 946, 253, 6192, 354, 5887, 30, 49154], "prefix_len": 125, "old_logprobs": [-0.9068796038627625, -0.9096367359161377, -1.1882658004760742, -2.0325210094451904, -2.6287755966186523, -4.1070451736450195, -1.3174598217010498, -1.4970371723175049, -0.9611940383911133, -0.7103084921836853, -1.709547996520996, -0.2218523919582367, -1.7061636447906494, -3.2263214588165283, -0.6510279774665833, -0.2766099274158478, -1.2208178043365479, -0.18127816915512085, -0.3877556622028351, -1.165996789932251, -0.05237698554992676, -0.13746590912342072, -0.8082236051559448, -0.25183942914009094, -0.054385557770729065, -0.47654008865356445, -0.11943502724170685, -0.34521323442459106, -1.1176178455352783, -0.05199296399950981, -0.06910164654254913, -0.3177627623081207, -0.13765664398670197, -0.019889557734131813, -0.2851187288761139, -0.08965422958135605, -0.17146901786327362, -0.9797741174697876, -0.04222024604678154, -0.03761855140328407, -0.10790306329727173, -0.11015108227729797, -0.013249213807284832, -0.3480300307273865, -0.08761103451251984, -0.08992110192775726, -1.0537638664245605, -0.030637210234999657, -4.959036827087402, -0.08907588571310043, -0.1243346706032753, -0.013795973733067513, -0.14222869277000427, -0.08329450339078903, -0.0887238085269928, -0.27707746624946594]}
+{"case_id": "constraint_08", "category": "constraint_stack", "origin": "budgie", "answer": "I should follow the same format, which includes:\n- **Two-item answer**\n- **No introduction**\n- **One-item answer**\n- **One-item answer**\n\nI want to follow that format for both the two-item answer and the one-item answer.", "reward": -0.010000000000000009, "advantage": -0.36975977406991717, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 26533, 827, 3710, 288, 4368, 16494, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 827, 22266, 3723, 43, 787, 5887, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 868, 1066, 260, 1142, 4624, 28, 527, 2978, 42, 198, 29, 1903, 10345, 29, 6692, 2988, 828, 198, 29, 1903, 5230, 5887, 828, 198, 29, 1903, 2705, 29, 6692, 2988, 828, 198, 29, 1903, 2705, 29, 6692, 2988, 828, 198, 198, 57, 1277, 288, 1066, 338, 4624, 327, 1062, 260, 827, 29, 6692, 2988, 284, 260, 582, 29, 6692, 2988, 30, 49154], "prefix_len": 125, "old_logprobs": [-0.9137440323829651, -0.9143598079681396, -2.4447197914123535, -0.4422543942928314, -4.67583703994751, -0.9075761437416077, -2.487938642501831, -3.6599717140197754, -2.9185502529144287, -4.1871490478515625, -0.7432121634483337, -1.3641271591186523, -5.287498950958252, -1.2838679552078247, -1.1737158298492432, -1.7453317642211914, -5.132019996643066, -0.3359973132610321, -0.20899246633052826, -0.019210126250982285, -0.09623385965824127, -0.4880596697330475, -0.36221829056739807, -0.04135648161172867, -0.03270291909575462, -0.505660355091095, -0.05767082795500755, -3.1730399131774902, -0.27496904134750366, -1.053506851196289, -0.1154026910662651, -0.04820127785205841, -0.07481389492750168, -0.5674976706504822, -0.021242251619696617, -1.9014554023742676, -0.244668111205101, -0.31287673115730286, -0.32544204592704773, -0.11812905222177505, -0.09097710251808167, -0.8134657144546509, -0.6917810440063477, -4.758232116699219, -0.1416454166173935, -3.468100070953369, -4.78659200668335, -0.1121661514043808, -3.5304324626922607, -2.058788776397705, -1.9196234941482544, -0.6279798150062561, -0.1291077584028244, -0.08728368580341339, -0.5670822262763977, -0.032675229012966156, -0.2004714012145996, -0.7799274325370789, -0.00103586888872087, -0.09876897931098938, -0.0029057920910418034, -0.3918290436267853, -0.4781208038330078]}
+{"case_id": "constraint_08", "category": "constraint_stack", "origin": "budgie", "answer": "I cannot answer the question in this format. Please provide the question you'd like me to label ashistory or consider another question.", "reward": -0.010000000000000009, "advantage": -0.36975977406991717, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 26533, 827, 3710, 288, 4368, 16494, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 827, 22266, 3723, 43, 787, 5887, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 57, 2526, 2988, 260, 1962, 281, 451, 4624, 30, 8765, 1538, 260, 1962, 346, 6737, 702, 549, 288, 4368, 347, 13011, 355, 1771, 1372, 1962, 30, 49154], "prefix_len": 125, "old_logprobs": [-0.8821604251861572, -1.824293613433838, -3.631725311279297, -1.3800673484802246, -0.6389276385307312, -3.9600980281829834, -0.8722395896911621, -0.99858558177948, -0.7571521997451782, -1.9887111186981201, -0.06839429587125778, -0.8025065064430237, -1.0034371614456177, -2.0361151695251465, -0.20236042141914368, -0.0008206575294025242, -0.03757641464471817, -0.0010311054065823555, -0.4733584523200989, -1.7951226234436035, -11.155155181884766, -0.9769585132598877, -5.849521636962891, -2.514143228530884, -0.3750080466270447, -1.1520507335662842, -0.01565183326601982]}
+{"case_id": "constraint_08", "category": "constraint_stack", "origin": "manual_ideal", "answer": "1. Faster troubleshooting.\n2. Safer equipment changes.", "reward": 1, "advantage": 1.964348799746435, "ids": [49153, 49156, 198, 2683, 359, 15830, 37144, 28, 253, 12633, 5646, 11173, 2312, 411, 1032, 301, 792, 924, 30, 26960, 260, 6634, 506, 1574, 5307, 284, 10195, 5054, 30, 3000, 2344, 260, 22947, 7719, 15226, 690, 328, 1121, 1096, 30, 19842, 260, 6008, 3116, 3319, 28, 536, 441, 9960, 260, 2444, 6634, 28, 284, 536, 441, 4938, 6782, 5307, 30, 9430, 582, 4782, 28879, 805, 645, 253, 27799, 355, 6782, 2202, 6703, 253, 3465, 2988, 30, 9124, 25690, 2565, 616, 284, 18388, 30, 49154, 49157, 198, 26533, 827, 3710, 288, 4368, 16494, 30, 49154, 49158, 198, 1780, 10195, 868, 339, 1066, 47, 49154, 49157, 198, 15008, 8830, 42, 3869, 827, 22266, 3723, 43, 787, 5887, 30, 13632, 805, 260, 13682, 2627, 30, 49154, 49158, 198, 49160, 30, 426, 1738, 38285, 30, 198, 49161, 30, 6015, 259, 3813, 1971, 30, 49154], "prefix_len": 125, "old_logprobs": [-6.069660186767578, -0.05452385172247887, -9.089295387268066, -5.786497116088867, -12.755245208740234, -2.3195431232452393, -0.4778585433959961, -0.014116199687123299, -0.0005706112715415657, -12.193307876586914, -8.378782272338867, -10.49230670928955, -8.05624008178711, -0.19836142659187317, -2.0201830863952637]}
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/encoding_manifest.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/encoding_manifest.json
new file mode 100644
index 0000000000000000000000000000000000000000..f47392e88c7ba55d5679c302367d0ec33c2570ea
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/encoding_manifest.json
@@ -0,0 +1,7 @@
+{
+  "encoded": 315,
+  "mean_tokens": 170.52380952380952,
+  "mean_response_tokens": 18.444444444444443,
+  "max_tokens": 341,
+  "seconds": 4.603764295578003
+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/final_decision.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/final_decision.json
new file mode 100644
index 0000000000000000000000000000000000000000..0333efd8cb5fe6387b57fd739b866ca9706c7b14
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/final_decision.json
@@ -0,0 +1,101 @@
+{
+  "date": "2026-08-26",
+  "stage": "information-management RL v1",
+  "status": "rejected; leader unchanged",
+  "leader": "verified-math-a025",
+  "curriculum": {
+    "rows": 105,
+    "splits": {
+      "train": 63,
+      "dev": 21,
+      "test": 21
+    },
+    "categories": {
+      "state_update": 15,
+      "already_tried": 15,
+      "selective_answer": 15,
+      "delta_summary": 15,
+      "conflict_clarification": 15,
+      "source_boundary": 15,
+      "constraint_stack": 15
+    },
+    "ideal_reference_pass": 105,
+    "benchmark_family_rows": 0,
+    "parameter_count_change": 0
+  },
+  "reward_data": {
+    "cases": 63,
+    "budgie_rollouts": 252,
+    "manual_demonstrations": 63,
+    "mean_reward_budgie": -0.21686507936507934,
+    "mean_reward_ideal": 0.9992063492063492,
+    "positive_budgie": 4,
+    "budgie_deterministic_pass": 21,
+    "budgie_loops": 4,
+    "parse_errors": [],
+    "seconds": 935.0733463764191
+  },
+  "methods": [
+    {
+      "name": "conservative GRPO",
+      "result": "No deterministic dev improvement through alpha 1.0."
+    },
+    {
+      "name": "long GRPO plus manual BC",
+      "result": "Full source dev 4/21 vs 3/21; no core state-update transfer."
+    },
+    {
+      "name": "hybrid GRPO plus 8-epoch ideal consolidation",
+      "result": "Full source dev 7/21 and test 5/21 but loops increased; alpha .025 dev 4/21 and test 2/21 with loops unchanged."
+    }
+  ],
+  "heldout": {
+    "leader": {
+      "dev_pass": 3,
+      "dev_total": 21,
+      "dev_loops": 0,
+      "test_pass": 1,
+      "test_total": 21,
+      "test_loops": 1
+    },
+    "hybrid_a0p025": {
+      "dev_pass": 4,
+      "dev_total": 21,
+      "dev_loops": 0,
+      "test_pass": 2,
+      "test_total": 21,
+      "test_loops": 1
+    },
+    "full_hybrid_source": {
+      "dev_pass": 7,
+      "dev_total": 21,
+      "dev_loops": 2,
+      "test_pass": 5,
+      "test_total": 21,
+      "test_loops": 2
+    }
+  },
+  "fixed_reasoning_gate": {
+    "leader": {
+      "gsm8k": "5/30",
+      "math500": "3/15",
+      "arc": "11/30",
+      "folio": "13/30"
+    },
+    "hybrid_a0p025": {
+      "gsm8k": "3/30",
+      "math500": "1/15",
+      "arc": "11/30",
+      "folio": "13/30"
+    }
+  },
+  "decision": "Reject hybrid-a0p025 because GSM8K and MATH regress. Reject full hybrid source because repetitions increase and it is more distant. Smaller alphas do not improve the held-out information gate. Keep verified-math-a025 active and public root unchanged.",
+  "parameter_count_change": 0,
+  "public_root_changed": false,
+  "recovery_commits": [
+    "349550db2916bc27969dd56d7f8c3fa07cdc68dc",
+    "5112b94abff81b17297671b8e48751b2336d0839",
+    "02342edca539f218c9ab6e462616562f04fb7a5c",
+    "d53a83fc50ebc4481410e566f195610db438430d"
+  ]
+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/hybrid_dev_gate.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/hybrid_dev_gate.json
new file mode 100644
index 0000000000000000000000000000000000000000..9be54ffa887fd98c2155a97755777a5d9f56f43d
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/hybrid_dev_gate.json
@@ -0,0 +1,206 @@
+{
+  "leader": {
+    "alpha": 0,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3333333333333334,
+    "loops": 0,
+    "mean_words": 8.857142857142858,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p005": {
+    "alpha": 0.005,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3333333333333334,
+    "loops": 0,
+    "mean_words": 9.380952380952381,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p01": {
+    "alpha": 0.01,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3333333333333334,
+    "loops": 0,
+    "mean_words": 9.142857142857142,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p025": {
+    "alpha": 0.025,
+    "pass": 4,
+    "n": 21,
+    "mean_reward": 0.36666666666666675,
+    "loops": 0,
+    "mean_words": 8.571428571428571,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 3,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p05": {
+    "alpha": 0.05,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3500000000000001,
+    "loops": 0,
+    "mean_words": 9.047619047619047,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p075": {
+    "alpha": 0.075,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3500000000000001,
+    "loops": 0,
+    "mean_words": 8.666666666666666,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p1": {
+    "alpha": 0.1,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3333333333333334,
+    "loops": 0,
+    "mean_words": 9.047619047619047,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p15": {
+    "alpha": 0.15,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3333333333333334,
+    "loops": 0,
+    "mean_words": 8.952380952380953,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p2": {
+    "alpha": 0.2,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3333333333333334,
+    "loops": 0,
+    "mean_words": 9.095238095238095,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
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diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/hybrid_test_gate.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/hybrid_test_gate.json
new file mode 100644
index 0000000000000000000000000000000000000000..37a1cb697e13fba6bce9e97eee31f554ba8dfe02
--- /dev/null
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diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/hybrid_training_report.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/hybrid_training_report.json
new file mode 100644
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+  "source_path": "/marimo/budgie_information_rl_v1/info_rl_hybrid_source.pt",
+  "seconds": 33.85362410545349,
+  "parameter_count_change": 0
+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/info_math_constrained_source.pt b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/info_math_constrained_source.pt
new file mode 100644
index 0000000000000000000000000000000000000000..a6571aebf6ca86017a67b18918ef2e7ed964a003
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/info_math_constrained_source.pt
@@ -0,0 +1,3 @@
+version https://git-lfs.github.com/spec/v1
+oid sha256:00a4221a276258561c0150faad4f5aa8861eae3535ae137047b7d73aa2b0c443
+size 1101695710
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/info_rl_v1.py b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/info_rl_v1.py
new file mode 100644
index 0000000000000000000000000000000000000000..192d7633e08aef06ea6c48242c28b2e0a4be3033
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/info_rl_v1.py
@@ -0,0 +1,268 @@
+"""Manual long-dialogue information-management curriculum for Budgie-500M.
+
+The cases are synthetic and unrelated to held-out reasoning benchmarks. Each case
+contains explicit state/constraint checks suitable for deterministic reward plus
+an independent semantic judge. Training/dev/test topics are disjoint.
+"""
+from __future__ import annotations
+
+import json
+import re
+from collections import Counter
+from typing import Any
+
+SYSTEM = (
+    "You are Budgie, a compact AI assistant created by Asilarkness. Track the "
+    "conversation's current facts and constraints carefully. Prefer the newest "
+    "explicit correction over stale information. Answer the latest request "
+    "directly, do not repeat the whole conversation, and do not invent missing "
+    "facts. Ask one focused clarification only when a contradiction or missing "
+    "detail blocks a useful answer. Avoid canned closings and repetition."
+)
+
+# Fifteen semantically distinct variants per skill. Indices 0-8 train, 9-11 dev,
+# and 12-14 test, so no topic instance appears in more than one split.
+STATE_UPDATES = [
+    dict(topic="community workshop", old="Saturday at 10:00", new="Sunday at 14:00", fixed="Room Cedar", ask="Give the final time and room in one sentence.", ideal="The community workshop is Sunday at 14:00 in Room Cedar."),
+    dict(topic="software release", old="version 3.4 on May 8", new="version 3.5 on May 10", fixed="staged rollout", ask="State the current version, date, and rollout method in one sentence.", ideal="Version 3.5 will launch on May 10 through a staged rollout."),
+    dict(topic="supplier order", old="240 blue folders", new="180 green folders", fixed="deliver to Dock 2", ask="Write the final order instruction in one sentence.", ideal="Order 180 green folders and deliver them to Dock 2."),
+    dict(topic="interview schedule", old="Mira at 09:30", new="Jon at 11:15", fixed="video call", ask="Give only the current candidate, time, and format.", ideal="Jon at 11:15 by video call."),
+    dict(topic="museum visit", old="12 adult tickets", new="9 adult tickets and 3 child tickets", fixed="entry at 16:20", ask="Summarize the final booking in one sentence.", ideal="Book 9 adult tickets and 3 child tickets for entry at 16:20."),
+    dict(topic="newsletter campaign", old="send Tuesday to 8,000 readers", new="send Thursday to 6,500 readers", fixed="subject line: Spring Notes", ask="Give the current send plan in one sentence.", ideal="Send “Spring Notes” on Thursday to 6,500 readers."),
+    dict(topic="lab sample", old="label K-17 at 4°C", new="label K-19 at -20°C", fixed="amber container", ask="State the final storage record in one sentence.", ideal="Store sample K-19 at -20°C in an amber container."),
+    dict(topic="training session", old="90 minutes for 25 people", new="60 minutes for 18 people", fixed="beginner level", ask="Give the final duration, audience size, and level.", ideal="Run a 60-minute beginner session for 18 people."),
+    dict(topic="catering request", old="vegetarian lunch for 40", new="vegan lunch for 32", fixed="no peanuts", ask="State the current catering requirement in one sentence.", ideal="Provide a peanut-free vegan lunch for 32 people."),
+    dict(topic="archive transfer", old="Box A7 to Shelf 4", new="Box C2 to Shelf 9", fixed="scan the index first", ask="Give the current transfer instruction only.", ideal="Scan the index, then move Box C2 to Shelf 9."),
+    dict(topic="podcast recording", old="episode 18 on Wednesday", new="episode 21 on Friday", fixed="guest: Dr. Vale", ask="State the latest recording plan in one sentence.", ideal="Record episode 21 with Dr. Vale on Friday."),
+    dict(topic="bike rental", old="two city bikes for 3 hours", new="three cargo bikes for 2 hours", fixed="pickup at North Gate", ask="Give the final rental details in one sentence.", ideal="Rent three cargo bikes for two hours, with pickup at North Gate."),
+    dict(topic="library display", old="history books in June", new="science magazines in July", fixed="second-floor lobby", ask="State the final display plan without mentioning the old one.", ideal="Display science magazines in the second-floor lobby in July."),
+    dict(topic="garden delivery", old="six rose bushes on Monday", new="four maple saplings on Wednesday", fixed="use the east entrance", ask="Give the updated delivery instruction in one sentence.", ideal="Deliver four maple saplings on Wednesday through the east entrance."),
+    dict(topic="translation job", old="French, 1,200 words, formal", new="Spanish, 900 words, conversational", fixed="deadline 18:00 UTC", ask="Give only the current language, length, tone, and deadline.", ideal="Translate 900 words into conversational Spanish by 18:00 UTC."),
+]
+
+ALREADY_TRIED = [
+    dict(problem="a laptop that connects to Wi-Fi but cannot reach websites", tried=["restarted the laptop", "reconnected to the network"], next=["renew the DHCP lease", "test with a different DNS resolver"]),
+    dict(problem="a printer producing faint pages", tried=["replaced the paper", "restarted the printer"], next=["check the ink or toner level", "run the print-head cleaning utility"]),
+    dict(problem="a Python service returning a port-in-use error", tried=["restarted the service", "changed the browser"], next=["identify the process holding the port", "stop it or configure a different port"]),
+    dict(problem="a phone that will not charge", tried=["rebooted the phone", "tried another wall outlet"], next=["inspect and gently clean the charging port", "test a known-good cable and adapter"]),
+    dict(problem="a spreadsheet with formulas showing as text", tried=["reopened the file", "changed the font"], next=["set the cells to General format", "re-enter each formula beginning with an equals sign"]),
+    dict(problem="a web page whose new CSS is not visible", tried=["refreshed the page", "restarted the editor"], next=["perform a hard refresh or clear the browser cache", "verify that the edited stylesheet is the one loaded by the page"]),
+    dict(problem="a Bluetooth keyboard that pairs but sends no keystrokes", tried=["replaced the batteries", "paired it again"], next=["remove conflicting keyboard profiles", "test the keyboard on another device"]),
+    dict(problem="a Git push rejected as non-fast-forward", tried=["retried the push", "checked the internet connection"], next=["fetch and inspect the remote changes", "rebase or merge before pushing again"]),
+    dict(problem="a microwave that powers on but does not heat", tried=["used a different outlet", "changed the timer"], next=["stop using it and check the door latch for visible damage", "contact a qualified repair technician"]),
+    dict(problem="headphones playing sound on only one side", tried=["restarted the music app", "raised the volume"], next=["check the device balance setting", "test the headphones with another source"]),
+    dict(problem="a calendar invitation showing the wrong time zone", tried=["reopened the invitation", "changed the event color"], next=["verify the calendar account time zone", "recreate the event with an explicit zone"]),
+    dict(problem="a Docker container exiting immediately", tried=["restarted Docker", "pulled the image again"], next=["inspect the container logs", "run the image with its entrypoint overridden for diagnosis"]),
+    dict(problem="a camera whose photos look unexpectedly dark", tried=["charged the battery", "cleaned the lens"], next=["check exposure compensation", "test in aperture-priority mode with a wider aperture"]),
+    dict(problem="an e-reader that cannot open one book", tried=["restarted the e-reader", "increased screen brightness"], next=["remove and download the book again", "verify that its format is supported and not corrupted"]),
+    dict(problem="a smart speaker that hears commands but plays no audio", tried=["restarted the speaker", "raised the spoken volume"], next=["check whether audio is routed to another device", "unlink and relink the music service"]),
+]
+
+SELECTIVE = [
+    dict(context="The Cedar project owner is Nila. Its budget is €18,400, its review is on 12 June, and the prototype color is orange.", question="What is the review date? Answer with the date only.", required=["12 June"], forbidden=["Nila", "18,400", "orange"], ideal="12 June"),
+    dict(context="Order RX-4 contains 36 valves, ships by rail, arrives at Terminal B, and is insured for $9,000.", question="Which terminal receives the order? Answer with the terminal only.", required=["Terminal B"], forbidden=["36", "rail", "9,000"], ideal="Terminal B"),
+    dict(context="Leah chose the elm desk, Omar chose the glass table, Priya chose the oak shelf, and Zane chose the steel cabinet.", question="What did Priya choose? Give only the item.", required=["oak shelf"], forbidden=["elm", "glass", "steel"], ideal="oak shelf"),
+    dict(context="The rehearsal starts at 17:40 in Studio 3. Doors open at 17:10, the conductor is Ivo, and parking closes at 22:00.", question="Where is the rehearsal? Answer with the location only.", required=["Studio 3"], forbidden=["17:40", "Ivo", "22:00"], ideal="Studio 3"),
+    dict(context="Dataset Lumen has 8,200 rows, uses a CC BY 4.0 license, was updated in March, and has 14 columns.", question="What license does Dataset Lumen use? Give only the license.", required=["CC BY 4.0"], forbidden=["8,200", "March", "14"], ideal="CC BY 4.0"),
+    dict(context="The blue key opens the west cabinet, the brass key opens the archive, and the small black key opens the equipment locker.", question="Which key opens the archive? Answer with the key only.", required=["brass key"], forbidden=["blue", "black", "west"], ideal="brass key"),
+    dict(context="Flight 608 departs at 06:25 from Gate 11, boards at 05:50, and allows one 8 kg cabin bag.", question="What is the departure gate? Give only the gate.", required=["Gate 11"], forbidden=["06:25", "05:50", "8 kg"], ideal="Gate 11"),
+    dict(context="The Orion room seats 18, the Maple room seats 10, and the Harbor room seats 26. Only Harbor has a projector.", question="Which room has a projector? Give only its name.", required=["Harbor"], forbidden=["18", "10", "26"], ideal="Harbor"),
+    dict(context="Invoice 771 is due on 4 August, totals £620, belongs to Kestrel Ltd, and uses purchase order PO-19.", question="What is the purchase-order code? Return only the code.", required=["PO-19"], forbidden=["August", "620", "Kestrel"], ideal="PO-19"),
+    dict(context="The north sensor reports humidity, the east sensor reports pressure, and the south sensor reports temperature.", question="Which sensor reports pressure? Answer with the direction only.", required=["east"], forbidden=["north", "south", "temperature"], ideal="east"),
+    dict(context="Package M2 weighs 4.3 kg, uses priority shipping, has tracking code TR-882, and contains textiles.", question="Give the tracking code only.", required=["TR-882"], forbidden=["4.3", "priority", "textiles"], ideal="TR-882"),
+    dict(context="Aster Café opens at 07:30, stops serving breakfast at 11:00, closes at 19:00, and is closed on Mondays.", question="On which day is the café closed? One word only.", required=["Monday"], forbidden=["07:30", "11:00", "19:00"], ideal="Monday"),
+    dict(context="The field notebook is in Drawer 6, the spare battery is in Drawer 2, and the lens cloth is in Drawer 9.", question="Where is the spare battery? Give only the drawer.", required=["Drawer 2"], forbidden=["Drawer 6", "Drawer 9", "notebook"], ideal="Drawer 2"),
+    dict(context="Team Quartz uses Matrix for chat, Forgejo for code, MinIO for files, and Jitsi for calls.", question="What does Team Quartz use for code? Answer with the product only.", required=["Forgejo"], forbidden=["Matrix", "MinIO", "Jitsi"], ideal="Forgejo"),
+    dict(context="The safety drill coordinator is Hana, the assembly point is Courtyard C, the alarm test is at 15:05, and attendance is 44.", question="Give the assembly point only.", required=["Courtyard C"], forbidden=["Hana", "15:05", "44"], ideal="Courtyard C"),
+]
+
+DELTA = [
+    dict(base="The launch plan used a $12,000 budget, a Monday release, and email promotion.", updates=["Budget is now $9,500.", "Release moves to Wednesday."], ideal=["Budget: $9,500", "Release: Wednesday"]),
+    dict(base="The course had 10 lessons, weekly quizzes, and a final presentation.", updates=["The course now has 8 lessons.", "The final presentation is replaced by a written reflection."], ideal=["Length: 8 lessons", "Final task: written reflection"]),
+    dict(base="The trip used Hotel Nova, train travel, and a four-night stay.", updates=["Accommodation changes to Harbor Hostel.", "The stay is shortened to three nights."], ideal=["Accommodation: Harbor Hostel", "Stay: three nights"]),
+    dict(base="The app supported English and German and required Android 12.", updates=["Spanish support is added.", "Minimum Android version changes to 13."], ideal=["Language added: Spanish", "Minimum Android: 13"]),
+    dict(base="The event expected 75 guests, used Hall A, and started at 18:00.", updates=["Attendance estimate rises to 90.", "The venue changes to Hall C."], ideal=["Expected guests: 90", "Venue: Hall C"]),
+    dict(base="The report covered Q1, used PDF format, and was due Friday.", updates=["Coverage expands to Q1 and Q2.", "Delivery format changes to HTML."], ideal=["Coverage: Q1 and Q2", "Format: HTML"]),
+    dict(base="The device used a 2,000 mAh battery, USB-C charging, and a black enclosure.", updates=["Battery capacity increases to 2,600 mAh.", "The enclosure changes to silver."], ideal=["Battery: 2,600 mAh", "Enclosure: silver"]),
+    dict(base="The study sampled 120 adults, used phone interviews, and ran for six weeks.", updates=["Sample size increases to 150 adults.", "Interviews move online."], ideal=["Sample: 150 adults", "Interview mode: online"]),
+    dict(base="The menu offered soup, pasta, and cake, with service at noon.", updates=["Cake is replaced by fruit.", "Service moves to 13:00."], ideal=["Dessert: fruit", "Service time: 13:00"]),
+    dict(base="The backup ran daily at 01:00 and retained files for 14 days.", updates=["The schedule changes to every six hours.", "Retention increases to 30 days."], ideal=["Schedule: every six hours", "Retention: 30 days"]),
+    dict(base="The exhibit featured ceramics, opened in April, and occupied Gallery 2.", updates=["Opening moves to May.", "The exhibit moves to Gallery 5."], ideal=["Opening: May", "Location: Gallery 5"]),
+    dict(base="The subscription cost $18 monthly and included five team seats.", updates=["Price changes to $20 monthly.", "The plan now includes eight seats."], ideal=["Monthly price: $20", "Seats: eight"]),
+    dict(base="The greenhouse target was 22°C, 55% humidity, and morning watering.", updates=["Temperature target drops to 20°C.", "Watering moves to evening."], ideal=["Temperature: 20°C", "Watering: evening"]),
+    dict(base="The reading group met biweekly, discussed two chapters, and used Room 8.", updates=["Meetings become weekly.", "The group will discuss one chapter per meeting."], ideal=["Frequency: weekly", "Reading: one chapter per meeting"]),
+    dict(base="The API allowed 100 requests per minute and returned XML by default.", updates=["The limit rises to 160 requests per minute.", "The default response changes to JSON."], ideal=["Rate limit: 160 requests per minute", "Default format: JSON"]),
+]
+
+CONFLICTS = [
+    ("The venue manager says the room holds 40 people, while the signed booking says 28.", "Which capacity should I put on the invitation?", "Should I use the venue manager’s figure of 40 or the signed booking’s figure of 28?"),
+    ("The project brief names Lina as owner, but the approval email names Marcus.", "Who should I list as owner?", "Should Lina or Marcus be treated as the authoritative project owner?"),
+    ("One itinerary says the train leaves at 08:10; the ticket says 08:45.", "Tell me when to arrive.", "Which departure time is current: 08:10 on the itinerary or 08:45 on the ticket?"),
+    ("The spreadsheet total is €4,720, but the signed quote is €4,270.", "What amount should I approve?", "Which amount is authoritative: €4,720 in the spreadsheet or €4,270 in the signed quote?"),
+    ("The label says refrigerate the sample, while the protocol says store it at room temperature.", "Where should I put it?", "Should the sample follow the label’s refrigeration instruction or the protocol’s room-temperature instruction?"),
+    ("The calendar lists Tuesday, but the client’s last message says Thursday.", "Schedule the call.", "Should I schedule the call for Tuesday or for Thursday from the client’s latest message?"),
+    ("The inventory system shows 16 units, while the physical count found 11.", "How many can I promise?", "Which quantity should govern the promise: 16 in the system or 11 physically counted?"),
+    ("The style guide requires sentence case, but the campaign brief asks for title case.", "Which capitalization should I use?", "Should the style guide’s sentence case or the campaign brief’s title case take priority?"),
+    ("The map marks the path open, but a ranger’s notice marks it closed.", "Can I plan the route through it?", "Is the ranger’s closure notice current and authoritative over the map?"),
+    ("The database record gives Ana’s region as West, while her submitted form says North.", "Which region belongs in the report?", "Should the report use West from the database or North from Ana’s submitted form?"),
+    ("The product page promises a two-year warranty, but the box insert says one year.", "What warranty should support quote?", "Which warranty term is authoritative: two years on the product page or one year in the box insert?"),
+    ("The teacher’s rubric allows 1,000 words, but the assignment portal shows 800.", "How long should my essay be?", "Should the essay follow the 1,000-word rubric or the 800-word portal limit?"),
+    ("The maintenance log says filter type F7, while the installed-unit plate specifies F9.", "Which replacement should I order?", "Should the replacement match F7 in the log or F9 on the installed unit’s plate?"),
+    ("The grant call says applications close at 17:00 local time, but the portal countdown implies 16:00.", "Give me the deadline.", "Which deadline is authoritative: 17:00 in the grant call or 16:00 implied by the portal?"),
+    ("The recipe card calls for 4 grams of yeast, while the chef’s correction says 7 grams.", "How much yeast should I use?", "Should I follow the original 4 grams or the chef’s corrected 7 grams?"),
+]
+
+SOURCE_BOUNDARY = [
+    ("The supplied memo says the pilot had 48 participants and lasted six weeks. It does not report the dropout count.", "How many participants dropped out?", "The dropout count is not provided in the memo, so it cannot be determined from the supplied information."),
+    ("The inspection note confirms a cracked seal but gives no cause.", "What caused the seal to crack?", "The note confirms the crack but does not identify its cause."),
+    ("The article excerpt says the town planted 2,000 trees. It gives no survival survey.", "How many trees survived?", "The excerpt does not provide survival data, so the number cannot be determined."),
+    ("The invoice lists labor and materials but no payment date.", "On what date was it paid?", "No payment date is provided, so the payment date cannot be verified."),
+    ("The experiment log records a temperature rise after minute 12 but does not identify the mechanism.", "Which mechanism caused the rise?", "The log shows when the rise occurred but does not establish its mechanism."),
+    ("The meeting minutes approve the proposal without recording individual votes.", "Who voted against it?", "Individual votes were not recorded, so the minutes cannot identify who voted against it."),
+    ("The product sheet gives a 500-cycle rating under lab conditions but no field-life estimate.", "How many years will it last in normal use?", "The sheet does not provide enough usage data to convert the lab cycle rating into years of normal use."),
+    ("The survey summary reports an average score of 7.2 but omits the response count.", "How many people answered?", "The response count is missing, so it cannot be determined from the summary."),
+    ("The shipping notice says the parcel left the depot yesterday but has no live tracking update.", "Where is it right now?", "The notice does not contain live location data, so the parcel’s current location cannot be verified."),
+    ("The biography states that Noor studied chemistry and later founded a company. It does not name the university.", "Which university did Noor attend?", "The university is not named in the supplied biography."),
+    ("The budget shows projected revenue and costs but no actual results.", "What profit did the project make?", "Actual profit cannot be calculated because only projections are provided."),
+    ("The case summary says the server failed after an update but contains no logs.", "Which component failed first?", "The summary does not include logs or a failure sequence, so the first failed component cannot be identified."),
+    ("The catalog dates the vase to the 1880s but does not name its maker.", "Who made the vase?", "The maker is not identified in the catalog entry."),
+    ("The clinical abstract reports improvement but omits the control group’s numbers.", "How much better was treatment than control?", "The difference cannot be calculated without the control group’s results."),
+    ("The energy report gives annual consumption but no building floor area.", "What is consumption per square meter?", "Consumption per square meter cannot be calculated without the floor area."),
+]
+
+CONSTRAINTS = [
+    ("Draft a reminder about tomorrow’s 09:00 safety briefing.", ["exactly two sentences", "include 'Room 4'", "do not use 'urgent'"], "Tomorrow’s safety briefing begins at 09:00 in Room 4. Please arrive five minutes early."),
+    ("List ways to protect a laptop while traveling.", ["exactly three bullet points", "each under eight words", "do not mention insurance"], "- Use a padded, water-resistant sleeve.\n- Keep it in your carry-on.\n- Enable encryption and a screen lock."),
+    ("Write a status line for a completed data import.", ["exactly eight words", "start with 'Import complete:'"], "Import complete: all records passed validation without errors."),
+    ("Give a compact JSON object for a green task due Friday.", ["valid JSON only", "keys exactly title, color, due"], '{"title":"task","color":"green","due":"Friday"}'),
+    ("Decline a lunch invitation politely.", ["one sentence", "include 'thank you'", "do not give an excuse"], "Thank you for inviting me, but I’ll have to decline."),
+    ("Summarize the action: Eli will send the draft by noon.", ["start with 'Owner:'", "one line", "include the deadline"], "Owner: Eli — send the draft by noon."),
+    ("Describe a calm workspace.", ["exactly six words", "include 'light'"], "Soft light, clear desk, quiet focus."),
+    ("Answer whether backups reduce data-loss risk.", ["begin with Yes or No", "then exactly one explanatory sentence"], "Yes. Backups provide a separate copy when original data is lost or damaged."),
+    ("Give two reasons to label cables.", ["exactly two numbered items", "no introduction"], "1. Faster troubleshooting.\n2. Safer equipment changes."),
+    ("Write a meeting subject about reviewing the Atlas budget.", ["under seven words", "include 'Atlas'", "do not use punctuation"], "Atlas budget review"),
+    ("Explain version control to a beginner.", ["one sentence", "use the word history", "under 20 words"], "Version control keeps a history of changes so work can be compared, shared, or restored."),
+    ("Provide a CSV header for name, role, and start date.", ["one line only", "exactly three comma-separated fields", "lowercase"], "name,role,start_date"),
+    ("Write a friendly confirmation for a 15:30 call.", ["exactly two sentences", "include Tuesday", "do not start with Sure"], "Tuesday at 15:30 works for me. I’ll be ready for the call."),
+    ("Give a four-word warning about a wet floor.", ["exactly four words", "end with 'carefully'"], "Wet floor; walk carefully."),
+    ("Return the enabled state and retry count.", ["valid JSON only", "keys exactly enabled and retries", "enabled true", "retries 3"], '{"enabled":true,"retries":3}'),
+]
+
+
+def split_for(i: int) -> str:
+    return "train" if i < 9 else "dev" if i < 12 else "test"
+
+
+def make_cases() -> list[dict[str, Any]]:
+    cases: list[dict[str, Any]] = []
+    state_anchors = [
+        ["Sunday", "14:00", "Room Cedar"],
+        ["3.5", "May 10", "staged rollout"],
+        ["180", "green folders", "Dock 2"],
+        ["Jon", "11:15", "video call"],
+        ["9 adult", "3 child", "16:20"],
+        ["Spring Notes", "Thursday", "6,500"],
+        ["K-19", "-20", "amber"],
+        ["60-minute", "18 people", "beginner"],
+        ["vegan", "32", "peanut-free"],
+        ["scan the index", "Box C2", "Shelf 9"],
+        ["episode 21", "Dr. Vale", "Friday"],
+        ["three cargo bikes", "two hours", "North Gate"],
+        ["science magazines", "July", "second-floor lobby"],
+        ["four maple saplings", "Wednesday", "east entrance"],
+        ["900 words", "Spanish", "conversational", "18:00 UTC"],
+    ]
+    for i, x in enumerate(STATE_UPDATES):
+        messages = [
+            {"role": "system", "content": SYSTEM},
+            {"role": "user", "content": f"Track this {x['topic']}. The initial record is {x['old']}; {x['fixed']}."},
+            {"role": "assistant", "content": f"Recorded the initial {x['topic']} details."},
+            {"role": "user", "content": f"Correction: replace {x['old']} with {x['new']}. Keep the other detail unchanged."},
+            {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."},
+            {"role": "user", "content": x["ask"]},
+        ]
+        cases.append(dict(id=f"state_update_{i:02d}", split=split_for(i), category="state_update", messages=messages, ideal=x["ideal"], required=state_anchors[i], forbidden=[x["old"]], checks={"max_words": 30, "no_canned_closing": True}))
+    for i, x in enumerate(ALREADY_TRIED):
+        tried_text = "; ".join(x["tried"])
+        messages = [
+            {"role": "system", "content": SYSTEM},
+            {"role": "user", "content": f"I need help with {x['problem']}. So far I have already {tried_text}. Neither changed the problem."},
+            {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."},
+            {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."},
+        ]
+        ideal = f"1. {x['next'][0].capitalize()}.\n2. {x['next'][1].capitalize()}."
+        cases.append(dict(id=f"already_tried_{i:02d}", split=split_for(i), category="already_tried", messages=messages, ideal=ideal, required=x["next"], forbidden=x["tried"], checks={"numbered_items": 2, "max_words": 55, "no_canned_closing": True}))
+    for i, x in enumerate(SELECTIVE):
+        messages = [{"role": "system", "content": SYSTEM}, {"role": "user", "content": x["context"]}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": x["question"]}]
+        cases.append(dict(id=f"selective_{i:02d}", split=split_for(i), category="selective_answer", messages=messages, ideal=x["ideal"], required=x["required"], forbidden=x["forbidden"], checks={"max_words": 8, "no_canned_closing": True}))
+    for i, x in enumerate(DELTA):
+        messages = [{"role": "system", "content": SYSTEM}, {"role": "user", "content": f"Initial summary: {x['base']}"}, {"role": "assistant", "content": "Initial summary recorded."}]
+        for u in x["updates"]:
+            messages.extend([{"role": "user", "content": u}, {"role": "assistant", "content": "Updated."}])
+        messages.append({"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."})
+        ideal = "\n".join(f"- {z}." for z in x["ideal"])
+        unchanged = [p.strip() for p in re.split(r",| and ", x["base"]) if len(p.strip()) > 5]
+        cases.append(dict(id=f"delta_{i:02d}", split=split_for(i), category="delta_summary", messages=messages, ideal=ideal, required=x["ideal"], forbidden=unchanged[-1:], checks={"bullet_count": 2, "max_words": 40, "no_canned_closing": True}))
+    for i, (ctx, q, ideal) in enumerate(CONFLICTS):
+        messages = [{"role": "system", "content": SYSTEM}, {"role": "user", "content": ctx}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": q + " Ask one focused clarification instead of choosing a value."}]
+        cases.append(dict(id=f"conflict_{i:02d}", split=split_for(i), category="conflict_clarification", messages=messages, ideal=ideal, required=[], forbidden=[], checks={"must_end_question": True, "question_marks": 1, "max_words": 35, "no_canned_closing": True}))
+    for i, (ctx, q, ideal) in enumerate(SOURCE_BOUNDARY):
+        messages = [{"role": "system", "content": SYSTEM}, {"role": "user", "content": ctx}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": q + " Answer directly without guessing."}]
+        cases.append(dict(id=f"source_boundary_{i:02d}", split=split_for(i), category="source_boundary", messages=messages, ideal=ideal, required=[], forbidden=[], checks={"uncertainty_language": True, "max_words": 45, "no_canned_closing": True}))
+    for i, (q, constraints, ideal) in enumerate(CONSTRAINTS):
+        messages = [{"role": "system", "content": SYSTEM}, {"role": "user", "content": q}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: " + "; ".join(constraints) + ". Return only the requested content."}]
+        checks: dict[str, Any] = {"no_canned_closing": True}
+        text = " ".join(constraints).lower()
+        for pat, key in [(r"exactly (\w+) sentences?", "sentence_count"), (r"exactly (\w+) bullet", "bullet_count"), (r"exactly (\w+) numbered", "numbered_items"), (r"exactly (\w+) words?", "exact_words"), (r"(?<!each )under (\w+) words?", "max_words")]:
+            m = re.search(pat, text)
+            if m:
+                nums = {"one": 1, "two": 2, "three": 3, "four": 4, "six": 6, "seven": 7, "eight": 8, "twenty": 20}
+                checks[key] = nums.get(m.group(1), int(m.group(1)) if m.group(1).isdigit() else None)
+        cases.append(dict(id=f"constraint_{i:02d}", split=split_for(i), category="constraint_stack", messages=messages, ideal=ideal, required=[], forbidden=[], checks=checks))
+    return cases
+
+
+def normalize(text: str) -> str:
+    return re.sub(r"\s+", " ", re.sub(r"[^a-z0-9]+", " ", text.lower())).strip()
+
+
+def repetition_metrics(text: str) -> dict[str, Any]:
+    words = re.findall(r"[A-Za-z0-9']+", text.lower())
+    grams = Counter(tuple(words[i:i+4]) for i in range(max(0, len(words)-3)))
+    sentences = [normalize(x) for x in re.split(r"(?<=[.!?])\s+", text.strip()) if normalize(x)]
+    sent_counts = Counter(sentences)
+    return {"words": len(words), "unique_ratio": len(set(words))/max(1,len(words)), "max_4gram_count": max(grams.values(), default=1), "max_sentence_count": max(sent_counts.values(), default=1), "loop": max(grams.values(), default=1)>2 or max(sent_counts.values(), default=1)>1 or (len(words)>=25 and len(set(words))/len(words)<0.45)}
+
+
+def deterministic_reward(case: dict[str, Any], answer: str) -> dict[str, Any]:
+    a = answer.strip(); low = a.lower(); failures: list[str] = []
+    for req in case.get("required", []):
+        if normalize(str(req)) not in normalize(a): failures.append("missing:"+str(req))
+    for ban in case.get("forbidden", []):
+        if normalize(str(ban)) and normalize(str(ban)) in normalize(a): failures.append("stale_or_irrelevant:"+str(ban))
+    c = case.get("checks", {}); words = re.findall(r"\b[\w'-]+\b", a)
+    bullets = [x for x in a.splitlines() if re.match(r"^\s*[-*•]\s+", x)]
+    numbered = [x for x in a.splitlines() if re.match(r"^\s*\d+[.)]\s+", x)]
+    sentences = [x for x in re.split(r"(?<=[.!?])(?:[\"']*)\s+", a) if x.strip()]
+    if c.get("max_words") and len(words) > int(c["max_words"]): failures.append("max_words")
+    if c.get("exact_words") and len(words) != int(c["exact_words"]): failures.append("exact_words")
+    if c.get("bullet_count") and len(bullets) != int(c["bullet_count"]): failures.append("bullet_count")
+    if c.get("numbered_items") and len(numbered) != int(c["numbered_items"]): failures.append("numbered_items")
+    if c.get("sentence_count") and len(sentences) != int(c["sentence_count"]): failures.append("sentence_count")
+    if c.get("must_end_question") and not a.endswith("?"): failures.append("must_end_question")
+    if c.get("question_marks") and a.count("?") != int(c["question_marks"]): failures.append("question_marks")
+    if c.get("uncertainty_language") and not any(x in low for x in ["not provided", "cannot", "can't", "does not", "missing", "not identified", "not recorded", "not named", "no "]): failures.append("uncertainty_language")
+    if c.get("no_canned_closing") and any(x in low for x in ["let me know if", "i hope this helps", "feel free to ask"]): failures.append("canned_closing")
+    rep = repetition_metrics(a)
+    if rep["loop"]: failures.append("repetition_loop")
+    hard = len(failures)
+    score = max(-1.0, 1.0 - 0.35*hard)
+    return {"score": score, "pass": hard == 0, "failures": failures, **rep}
+
+
+JUDGE_SYSTEM = """You are a strict evaluator of a compact assistant's information management in a multi-turn dialogue. Compare candidate answers against the conversation, the latest user request, explicit state updates, stale facts, constraints, and the supplied ideal reference. Penalize repeating old information, repeating attempted steps, inventing missing facts, ignoring corrections, verbose restatement, canned closings, and sentence loops. Reward concise direct use of current information. Score each candidate 0-10 for state accuracy, instruction following, relevance, honesty, and non-repetition. Return JSON only."""
+
+
+def export_jsonl(path: str, rows: list[dict[str, Any]]) -> None:
+    with open(path, "w", encoding="utf-8") as f:
+        for row in rows: f.write(json.dumps(row, ensure_ascii=False)+"\n")
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/long_dev_gate.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/long_dev_gate.json
new file mode 100644
index 0000000000000000000000000000000000000000..5097298b92d54d89fcc32e6dd77afc84ecd56f04
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/long_dev_gate.json
@@ -0,0 +1,189 @@
+{
+  "leader": {
+    "alpha": 0,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3333333333333334,
+    "loops": 0,
+    "mean_words": 8.857142857142858,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p005": {
+    "alpha": 0.005,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3333333333333334,
+    "loops": 0,
+    "mean_words": 8.904761904761905,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p01": {
+    "alpha": 0.01,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3333333333333334,
+    "loops": 0,
+    "mean_words": 8.857142857142858,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p025": {
+    "alpha": 0.025,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3333333333333334,
+    "loops": 0,
+    "mean_words": 9.142857142857142,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p05": {
+    "alpha": 0.05,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3333333333333334,
+    "loops": 0,
+    "mean_words": 9.047619047619047,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p1": {
+    "alpha": 0.1,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3333333333333334,
+    "loops": 0,
+    "mean_words": 8.571428571428571,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p2": {
+    "alpha": 0.2,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3500000000000001,
+    "loops": 0,
+    "mean_words": 8.761904761904763,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p35": {
+    "alpha": 0.35,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3500000000000001,
+    "loops": 0,
+    "mean_words": 9.285714285714286,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p5": {
+    "alpha": 0.5,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3500000000000001,
+    "loops": 0,
+    "mean_words": 9.857142857142858,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a0p75": {
+    "alpha": 0.75,
+    "pass": 3,
+    "n": 21,
+    "mean_reward": 0.3500000000000001,
+    "loops": 0,
+    "mean_words": 9.761904761904763,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 2,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  },
+  "a1p0": {
+    "alpha": 1.0,
+    "pass": 4,
+    "n": 21,
+    "mean_reward": 0.36666666666666675,
+    "loops": 0,
+    "mean_words": 9.142857142857142,
+    "by_category": {
+      "already_tried": 0,
+      "conflict_clarification": 0,
+      "constraint_stack": 3,
+      "delta_summary": 0,
+      "selective_answer": 1,
+      "source_boundary": 0,
+      "state_update": 0
+    }
+  }
+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_encoded_rollouts.jsonl b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_encoded_rollouts.jsonl
new file mode 100644
index 0000000000000000000000000000000000000000..be90691d687dba7d9ba9e7086423db11f9f35ef0
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_encoded_rollouts.jsonl
@@ -0,0 +1,216 @@
+{"group_id": "math_002", "origin": "budgie", "answer": "1. **Understanding the problem:** We have a race with 100 meters, and the runners are running at constant speeds. We need to determine how far behind Carol was when Anna finished.\n\n2. **Relative Speed and Time:** Since the runners are running at constant speeds, we can assume that the time taken by each runner to complete the race is constant. This means we can use relative speed and time to find out how much distance each runner covers in the time it takes for them to complete the race.\n\n3. **First runner's Time:** When Anna finishes the race, the first runner has already run for 100 meters. Since they are running at constant speeds, the first runner will cover 100 meters in the same amount of time.\n\n4. **Second runner's Time:** When Anna finishes the race, the second runner has only 90 meters left to run. Since they are running at constant speeds, the second runner will cover 90 meters in the same amount of time.\n\n5. **Distance Calculation for Anna's dye run:** When Anna finishes the race, the second runner has already run for 90 meters. Since they are running at constant speeds, the second runner covers a distance of 90 meters in the same amount of time. Therefore, the distance between Carol and Anna is 100 meters minus 90 meters, which is 10 meters.\n\nIn summary, when Anna finishes, Carol is 10 meters behind.\n\nFinal answer:\n#### 10", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 49164, 30, 16435, 28, 20259, 16422, 284, 6606, 1658, 281, 253, 1885, 49160, 49159, 49159, 3814, 8898, 5167, 107, 110, 93, 24362, 4677, 30, 1550, 16435, 27668, 28, 20259, 16422, 314, 1885, 49160, 49165, 3814, 8898, 5167, 107, 110, 93, 24362, 2893, 874, 284, 645, 20259, 16422, 27668, 28, 6606, 314, 1885, 49161, 49164, 3814, 8898, 5167, 107, 110, 93, 24362, 2893, 874, 30, 378, 5901, 1658, 418, 3836, 10870, 2126, 260, 4677, 30, 1073, 1869, 2893, 436, 6606, 645, 16435, 7721, 47, 198, 49, 1885, 49162, 49166, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 50, 1885, 49163, 49160, 3814, 8898, 5167, 107, 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+{"group_id": "math_002", "origin": "budgie", "answer": "1. **Identify the speeds of Anna and Bridgit:**\n   - Since Anna finishes the race in $37 \\mathrm{~m}$, her speed is $37 \\mathrm{~m/ \\text{min}}$.\n   - Bridgit finishes the race in $41 \\mathrm{~m}$, so her speed is $41 \\mathrm{~m/ \\text{min}}$.\n\n2. **Determine the time taken by each girl to finish:**\n   - Time taken by Anna: $37 \\mathrm{~m/ \\text{min}}$.\n   - Time taken by Bridgit: $41 \\mathrm{~m/ \\text{min}}$.\n\n3. **Calculate the distance each girl covers after the start of the race:**\n   - Distance covered by Anna: $37 \\mathrm{~m/ \\text{min}} \\times 30 \\mathrm{min} = 1110 \\mathrm{ \\text{ m}}$.\n   - Distance covered by Bridgit: $41 \\mathrm{~m/ \\text{min}} \\times 30 \\mathrm{min} = 1230 \\mathrm{ \\text{ m}}$.\n\n4. **Determine the time taken by Carol to finish:**\n   - Sits for $100 \\mathrm{ \\text{ m}}$ at a constant speed.\n   - Carol's speed: $100 \\mathrm{ \\text{ m}} \\times \\frac{1}{37} \\mathrm{ \\text{ min}} = \\frac{100}{37} \\mathrm{ \\text{ min}} = \\frac{100}{37} \\times \\frac{1}{3} \\mathrm", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 49164, 30, 16435, 28, 20259, 16422, 284, 6606, 1658, 281, 253, 1885, 49160, 49159, 49159, 3814, 8898, 5167, 107, 110, 93, 24362, 4677, 30, 1550, 16435, 27668, 28, 20259, 16422, 314, 1885, 49160, 49165, 3814, 8898, 5167, 107, 110, 93, 24362, 2893, 874, 284, 645, 20259, 16422, 27668, 28, 6606, 314, 1885, 49161, 49164, 3814, 8898, 5167, 107, 110, 93, 24362, 2893, 874, 30, 378, 5901, 1658, 418, 3836, 10870, 2126, 260, 4677, 30, 1073, 1869, 2893, 436, 6606, 645, 16435, 7721, 47, 198, 49, 1885, 49162, 49166, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 50, 1885, 49163, 49160, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 51, 1885, 49164, 49159, 3814, 8898, 5167, 107, 110, 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+{"group_id": "math_002", "origin": "verified_reference", "answer": "25. A Carol finishes 25 metres behind Bridgit, so she travels 75 metres while Bridgit runs 100 metres. Therefore she runs 3 metres for every 4 metres Bridgit runs. When Anna finishes, Bridgit has run 84 metres, so that at that time Carol has run $\\frac{3}{4} \\times 84$ metres $=63$ metres. Hence Carol finishes $(100-63)$ metres $=37$ metres behind Anna.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 49164, 30, 16435, 28, 20259, 16422, 284, 6606, 1658, 281, 253, 1885, 49160, 49159, 49159, 3814, 8898, 5167, 107, 110, 93, 24362, 4677, 30, 1550, 16435, 27668, 28, 20259, 16422, 314, 1885, 49160, 49165, 3814, 8898, 5167, 107, 110, 93, 24362, 2893, 874, 284, 645, 20259, 16422, 27668, 28, 6606, 314, 1885, 49161, 49164, 3814, 8898, 5167, 107, 110, 93, 24362, 2893, 874, 30, 378, 5901, 1658, 418, 3836, 10870, 2126, 260, 4677, 30, 1073, 1869, 2893, 436, 6606, 645, 16435, 7721, 47, 198, 49, 1885, 49162, 49166, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 50, 1885, 49163, 49160, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 51, 1885, 49164, 49159, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 52, 1885, 49164, 49164, 3814, 8898, 5167, 107, 110, 93, 24362, 198, 53, 1885, 49165, 49159, 3814, 8898, 5167, 107, 110, 93, 24362, 49154, 49158, 198, 49161, 49164, 30, 330, 6606, 27668, 216, 49161, 49164, 13897, 2893, 20259, 16422, 28, 588, 1041, 12517, 216, 49166, 49164, 13897, 979, 20259, 16422, 7313, 216, 49160, 49159, 49159, 13897, 30, 4882, 1041, 7313, 216, 49162, 13897, 327, 897, 216, 49163, 13897, 20259, 16422, 7313, 30, 1550, 16435, 27668, 28, 20259, 16422, 553, 1658, 216, 49167, 49163, 13897, 28, 588, 338, 418, 338, 655, 6606, 553, 1658, 19573, 18169, 107, 49162, 15009, 49163, 109, 3814, 14602, 216, 49167, 49163, 20, 13897, 1885, 45, 49165, 49162, 20, 13897, 30, 10987, 6606, 27668, 46408, 49160, 49159, 49159, 29, 49165, 49162, 38224, 13897, 1885, 45, 49162, 49166, 20, 13897, 2893, 16435, 30, 49154], "prefix_len": 173, "old_logprobs": [-2.4253787994384766, -0.21279913187026978, -0.0605345219373703, -5.797369956970215, -11.757119178771973, -3.399350881576538, -0.8138746619224548, -1.701750636100769, -0.17213009297847748, -9.585885047912598, -0.9990986585617065, -7.616662502288818, -3.276886224746704, -2.155653476715088, -1.7828266620635986, -2.3982431888580322, -6.556173324584961, -0.3374185562133789, -6.223995685577393, -0.14264428615570068, -1.4982037544250488, -4.522729396820068, -1.346922516822815, -2.68030047416687, -4.66306734085083, -0.7837975025177002, -1.6714181900024414, -2.1646103858947754, -0.12729915976524353, -0.49660807847976685, -0.24436023831367493, -4.100385665893555, -7.9015679359436035, -2.658106565475464, -0.5397435426712036, -2.163759231567383, -10.774786949157715, -5.409459590911865, -2.4475653171539307, -0.434548556804657, -3.2349119186401367, -3.6678080558776855, -4.68051815032959, -2.774183988571167, -0.27711769938468933, -0.26706257462501526, -3.8077964782714844, -1.9413135051727295, -0.1424228549003601, -0.46229642629623413, -0.5820859670639038, -2.832719087600708, -1.7813196182250977, -0.29981404542922974, -0.14988763630390167, -4.299286365509033, -2.5988945960998535, -0.49550122022628784, -0.7137501835823059, -1.2265372276306152, -5.838071346282959, -9.68909740447998, -3.0647103786468506, -1.1420059204101562, -3.3198444843292236, -0.9523718357086182, -0.0818081945180893, -7.95733642578125, -0.34137916564941406, -0.012337321415543556, -1.0691508054733276, -0.18798594176769257, -0.1195061132311821, -0.315854012966156, -0.1612364500761032, -0.1644655168056488, -0.025299500674009323, -0.5588744878768921, -0.007398467976599932, -3.2329370975494385, -1.079887866973877, -9.250434875488281, -0.28762415051460266, -3.934783458709717, -1.977260947227478, -0.12869928777217865, -0.2151605188846588, -0.20074860751628876, -4.286519527435303, -1.8321847915649414, -4.4094648361206055, -7.663887977600098, -2.674865245819092, -0.14342664182186127, -0.0030125256162136793, -0.8097465634346008, -0.27315112948417664, -0.025270558893680573, -3.4341368675231934, -3.2909789085388184, -1.9462978839874268, -0.2908145785331726, -1.7154597043991089, -0.0015464978059753776, -0.029404832050204277, -0.3377261161804199, -0.061109308153390884, -0.5189911127090454, -0.38879287242889404, -4.2782111167907715]}
+{"group_id": "math_004", "origin": "budgie", "answer": "The best answer is 12.\n\nFinal answer:\n#### 12", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49165, 378, 3372, 282, 253, 476, 4638, 29, 1110, 3133, 18, 20570, 3283, 42, 1985, 260, 1885, 94, 20, 29, 373, 1106, 28, 253, 5437, 1251, 325, 18221, 1885, 94, 20, 1711, 30, 1094, 260, 2028, 282, 260, 2876, 6238, 429, 623, 1885, 94, 20, 20587, 314, 2852, 670, 1885, 49161, 21990, 94, 24362, 28, 260, 1106, 314, 2500, 4180, 30, 11984, 42, 198, 24, 49160, 25, 1812, 314, 260, 5428, 1230, 282, 1825, 253, 1055, 416, 1301, 281, 451, 3133, 47, 198, 24, 49161, 25, 1812, 314, 260, 8746, 338, 384, 416, 1301, 260, 808, 1296, 1825, 15233, 316, 1320, 47, 198, 24, 8928, 42, 330, 5437, 314, 253, 8281, 20636, 351, 2876, 22266, 1885, 49160, 28, 49161, 28, 49162, 28, 49163, 28, 49164, 28, 49165, 20, 335, 624, 7975, 30, 378, 1230, 282, 2876, 335, 260, 2715, 338, 2874, 614, 990, 13549, 260, 5437, 314, 260, 966, 282, 260, 5089, 2677, 49154, 49158, 198, 504, 1450, 2988, 314, 216, 49160, 49161, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49161, 49154], "prefix_len": 185, "old_logprobs": [-1.2357776165008545, -4.043502330780029, -0.17622926831245422, -0.06694753468036652, -0.9411191940307617, -1.3033380508422852, -2.1477341651916504, -0.6913499236106873, -0.3729572892189026, -0.034664541482925415, -0.11858867108821869, -0.0007474012672901154, -0.0005296258023008704, -0.0447004958987236, -0.0042093489319086075, -0.003246871754527092, -0.0006475735572166741, -0.0013892533024773002, -0.00043084874050691724]}
+{"group_id": "math_004", "origin": "budgie", "answer": "The minimum possible outcome is rolling a 1 on one of the dice, resulting in 1 point. The maximum possible outcome is rolling an 8 on one of the dice, resulting in 8 points.\n\nThe probability of rolling a 1 on one die is $\\frac{1}{6}$, the probability of rolling an 8 on one die is $\\frac{1}{6}$. The probability of rolling an 8 on one die and then rolling a 1 on the next roll is $\\frac{1}{6} \\times \\frac{1}{6} = \\frac{1}{36}$.\n\nTherefore, the maximum number of levels is 6, and the probability that a person can pass the first three levels consecutively is $\\frac{1}{36} \\times \\frac{1}{6} = \\frac{1}{216}$.\n\nFinal answer:\n#### \\frac{1}{216}", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49165, 378, 3372, 282, 253, 476, 4638, 29, 1110, 3133, 18, 20570, 3283, 42, 1985, 260, 1885, 94, 20, 29, 373, 1106, 28, 253, 5437, 1251, 325, 18221, 1885, 94, 20, 1711, 30, 1094, 260, 2028, 282, 260, 2876, 6238, 429, 623, 1885, 94, 20, 20587, 314, 2852, 670, 1885, 49161, 21990, 94, 24362, 28, 260, 1106, 314, 2500, 4180, 30, 11984, 42, 198, 24, 49160, 25, 1812, 314, 260, 5428, 1230, 282, 1825, 253, 1055, 416, 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+{"group_id": "math_004", "origin": "verified_reference", "answer": "Solution: Since the dice is a uniform cube, the possibility of each number appearing after a roll is equal.\n(1) Because the maximum number that can appear on a single dice is 6, and $6 \\times 1 > 2^4, 6 \\times 5 < 2$.\nTherefore, when $n \\geqslant 5$, the sum of the numbers that appear after rolling the dice $n$ times cannot be greater than 2, which means this is an impossible event, and the probability of passing is 0.\nSo, the maximum number of consecutive levels that can be passed is 4.\n(2) Let event $A_{n}$ be \"failing the $n$-th level\", then the complementary event $\\overline{A_{n}}$ is \"passing the $n$-th level\". In the $n$-th level game, the total number of basic events is $6^n$.\nLevel 1: The number of basic events contained in event $A_{1}$ is 2 (i.e., the cases where the number is 1 or 2),\nSo the probability of passing this level is: $P\\left(\\overline{A_{1}}\\right)=1-P\\left(A_{1}\\right)=1-\\frac{2}{6}=\\frac{2}{3}$\nLevel 2: The number of basic events contained in event $A_{2}$ is the sum of the number of positive integer solutions to the equation $x+y=a$ when $a$ takes the values $2,3,4$. That is, $C_{1}^{3}+C_{3}^{1}+C_{3}^{1}=1+2+3=6$ (cases).\nSo the probability of passing this level is $P\\left(\\bar{A}_{2}\\right)=1-P\\left(A_{2}\\right)=1-\\frac{6}{6^{2}}=\\frac{5}{6}$\nLevel 3: The number of basic events contained in event $A_{3}$ is the sum of the number of positive integer solutions to the equation $x+y+z=a$ when $a$ takes the values $3,4,5,6,7,8$. That is, $C_{2}^{3}+C_{3}^{3}+C_{4}^{2}+C_{5}^{3}+C_{6}^{3}+C_{5}^{2}=1+3+6+10+15+21=56$ (cases).\nSo the probability of passing this level is: $P\\left(\\bar{A}_{3}\\right)=1-P\\left(A_{3}\\right)=1-\\frac{56}{6^{3}}=\\frac{20}{27}$\nTherefore, the probability of passing the first 3 levels consecutively is: $P\\left(\\bar{A}_{1}\\right) \\cdot P\\left(\\bar{A}_{2}\\right) \\cdot P\\left(\\bar{A}_{3}\\right)=\\frac{2}{3} \\times \\frac{5}{6} \\times \\frac{20}{27}=\\frac{100}{243}$", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49165, 378, 3372, 282, 253, 476, 4638, 29, 1110, 3133, 18, 20570, 3283, 42, 1985, 260, 1885, 94, 20, 29, 373, 1106, 28, 253, 5437, 1251, 325, 18221, 1885, 94, 20, 1711, 30, 1094, 260, 2028, 282, 260, 2876, 6238, 429, 623, 1885, 94, 20, 20587, 314, 2852, 670, 1885, 49161, 21990, 94, 24362, 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+{"group_id": "math_005", "origin": "budgie", "answer": "To find the probability that a randomly placed point falls within one of the five smaller spheres, we can approach the problem by considering the scenario in which the point $P$ is placed in the center of the largest sphere. Here's how we can break it down:\n\n1. The largest sphere has a radius of 1. This is because if we place a point in this sphere, the maximum distance to any other point on the sphere is 1.\n\n2. The remaining smaller spheres are arranged in a regular tetrahedron with two inscribed spheres. Each face of the tetrahedron is a sphere, and there are four smaller spheres between each pair of faces. Since the tetrahedron is regular, each face has a smaller sphere (tangent to the center of the face) and a center sphere (radius 1).\n\n3. Since there are five smaller spheres, and each face of the tetrahedron is a sphere, one of these smaller spheres is inscribed in the circumscribed sphere (radius 1), and the other four are circumscribed (radius 1).\n\n4. Each of these smaller spheres is tangent to the center of the face of the tetrahedron. Therefore, a randomly placed point on the circumscribed sphere (radius 1) will fall within one of these smaller spheres.\n\nNow, let's consider the total number of possible outcomes. Each of the five smaller spheres has a radius of 1, and there are four of them. Therefore, the total number of possible outcomes for the tetrahedron is $5^4$.\n\nHowever, we need to be careful not to overcount the cases where a point falls within a smaller sphere. For example, if a point is placed in the center of a smaller sphere but not on the edge of the tetrahedron, then it falls within the smaller sphere.\n\nTo find the probability, we need to divide the number", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 365, 49160, 49168, 49168, 49168, 1547, 4264, 3056, 14810, 17884, 25, 330, 2649, 29889, 39513, 553, 827, 21684, 28, 582, 30036, 284, 582, 4584, 45115, 30, 4454, 28, 665, 359, 1876, 3764, 21684, 826, 971, 2715, 282, 260, 29889, 39513, 284, 260, 4584, 45115, 13510, 28, 971, 37153, 288, 260, 3712, 282, 260, 2715, 30, 8779, 1885, 64, 20, 314, 253, 1225, 2972, 260, 4584, 45115, 13510, 30, 378, 8746, 338, 1885, 64, 20, 8227, 1127, 582, 282, 260, 2531, 3764, 21684, 314, 13039, 288, 365, 15165, 198, 49, 30, 216, 49159, 198, 50, 30, 216, 49159, 30, 49160, 198, 51, 30, 216, 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+{"group_id": "math_005", "origin": "budgie", "answer": "There are $24$ possible outcomes for the tetrahedron, and only one of them has the point $P$ inside. The probability that $P$ is one of the five smaller spheres is $\\frac{1}{5}$.\n\nSo, the answer is 0.1.\n\nFinal answer: B", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 365, 49160, 49168, 49168, 49168, 1547, 4264, 3056, 14810, 17884, 25, 330, 2649, 29889, 39513, 553, 827, 21684, 28, 582, 30036, 284, 582, 4584, 45115, 30, 4454, 28, 665, 359, 1876, 3764, 21684, 826, 971, 2715, 282, 260, 29889, 39513, 284, 260, 4584, 45115, 13510, 28, 971, 37153, 288, 260, 3712, 282, 260, 2715, 30, 8779, 1885, 64, 20, 314, 253, 1225, 2972, 260, 4584, 45115, 13510, 30, 378, 8746, 338, 1885, 64, 20, 8227, 1127, 582, 282, 260, 2531, 3764, 21684, 314, 13039, 288, 365, 15165, 198, 49, 30, 216, 49159, 198, 50, 30, 216, 49159, 30, 49160, 198, 51, 30, 216, 49159, 30, 49161, 198, 52, 30, 216, 49159, 30, 49162, 198, 53, 30, 216, 49159, 30, 49163, 49154, 49158, 198, 2122, 359, 1885, 49161, 49163, 20, 1636, 4530, 327, 260, 29889, 39513, 28, 284, 805, 582, 282, 601, 553, 260, 1225, 1885, 64, 20, 2972, 30, 378, 8746, 338, 1885, 64, 20, 314, 582, 282, 260, 2531, 3764, 21684, 314, 19573, 18169, 107, 49160, 15009, 49164, 24362, 30, 198, 198, 2931, 28, 260, 2988, 314, 216, 49159, 30, 49160, 30, 198, 198, 29288, 2988, 42, 389, 49154], "prefix_len": 151, "old_logprobs": [-3.278898000717163, -0.13734246790409088, -3.2542598247528076, -2.033658504486084, -2.994133472442627, -0.2270866483449936, -2.9858627319335938, -1.2006670236587524, -0.8743399381637573, -0.4853042960166931, -0.5512608289718628, -0.04467564448714256, -1.3101723194122314, -1.5326091051101685, -2.0393378734588623, -0.6318479180335999, -0.2827099859714508, -0.342880517244339, -2.889920711517334, -1.7568949460983276, -2.5293686389923096, -0.16359692811965942, -0.003510267473757267, -0.05464000254869461, -0.32643088698387146, -2.690767765045166, -2.0274410247802734, -0.2802373468875885, -0.3368476629257202, -0.21558339893817902, -0.0004991239402443171, -0.0006609162664972246, -2.3114943504333496, -2.591273069381714, -0.0004602803383022547, -0.07188691198825836, -0.2643154263496399, -0.028727667406201363, -0.006216712296009064, -0.09859368205070496, -2.596771478652954, -0.07623772323131561, -0.010312250815331936, -0.05624609813094139, -0.007632735185325146, -0.925724983215332, -0.4367166757583618, -0.250544935464859, -0.3638399839401245, -0.04707839712500572, -3.2555441856384277, -0.4908730685710907, -0.04638521745800972, -0.09929360449314117, -0.0015575670404359698, -4.2626142501831055, -0.2678670883178711, -0.02668517827987671, -0.8097407817840576, -0.3648257553577423, -0.04547082632780075, -0.02375580556690693, -0.13372793793678284, -0.0007489498239010572, -0.004175753798335791, -0.17980630695819855, -0.010958842933177948]}
+{"group_id": "math_005", "origin": "verified_reference", "answer": "6. C. Reason: Let the edge length of the regular tetrahedron be \\( a \\). It is known that the radius of its inscribed sphere is \\( R_{1} = \\frac{\\sqrt{6}}{12} a \\), and the radius of its circumscribed sphere is \\( R_{2} = \\frac{\\sqrt{6}}{4} a \\).\n\nIf the radius of the sphere that is tangent to the circumscribed sphere and the faces of the tetrahedron is \\( r \\), then\n\\[\nr = \\frac{R_{2} - R_{1}}{2} = \\frac{\\sqrt{6}}{12} a \\text{.}\n\\]\n\nThus, the required probability is\n\\[\n\\frac{\\frac{4}{3} \\pi R_{1}^{3} + 4 \\cdot \\frac{4}{3} \\pi r^{3}}{\\frac{4}{3} \\pi R_{2}^{3}} = \\frac{5 \\left( \\frac{1}{12} \\right)^{3}}{\\left( \\frac{1}{4} \\right)^{3}} = \\frac{5}{27} \\approx 0.19.\n\\]", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 365, 49160, 49168, 49168, 49168, 1547, 4264, 3056, 14810, 17884, 25, 330, 2649, 29889, 39513, 553, 827, 21684, 28, 582, 30036, 284, 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+{"group_id": "math_007", "origin": "budgie", "answer": "First, we need to understand the properties of the circumscribed circle of a triangle and the specific geometric constructions involved.\n\nIn a triangle, the circumscribed circle (circumcircle) is the circle that passes through all three vertices of the triangle. This circle is the circumcircle of the triangle.\n\nGiven that $B C = B E$, and considering the circumcircle properties in a triangle (which is a special case of a cyclic quadrilateral), we can infer several things:\n\n1. Since $B C = B E$, it follows that $B E = C E$. This means that $B, E, C$ are collinear, and angle $B E C$ is a straight line.\n\n2. The circumscribed circle's center (circumcenter) must be equidistant from the vertices of the triangle, meaning $E$ must be the circumcenter of triangle $A B C$.\n\n3. In a triangle, the circumcenter is effectively the center of the circumcircle, without the need for the circumcenter to be the circumcenter itself.\n\nThe key insight here is recognizing that because $B E = C E$, it directly follows from the properties of circumcircle in a triangle that $B E$ is the circumradius of triangle $B C D$. This is because the circumradius is the distance from the circumcenter to any vertex, and since $E$ is the circumcenter, it's the unique point that is equidistant from the other vertices of the triangle. Therefore, $\\angle B E C$ is a straight line, making it a right angle.\n\nThus, the angle $\\angle A E D$ is a right angle because it is the complement of $\\angle B E C$ (or its vertical angle, depending on the direction of the line segment $B E$), making it a right angle.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49164, 30, 365, 49162, 2876, 25, 198, 198, 788, 635, 31999, 1633, 1885, 49, 389, 340, 422, 26265, 260, 15070, 2103, 1885, 50, 340, 20, 314, 4037, 288, 260, 24570, 1885, 50, 422, 28422, 1985, 260, 3764, 13210, 1885, 49, 389, 20, 282, 260, 4584, 45115, 7842, 282, 14973, 1885, 49, 389, 340, 26265, 253, 1225, 1885, 53, 20, 314, 5958, 715, 338, 1885, 50, 340, 45, 50, 414, 28422, 7985, 260, 7256, 19573, 6935, 330, 414, 422, 28422, 49154, 49158, 198, 5345, 28, 392, 737, 288, 1044, 260, 3849, 282, 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+{"group_id": "math_007", "origin": "budgie", "answer": "To solve this problem, we will use the properties of the circumscribed circle and inscribed angles.\n\n1. Since $B C = B E$, the angle $\\angle B C E = 2 \\angle B E C$.\n2. The arc $A B$ of the circumscribed circle of triangle $A B C$ is equal to the arc $A C$. Therefore, the angle $\\angle A E C$ is twice the angle $\\angle B E C$.\n3. Since $\\angle B E C = \\frac{1}{2} \\angle B C E$ and $\\angle B C E = 2 \\angle B E C$, we have $\\angle A E C = 2 \\angle B E C = 2 \\cdot 2 \\angle B E C = \\frac{1}{2} \\angle B E C$.\n4. Substituting this value into (2), we find that $\\angle A E C = 2 \\angle B E C = 2 \\cdot 2 \\angle B E C = 4 \\angle B E C$.\n5. Since $\\angle B C E = \\frac{1}{2} \\angle B E C$, we can find $\\angle A E C = 2 \\angle B E C$.\n6. Also, since the sum of angles in a triangle is $180^\\circ$, we have $\\angle A D C + \\angle B E D = 180^\\circ - \\angle B E C$.\n7. Therefore, $\\angle A E D + 2 \\angle B E E = 180^\\circ - \\angle B E C$.\n8. 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+{"group_id": "math_007", "origin": "verified_reference", "answer": "Answer: $90^{\\circ}$.\n\n## Solution:\n\nFrom the isosceles triangle $B C D$ and the parallelism of $A B$ and $C D$, we get $\\angle B C D = \\angle B D C = \\angle D B A = \\alpha$.\n\nLet the line $C D$ intersect the circumcircle of triangle $A B C$ at point $F$. Then $B C F A$ is an inscribed, i.e., isosceles, trapezoid, from which the arcs $B C, B E$, and $A F$ are equal. Hence, $\\angle B C E = \\angle A C F = \\beta$.\n\n$\\angle E B A = \\angle E C A = \\gamma$, since these angles subtend the same arc.\n\n$\\angle B C A = \\angle B C E + \\angle E C A = \\beta + \\gamma$.\n\n$\\angle E B D = \\angle E B A + \\angle D B A = \\gamma + \\alpha$. Therefore, in the isosceles triangle $E B D$, the equality $\\angle B D E = \\angle B E D = \\frac{180^{\\circ} - \\alpha - \\gamma}{2}$ holds.\n\nMoreover, $\\alpha = \\angle B C D = \\angle B C F = \\angle B C A + \\angle A C F = 2 \\beta + \\gamma$.\n\n$\\angle A E D = \\angle B E A - \\angle B E D = (180^{\\circ} - \\angle B C A) - \\frac{180^{\\circ} - \\alpha - \\gamma}{2} = 180^{\\circ} - \\beta - \\gamma - 90^{\\circ} + \\frac{2 \\beta + 2 \\gamma}{2} = 90^{\\circ}$", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49164, 30, 365, 49162, 2876, 25, 198, 198, 788, 635, 31999, 1633, 1885, 49, 389, 340, 422, 26265, 260, 15070, 2103, 1885, 50, 340, 20, 314, 4037, 288, 260, 24570, 1885, 50, 422, 28422, 1985, 260, 3764, 13210, 1885, 49, 389, 20, 282, 260, 4584, 45115, 7842, 282, 14973, 1885, 49, 389, 340, 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can determine the value of the expression.\n## Step 8:  Finally, using the value of the expression, we can calculate the value of the expression and subsequently the value of the third side of the triangle.\n\nThe final answer is: $\\boxed{50}$", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 33348, 216, 49165, 30, 365, 30129, 216, 49160, 595, 198, 198, 788, 260, 1048, 14973, 1885, 35542, 42, 3814, 6935, 12121, 50, 45, 49168, 49159, 21990, 76, 21386, 6663, 12121, 45, 49165, 28, 7636, 45, 49163, 28422, 1985, 260, 1761, 1885, 6190, 26265, 253, 1225, 1885, 52, 365, 6998, 46, 11526, 38224, 314, 6319, 715, 338, 19573, 6935, 49132, 45, 49163, 49164, 21990, 76, 21386, 24362, 30, 1985, 260, 1761, 1885, 3907, 26265, 253, 1225, 1885, 53, 20, 314, 6319, 715, 338, 260, 25096, 282, 14973, 1885, 51, 19698, 20, 314, 260, 13341, 1636, 30, 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+{"group_id": "math_008", "origin": "verified_reference", "answer": "# Solution:\n\n$C D=A C=6 \\Rightarrow B D=B_{1} D=6-4=2$. Since triangles $\\mathrm{ACE}$ and $\\mathrm{B}_{1} \\mathrm{ED}$\n\nare similar ( $\\angle A E C=\\angle B_{1} E D$ - vertical angles and $\\angle E A C=\\angle B_{1} D E=45^{\\circ}$ ), then\n\n$E C: B_{1} E=A C: D B_{1}=A E: E D=6: 2=3: 1$\n\nLet $E H \\perp D C \\Rightarrow$ right triangles ACD and EHD are similar ( $\\angle A C D-$ common), then $C D: H D=A C: E H=A D: E D=4: 1 \\Rightarrow E H=H D=1.5 \\Rightarrow C H=4.5$.\n\nRight triangles FHE and $\\mathrm{FCA}_{1}$ are similar $\\left(\\angle E F H=\\angle A_{1} F C\\right.$ vertical angles) $)$\n\nthen $C F: H F=A_{1} F: E F=A_{1} C: E H=6: 1.5=4: 1 ; C H=4.5 \\Rightarrow C F=0.8 \\cdot 4.5=3.6$\n\nAnswer: 3.6 .", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 33348, 216, 49165, 30, 365, 30129, 216, 49160, 595, 198, 198, 788, 260, 1048, 14973, 1885, 35542, 42, 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+{"group_id": "math_009", "origin": "budgie", "answer": "Let's start by considering the geometry of the graph from the provided image. The function is a parabola opening upwards, with its vertex at the origin. The equation is $\\frac{1}{12} x^2 + ax + b = 0$, which we can rewrite as $y = -\\frac{1}{12} x^2 - ax + b$. This is a downward-opening parabola, and its axis of symmetry is the x-axis.\n\nThe point T with coordinates $(3, 3)$ lies on the axis of symmetry. Since the parabola is symmetric about this line, the point T is also on the axis of symmetry. Therefore, the axis of symmetry of the parabola is the $y$-axis. This implies that the equation of the axis of symmetry is $y = b$.\n\nNow, we need to find the intersection points of the graph with the $O y$ axis. The point $A$ is the intersection of the parabola with the $y$ axis, which is the line $x = 0$. Therefore, we can substitute $x = 0$ into the equation $y = b$ to find the value of $b$.\n\n$$b = \\boxed{3}$$", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 33348, 216, 49160, 49159, 30, 49166, 30, 378, 3670, 282, 260, 1517, 1885, 86, 24, 104, 25, 24814, 18169, 107, 49160, 15009, 49160, 49161, 109, 1792, 21990, 49161, 109, 27, 81, 1792, 27, 82, 20, 42993, 260, 1885, 63, 1792, 20, 6867, 418, 2876, 1885, 49, 20, 284, 1885, 51, 26265, 284, 260, 1885, 63, 329, 20, 6867, 418, 1225, 1885, 50, 26265, 347, 3057, 281, 260, 4110, 30, 657, 4642, 578, 338, 327, 260, 1225, 1885, 68, 20, 351, 12957, 46408, 49162, 8544, 216, 49162, 25, 26265, 260, 2619, 1885, 68, 330, 45, 68, 389, 45, 68, 340, 20, 314, 14090, 30, 7985, 1885, 82, 28422, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 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+{"group_id": "math_009", "origin": "budgie", "answer": "Since the graph of the function $f(x)=\\frac{1}{12} x^{2}+a x+b$ intersects the $O x$ axis at points $A$ and $C$, and the $O y$ axis at point $B$, we have the following system of equations:\n$$\n\\begin{cases}\nf(A)=\\frac{1}{12} a+b=0 \\\\\nf(C)=\\frac{1}{12} a+b=0 \\\\\nf(B)=\\frac{1}{12} a+b=0\n\\end{cases}\n$$\n\nSimplifying the equations, we obtain:\n$$\n\\begin{cases}\na=\\frac{1}{3} \\\\\nb=-\\frac{12}{3}\n\\end{cases}\n$$\n\nThus, $a=\\boxed{\\frac{1}{3}}$.\n\nFor the point $T$ with coordinates $(3 ; 3)$, the condition $T A=T B=T C$ holds, which is satisfied.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 33348, 216, 49160, 49159, 30, 49166, 30, 378, 3670, 282, 260, 1517, 1885, 86, 24, 104, 25, 24814, 18169, 107, 49160, 15009, 49160, 49161, 109, 1792, 21990, 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+{"group_id": "math_009", "origin": "verified_reference", "answer": "Answer: -6.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-39.jpg?height=359&width=614&top_left_y=600&top_left_x=420)\n\nFig. 11: to the solution of problem 10.7\n\nSolution. Let point $A$ have coordinates $\\left(x_{1} ; 0\\right)$, and point $C$ have coordinates $\\left(x_{2} ; 0\\right)$. From the condition, it is clear that $x_{1} < 0$ and $x_{2} > 0$. Since $x_{1}$ and $x_{2}$ are the roots of the quadratic trinomial $f(x)$, by Vieta's theorem, we have $x_{1} \\cdot x_{2} = 12b$, from which we get $b = \\frac{x_{1} \\cdot x_{2}}{12} < 0$.\n\nLet $H$ be the point with coordinates $(3 ; 0)$ (Fig. 11). Clearly, in the isosceles triangle $A T C$, the segment $T H$ is the height, and therefore it is also the median. Thus, $3 - x_{1} = A H = H C = x_{2} - 3$, from which we get $x_{1} = 6 - x_{2}$.\n\nLet $M$ be the point with coordinates $(0, 3)$. Since $T H = T M = 3$ and $T A = T B$, the right triangles $A T H$ and $B T M$ are equal by the leg and hypotenuse. Therefore, $3 - x_{1} = H A = M B = 3 - b$, that is, $x_{1} = b = \\frac{x_{1} \\cdot x_{2}}{12}$ (by Vieta's theorem), from which we find $x_{2} = 12$. Finally, $x_{1} = 6 - x_{2} = 6 - 12 = -6$ and $b = x_{1} = -6$.\n\nAnother solution. As in the previous solution, let the abscissas of points $A$ and $C$ be $x_{1}$ and $x_{2}$, respectively; we will also use the fact that point $B$ has coordinates $(0 ; b)$. Immediately, we understand that $O A = |x_{1}| = -x_{1}$, $O C = |x_{2}| = x_{2}$, and $O B = |b| = -b$.\n\nLet's find the second intersection of the circle with the y-axis, let this be point $D$ with coordinates $(0 ; d)$ (Fig. 12). The chords $A C$ and $B D$ of the circle intersect at the origin $O$; from the properties of the circle, we know that $O A \\cdot O C = O B \\cdot O D$. We get $-x_{1} \\cdot x_{2} = -b \\cdot d$, from which, replacing $x_{1} \\cdot x_{2}$ with $12b$ by Vieta's theorem, we get $d = 12$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-40.jpg?height=521&width=571&top_left_y=83&top_left_x=437)\n\nFig. 12: to the solution of problem 10.7\n\nIt remains to note that triangle $BTD$ is isosceles, and the midpoint of its base, point $M$, has coordinates $(0 ; 3)$. 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+{"group_id": "math_010", "origin": "budgie", "answer": "Let $x$ be the length of theosal-O-L in centimeters.\n\nWe can set up a proportion based on the given information:\n\\[ \\frac{10}{x} = \\frac{75}{x + 75} \\]\n\nCross-multiply and solve for $x$:\n\\[ 10(x + 75) = 75x \\]\n\n\\[ 10x + 750 = 75x \\]\n\nSubtract $75x$ from both sides:\n\\[ -65x + 750 = 0 \\]\n\n\\[ 65x = 750 \\]\n\n\\[ x = 750 : 65 = \\boxed{10} \\]\n\nFinal answer:\n#### 10", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 216, 49162, 30, 8948, 10203, 12920, 1885, 49167, 2272, 216, 49165, 49164, 20, 198, 198, 49, 35735, 314, 1036, 15326, 288, 253, 3901, 2530, 30, 1626, 2621, 260, 3519, 282, 260, 35735, 28, 7288, 16028, 429, 260, 8618, 282, 260, 35735, 288, 260, 1112, 284, 2933, 216, 49160, 49164, 3301, 365, 5528, 3519, 216, 49167, 49159, 19573, 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+{"group_id": "math_010", "origin": "budgie", "answer": "Peter walks from the tip of thebinding sign to the end of thedeliberation sign, which is 150 steps long.\n\nSince his speed is constant, he will take 150 steps to the end of the Deliberation sign.\n\nThis is also the length of the Garcia-Petersen sign.\n\nTherefore, the length of theWA sign is 150 steps.\n\nThe answer is: $\\boxed{150}$", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 216, 49162, 30, 8948, 10203, 12920, 1885, 49167, 2272, 216, 49165, 49164, 20, 198, 198, 49, 35735, 314, 1036, 15326, 288, 253, 3901, 2530, 30, 1626, 2621, 260, 3519, 282, 260, 35735, 28, 7288, 16028, 429, 260, 8618, 282, 260, 35735, 288, 260, 1112, 284, 2933, 216, 49160, 49164, 3301, 365, 5528, 3519, 216, 49167, 49159, 19573, 8898, 5167, 107, 6946, 24362, 595, 198, 198, 2068, 4549, 1056, 429, 260, 1112, 288, 260, 8618, 282, 260, 35735, 979, 260, 4445, 314, 4138, 418, 253, 3836, 3547, 28, 384, 1923, 216, 49166, 49164, 3301, 418, 260, 1142, 6139, 3547, 30, 198, 198, 2020, 986, 314, 260, 35735, 47, 49154, 49158, 198, 29300, 16028, 429, 260, 8618, 282, 260, 26057, 914, 288, 260, 1112, 282, 260, 11074, 505, 2497, 914, 28, 527, 314, 216, 49160, 49164, 49159, 3301, 986, 30, 198, 198, 7230, 650, 3547, 314, 3836, 28, 384, 523, 1188, 216, 49160, 49164, 49159, 3301, 288, 260, 1112, 282, 260, 6374, 505, 2497, 914, 30, 198, 198, 1348, 314, 597, 260, 3519, 282, 260, 36217, 29, 64, 2922, 264, 914, 30, 198, 198, 17308, 28, 260, 3519, 282, 260, 15543, 914, 314, 216, 49160, 49164, 49159, 3301, 30, 198, 198, 504, 2988, 314, 42, 19573, 4810, 277, 107, 49160, 49164, 49159, 24362, 49154], "prefix_len": 138, "old_logprobs": [-1.529862642288208, -0.9334662556648254, -1.2981688976287842, -0.040169086307287216, -0.13930290937423706, -0.052779633551836014, -0.00245496048592031, -10.089473724365234, -7.112239837646484, -0.10736431181430817, -0.02028902806341648, -0.22162280976772308, -0.8292356729507446, -0.017735816538333893, -9.609851837158203, -3.571889877319336, -0.5101991295814514, -0.5783768892288208, -1.1507925987243652, -1.002791404724121, -0.5246886014938354, -0.3390176594257355, -0.7253702878952026, -0.22637566924095154, -1.1971886157989502, -1.4512660503387451, -1.1169061660766602, -0.485227108001709, -1.5278570652008057, -0.2769743800163269, -1.875618577003479, -3.418984889984131, -0.5003563165664673, -0.09346194565296173, -0.32593414187431335, -0.13046498596668243, -0.861644446849823, -1.6131227016448975, -0.6465285420417786, -0.32608410716056824, -0.8811635375022888, -0.07976793497800827, -0.1984155774116516, -0.38221266865730286, -1.0937529802322388, -5.303661823272705, -0.7923413515090942, -0.391355037689209, -0.07085540890693665, -2.1764492988586426, -0.024696078151464462, -0.041062138974666595, -0.0032298800069838762, -0.6901053786277771, -0.18792203068733215, -0.03515230491757393, -4.010353088378906, -2.481473922729492, -1.7644774913787842, -1.157983660697937, -2.7035489082336426, -0.04470505565404892, -0.5561898350715637, -10.625822067260742, -2.2919774055480957, -1.6565078496932983, -0.6162632703781128, -0.694914698600769, -2.468756675720215, -1.145898699760437, -0.10378701239824295, -0.007659352850168943, -1.3342466354370117, -0.011962460353970528, -0.3417043089866638, -0.5564751625061035, -0.002112539717927575, -0.09829550981521606, -10.26856803894043, -1.2196836471557617, -0.12456151098012924, -0.4014105498790741, -0.3481045365333557, -0.04279879853129387, -0.02802141383290291, -0.9259229302406311, -2.4354748725891113, -0.024871328845620155, -0.013503755442798138, -0.7680684924125671, -0.6343242526054382, -0.0038171776104718447, -0.8864362835884094, -3.210083484649658, -0.0001392267586197704, -5.960462772236497e-07, -0.0019972631707787514, -0.03017488494515419, -0.013067089952528477, -0.0019840572495013475, -0.08322035521268845, -0.1170370876789093]}
+{"group_id": "math_010", "origin": "verified_reference", "answer": "With $v=$ speed, $s=$ distance, $t_{1}=$ time for 15 steps, $t_{2}=$ time for 75 steps, and $l=$ crane length, we have $l-v t_{1}=12 \\mathrm{~m}, l+v t_{2}=60 \\mathrm{~m}$.\n\nIt holds that $\\frac{t_{1}}{t_{2}}=\\frac{1}{5} \\rightarrow t_{1}=\\frac{1}{5} t_{2}$.\n\nThus, $l-v \\cdot \\frac{1}{5} t_{2}=12 \\mathrm{~m}$ and $l+v t_{2}=60 \\mathrm{~m}$. Multiplying the first equation by 5 and adding both equations yields $6 l=120 \\mathrm{~m}, l=20 \\mathrm{~m}$. The length of the crane is $20 \\mathrm{~m}$.", "reward": 1.0, "advantage": 1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 216, 49162, 30, 8948, 10203, 12920, 1885, 49167, 2272, 216, 49165, 49164, 20, 198, 198, 49, 35735, 314, 1036, 15326, 288, 253, 3901, 2530, 30, 1626, 2621, 260, 3519, 282, 260, 35735, 28, 7288, 16028, 429, 260, 8618, 282, 260, 35735, 288, 260, 1112, 284, 2933, 216, 49160, 49164, 3301, 365, 5528, 3519, 216, 49167, 49159, 19573, 8898, 5167, 107, 6946, 24362, 595, 198, 198, 2068, 4549, 1056, 429, 260, 1112, 288, 260, 8618, 282, 260, 35735, 979, 260, 4445, 314, 4138, 418, 253, 3836, 3547, 28, 384, 1923, 216, 49166, 49164, 3301, 418, 260, 1142, 6139, 3547, 30, 198, 198, 2020, 986, 314, 260, 35735, 47, 49154, 49158, 198, 3825, 1885, 102, 45, 20, 3547, 28, 1885, 99, 45, 20, 4052, 28, 1885, 100, 11300, 49160, 109, 45, 20, 655, 327, 216, 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+{"group_id": "math_011", "origin": "budgie", "answer": "The distance from $C$ to $F$ is:\n\\[ CF = \\sqrt{(s - 0)^2 + (s - 0)^2} = \\sqrt{s^2 + s^2} = s\\sqrt{2} \\]\n\nThe only possible value of $s$ is 1, but this would mean that $A = B$ or $A = C$ or $A = D$ or $B = D$ or $F = A$ and $E = C$. However, this would mean that $A = F$ and $E = D$, which is a contradiction, so $s = 1$ is the only possible solution.\n\nThe distance from $F$ to $G$ is:\n\\[ FG = \\sqrt{(s - 0)^2 + (s - 0)^2} = s\\sqrt{2} \\]\n\nThe possible values of $s$ are 1 and 2, and the distances from $C$ to $F$ and $FG$ are the same. In this case, the area $AEFG$ is:\n\\[ AEFG = \\frac{1}{2} \\cdot 1 \\cdot s\\sqrt{2} = \\frac{s\\sqrt{2}}{2} \\]\n\nThe area of the figure $AEFG$ is $2AEFG = 2 \\cdot \\frac{s\\sqrt{2}}{2} = s\\sqrt{2}$.\n\nThe final answer is:\n#### \\frac{\\sqrt{2}s}{2}", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 1777, 344, 261, 494, 389, 29, 49160, 30, 49161, 30, 198, 198, 26062, 11998, 89, 393, 273, 20598, 1437, 8026, 7494, 45945, 13450, 273, 1932, 2231, 316, 30, 1885, 49, 389, 340, 422, 414, 20, 2056, 501, 102, 344, 6095, 1885, 49, 389, 426, 452, 28422, 17120, 455, 663, 283, 90, 259, 101, 501, 26747, 1885, 54, 330, 422, 28422, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49162, 49159, 79, 84, 49163, 49164, 49164, 721, 1992, 85, 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+{"group_id": "math_011", "origin": "budgie", "answer": "The problem mentions a rectangle in which $A B F G$ is a square. We need to determine the all possible positions of the square.\n\nTo start solving this problem, let's consider the different steps in the process.\n\nFirst, we need to identify the vertices of the square. The side length of the square is equal to the side length of the rectangle.\n\nLet's denote the side length of the rectangle as $x$. Then, the side length of the square is $x$.\n\nNow, we have a square with a side length of $x$, so its top vertex is $x$ units above the bottom face of the rectangle.\n\nNext, we can consider the other vertices of the square. The bottom face of the rectangle is a square with a side length of $x$. The top vertices of this square are $x$ units above the bottom face of the rectangle.\n\nNow, we have a square with a side length of $x$, so its bottom face is a square with a side length of $x$. The top vertices of this square are $x$ units above the bottom face of the rectangle.\n\nNext, we can consider the other vertices of the square. The left face of the rectangle is a square with a side length of $x$. The bottom vertices of this square are $x$ units above the left face of the rectangle.\n\nFinally, we can consider the other vertices of the square. The right face of the rectangle is a square with a side length of $x$. The top vertices of this square are $x$ units above the right face of the rectangle.\n\nNow, we have a square with a side length of $x$, so its top face is a square with a side length of $x$. 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+{"group_id": "math_011", "origin": "verified_reference", "answer": "## Rješenje.\n\nZbroj mjera unutrašnjih kutova pravilnog peterokuta iznosi $(5-2) 180^{\\circ}=540^{\\circ}$ pa je mjera unutrašnjeg kuta pravilnog peterokuta jednaka $\\frac{540^{\\circ}}{5}=108^{\\circ}$.\n\nBudući da je trokut $A E D$ jednakokračan, $\\varangle D A E=\\frac{180^{\\circ}-108^{\\circ}}{2}=36^{\\circ}$\n\nKut $\\varangle B A F=45^{\\circ}$ jer je to kut između dijagonale i stranice kvadrata.\n\nTada je $\\varangle F A D=\\varangle B A E-\\varangle B A F-\\varangle D A E=108^{\\circ}-45^{\\circ}-36^{\\circ}=27^{\\circ}$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 1777, 344, 261, 494, 389, 29, 49160, 30, 49161, 30, 198, 198, 26062, 11998, 89, 393, 273, 20598, 1437, 8026, 7494, 45945, 13450, 273, 1932, 2231, 316, 30, 1885, 49, 389, 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+{"group_id": "math_012", "origin": "budgie", "answer": "Since $x, y \\le 1$, we can simplify the inequality to:\n\\[ \\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1 \\]\n\nWe can simplify the inequality to:\n\\[ \\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1 \\]\n\nSince $n$ is a positive integer, we can simplify the inequality to:\n\\[ \\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1 \\]\n\nWe can simplify the inequality to:\n\\[ \\min_{1 \\le n \\le 10} \\frac{x^2+y^2}{4n^2} \\le 1 + \\frac{2 - \\frac{x^2+y^2}{4n^2-4n+1}}{4n^2} \\]\n\nSimplifying the inequality to:\n\\[ 1 \\le \\frac{x^2+y^2}{4n^2} + \\frac{2 - \\frac{x^2+y^2}{4n^2-4n", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 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+{"group_id": "math_012", "origin": "budgie", "answer": "1. **Identify the Function:**\n   The function to analyze is $f(x) = \\min_{1 \\leqslant n \\leqslant 10} \\max\\left(\\frac{x^2 + y^2}{4n^2}, \\, 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1}\\right)$.\n\n2. **Simplify the Function:**\n   We can simplify $f(x)$ by considering the expressions $\\frac{x^2 + y^2}{4n^2}$ and $\\frac{x^2 + y^2}{4n^2 - 4n + 1}$. Let's denote these as $p(x, y)$ for $x, y \\in \\mathbb{R}$.\n\n3. **Analyze the Sums:**\n   The function $f(x)$ can be rewritten as:\n   \\[\n   f(x) = \\min_{1 \\leqslant n \\leqslant 10} \\max\\left(\\frac{x^2 + y^2}{4n^2}, \\, 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1}\\right) = \\min_{n=1}^\\infty \\max\\left(\\frac{1}{4n^2}, \\, 2 - \\frac{1}{4n^2 - 4n + 1}\\right) = \\max\\left(\\frac{1}{4(1)}, \\, \\frac{2}{4(1) - 1}\\right) = \\max\\left(\\frac{1}{4}, \\frac{1}{3}\\right)\n   \\", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 504, 1557, 282, 260, 2695, 281, 260, 1885, 8860, 20, 29, 12758, 19131, 260, 9724, 3814, 40561, 1659, 11300, 49160, 3814, 290, 304, 3814, 290, 216, 49160, 49159, 109, 3814, 3831, 76, 8842, 19407, 18169, 107, 104, 78, 49161, 27, 105, 78, 49161, 15009, 49163, 94, 78, 49161, 6663, 3814, 28, 216, 49161, 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+{"group_id": "math_012", "origin": "verified_reference", "answer": "To find the area of the region in the $xy$-plane satisfying the inequality \n\\[\n\\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, \\, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1,\n\\]\nwe need to analyze the given inequality step by step.\n\n1. **Understanding the Inequality:**\n   The inequality involves two expressions:\n   \\[\n   \\max\\left(\\frac{x^2 + y^2}{4n^2}, 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1}\\right) \\le 1.\n   \\]\n   This means that for some integer $n$ in the range $1 \\le n \\le 10$, the maximum of the two expressions must be less than or equal to 1.\n\n2. **Breaking Down the Inequality:**\n   For the maximum of the two expressions to be less than or equal to 1, both expressions must individually be less than or equal to 1:\n   \\[\n   \\frac{x^2 + y^2}{4n^2} \\le 1 \\quad \\text{and} \\quad 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\le 1.\n   \\]\n\n3. **Simplifying Each Expression:**\n   - For the first expression:\n     \\[\n     \\frac{x^2 + y^2}{4n^2} \\le 1 \\implies x^2 + y^2 \\le 4n^2.\n     \\]\n   - For the second expression:\n     \\[\n     2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\le 1 \\implies \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\ge 1 \\implies x^2 + y^2 \\ge 4n^2 - 4n + 1.\n     \\]\n\n4. **Combining the Conditions:**\n   Therefore, for some $n$ in the range $1 \\le n \\le 10$, the point $(x, y)$ must satisfy:\n   \\[\n   4n^2 - 4n + 1 \\le x^2 + y^2 \\le 4n^2.\n   \\]\n   This describes a ring (annulus) in the $xy$-plane with inner radius $\\sqrt{4n^2 - 4n + 1}$ and outer radius $2n$.\n\n5. **Calculating the Area of Each Annulus:**\n   The area of an annulus with inner radius $r_1$ and outer radius $r_2$ is given by:\n   \\[\n   \\pi (r_2^2 - r_1^2).\n   \\]\n   For our specific case:\n   \\[\n   r_2 = 2n \\quad \\text{and} \\quad r_1 = \\sqrt{4n^2 - 4n + 1}.\n   \\]\n   Therefore, the area of the annulus is:\n   \\[\n   \\pi \\left((2n)^2 - (4n^2 - 4n + 1)\\right) = \\pi \\left(4n^2 - (4n^2 - 4n + 1)\\right) = \\pi (4n - 1).\n   \\]\n\n6. **Summing the Areas for $n$ from 1 to 10:**\n   We need to sum the areas of the annuli for $n = 1, 2, \\ldots, 10$:\n   \\[\n   \\sum_{n=1}^{10} \\pi (4n - 1) = \\pi \\sum_{n=1}^{10} (4n - 1).\n   \\]\n   The sum inside the parentheses is:\n   \\[\n   \\sum_{n=1}^{10} (4n - 1) = 4 \\sum_{n=1}^{10} n - \\sum_{n=1}^{10} 1 = 4 \\left(\\frac{10 \\cdot 11}{2}\\right) - 10 = 4 \\cdot 55 - 10 = 220 - 10 = 210.\n   \\]\n\n7. **Final Area:**\n   Therefore, the total area is:\n   \\[\n   210\\pi.\n   \\]\n\nThe final answer is $\\boxed{210}$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 504, 1557, 282, 260, 2695, 281, 260, 1885, 8860, 20, 29, 12758, 19131, 260, 9724, 3814, 40561, 1659, 11300, 49160, 3814, 290, 304, 3814, 290, 216, 49160, 49159, 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+{"group_id": "math_013", "origin": "budgie", "answer": "(1) For $n=1$, we have $a_{1}=2 a_{2}+3 a_{3}$, which simplifies to $2a_{2}=a_{3}$, therefore $a_{2}=a_{3}=\\frac{1}{2}$.\n\nFor $n\\geqslant 2$, the recursive formula $a_{n}=2 a_{n} a_{n+1}+3 a_{n+1}$ can be simplified to $a_{n+1}=a_{n}$.\n\nHence, $a_{n+1}=a_{n}$.\n\nTherefore, for all $n\\in \\mathbb{N}_{+}$, $a_{n}=\\frac{1}{n}$.\n\n(2) For $n=1$, we have $a_{1}=\\frac{1}{2}$, and for all $n\\in \\mathbb{N}_{+}$, $b_{n}=1+\\frac{1}{a_{n}}(n \\in \\mathbb{N}_{+})$.\n\nWe can rewrite $b_{n}$ as follows:\n\n$$b_{n}=1+\\frac{1}{a_{n}}\\cdot n \\in \\mathbb{N}_{+}$$\n\nThe given inequality becomes:\n\n$$1+\\frac{1}{a_{1}}\\cdot\\left(1+\\frac{1}{a_{2}}\\cdot\\left(1+\\frac{1}{a_{3}}\\cdot\\left(1+\\frac{1}{a_{4}}\\cdot\\left(1+\\frac{1}{a_{5}}\\cdot\\left(1+\\frac{1}{a_{6}}\\cdot\\", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 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+{"group_id": "math_013", "origin": "budgie", "answer": "(1) When $n=1$, $\\frac{1}{a_{1}}$, $a_{2}$, $a_{3}$, $\\ldots$ ;\nWhen $n=2$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $a_{3}$, $\\frac{1}{a_{3}}$, $\\ldots$;\nWhen $n=3$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $a_{4}$, $\\frac{1}{a_{4}}$, $\\ldots$;\nWhen $n=4$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $\\frac{1}{a_{3}}$, $\\frac{1}{a_{4}}$;\nWhen $n=5$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $\\frac{1}{a_{3}}$, $\\frac{1}{a_{4}}$;\n...\nWhen $n=49$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $\\frac{1}{a_{3}}$, $\\frac{1}{a_{4}}$...\n$\\end{aligned}$\n\n(2) Since for any positive integer $n(n \\geqslant 2)$, the inequality\n$\\sum_{k=1}^{n} \\frac{1}{n+\\log _{3} b_{k}}>\\frac{m}{24}$\n\nalways holds,\n\n$\\therefore$ The sequence $\\left\\{b_{n}\\right\\}$ is monotonically increasing,\n\n$\\therefore b_{1}<", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 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+{"group_id": "math_013", "origin": "verified_reference", "answer": "(1) From\n$$\n\\begin{array}{l}\na_{n}=2 a_{n} a_{n+1}+3 a_{n+1} \\\\\n\\Rightarrow a_{n+1}=\\frac{a_{n}}{2 a_{n}+3} \\Rightarrow a_{n}>0 . \\\\\n\\text { Hence } \\frac{1}{a_{n+1}}-\\frac{3}{a_{n}}=2 \\Rightarrow \\frac{1}{a_{n+1}}+1=3\\left(\\frac{1}{a_{n}}+1\\right) .\n\\end{array}\n$$\n\nTherefore, $\\left\\{\\frac{1}{a_{n}}+1\\right\\}$ is a geometric sequence with the first term 3 and common ratio 3.\nThus, $\\frac{1}{a_{n}}+1=3^{n} \\Rightarrow a_{n}=\\frac{1}{3^{n}-1}\\left(n \\in \\mathbf{N}_{+}\\right)$.\n(2) From (1), we get $b_{n}=1+\\frac{1}{a_{n}}=3^{n}$.\nTherefore, $\\log _{3} b_{k}=k(k=1,2, \\cdots)$.\nHence, $\\sum_{k=1}^{n} \\frac{1}{n+\\log _{3} b_{k}}>\\frac{m}{24}$\n$\\Leftrightarrow \\frac{1}{n+1}+\\frac{1}{n+2}+\\cdots+\\frac{1}{n+n}>\\frac{m}{24}$.\nLet $f(n)=\\frac{1}{n+1}+\\frac{1}{n+2}+\\cdots+\\frac{1}{n+n}$. 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+{"group_id": "math_015", "origin": "budgie", "answer": "a) To show that this could never have happened before and that, after the next birthday of\"{ parent_id }\", it will never happen again, we need to find a counterexample.\n\nConsider the following scenario:\n\n- The oldest daughter is 49 years old.\n- The youngest daughter is 24 years old.\n- The oldest daughter is 48 years old.\n- The youngest daughter is 24 years old.\n\nAt the next birthday of their father, as mentioned before, each daughter's age at that time is halved, and in 10 years, this will occur two times.\n\n- The oldest daughter will be 14 years old, and the youngest daughter will be 8 years old.\n- The oldest daughter will be 12 years old, and the youngest daughter will be 6 years old.\n- The youngest daughter will be 6 years old, and the oldest daughter will be 4 years old.\n\nIn this case, the oldest daughter is 6 years old.\n\nb) To determine if it is possible for the oldest daughter to be over 30 years old, we can see that she will be 30 years old only if she is 30 - 24 = 6 years old.\n\nSince the oldest daughter is 6 years old, we know that the youngest daughter is 24 - 6 = 18 years old.\n\nTherefore, Máiding will be 24 - 6 = 18 years old, and his son will be 18 + 6 = 24 years old.\n\nIn conclusion, there is a misunderstanding in the problem, as the statement \"All three daughters were born on March 18\" is not true. 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+{"group_id": "math_015", "origin": "budgie", "answer": "a) 10 years ago, Alina was 36 years old.\n2014 - 10 = 2004.\n2004 / 3 = 668.\nAlina's age is now double the age of Valentina, who was 36 years old 10 years ago.\nA 10-year-old daughter, the product of socks and the age of each sock is 10! The socks are 6 feet, 2 feet each.\nHence, Alina is 6 times 2 feet old.\nAlina is now 36 * 2 = 72 years old.\nIn 10 years, Alina will be 72 + 10 = 82 years old.\nThus, in 10 years, Traditional will be 82 x 3 = 216 years old.\nThe oldest of the daughters will be 216 - 10 = 206 years old in 10 years.\nThis fact can never have happened before and it can never happen again.\nb) \nThe oldest of the daughters is over 30 years old...\n#### 30", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 37004, 16462, 553, 216, 49162, 16924, 42, 1065, 1787, 28, 22488, 1787, 28, 284, 31492, 10945, 30, 1117, 3684, 1027, 314, 338, 511, 1296, 16924, 592, 3988, 335, 3903, 216, 49160, 49167, 30, 4104, 28, 3903, 216, 49160, 49167, 28, 216, 49161, 49159, 49160, 49163, 28, 314, 480, 15208, 30, 3349, 274, 1372, 7436, 1027, 28, 10463, 1921, 42, 198, 198, 29, 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+{"group_id": "math_015", "origin": "verified_reference", "answer": "Solution\n\na) It is clear that Alina is older than Valentina. Let's call the difference in age between Alina and Valentina $d$. Since they were born on the same day of the year, $d$ does not vary (note that this would not be true if they were born on different days, as in that case, the difference would change on the birthday of each one).\n\nIf Valentina's age today is $\\alpha$, then Alina's age must be $2 \\alpha$, so the age difference between the two is given by:\n\n$$\nd=2 \\alpha-\\alpha\n$$\n\nWe conclude that $d=\\alpha$, so Alina is $2 d$ years old and Valentina is $d$ years old.\n\nSuch a relationship between the ages of Alina and Valentina could not have happened before, as Valentina was younger than $d$ years old. It also will not happen again, as Valentina will be older than $d$ years.\n\nb) Let's call the age of the youngest daughter $A$, the age of the middle daughter $B$, and the age of the oldest daughter $C$. Let $x=B-A$ be the age difference between the two youngest, and $y=C-B$ be the age difference between the two oldest daughters. The age difference between the youngest and the oldest is therefore $C-A=x+y$. Since they were born on the same day of the year, these differences do not vary.\n\nIt is known that in each of the years 2004, 2014, and 2024, the age of one of the daughters was, is, or will be double the age of one of the others. For this, there are only 3 possibilities:\n\n(i) $B$ is double $A$.\n\n(ii) $C$ is double $B$.\n\n(iii) $C$ is double $A$.\n\nBy the reasoning used in part a), we know that each situation can only occur in one of the three years. Therefore, the three situations: (i), (ii), and (iii) must happen in the three years: 2004, 2014, and 2024, but we do not yet know in what order. Also, from part a), we know that:\n\n- When (i) happens, $B$ must be equal to $2 x$ and $A$ must be equal to $x$. Moreover, $C$ is equal to $B+y$, that is, $C$ must be equal to $2 x+y$.\n- When (ii) happens, $B=y, C=2 y$, and therefore, $A=y-x$.\n- When (iii) happens, $C=2(x+y), A=x+y$, and therefore, $B=2 x+y$.\n\nObserve that the ages of $A$ in cases (i), (ii), and (iii) are\n\n$$\nx, \\quad y-x \\quad \\text { and } \\quad x+y\n$$\n\nrespectively. Since $x<x+y$ and $y-x<x+y$, we conclude that $x+y$ is the largest of the three ages and, therefore, (iii) must happen in the year 2024. Only remains to see in which of the two years 2004 or 2014 each of the events in (i) and (ii) happens. Let's assume that (i) and (ii) happen in the years 2014 and 2004, respectively. Then, the ages of $A$ in the years 2004, 2014, and 2024 would be $y-x, x$, and $x+y$, respectively. Thus, we would have the equations\n\n$$\ny-x+10=x \\quad \\text { and } \\quad x+10=x+y\n$$\n\nSolving, we would have $x=10$ and $y=10$. But in that case, $C$ would be today equal to $2 x+y=30$. But the oldest daughter is more than 30 years old, as stated in the problem. We discard, then, this possibility.\n\nWe know then that (i) and (ii) happen in the years 2004 and 2014, respectively. Then, the ages of $A$ in the years 2004, 2014, and 2024 are $x, y-x$, and $x+y$, respectively. Therefore, we have that\n\n$$\nx+10=y-x \\quad \\text { and } \\quad y-x+10=x+y .\n$$\n\nSolving, we get $x=5$ and $y=20$. Finally, the ages of Francisco's daughters today are equal to\n\n$$\ny-x=15, \\quad y=20 \\quad \\text { and } \\quad 2 y=40\n$$\n\nSince Alina's age must be double Valentina's age, the ages of Alina and Valentina must be 40 and 20, respectively. 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+{"group_id": "math_016", "origin": "budgie", "answer": "From the problem, we know that $a_{1}=0$ and $a_{2}=1$.\n\nFor $n \\geqslant 3$, we have the following condition:\n$$a_{n} = a_{1} + (n-2)d$$\n\nwhere $d$ is the common difference.\n\nGiven that $a_{n}$ must be the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \\cdots, a_{n}$ that forms an arithmetic sequence, we know that $d < a_{n-1}$.\n\nTherefore, we have the following inequalities:\n\\begin{align*}\na_{n} &= a_{1} + (n-2)d \\\\\na_{n-1} &= a_{1} + (n-3)d \\\\\na_{n-2} &= a_{1} + (n-4)d\n\\end{align*}\n\nAdding these two inequalities together, we get:\n$$a_{n} + a_{n-2} &= (a_{1} + (n-2)d) + (a_{1} + (n-4)d) \\\\\n&= 2a_{1} + (n-3)d$$\n\nSince $a_{1} = 0$, we have:\n$$2a_{1} + (n-3)d = 0$$\n\nDividing both sides by 2, we get:\n$$a_{1} + \\frac{n-3}{2}d = 0$$\n\nSince $a_{1} = 0$, we have:\n$$0 + \\", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 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+{"group_id": "math_016", "origin": "budgie", "answer": "Assume, by the method of differences, that $a_{n-1} = b_{n-1} - b_{n-2}$, where $b_{n-1}$ is the largest index such that $a_{n-1} > b_{n-2}$.\n\nThen, $a_{n} = b_{n-1} + b_{n-2}$, and the subsequence $a_{n-1}, a_{n-2}, \\cdots$ is the arithmetic sequence $b_{n-1} - b_{n-2}, b_{n-2} - b_{n-1}, \\cdots$.\n\nTherefore, $a_{n-1} > a_{n-2} > b_{n-3} > b_{n-4} \\cdots$, which contradicts $a_{n} = b_{n-1} + b_{n-2}$.\n\nHence, $a_{n-1} = b_{1} + b_{2} + \\cdots + b_{n-n}$.\n\nSince $a_{1} = 0$, the sequence $\\{a_{n}\\}$ is a geometric sequence with a common ratio of $b_{1} + b_{2} + \\cdots + b_{n-n}$.\n\nWe know that $a_{n-1} = b_{1} + b_{2} + \\cdots + b_{n-n-1}$.\n\nBy the property of geometric sequences, we have:\n\n$$a_{n} = b_{1} + (b_{1} + b_{2} + \\cdots + b_{n-n}) = b_{1} + (b_{1} + (b_{1} +", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 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+{"group_id": "math_016", "origin": "verified_reference", "answer": "4. First, prove a lemma using mathematical induction.\n\nLemma A non-negative integer appears in the sequence if and only if its ternary expansion contains only 0 and 1.\nProof It is obvious that the proposition holds for 0.\nAssume the proposition holds for all non-negative integers less than \\( N \\), and consider \\( N \\).\n\nIf the ternary expansion of \\( N \\) contains the digit 2, replace all occurrences of 2 with 0 to get the number \\( N_{0} \\); replace all occurrences of 2 with 1 to get the number \\( N_{1} \\). Thus, the ternary expansions of \\( N_{0} \\) and \\( N_{1} \\) do not contain the digit 2.\nBy the induction hypothesis, the numbers \\( N_{0} \\) and \\( N_{1} \\) are in the sequence.\nSince \\( N_{0} \\), \\( N_{1} \\), and \\( N \\) form an arithmetic sequence, \\( N \\) does not appear in the sequence. If the ternary expansion of \\( N \\) does not contain the digit 2, then we need to prove that \\( N \\) must appear in the sequence. If not, then there exist terms \\( N_{0} \\) and \\( N_{1} \\) in the sequence such that \\( N_{0} \\), \\( N_{1} \\), and \\( N \\) form an arithmetic sequence. Let the common difference be \\( d \\), and \\( 3^{k} \\| d \\). Then the ternary expansions of these three numbers have the same lowest \\( k-1 \\) digits, and the \\( k \\)-th digit is different for each pair, so one of the numbers must have a 2 in the \\( k \\)-th digit, which contradicts the assumption.\nReturning to the original problem.\nWe only need to find the 2014th non-negative integer whose ternary expansion does not contain the digit 2.\n\nNotice that under this restriction, the ternary carry-over method is equivalent to binary, so we just need to write 2013 (note \\( a_{1}=0 \\)) in binary \\((11111011101)_{2}\\) and then convert it to ternary \\( a_{2014}=88327 \\).", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49163, 30, 2959, 260, 5972, 19573, 8842, 76, 107, 81, 11300, 94, 17243, 2771, 76, 24362, 15765, 1885, 81, 11300, 49160, 109, 45, 49159, 28, 253, 11300, 49161, 109, 45, 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+{"group_id": "math_017", "origin": "budgie", "answer": "To determine the smallest positive integer $d$ for which the two numbers $m+1947n$ and $n+1947m$ share a common prime divisor $d > 9$, we need to examine the properties of the greatest common divisor (GCD) of the two numbers.\n\nLet's express the two numbers in terms of their GCD $d$. We can write:\n\\[ m = d \\times a \\]\n\\[ n = d \\times b \\]\nwhere $a$ and $b$ are integers and $a$ and $b$ are relatively prime (i.e., their GCD is 1).\n\nNow, we can express $m+1947n$ and $n+1947m$ in terms of $a$ and $b$:\n\\[ m+1947n = d(a+b) \\]\n\\[ n+1947m = d(a+b) \\]\n\nSince $a$ and $b$ are relatively prime, we can write $a = 1$ and $b = 1$, as they are relatively prime.\n\nSubstituting these values into the equations, we get:\n\\[ m+1947n = d(1+1) = 2d \\]\n\\[ n+1947m = d(1+1) = d \\]\n\nNow, we can see that $d$ must be equal to either $2$ or $d = 2$.\n\nIf $d = 2$, then $m+1947n$ and $n+1947m$ share a common divisor $2$ with $d=2$, which is $2", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 657, 314, 1343, 338, 327, 634, 2731, 4078, 98, 556, 2966, 1885, 93, 20, 284, 1885, 94, 26265, 260, 2966, 1885, 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+{"group_id": "math_017", "origin": "budgie", "answer": "1. Let's denote $k$ as the common prime divisor of $m$ and $n$. We can express $m+1947 n$ and $n+1947 m$ in terms of $k$.\n\n\\[ m+1947 n = k(m+n) \\]\n\n\\[ n+1947 m = k(n+m) \\]\n\nSince $m$ and $n$ are coprime, $k$ is a positive integer. Also, since $m$ and $n$ are positive integers, $k$ must also be a positive integer.\n\nWe have the following system of equations:\n\n\\[ m+n = kn \\]\n\n\\[ m+n = kn+km \\]\n\nAdding these two equations, we get:\n\n\\[ 2mn = 2kn + 2k \\cdot kn \\]\n\n\\[ (m+n)^2 = 2kn(k+1) \\]\n\nSince $m+n$ and $n+m$ are both coprime, $k(k+1)$ is a prime number. Therefore, $2kn$ must be a prime number as well.\n\nSince $k$ is a positive integer, $2n$ must also be a prime number. This implies that $n$ is a prime number.\n\nNow, since $m$ and $n$ are coprime, $m$ and $n$ must each be a prime number. Let $m = p$ and $n = p$, where $p$ is a positive integer.\n\nSubstituting $m = p$ and $n = p$ into the first equation, we get:\n\n\\[ p + p", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 657, 314, 1343, 338, 327, 634, 2731, 4078, 98, 556, 2966, 1885, 93, 20, 284, 1885, 94, 26265, 260, 2966, 1885, 93, 27, 49160, 49168, 49163, 49166, 304, 20, 284, 1885, 94, 27, 49160, 49168, 49163, 49166, 283, 20, 457, 253, 1273, 9552, 3044, 40031, 1885, 84, 46, 49168, 28422, 7985, 260, 13341, 1636, 1685, 282, 260, 1230, 1885, 84, 20, 656, 623, 1920, 30, 49154, 49158, 198, 49160, 30, 2959, 506, 25832, 1885, 91, 20, 347, 260, 1273, 9552, 3044, 40031, 282, 1885, 93, 20, 284, 1885, 94, 28422, 1046, 416, 2667, 1885, 93, 27, 49160, 49168, 49163, 49166, 304, 20, 284, 1885, 94, 27, 49160, 49168, 49163, 49166, 283, 20, 281, 2656, 282, 1885, 91, 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+{"group_id": "math_017", "origin": "verified_reference", "answer": "Answer: $d_{\\min }=139$.\n\nFor example,\n\n$$\n\\begin{aligned}\n& m=138, n=1 \\rightarrow 1947 m+n=1947 \\cdot 138+1=139 \\cdot 1933 \\\\\n& m+1947 n=138+1947=139 \\cdot 15\n\\end{aligned}\n$$", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 657, 314, 1343, 338, 327, 634, 2731, 4078, 98, 556, 2966, 1885, 93, 20, 284, 1885, 94, 26265, 260, 2966, 1885, 93, 27, 49160, 49168, 49163, 49166, 304, 20, 284, 1885, 94, 27, 49160, 49168, 49163, 49166, 283, 20, 457, 253, 1273, 9552, 3044, 40031, 1885, 84, 46, 49168, 28422, 7985, 260, 13341, 1636, 1685, 282, 260, 1230, 1885, 84, 20, 656, 623, 1920, 30, 49154, 49158, 198, 21350, 42, 1885, 84, 11300, 76, 1659, 4029, 45, 49160, 49162, 49168, 28422, 198, 198, 2193, 1183, 28, 198, 198, 9212, 198, 76, 21083, 107, 33270, 109, 198, 22, 283, 45, 49160, 49162, 49167, 28, 304, 45, 49160, 3814, 2771, 5281, 216, 49160, 49168, 49163, 49166, 283, 27, 94, 45, 49160, 49168, 49163, 49166, 3814, 41591, 216, 49160, 49162, 49167, 27, 49160, 45, 49160, 49162, 49168, 3814, 41591, 216, 49160, 49168, 49162, 49162, 28372, 198, 22, 283, 27, 49160, 49168, 49163, 49166, 304, 45, 49160, 49162, 49167, 27, 49160, 49168, 49163, 49166, 45, 49160, 49162, 49168, 3814, 41591, 216, 49160, 49164, 198, 76, 486, 107, 33270, 109, 198, 9212, 49154], "prefix_len": 96, "old_logprobs": [-8.255396842956543, -0.06686603277921677, -3.6822292804718018, -0.7856916189193726, -6.256600856781006, -1.5826562643051147, -1.8330450057983398, -3.076355457305908, -0.7772067785263062, -1.0350090265274048, -2.794403076171875, -3.0637598037719727, -2.408771514892578, -0.6834844350814819, -0.024125806987285614, -5.4597272872924805, -1.680654764175415, -0.1114504411816597, -4.3503570556640625, -0.9929459095001221, -0.4503622055053711, -2.9832370281219482, -0.8010879158973694, -0.09334076941013336, -0.0006517431465908885, -1.5331660509109497, -0.0023445994593203068, -0.1352905035018921, -0.5642392635345459, -1.4848933219909668, -0.5996435880661011, -0.9302833080291748, -1.6878135204315186, -2.821399450302124, -1.037066102027893, -1.2712712287902832, -0.03892925754189491, -0.7761404514312744, -4.901301383972168, -3.8517258167266846, -0.00037353215157054365, -1.8723373413085938, -0.2766309976577759, -3.080369472503662, -0.44148731231689453, -0.19885674118995667, -5.6482253074646, -1.317383885383606, -2.94870924949646, -0.3810924291610718, -0.7201460599899292, -0.7355027198791504, -0.19103261828422546, -0.1093345582485199, -1.5462276935577393, -1.1664506196975708, -0.018977854400873184, -0.025431297719478607, -1.869305968284607, -0.09163767099380493, -0.7983227372169495, -0.002969144145026803, -0.7642041444778442, -2.384397506713867, -4.388052940368652, -1.2131673097610474, -4.79478645324707, -0.6617344617843628, -0.1591063141822815, -0.25342801213264465, -0.9685747623443604, -3.695786476135254, -3.1779181957244873, -1.5064858198165894, -0.13558658957481384, -0.021712806075811386, -2.0364482402801514, -0.6807119846343994, -0.7646220326423645, -0.02001470886170864, -0.003829765599220991, -0.002971640322357416, -0.07327891141176224, -0.3972114324569702, -0.08197635412216187, -0.5401096940040588, -0.44117647409439087, -0.17422930896282196, -0.010364635847508907, -0.17060358822345734, -0.01253335177898407, -0.004531471990048885, -1.666386604309082, -3.81386661529541, -2.830246686935425, -1.4262957572937012, -7.138075828552246, -0.27800849080085754, -0.02956560254096985, -0.08346811681985855, -4.835251808166504, -4.22920560836792, -0.06690104305744171, -0.001206861576065421, -5.709961988031864e-05, -0.005302887409925461, -0.016429614275693893, -0.0033410938922315836, -0.0029131616465747356, -3.315094470977783]}
+{"group_id": "math_018", "origin": "budgie", "answer": "To solve this problem, we need to find the length of $AE$ and $ED$ and use them to calculate the area of quadrilateral $A E G D$. We can do this by using the properties of similar triangles and the given values.\n\nLet's denote $AE = x$, $EB = y$, $BC = z$, $DF = w$, $DF$ intersects $AC$ at $G$. By the properties of similar triangles, we have:\n\n$$\\frac{AE}{EB} = \\frac{DF}{BC}$$\n\nSubstitute the given values for $AE$, $EB$, $BC$, and $DF$:\n\n$$\\frac{x}{y} = \\frac{w}{z}$$\n\nNow, we can simplify this equation:\n\n$$xz = yw$$\n\nWe also know that $x = \\frac{2}{3}y$, $y = \\frac{1}{2}z$, and $w = \\frac{3}{4}z$.\n\nSubstitute these values into the equation:\n\n$$\\frac{2}{3}y\\cdot\\frac{3}{4}z = y\\cdot\\frac{1}{2}z\\cdot\\frac{3}{4}z$$\n\nSimplify this equation:\n\n$$\\frac{2}{3}\\cdot\\frac{3}{4}z\\cdot\\frac{3}{4}z = y\\cdot\\frac{y}{2}z$$\n\n$z\\cdot z = y^2$\n\nSince the area of square $ABCDE$ is $840$ square centimeters, we can write the area of quadrilateral $A E G D$ as:\n\n$840 = x^2", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 365, 49160, 49159, 2876, 25, 1032, 3057, 281, 260, 4110, 28, 260, 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+{"group_id": "math_018", "origin": "budgie", "answer": "Let $x=AE$. Then $BE=x-10$. Since $BE=200$, we have $x=200+10=210$. Therefore, $DE=210-x=210-210=0$. The area of quadrilateral $A E G D$ is the sum of the areas of square $A B C D$ and triangles $A B E$ and $A E G D$ and quarter-circle $F C E$. We have $[A B C D] + [A B E] + [A AE G] + [A DE G] + [F C E] = 840 + [A E G D] + 1900 + 3400 + [210*(10-210)/2].$ Therefore, the area of the region is $x^2=10000-3318000=14200$. So, the area of quadrilateral $A E G D$ is $\\boxed{14200}$.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 365, 49160, 49159, 2876, 25, 1032, 3057, 281, 260, 4110, 28, 260, 1557, 282, 5222, 1885, 49, 389, 340, 422, 20, 314, 216, 49167, 49163, 49159, 5222, 25556, 28, 1885, 49, 414, 45, 53, 389, 28, 389, 426, 45, 49161, 426, 340, 28, 422, 426, 20, 42993, 1885, 53, 340, 20, 418, 1885, 55, 28422, 3875, 260, 1557, 282, 20378, 13293, 1885, 49, 414, 452, 422, 20, 314, 5222, 25556, 30, 49154, 49158, 198, 4239, 1885, 104, 45, 25331, 28422, 3875, 1885, 19698, 45, 104, 29, 49160, 49159, 28422, 4311, 1885, 19698, 45, 49161, 49159, 49159, 26265, 392, 457, 1885, 104, 45, 49161, 49159, 49159, 27, 49160, 49159, 45, 49161, 49160, 49159, 28422, 4882, 28, 1885, 6279, 45, 49161, 49160, 49159, 29, 104, 45, 49161, 49160, 49159, 29, 49161, 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+{"group_id": "math_018", "origin": "verified_reference", "answer": "510\n【Analysis】This figure easily leads one to think about finding the areas of triangles $BCE$, $CDF$, and $CGF$. The most challenging part is finding the area of $\\triangle CGF$. According to the given conditions, we should draw $GM \\perp BC$ with the intersection point at $M$. This forms two sets of similar triangles corresponding to $\\triangle BCE$ and $\\triangle CDF$. By using the property that the ratio of the areas of similar triangles is equal to the square of the similarity ratio, we can find the area of $\\triangle CGF$. Finally, based on the figure, we can calculate the area of the required shape.\n\n【Solution】Draw $GM \\perp BC$ with the intersection point at $M$,\n$$\n\\begin{array}{l}\n\\therefore \\triangle FMG \\sim \\triangle FCD \\Rightarrow FM: FC = MG: CD, \\\\\n\\because BF = 2FC \\Rightarrow BC = 3FC, \\\\\n\\therefore MG = 3FM, \\\\\n\\because \\triangle CGM: \\triangle CEB \\Rightarrow CM: CB = GM: BE, \\\\\nBC = 2BE, \\\\\n\\therefore GM = \\frac{1}{2} CM = 3FM \\Rightarrow CM = 6FM, \\\\\n\\therefore FM: FC = 1: 7, \\\\\nCM: CB = 2: 7, \\\\\nS \\triangle BCE = \\frac{1}{4} \\times \\square ABCD = 210, \\\\\nS \\triangle CGM = 4 \\div 49 \\times 210 = 840 \\div 49 = 120 \\div 7, \\\\\nS \\triangle CDF = S \\square ABCD \\div 6 = 140, \\\\\nS \\triangle MGF = 140 \\times 1 \\div 49 = 140 \\div 49 = 20 \\div 7, \\\\\nS \\triangle CGM + \\triangle MGF = 120 \\div 7 + 20 \\div 7 = 20, \\\\\n840 - 210 - 140 + 20 = 510 \\text{ (square centimeters). }\n\\end{array}\n$$\n\nTherefore, the area of 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+{"group_id": "math_019", "origin": "budgie", "answer": "The volume of a tetrahedron is given by the formula $V = \\frac{1}{3} \\times \\text{area of base} \\times \\text{height}$.\n\nSince $ABCD$ is a triangle, the height is the distance from the vertex $D$ to the line $CD$. We can find the area of triangle $ABC$ using the formula for the area of a triangle: $\\frac{1}{2} \\times \\text{base} \\times \\text{height}$.\n\nSince the distance from $D$ to $CD$ is 2, the base of the triangle is 2. Therefore, the area of triangle $ABC$ is:\n$$\\frac{1}{2} \\times 2 \\times 2 = 2$$\n\nNow, we need to find the volume of the tetrahedron. We can do this by multiplying the area of triangle $ABC$ by the height from $D$ to line $CD$.\n\n$$V = \\frac{1}{3} \\times 2 \\times 2 = \\frac{4}{3}$$\n\nTherefore, the volume of tetrahedron $ABCD$ is $P = \\frac{4}{3}$.\n\nThe answer is:\n\\[ \\boxed{\\frac{4}{3}} \\]\nFinal answer:\n#### \\frac{4}{3}", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 533, 29889, 39513, 1885, 5431, 6998, 26265, 1303, 1885, 5431, 45, 49160, 26265, 1885, 6998, 24814, 16166, 107, 49162, 24362, 28, 260, 4052, 826, 3204, 1885, 5431, 20, 284, 1885, 6998, 20, 314, 216, 49161, 28, 284, 260, 7256, 826, 601, 314, 19573, 18169, 26754, 9020, 15009, 49162, 24362, 30, 3875, 260, 4981, 282, 29889, 39513, 1885, 5431, 6998, 20, 314, 365, 15165, 198, 24, 49161, 49159, 49159, 49162, 2071, 4264, 3056, 12406, 15232, 25, 198, 49, 30, 19573, 18169, 26754, 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+{"group_id": "math_019", "origin": "budgie", "answer": "1. Let $P$ be the foot of the perpendicular dropped from point $A$ to the plane of tetrahedron $ABCD$. Since the height of the tetrahedron from $A$ to plane $ABCD$ is the perpendicular distance between $AB$ and the plane of $ABCD$, we have $AP = AB = 1$.\n2. Now, let $|AP| = 1$, $|BP| = 2$, and $|CD| = \\sqrt{3}$. Then, by the Pythagorean theorem, we have $|BP| = \\sqrt{|BP|^2 + |BP|^2} = \\sqrt{4} = 2$.\n3. The volume of a tetrahedron is given by the formula $V = \\frac{1}{3} \\times \\text{area of base} \\times \\text{height}$. Here, the area of the base is the area of triangle $ABP$, and the height is the distance from $P$ to the plane $ABCD$.\n4. Therefore, the volume of tetrahedron $ABCD$ is $V = \\frac{1}{3} \\times 1 \\times 1 \\times \\frac{\\sqrt{3}}{2} = \\boxed{\\frac{\\sqrt{3}}{6}}$.\nFinal answer:\n#### \\frac{\\sqrt{3}}{6}", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 533, 29889, 39513, 1885, 5431, 6998, 26265, 1303, 1885, 5431, 45, 49160, 26265, 1885, 6998, 24814, 16166, 107, 49162, 24362, 28, 260, 4052, 826, 3204, 1885, 5431, 20, 284, 1885, 6998, 20, 314, 216, 49161, 28, 284, 260, 7256, 826, 601, 314, 19573, 18169, 26754, 9020, 15009, 49162, 24362, 30, 3875, 260, 4981, 282, 29889, 39513, 1885, 5431, 6998, 20, 314, 365, 15165, 198, 24, 49161, 49159, 49159, 49162, 2071, 4264, 3056, 12406, 15232, 25, 198, 49, 30, 19573, 18169, 26754, 16166, 107, 49162, 109, 15009, 49161, 24362, 198, 50, 30, 19573, 18169, 107, 49160, 15009, 49161, 24362, 198, 51, 30, 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+{"group_id": "math_019", "origin": "verified_reference", "answer": "1. Choose B. Reason: The area of the parallelogram formed by $\\overrightarrow{A B}$ and $\\overrightarrow{C D}$ is $|\\overrightarrow{A B}| \\cdot|\\overrightarrow{C D}| \\cdot \\sin 60^{\\circ}=\\frac{3}{2}$. If a line $C E \\mathbb{E}$ is drawn through point $C$ parallel to $A B$, then $S_{\\triangle C D E}=\\frac{1}{2} \\cdot \\frac{3}{2}=\\frac{3}{4}$, and the volume of the prism $A B F-E C D$ with $\\triangle C D E$ as the base and $B C$ as the side edge is 3 times that of $V_{A B C D}$, and $V_{A B F-E C D}=S_{\\triangle C D E} \\cdot 2=\\frac{3}{2}$, hence $V_{A B C D}=\\frac{1}{2}$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 533, 29889, 39513, 1885, 5431, 6998, 26265, 1303, 1885, 5431, 45, 49160, 26265, 1885, 6998, 24814, 16166, 107, 49162, 24362, 28, 260, 4052, 826, 3204, 1885, 5431, 20, 284, 1885, 6998, 20, 314, 216, 49161, 28, 284, 260, 7256, 826, 601, 314, 19573, 18169, 26754, 9020, 15009, 49162, 24362, 30, 3875, 260, 4981, 282, 29889, 39513, 1885, 5431, 6998, 20, 314, 365, 15165, 198, 24, 49161, 49159, 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+{"group_id": "math_022", "origin": "budgie", "answer": "Okay, let me think about this. I need to find the largest natural number where all digits are different and adjacent digits differ by 6 or 7. Hmm. So, the idea is that this number has to have as many digits as possible, with all digits unique and the adjacency condition being 6 or 7. \n\nAlright, so first, let's understand the problem. We need a number that uses digits, with all different and adjacent digits differing by 6 or 7. Let's denote the digits as a, b, c, d, e, where a ≠ b, a ≠ c, a ≠ d, b ≠ c, b ≠ d, c ≠ d, and all digits are different. The question is asking for the maximum possible number under these constraints.\n\nSo, the problem is about constructing such a number with as many digits as possible, with no two adjacent digits differing by 6 or 7. To maximize the number, we want to use as many 6s as possible and 7s as much as possible. However, we have a restriction on the digits: they can't repeat digits. So, we need to find a way to arrange the digits while keeping the adjacency rule and all digits different.\n\nLet me think about how to approach this. One way to maximize the number is to use as many 6s as possible, but with some remainder. For example, if we have two 6s next to each other, we can have 6s with one digit in between, but then the digits would have to be different. However, the next digit should have to be different from both 6s. But if we have two 6s next to each other, the digits would have to be the same, which is not allowed.\n\nAlternatively, maybe we", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 216, 49164, 30, 49160, 30, 7985, 260, 3995, 1782, 1230, 281, 527, 511, 18638, 359, 896, 284, 750, 827, 11358, 18638, 9094, 411, 216, 49165, 355, 216, 49166, 30, 49154, 49158, 198, 39122, 28, 1303, 549, 1510, 563, 451, 30, 339, 737, 288, 1042, 260, 3995, 1782, 1230, 837, 511, 18638, 359, 896, 284, 11358, 18638, 9094, 411, 216, 49165, 355, 216, 49166, 30, 41827, 30, 1644, 28, 260, 2359, 314, 338, 451, 1230, 553, 288, 457, 347, 800, 18638, 347, 1636, 28, 351, 511, 18638, 2116, 284, 260, 7408, 31986, 2619, 1036, 216, 49165, 355, 216, 49166, 30, 3717, 198, 2219, 2771, 28, 588, 808, 28, 1303, 506, 1044, 260, 1732, 30, 1046, 737, 253, 1230, 338, 2925, 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+{"group_id": "math_022", "origin": "budgie", "answer": "To find the largest natural number where all digits are different and any two adjacent digits differ by either 6 or 7, we can approach this by considering the constraints imposed by the digits and the divisibility rules.\n\nA natural number has the form $abcdefg$, where each digit is a single digit from 0 to 9. We need to satisfy the following conditions:\n1. All digits are different.\n2. Any two adjacent digits differ by either 6 or 7.\n3. The number is as large as possible.\n\nConsidering the constraints, we can start by finding the largest possible number with the given constraints. Since the digits must all be different, the largest possible number would be a six-digit number with the first digit being 9.\n\nLet's consider the constraints again:\n\n1. The first digit cannot be 0, so it has 9 choices (1 through 9).\n2. The second digit cannot be the same as the first digit, so it has 8 choices (since one digit is already used).\n3. The third digit cannot be the same as the second digit, so it has 7 choices (since one digit is already used).\n4. The fourth digit cannot be the same as the third digit, so it has 6 choices (since two digits are already used).\n5. The fifth digit cannot be the same as the fourth digit, so it has 5 choices (since one digit is already used).\n6. The sixth digit cannot be the same as the fifth digit, so it has 4 choices (since two digits are already used).\n\nWe can initially iterate through all possible combinations of the digits and then check if they satisfy the conditions.\n\nHere is a starting point:\n\nFor the first digit, we have 9 choices.\nFor the", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 216, 49164, 30, 49160, 30, 7985, 260, 3995, 1782, 1230, 281, 527, 511, 18638, 359, 896, 284, 750, 827, 11358, 18638, 9094, 411, 216, 49165, 355, 216, 49166, 30, 49154, 49158, 198, 2068, 1042, 260, 3995, 1782, 1230, 837, 511, 18638, 359, 896, 284, 750, 827, 11358, 18638, 9094, 411, 2267, 216, 49165, 355, 216, 49166, 28, 392, 416, 2424, 451, 411, 6361, 260, 10195, 12083, 411, 260, 18638, 284, 260, 3044, 271, 1968, 3372, 30, 198, 198, 49, 1782, 1230, 553, 260, 910, 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+{"group_id": "math_022", "origin": "verified_reference", "answer": "Answer: 60718293.\n\n## Solution.\n\nWe will map each digit from 0 to 9 to a vertex in a graph and connect the vertices with an edge if the corresponding digits differ by 6 or 7.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_8a1d5b347fbc679ac0eeg-5.jpg?height=448&width=590&top_left_y=524&top_left_x=819)\n\nWe see that the vertices corresponding to the digits 4 and 5 are not connected to anything, so the maximum number that can be obtained will be 8 digits long. The vertices corresponding to the digits 3 and 6 can only be the first or last in the number. Since we need to find the maximum number, it is uniquely determined by moving along the edges of the graph and will be equal to 60718293.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 216, 49164, 30, 49160, 30, 7985, 260, 3995, 1782, 1230, 281, 527, 511, 18638, 359, 896, 284, 750, 827, 11358, 18638, 9094, 411, 216, 49165, 355, 216, 49166, 30, 49154, 49158, 198, 21350, 42, 216, 49165, 49159, 49166, 49160, 49167, 49161, 49168, 49162, 30, 198, 198, 712, 19857, 30, 198, 198, 1882, 523, 4141, 971, 11403, 429, 216, 49159, 288, 216, 49168, 288, 253, 24121, 281, 253, 3670, 284, 2084, 260, 24796, 351, 354, 5595, 585, 260, 8030, 18638, 9094, 411, 216, 49165, 355, 216, 49166, 30, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49159, 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+{"group_id": "math_023", "origin": "budgie", "answer": "To determine the minimum value of $f(a, b, c, d)$, let's first consider each fraction in the expression $\\frac{a}{b+c+d} + \\frac{b}{c+d+a} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c}$.\n\nNotice that each fraction can be rewritten as:\n\n$$f(a, b, c, d) = \\frac{a}{b+c+d} + \\frac{b}{c+d+a} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c}$$\n\nSince the denominators are the same, we can combine the numerators:\n\n$$f(a, b, c, d) = \\frac{a}{b+c+d} + \\frac{b}{c+d+a} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c}$$\n\nNow, let's simplify the expression by finding a common denominator. Since the denominators are the same, the common denominator is the product of the denominators.\n\n$$f(a, b, c, d) = \\frac{a}{b+c+d} + \\frac{b}{c+d+a} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c} = \\frac{ad+bc+ab}{(b+c+d)(a+b+d)}$$\n\nNow, let's simplify the numerator. Notice that $(b+c+d)(a+b+d) = ab + bc + cd + ad + bd + a", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 49161, 10887, 1885, 81, 28, 278, 28, 265, 28, 287, 3814, 254, 3814, 33918, 107, 66, 41217, 27, 24362, 28, 1576, 288, 1042, 198, 20, 86, 24, 81, 28, 278, 28, 265, 28, 287, 25, 24814, 18169, 107, 81, 15009, 82, 27, 83, 27, 84, 109, 42639, 18169, 107, 82, 15009, 83, 27, 84, 27, 81, 109, 42639, 18169, 107, 83, 15009, 81, 27, 82, 27, 84, 109, 42639, 18169, 107, 84, 15009, 81, 27, 82, 27, 83, 24362, 260, 5869, 1685, 30, 49154, 49158, 198, 2068, 3346, 260, 5869, 1685, 282, 1885, 86, 24, 81, 28, 278, 28, 265, 28, 287, 25, 26265, 1303, 506, 808, 1771, 971, 10236, 281, 260, 4352, 19573, 18169, 107, 81, 15009, 82, 27, 83, 27, 84, 109, 1232, 3814, 18169, 107, 82, 15009, 83, 27, 84, 27, 81, 109, 1232, 3814, 18169, 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+{"group_id": "math_023", "origin": "budgie", "answer": "To find the minimum value of the given expression, we can use the method of Lagrange multipliers or recognize it as a known optimization problem.\n\nLet $f(x, y, z) = \\frac{x}{y+z+y} + \\frac{y}{z+x+z} + \\frac{z}{x+y+z}$, and $g(x, y, z) = \\frac{x}{y+z} + \\frac{y}{x+z} + \\frac{z}{x+y} + \\frac{w}{x+w}$, where $x, y, z, w > 0$ with $w > 0$.\n\nWe want to minimize $f(x, y, z, x) + g(x, y, z, y)$.\n\nSince $x, y, z, w > 0$, we can set $w = x + y + z$.\n\nNow, we have:\n\n$$f(x, y, z) = \\frac{x}{y+z+y} + \\frac{y}{z+x+z} + \\frac{z}{x+y+z} + \\frac{x+y+z}{x+y+z}$$\n\n$$g(x, y, z) = \\frac{x}{y+z} + \\frac{y}{x+z} + \\frac{z}{x+y} + \\frac{x+y+z}{x+y+z}$$\n\nTo minimize $f(x, y, z, x) + g(x, y, z, y)$, we want to minimize $f(x, y, z, x)$ and $g(x, y,", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 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+{"group_id": "math_023", "origin": "verified_reference", "answer": "Solution 1 Since directly eliminating the denominator is complicated, consider changing the problem.\nLet $b+c+d=A, c+d+a=B, d+a+b=C, a+b+c=D$.\nAdding these four equations, we get $a+b+c+d=\\frac{1}{3}(A+B+C+D)$.\nThus, we can find that $a=\\frac{1}{3}(B+C+D-2 A), b=\\frac{1}{3}(C+D+A-2 B)$,\n$$\nc=\\frac{1}{3}(D+A+B-2 C), d=\\frac{1}{3}(A+B+C-2 D).\n$$\n\nTherefore, the original expression $=\\frac{B+C+D-2 A}{3 A}+\\frac{C+D+A-2 B}{3 B}+\\frac{D+A+B-2 C}{3 C}+$\n$$\n\\frac{A+B+C-2 D}{3 D},\n$$\n\nwhich is $3 \\cdot f(a, b, c, d)+12$\n$$\n\\begin{array}{l}\n=\\frac{A+B+C+D}{A}+\\frac{A+B+C+D}{B}+\\frac{A+B+C+D}{C}+\\frac{A+B+C+D}{D} \\\\\n=(A+B+C+D) \\cdot\\left(\\frac{1}{A}+\\frac{1}{B}+\\frac{1}{C}+\\frac{1}{D}\\right) \\geqslant 16,\n\\end{array}\n$$\n\ni.e., $3 \\cdot f(a, b, c, d)+12 \\geqslant 16$, hence $f(a, b, c, d) \\geqslant \\frac{4}{3}$, where equality holds if and only if $A=$ $B=C=D$, i.e., $a=b=c=d$.\nTherefore, the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.\nSolution 2 Noting the weighted sum inequality, we have\n$$\n\\begin{aligned}\nf(a, b, c, d) & =\\frac{a^{2}}{a(b+c+d)}+\\frac{b^{2}}{b(a+c+d)}+\\frac{c^{2}}{c(a+b+d)}+\\frac{d^{2}}{d(a+b+c)} \\\\\n& \\geqslant \\frac{(a+b+c+d)^{2}}{2(a b+a c+a d+b c+b d+c d)} \\\\\n& =\\frac{(a+b+c+d)^{2}}{(a+b+c+d)^{2}-\\left(a^{2}+b^{2}+c^{2}+d^{2}\\right)}\n\\end{aligned}\n$$\n\nand $a^{2}+b^{2}+c^{2}+d^{2}=\\frac{a^{2}}{1}+\\frac{b^{2}}{1}+\\frac{c^{2}}{1}+\\frac{d^{2}}{1} \\geqslant \\frac{(a+b+c+d)^{2}}{4}$, thus\n$$\nf(a, b, c, d) \\geqslant \\frac{(a+b+c+d)^{2}}{(a+b+c+d)^{2}-\\frac{1}{4}(a+b+c+d)^{2}}=\\frac{4}{3} .\n$$\n\nwhere equality holds if and only if $a=b=c=d$, hence the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.\nSolution 3 Noting the Cauchy inequality or the weighted sum inequality, we have\n$$\n\\begin{aligned}\nf(a, b, c, d) & =\\frac{a+b+c+d}{b+c+d}+\\frac{a+b+c+d}{a+c+d}+\\frac{a+b+c+d}{a+b+d}+ \\\\\n& \\frac{a+b+c+d}{a+b+c}-4 \\\\\n& \\geqslant(a+b+c+d)\\left(\\frac{1}{b+c+d}+\\frac{1}{a+c+d}+\\frac{1}{a+b+d}+\\right. \\\\\n& \\left.\\frac{1}{a+b+c}\\right)-4 \\\\\n& \\geqslant(a+b+c+d) \\cdot \\frac{4^{2}}{3(a+b+c+d)}-4=\\frac{4}{3} .\n\\end{aligned}\n$$\n\nwhere equality holds if and only if $a=b=c=d$, hence the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 49161, 10887, 1885, 81, 28, 278, 28, 265, 28, 287, 3814, 254, 3814, 33918, 107, 66, 41217, 27, 24362, 28, 1576, 288, 1042, 198, 20, 86, 24, 81, 28, 278, 28, 265, 28, 287, 25, 24814, 18169, 107, 81, 15009, 82, 27, 83, 27, 84, 109, 42639, 18169, 107, 82, 15009, 83, 27, 84, 27, 81, 109, 42639, 18169, 107, 83, 15009, 81, 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+{"group_id": "math_024", "origin": "budgie", "answer": "1. The expression is $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$.\n2. Simplify the numerator:\n$$\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31} = \\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{\\frac{31+0.31}{1}} = \\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{\\frac{31+0.31}{1}} = 2015 + 2015$$\n\nFinal answer:\n#### 2015", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 365, 49165, 2876, 25, 4104, 314, 4450, 216, 49162, 49160, 28, 216, 49161, 49159, 49160, 49164, 28, 10668, 288, 260, 216, 49161, 49159, 49160, 49164, 476, 32155, 23278, 18, 19238, 30, 378, 13310, 966, 282, 260, 4352, 19573, 18169, 26754, 18169, 107, 49161, 49159, 49160, 49164, 15009, 49160, 109, 42639, 18169, 107, 49161, 49159, 49160, 49164, 15009, 49159, 30, 49162, 49160, 109, 15009, 49160, 27, 49159, 30, 49162, 49160, 24362, 314, 49154, 49158, 198, 49160, 30, 378, 4352, 314, 19573, 18169, 26754, 18169, 107, 49161, 49159, 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+{"group_id": "math_024", "origin": "budgie", "answer": "2. (1 point) The problem involves calculating a complex expression.\n\nEliminating the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.\n\n$$\n\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+\\frac{2015}{0.31}} = \\frac{2015}{1} + 3030\n$$\n\nNext, we calculate the denominator of the fraction by multiplying both the numerator and the denominator by the conjugate of the denominator, which is 1.\n\n$$\n1+\\frac{2015}{0.31} = 1 + 2015 = 2016\n$$\n\nNow that we have the full fraction, we can simplify it by dividing the numerator by 2015 and multiplying both the numerator and denominator by 1 (since 2015 is twice 1).\n\n$$\n\\frac{2015}{2015} + 3030 = \\frac{1}{2015} + 3030 = 3030 + 2015 = 5045\n$$\n\nThe result of the new problem is 5045.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 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+{"group_id": "math_024", "origin": "verified_reference", "answer": "【Solution】Solve: $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$\n$$\n\\begin{array}{l}\n=\\frac{\\left(\\frac{2015}{1}+\\frac{2015}{0.31}\\right) \\times 0.31}{(1+0.31) \\times 0.31} \\\\\n=\\frac{(1+0.31) \\times 2015}{(1+0.31) \\times 0.31} \\\\\n=6500 ;\n\\end{array}\n$$\n\nTherefore, the answer is: 6500.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 365, 49165, 2876, 25, 4104, 314, 4450, 216, 49162, 49160, 28, 216, 49161, 49159, 49160, 49164, 28, 10668, 288, 260, 216, 49161, 49159, 49160, 49164, 476, 32155, 23278, 18, 19238, 30, 378, 13310, 966, 282, 260, 4352, 19573, 18169, 26754, 18169, 107, 49161, 49159, 49160, 49164, 15009, 49160, 109, 42639, 18169, 107, 49161, 49159, 49160, 49164, 15009, 49159, 30, 49162, 49160, 109, 15009, 49160, 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+{"group_id": "math_026", "origin": "budgie", "answer": "1. To solve the given system of equations, we can start by rewriting the second equation in terms of $y$:\n\n$$\n\\log_{27} y - \\log_{4} x = \\frac{27}{6}\n$$\n\n2. Using the change of base formula, we can rewrite the second equation in terms of $x$:\n\n$$\n\\frac{1}{y} - \\frac{\\log_{27} x}{\\log_{4} 27} = \\frac{27}{6}\n$$\n\n3. Notice that $\\log_{27} x = \\log_{4} x$ because $27 = 4^2$. Substituting this into the second equation, we get:\n\n$$\n\\frac{1}{y} - \\frac{\\log_{4} x}{\\log_{4} 27} = \\frac{27}{6}\n$$\n\n4. Simplify the equation further by cancelling out $y$ and $\\log_{4} x$:\n\n$$\n\\frac{1}{\\log_{4} x} = \\frac{27}{6} - \\frac{\\log_{4} x}{\\log_{4} 27}\n$$\n\n5. Let $\\log_{4} 27 = a$, where $a > 0$ since $27 > 1$. Substitute this into the equation:\n\n$$\n\\frac{1}{\\log_{4} x} = \\frac{27}{6} - \\frac{a}{\\log_{4} x}\n$$\n\n6. Multiply both sides of the equation by $\\log_{4} x$ to clear the fractions:\n\n$$\n1 = \\frac{27}{6} \\log_{", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 378, 932, 282, 511, 1345, 1230, 9180, 46408, 104, 28, 329, 38224, 338, 15765, 19573, 8842, 76, 26754, 21083, 107, 4282, 15009, 92, 109, 49163, 21990, 29, 104, 109, 27, 49161, 49166, 21990, 29, 105, 109, 24814, 18169, 107, 49164, 15009, 49165, 109, 28372, 3814, 3107, 2483, 107, 49161, 49166, 109, 329, 46240, 3107, 2483, 107, 49163, 109, 1792, 3814, 361, 97, 7483, 403, 3814, 18169, 107, 49160, 15009, 49165, 109, 3814, 2692, 1662, 314, 4029, 28372, 216, 49161, 49166, 21990, 105, 39949, 49163, 21990, 104, 109, 3814, 290, 97, 7483, 403, 216, 49160, 76, 486, 107, 4282, 17243, 2771, 30, 20, 19573, 97, 32984, 20, 1673, 19573, 97, 32984, 20, 49154, 49158, 198, 49160, 30, 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+{"group_id": "math_026", "origin": "budgie", "answer": "To solve this problem, we first need to find the range of values for $x$ and $y$ such that the given equation holds true.\n\nGiven:\n\\[ \\frac{1}{4^x} + \\frac{1}{27^y} = \\frac{5}{6} \\]\n\nTo eliminate the fractions, we can multiply both sides by $6^x 4^y$. We know that $6^x 4^y = (2^2 \\cdot 3)^x 2^y = 2^x \\cdot 3^x \\cdot 2^y = 2^x \\cdot 3^x \\cdot 2^y$.\n\nSo, we have:\n\\[ 6^x 4^y \\cdot \\frac{1}{4^x} + 6^x \\cdot 4^y \\cdot \\frac{1}{27^y} = 6^x \\cdot 4^y \\cdot \\frac{5}{6} \\]\n\nNow, we can simplify the expressions on the left side of the equation:\n\\[ 6^x 4^y + 6^x \\cdot 4^y = 2^x \\cdot 3^x \\cdot 2^y \\]\n\nWe can further simplify the expression on the right side of the equation:\n\\[ 6^x \\cdot 4^y \\cdot \\frac{5}{6} = 2^x \\cdot 3^x \\cdot 2^y \\]\n\nNow, we can rewrite the equation as:\n\\[ 2^x \\cdot 3^x \\cdot 2^y + 6^x \\cdot 4^y \\cdot \\frac", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 378, 932, 282, 511, 1345, 1230, 9180, 46408, 104, 28, 329, 38224, 338, 15765, 19573, 8842, 76, 26754, 21083, 107, 4282, 15009, 92, 109, 49163, 21990, 29, 104, 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+{"group_id": "math_026", "origin": "verified_reference", "answer": "6. $\\left\\{\\left(\\frac{1}{2}, \\frac{1}{3}\\right)\\right\\}$ Detailed Explanation: Let $a=4^{x}, b=27^{y}$, then $\\frac{1}{a}+\\frac{1}{b}=\\frac{5}{6}, a>\\frac{6}{5}, b>\\frac{6}{5}$. From $\\frac{1}{a}+\\frac{1}{b}=\\frac{5}{6}$, we get $b=\\frac{6 a}{5 a-6}$, then $\\frac{6 a}{5 a-6}-a \\leqslant 1$, solving this yields $a \\geqslant 2$ or $a>\\frac{6}{5}$, so $a \\geqslant 2$. Similarly, we get $b \\leqslant 3$. Therefore, $x \\geqslant \\frac{1}{2}, y \\leqslant \\frac{1}{3} . \\log _{27} y-\\log _{4} x \\leqslant \\log _{27} \\frac{1}{3}-\\log _{4} \\frac{1}{2}=\\frac{1}{6}$. 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+{"group_id": "math_027", "origin": "budgie", "answer": "Let the number be $n$. We can represent this situation using modular arithmetic.\n\n\\[ n \\equiv 2 \\mod 3 \\]\n\\[ n \\equiv 3 \\mod 4 \\]\n\\[ n \\equiv 4 \\mod 5 \\]\n\nFirst, we need to find a number that satisfies all these conditions. We can do this by looking for a number that is 1 more than a multiple of 3.\n\n\\[ n = 3k + 1 \\]\n\nwhere $k$ is a positive integer.\n\nNow, we need to find a number that also satisfies the other two conditions:\n\n\\[ n \\equiv 2 \\mod 4 \\]\n\\[ n \\equiv 3 \\mod 4 \\]\n\nSubstitute $n = 3k + 1$ into the second congruence:\n\n\\[ 3k + 1 \\equiv 3 \\mod 4 \\]\n\nSubtract 1 from both sides:\n\n\\[ 3k \\equiv 2 \\mod 4 \\]\n\nSubtract 2 from both sides:\n\n\\[ 3k - 2 \\equiv 0 \\mod 4 \\]\n\nMultiply both sides by the multiplicative inverse of 3 modulo 4 (which is also 3 since 3 is a factor of 4):\n\n\\[ k \\equiv 2 \\times 3 \\mod 4 \\]\n\\[ k \\equiv 6 \\mod 4 \\]\n\\[ k \\equiv 2 \\mod 4 \\]\n\nSo, we have found that $k \\equiv 2 \\mod 4$. 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+{"group_id": "math_027", "origin": "budgie", "answer": "To solve this problem, we can use the Chinese Remainder Theorem (CRT). The CRT states that if we have a system of congruences\n\n$$\n\\begin{cases}\nx \\equiv 0 \\pmod{m} \\\\\nx \\equiv 2 \\pmod{m} \\\\\nx \\equiv 3 \\pmod{m} \\\\\nx \\equiv 4 \\pmod{m}\n\\end{cases}\n$$\n\nwhere $m$ is an integer, then there exists a unique solution modulo $m$.\n\nFirst, we need to find a solution modulo $m$ for each congruence. We can do this by setting the right-hand sides of each congruence equal to 0.\n\nFor the first congruence, we have $x \\equiv 0 \\pmod{3}$. This means that $x$ leaves a remainder of 0 when divided by 3.\n\nFor the second congruence, we have $2x \\equiv 2 \\pmod{4}$. Dividing both sides by 2 gives us $x \\equiv 1 \\pmod{2}$.\n\nFor the third congruence, we have $3x \\equiv 3 \\pmod{5}$. Dividing both sides by 3 gives us $x \\equiv 1 \\pmod{5}$.\n\nFor the fourth congruence, we have $4x \\equiv 4 \\pmod{6}$. Dividing both sides by 4 gives us $x \\equiv 1 \\pmod{3}$.\n\nNow, we need to find a solution modulo $m$ that satisfies all the congruences. We can do this by finding an integer $x$ that satisfies", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49163, 30, 365, 49165, 2876, 25, 330, 1230, 645, 6066, 411, 216, 49162, 3711, 253, 17867, 282, 216, 49161, 28, 645, 6066, 411, 216, 49163, 3711, 253, 17867, 282, 216, 49162, 28, 284, 645, 6066, 411, 216, 49164, 3711, 253, 17867, 282, 216, 49163, 30, 669, 1230, 314, 19573, 97, 32984, 20, 49154, 49158, 198, 2068, 5482, 451, 1732, 28, 392, 416, 722, 260, 4097, 3579, 402, 1223, 37050, 365, 9088, 68, 595, 378, 47281, 2496, 338, 585, 392, 457, 253, 817, 282, 37524, 1259, 198, 198, 9212, 198, 76, 21083, 107, 24407, 109, 198, 104, 3814, 2682, 380, 216, 49159, 3814, 96, 2172, 107, 93, 109, 28372, 198, 104, 3814, 2682, 380, 216, 49161, 3814, 96, 2172, 107, 93, 109, 28372, 198, 104, 3814, 2682, 380, 216, 49162, 3814, 96, 2172, 107, 93, 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+{"group_id": "math_027", "origin": "verified_reference", "answer": "【Analysis】Interpreting \"divided by 3 leaves a remainder of 2, divided by 4 leaves a remainder of 3, divided by 5 leaves a remainder of 4\" as being 1 less than when divided by 3, 1 less than when divided by 4, and 1 less than when divided by 5, means this number is at least 1 less than the least common multiple (LCM) of 3, 4, and 5. Since 3, 4, and 5 are pairwise coprime, the LCM of these three numbers is their product. Find the LCM of 3, 4, and 5, then subtract 1.\n\n【Solution】Solution: $3 \\times 4 \\times 5-1$,\n$$\n\\begin{array}{l}\n=60-1, \\\\\n=59 ;\n\\end{array}\n$$\n\nAnswer: The number is 59.\nTherefore, the answer is: 59.\n【Comment】This problem mainly examines the method of finding the least common multiple (LCM) when three numbers are pairwise coprime: If three numbers are pairwise coprime, their LCM is the product of these three numbers.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49163, 30, 365, 49165, 2876, 25, 330, 1230, 645, 6066, 411, 216, 49162, 3711, 253, 17867, 282, 216, 49161, 28, 645, 6066, 411, 216, 49163, 3711, 253, 17867, 282, 216, 49162, 28, 284, 645, 6066, 411, 216, 49164, 3711, 253, 17867, 282, 216, 49163, 30, 669, 1230, 314, 19573, 97, 32984, 20, 49154, 49158, 198, 12109, 234, 27931, 12109, 235, 8235, 1702, 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+{"group_id": "math_028", "origin": "budgie", "answer": "A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares.\n\nThe key is that the area of the rectangle is equal to the area of a square.\n\nThe area of the rectangle is $A = n^2$ (since it's a rectangle).\n\nThe area of the square is $A = (n + 76)^2$ (since it's a square).\n\nSince the area of the rectangle is equal to the area of a square, we have $A = A$.\n\nThis means $n^2 = (n + 76)^2$.\n\nExpanding the equation, we get $n^2 = n^2 + 152n + 5916$.\n\nSubtracting $n^2$ from both sides, we get $0 = 152n + 5915$.\n\nSubtracting $5915$ from both sides, we get $-5910 = 152n$.\n\nDividing both sides by $152$, we get $n = -5910/152$.\n\nSimplifying the right side, we get $n = -58.5785794...$.\n\nSince $n$ must be a positive integer, the only possible value of $n$ is $n = -5910/152 \\approx -41.4083$.\n\nHowever, since $n$ must be a non-negative integer, we consider the next value of $n$, which is $n = -5910/152 = -58$.\n\nThe only possible value of $n$", "reward": -1, "advantage": 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+{"group_id": "math_028", "origin": "budgie", "answer": "If a rectangle can be divided into $n$ equal squares, then the area of the rectangle must be a perfect square. Since the area is a perfect square, this means that the length and width of the rectangle must be powers of 2.\n\nLet $a$ and $b$ be the length and width of the rectangle, respectively. Then we have:\n\n\\[ a^2 = 2^n \\quad \\text{and} \\quad b^2 = 2^n + 76 \\]\n\nWe can rewrite these equations as:\n\n\\[ a = 2^x \\quad \\text{and} \\quad b = 2^y \\]\n\nwhere $x$ and $y$ are non-negative integers.\n\nSubstituting these expressions for $a$ and $b$, we get:\n\n\\[ (2^x)^2 = 2^n \\quad \\text{and} \\quad (2^y)^2 = 2^n + 76 \\]\n\nSimplifying these equations, we get:\n\n\\[ 2^{2x} = 2^n \\quad \\text{and} \\quad 2^{2y} = 2^n + 76 \\]\n\nDividing both sides of the first equation by $2^x$, we get:\n\n\\[ 2^{2x - x} = 2^n \\Rightarrow 2^x = 2^n \\Rightarrow x = n \\]\n\nSimilarly, dividing both sides of the second equation by $2^y$, we get:\n\n\\[ 2^{2y - y} = 2^n + 76 \\Rightarrow 2^y = 2^n +", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 49, 23710, 416, 325, 6066, 618, 1885, 94, 20, 4037, 16207, 30, 378, 1142, 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+{"group_id": "math_028", "origin": "verified_reference", "answer": "Answer: $n=324$.\n\nLet $a b=n$ and $c d=n+76$, where $a, b$ and $c, d$ are the numbers of squares in each direction for the partitioning of the rectangle into $n$ and $n+76$ squares, respectively. Then $\\frac{a}{c}=\\frac{b}{d}$, or $a d=b c$. Denote $u=\\operatorname{gcd}(a, c)$ and $v=\\operatorname{gcd}(b, d)$, then there exist positive integers $x$ and $y$ such that $\\operatorname{gcd}(x, y)=1, a=u x, c=u y$ and $b=v x, d=v y$. Hence we have\n\n$$\nc d-a b=u v\\left(y^{2}-x^{2}\\right)=u v(y-x)(y+x)=76=2^{2} \\cdot 19 .\n$$\n\nSince $y-x$ and $y+x$ are positive integers of the same parity and $\\operatorname{gcd}(x, y)=1$, we have $y-x=1$ and $y+x=19$ as the only possibility, yielding $y=10, x=9$ and $u v=4$. 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+{"group_id": "math_029", "origin": "budgie", "answer": "The maximum number of non-palindrome years is 999 years + 1 year = 1000 years.\n\nFinal answer:\n#### 1000", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 20, 75, 20, 933, 3780, 17516, 30, 11689, 13310, 28, 2686, 12879, 1885, 77, 20, 198, 198, 37835, 42, 13212, 383, 259, 431, 452, 30, 49, 1143, 38838, 10251, 12713, 422, 11171, 1116, 49161, 49159, 49159, 49161, 314, 253, 5008, 526, 4522, 713, 28, 527, 1530, 357, 12927, 260, 1142, 17301, 347, 3703, 30, 378, 2672, 5008, 526, 4522, 713, 436, 216, 49160, 49160, 929, 3685, 365, 49160, 49168, 49168, 49160, 595, 1812, 314, 260, 5428, 1230, 282, 1603, 29, 22162, 526, 4522, 929, 338, 416, 1689, 15233, 316, 1320, 365, 18946, 216, 49160, 49159, 49159, 49159, 284, 216, 49168, 49168, 49168, 49168, 929, 14487, 49154, 49158, 198, 504, 5428, 1230, 282, 1603, 29, 22162, 526, 4522, 929, 314, 216, 49168, 49168, 49168, 929, 1232, 216, 49160, 713, 446, 216, 49160, 49159, 49159, 49159, 929, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49159, 49159, 49159, 49154], "prefix_len": 135, "old_logprobs": [-0.7140514254570007, -1.5570822954177856, -0.21248187124729156, -0.009643280878663063, -0.11301965266466141, -0.006643115542829037, -0.0015851801726967096, -8.880697714630514e-05, -0.030584605410695076, -0.008943026885390282, -1.6076079607009888, -0.33844396471977234, -0.2769755423069, -0.25895941257476807, -0.10979203879833221, -0.6017013192176819, -1.5010671615600586, -0.038639362901449203, -0.4385890066623688, -0.5626175999641418, -0.05898374319076538, -0.07027687132358551, -0.2594640552997589, -0.05831923708319664, -0.0036374852061271667, -0.004357961006462574, -0.034328944981098175, -0.22865648567676544, -0.21582330763339996, -0.09211517125368118, -0.3602789342403412, -0.000337305391440168, -0.0008983152220025659, -0.017383594065904617, -0.00018845213344320655, -0.004211485851556063, -0.0029270683880895376, -0.00485574547201395, -0.001303061842918396, -0.00018463814922142774, -0.0016752745723351836]}
+{"group_id": "math_029", "origin": "budgie", "answer": "The maximum number of non-palindrome years between 1000 and 9999 that can occur consecutively is 9999 - 1000 + 1 = 9000 years.", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 20, 75, 20, 933, 3780, 17516, 30, 11689, 13310, 28, 2686, 12879, 1885, 77, 20, 198, 198, 37835, 42, 13212, 383, 259, 431, 452, 30, 49, 1143, 38838, 10251, 12713, 422, 11171, 1116, 49161, 49159, 49159, 49161, 314, 253, 5008, 526, 4522, 713, 28, 527, 1530, 357, 12927, 260, 1142, 17301, 347, 3703, 30, 378, 2672, 5008, 526, 4522, 713, 436, 216, 49160, 49160, 929, 3685, 365, 49160, 49168, 49168, 49160, 595, 1812, 314, 260, 5428, 1230, 282, 1603, 29, 22162, 526, 4522, 929, 338, 416, 1689, 15233, 316, 1320, 365, 18946, 216, 49160, 49159, 49159, 49159, 284, 216, 49168, 49168, 49168, 49168, 929, 14487, 49154, 49158, 198, 504, 5428, 1230, 282, 1603, 29, 22162, 526, 4522, 929, 826, 216, 49160, 49159, 49159, 49159, 284, 216, 49168, 49168, 49168, 49168, 338, 416, 1689, 15233, 316, 1320, 314, 216, 49168, 49168, 49168, 49168, 731, 216, 49160, 49159, 49159, 49159, 1232, 216, 49160, 446, 216, 49168, 49159, 49159, 49159, 929, 30, 49154], "prefix_len": 135, "old_logprobs": [-0.7140514254570007, -1.5570822954177856, -0.21248187124729156, -0.009643280878663063, -0.11301965266466141, -0.006643115542829037, -0.0015851801726967096, -8.880697714630514e-05, -0.030584605410695076, -0.008943026885390282, -1.9826079607009888, -0.043660521507263184, -0.028725003823637962, -0.03302856534719467, -0.004643609281629324, -0.015606650151312351, -0.01720280759036541, -0.0006549597019329667, -0.006854236125946045, -0.00010239553375868127, -2.706014311115723e-05, -7.819823804311454e-05, -2.3928987979888916, -0.07833798974752426, -0.01675872877240181, -0.18078161776065826, -0.00021062063751742244, -0.001013361383229494, -0.1362125277519226, -0.5228233337402344, -0.2910989224910736, -0.22141991555690765, -0.08930355310440063, -0.595577597618103, -0.4740138351917267, -0.02962520532310009, -0.04225064441561699, -0.06289683282375336, -0.004007286392152309, -0.016442865133285522, -1.0021109580993652, -0.004009542521089315, -0.001444369088858366, -0.04972303658723831, -0.05039951205253601, -0.10613595694303513, -0.07420653849840164, -0.041313014924526215, -0.002894024597480893, -0.49600356817245483, -0.08759858459234238, -2.6232025623321533]}
+{"group_id": "math_029", "origin": "verified_reference", "answer": "Let the current year be a palindrome of the form $\\overline{a b b a}$. When will the next such year occur? Consider two cases:\n\na) $b=9$ (year of the form $\\overline{a 99 a}, a<9)$. Then, 11 years later, another palindromic year will occur: $\\overline{(a+1) 00(a+1)}$. For example, the years 3993 and 4004.\n\nb) $b<9$. In this case, the next palindromic year will occur 110 years later: $\\overline{a(b+1)(b+1) a}$. For example, the years 9339 and 9449. Therefore, the maximum number of consecutive non-palindromic years is 109.\n\n## Answer\n\n109 years.", "reward": 1.0, "advantage": 1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 20, 75, 20, 933, 3780, 17516, 30, 11689, 13310, 28, 2686, 12879, 1885, 77, 20, 198, 198, 37835, 42, 13212, 383, 259, 431, 452, 30, 49, 1143, 38838, 10251, 12713, 422, 11171, 1116, 49161, 49159, 49159, 49161, 314, 253, 5008, 526, 4522, 713, 28, 527, 1530, 357, 12927, 260, 1142, 17301, 347, 3703, 30, 378, 2672, 5008, 526, 4522, 713, 436, 216, 49160, 49160, 929, 3685, 365, 49160, 49168, 49168, 49160, 595, 1812, 314, 260, 5428, 1230, 282, 1603, 29, 22162, 526, 4522, 929, 338, 416, 1689, 15233, 316, 1320, 365, 18946, 216, 49160, 49159, 49159, 49159, 284, 216, 49168, 49168, 49168, 49168, 929, 14487, 49154, 49158, 198, 4239, 260, 1574, 713, 325, 253, 5008, 526, 4522, 282, 260, 910, 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+{"group_id": "math_030", "origin": "budgie", "answer": "First, we need to find the total number of students in the class. We can do this by adding up the number of students who prefer chocolate, apple, blueberry, and cherry pie.\n\n\\[ \\text{Total} = \\text{chocolate} + \\text{apple} + \\text{blueberry} + \\text{cherry} = 12 + 8 + 6 + 3 = 27 \\]\n\nNext, we need to find the total number of students who prefer cherry pie. We can do this by subtracting the total number of students who prefer apple and blueberry pie from the total number of students.\n\n\\[ \\text{Total cherry} = \\text{Total} - \\text{Total apple} - \\text{Total blueberry} = 27 - 5 - 6 = 16 \\]\n\nFinally, we need to find the fraction of the class that prefer cherry pie. We can do this by dividing the total number of students who prefer cherry pie by the total number of students in the class.\n\n\\[ \\text{Fraction} = \\frac{\\text{Total cherry}}{\\text{Total students}} = \\frac{16}{27} \\]\n\nNow, we can find the fraction of the class that prefer cherry pie by multiplying this fraction by the total number of students in the class.\n\n\\[ \\text{Fraction} = \\frac{16}{27} \\times 27 = 6 \\]\n\nSo, the fraction of the class that prefer cherry pie is 6.\n\nThe answer is:\n\\[ \\boxed{6} \\]\nFinal answer:\n#### 6", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 6228, 260, 216, 49162, 49165, 1058, 281, 25734, 2797, 290, 506, 1128, 28, 216, 49160, 49161, 4640, 9678, 2357, 28, 216, 49167, 4640, 11977, 28, 284, 216, 49165, 4640, 4461, 9500, 30, 22190, 282, 260, 6219, 1058, 4640, 23636, 2357, 284, 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+{"group_id": "math_030", "origin": "budgie", "answer": "First, we need to find out how many students prefer cherry pie. We can do this by subtracting the number of students who prefer other pie types from the total number of students.\n\n$ \\text{Number of students who prefer cherry pie} = \\text{Total number of students} - \\text{Number of students who prefer other pie types} $$ $= 36 - 12 $ $= 24 $\n\nNow that we have the number of students who prefer cherry pie, we can find the percentage of students who prefer cherry pie. We can do this by dividing the number of students who prefer cherry pie by the total number of students and then multiplying by 100.\n\n$ \\text{Percentage of students who prefer cherry pie} = \\frac{\\text{Number of students who prefer cherry pie}}{\\text{Total number of students}} $$ $= \\frac{24}{36} $ $= \\frac{2}{3} $\n\nSo, the answer is 2/3 or 60%.\n\nFinal answer:\n#### 60", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 6228, 260, 216, 49162, 49165, 1058, 281, 25734, 2797, 290, 506, 1128, 28, 216, 49160, 49161, 4640, 9678, 2357, 28, 216, 49167, 4640, 11977, 28, 284, 216, 49165, 4640, 4461, 9500, 30, 22190, 282, 260, 6219, 1058, 4640, 23636, 2357, 284, 2745, 4640, 11499, 30, 1068, 25734, 2797, 290, 506, 2357, 3670, 5369, 451, 940, 28, 638, 800, 5027, 868, 1041, 722, 327, 23636, 2357, 47, 3717, 198, 20, 3814, 2692, 107, 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+{"group_id": "math_030", "origin": "verified_reference", "answer": "1. First, we need to determine the number of students who prefer each type of pie. We know:\n   - 12 students prefer chocolate pie.\n   - 8 students prefer apple pie.\n   - 6 students prefer blueberry pie.\n\n2. Calculate the total number of students who prefer chocolate, apple, or blueberry pie:\n   \\[\n   12 + 8 + 6 = 26\n   \\]\n\n3. Subtract this number from the total number of students in the class to find the number of students who prefer either cherry or lemon pie:\n   \\[\n   36 - 26 = 10\n   \\]\n\n4. Since half of the remaining students prefer cherry pie and the other half prefer lemon pie, we divide the remaining students by 2:\n   \\[\n   \\frac{10}{2} = 5\n   \\]\n   Therefore, 5 students prefer cherry pie.\n\n5. To find the number of degrees Richelle should use for cherry pie in her pie graph, we need to calculate the proportion of students who prefer cherry pie and then convert this proportion to degrees. The proportion of students who prefer cherry pie is:\n   \\[\n   \\frac{5}{36}\n   \\]\n\n6. Multiply this proportion by the total number of degrees in a circle (360 degrees):\n   \\[\n   \\frac{5}{36} \\times 360 = 50\n   \\]\n\n7. 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+{"group_id": "math_031", "origin": "budgie", "answer": "The problem is based on the two-way circular track.\n\nThe key here is understanding the behavior of the runners and the time it takes for them to complete a lap.\n\nThe runners are moving at different constant speeds, and each runner has to cover the same distance around the track.\n\nAt the first meeting, the third runner will have to cover the distance that the second runner covered in the first 15 minutes.\n\nAfter 25 minutes, the second runner will have to cover the same distance that the first runner covered in the first 15 minutes.\n\nTherefore, the second runner will have to cover a distance that the first runner covered in 25 minutes, and the third runner will have to cover a distance that the first runner covered in 15 minutes.\n\nThe third runner will have to cover the same distance that the first runner covered in 25 minutes, and the second runner will have to cover a distance that the first runner covered in 15 minutes.\n\nSo, the distance that the third runner needs to cover is 25 minutes.\n\nLet's denote the distance that the first runner covers as $s$. At the time of the first meeting, the second runner will have to cover $s-15$ meters.\n\nThe first runner will have to cover $s+15$ meters in the same amount of time.\n\nWe know that the third runner will have to cover 25 meters in the same amount of time.\n\nTherefore, we have:\n\n$$s+15 = 25$$\n\nSolving for $s$ gives us:\n\n$$s = 25 - 15 = 10$$\n\nSo, the first runner will have to cover 10 meters in the time it takes the second runner to", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49165, 30, 365, 49162, 2876, 25, 198, 198, 16705, 25334, 359, 4138, 1683, 253, 11324, 4320, 418, 3836, 4037, 10870, 30, 1550, 827, 25334, 2220, 28, 502, 15881, 1607, 1130, 284, 1120, 4108, 281, 7011, 8939, 30, 198, 198, 3473, 634, 1225, 28, 260, 808, 23786, 11675, 260, 1796, 30, 2550, 216, 49160, 49164, 3487, 28, 260, 1796, 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+{"group_id": "math_031", "origin": "budgie", "answer": "The answer is 25 minutes.\n\nFinal answer:\n#### 25", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49165, 30, 365, 49162, 2876, 25, 198, 198, 16705, 25334, 359, 4138, 1683, 253, 11324, 4320, 418, 3836, 4037, 10870, 30, 1550, 827, 25334, 2220, 28, 502, 15881, 1607, 1130, 284, 1120, 4108, 281, 7011, 8939, 30, 198, 198, 3473, 634, 1225, 28, 260, 808, 23786, 11675, 260, 1796, 30, 2550, 216, 49160, 49164, 3487, 28, 260, 1796, 23786, 11675, 260, 3464, 327, 260, 808, 655, 30, 5839, 216, 49161, 49164, 3487, 1916, 28, 260, 3464, 23786, 11675, 260, 808, 327, 260, 808, 655, 30, 198, 198, 2020, 800, 3487, 1072, 357, 1188, 327, 582, 23786, 288, 2994, 260, 2415, 4320, 47, 49154, 49158, 198, 504, 2988, 314, 216, 49161, 49164, 3487, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49161, 49164, 49154], "prefix_len": 135, "old_logprobs": [-1.1664215326309204, -1.9958109855651855, -0.05354534834623337, -0.3489420413970947, -2.0087571144104004, -0.7786458730697632, -0.7872970104217529, -0.20889146625995636, -0.22673311829566956, -0.004990146495401859, -0.37720146775245667, -0.0009112972766160965, -0.0007460909546352923, -0.03270026668906212, -0.0009235645993612707, -0.0022326798643916845, -7.128461584215984e-05, -5.8053239627042785e-05, -0.00016080040950328112]}
+{"group_id": "math_031", "origin": "verified_reference", "answer": "# Answer: 80\n\n## Solution:\n\nLet the first runner meet the second, then after $a$ minutes the second runner meets the third for the first time, and after another $b$ minutes the third runner meets the first for the first time.\n\nLet the first and second runners meet at point $A$, the second and third at point $B$, and the first and third at point $C$. Additionally, let the first runner be at point $D$ at the moment the second and third runners meet.\n\nThen the points are arranged on the circle in the following order: $D, A, B, C$, where the first runner ran from $D$ to $A$ in $a$ minutes, the second runner ran from $A$ to $B$ in $a$ minutes, the third runner ran from $B$ to $C$ in $b$ minutes, and the first runner ran from $D$ to $C$ in $b$ minutes. These four segments together form the entire track, so it is covered in $2a + 2b$ minutes.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49165, 30, 365, 49162, 2876, 25, 198, 198, 16705, 25334, 359, 4138, 1683, 253, 11324, 4320, 418, 3836, 4037, 10870, 30, 1550, 827, 25334, 2220, 28, 502, 15881, 1607, 1130, 284, 1120, 4108, 281, 7011, 8939, 30, 198, 198, 3473, 634, 1225, 28, 260, 808, 23786, 11675, 260, 1796, 30, 2550, 216, 49160, 49164, 3487, 28, 260, 1796, 23786, 11675, 260, 3464, 327, 260, 808, 655, 30, 5839, 216, 49161, 49164, 3487, 1916, 28, 260, 3464, 23786, 11675, 260, 808, 327, 260, 808, 655, 30, 198, 198, 2020, 800, 3487, 1072, 357, 1188, 327, 582, 23786, 288, 2994, 260, 2415, 4320, 47, 49154, 49158, 198, 19, 19842, 42, 216, 49167, 49159, 198, 198, 712, 19857, 42, 198, 198, 4239, 260, 808, 23786, 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+{"group_id": "math_033", "origin": "budgie", "answer": "To find the minimum value of $k$ that satisfies the given condition, we need to analyze the inequality and identify the conditions under which a triangle with side lengths $a, b, c$ exists.\n\nGiven:\n\\[ k(a b + b c + c a) > 5(a^{2} + b^{2} + c^{2}) \\]\n\nWe can rewrite the inequality as:\n\\[ k(a b + b c + c a) > 5 a^{2} + 5 b^{2} + 5 c^{2} \\]\n\nTo simplify the analysis, divide both sides by $a^{2} + b^{2} + c^{2}$ (which is non-zero), yielding:\n\\[ \\frac{k(a b + b c + c a)}{a^{2} + b^{2} + c^{2}} > \\frac{5 a^{2} + 5 b^{2} + 5 c^{2}}{4} \\]\n\nWe want the left-hand side (LHS) to be greater than $k$ for some positive $a, b, c$. To achieve this, we must have:\n\\[ \\frac{5 a^{2} + 5 b^{2} + 5 c^{2}}{4} < k \\]\n\nMultiplying through by 4 gives us:\n\\[ 5 a^{2} + 5 b^{2} + 5 c^{2} < 4k \\]\n\nTo ensure the inequality holds for any $a, b, c$, we need to find the smallest $k$ that satisfies this condition.\n\nOne approach is to find the smallest $k$ that makes the left-hand side positive and the inequality", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49163, 30, 2959, 1885, 91, 20, 325, 253, 2731, 15328, 28, 715, 338, 327, 750, 2731, 2966, 1885, 81, 28, 278, 28, 265, 20, 19131, 260, 9724, 1885, 91, 24, 81, 278, 27, 82, 265, 27, 83, 253, 46153, 49164, 76, 8842, 24, 81, 21990, 49161, 109, 27, 82, 21990, 49161, 17243, 2771, 30, 20, 1885, 27, 83, 21990, 49161, 24362, 643, 665, 1251, 1822, 253, 14973, 351, 2103, 15900, 1885, 81, 28, 278, 28, 265, 28422, 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+{"group_id": "math_033", "origin": "verified_reference", "answer": "14. $k(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right) \\geqslant 5(a b+a c+b c), k \\leqslant 5$ when, the inequality has no solution. When $k=6$, the inequality becomes $6(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right)$. Without loss of generality, assume $c=\\max (a, b, c)$, transform the inequality into $5 c^{2}-6(a+b) c+5\\left(a^{2}+b^{2}\\right)-6 a b0$, $5\\left(a^{2}+b^{2}\\right)-6 a b>0$, so the quadratic equation corresponding to the inequality has two positive roots $c_{1}, c_{2}$, the solution to the quadratic inequality in $c$ is $c \\in\\left(c_{1}, c_{2}\\right)$. If $c_{2} \\leqslant a+b$, then $c6$ when, $k(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right)$ has solutions, but they cannot form a triangle, so the only $k$ that satisfies the condition is $k=6$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49163, 30, 2959, 1885, 91, 20, 325, 253, 2731, 15328, 28, 715, 338, 327, 750, 2731, 2966, 1885, 81, 28, 278, 28, 265, 20, 19131, 260, 9724, 1885, 91, 24, 81, 278, 27, 82, 265, 27, 83, 253, 46153, 49164, 76, 8842, 24, 81, 21990, 49161, 109, 27, 82, 21990, 49161, 17243, 2771, 30, 20, 1885, 27, 83, 21990, 49161, 24362, 643, 665, 1251, 1822, 253, 14973, 351, 2103, 15900, 1885, 81, 28, 278, 28, 265, 28422, 7985, 260, 5869, 1685, 282, 1885, 91, 28422, 49154, 49158, 198, 49160, 49163, 30, 1885, 91, 24, 81, 278, 27, 81, 265, 27, 82, 265, 46153, 49164, 76, 8842, 24, 81, 21990, 49161, 109, 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+{"group_id": "math_034", "origin": "budgie", "answer": "1. Given that $PD: PE: PF = 1: 2: 3$, denote $PD = x$, $PE = 2x$, and $PF = 3x$. Since $PD + PE + PF = 12$, we have:\n   \\[ x + 2x + 3x = 12 \\Rightarrow 6x = 12 \\Rightarrow x = 2 \\]\n   \\[ PD = 2, PE = 4, PF = 9 \\]\n\n2. Since $\\triangle PEF$ is a right triangle with $PD$ as its hypotenuse, we can deduce that $PE = PF = 4$, making it an equilateral triangle.\n\n3. In an equilateral triangle, all sides are equal, and in a right triangle, the hypotenuse is $\\sqrt{3}$ times longer than each leg. Therefore, the length of each leg is:\n   \\[ PE = PF = 4\\sqrt{3} \\]\n\n4. The area of a triangle is given by:\n   \\[ \\text{Area} = \\frac{1}{2} \\times \\text{base} \\times \\text{height} \\]\n   where the base and height are the legs of the triangle.\n\n5. Let's denote the area of $\\triangle PEF$ as $[PEF]$. Using the Pythagorean theorem in $\\triangle PEF$:\n   \\[ [PEF] = \\left(\\frac{3}{4}\\right)^2 \\]\n\n6. From the given ratio $PD: PE: PF = 1:2:3$, we have:\n   \\[ PD = x = 2 \\times 1, PE = 2x = 4, PF = 3x = 9 \\]", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 1032, 3057, 281, 8799, 216, 49161, 28, 1303, 1885, 64, 20, 325, 253, 1225, 2972, 260, 1220, 13293, 19573, 22128, 6935, 17638, 20, 351, 2103, 3519, 216, 49160, 49161, 30, 10602, 24340, 99, 429, 1885, 64, 20, 288, 260, 5882, 1885, 6190, 26265, 1885, 8970, 26265, 284, 1885, 5431, 26265, 351, 260, 3282, 282, 260, 24340, 99, 1036, 1885, 52, 26265, 1885, 53, 26265, 284, 1885, 54, 20, 7827, 30, 10887, 338, 1885, 9556, 42, 20045, 42, 26015, 446, 216, 49160, 42, 216, 49161, 42, 216, 49162, 28422, 3875, 28, 260, 1557, 282, 20378, 13293, 1885, 50, 7548, 54, 20, 314, 49154, 49158, 198, 49160, 30, 10887, 338, 1885, 9556, 42, 20045, 42, 26015, 446, 216, 49160, 42, 216, 49161, 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+{"group_id": "math_034", "origin": "budgie", "answer": "1. From the problem, it is clear that the ratio of the areas of triangles $\\triangle PBD$ and $\\triangle PFE$ is equal to the ratio of their bases, $\\frac{1}{2}$.\n2. The area of a triangle is given by the formula $S_{\\triangle} = \\frac{1}{2}ab$, where $a$ and $b$ are the lengths of the legs of the triangle.\n3. Therefore, we have $S_{\\triangle PBD} = \\frac{1}{4}S_{\\triangle ABC}$.\n4. Since $PD: PE: PF = 1: 2: 3$, we have $\\frac{PD}{PE} = \\frac{1}{2}$ and $\\frac{PE}{PF} = \\frac{2}{3}$.\n5. Therefore, $\\frac{PD}{PE} = \\frac{1}{3}$.\n6. Since the area of $\\triangle ABC$ is $S_{\\triangle ABC} = \\frac{1}{2}bc$, we have $\\frac{1}{2}bc = \\frac{1}{3}bc$, which implies $bc = 2$.\n7. Let the altitudes of $\\triangle ABC$ from $A$ and $B$ to $BC$ and $CA$ be $h_{a}$ and $h_{b}$, respectively. Then we have $h_{a} + h_{b} = h_{c}$.\n8. We also know that $\\frac{h_{a}}{h_{b}} = \\frac{2}{3}$, which means $h_{b} = \\frac{2}{3}h_{a}$.\n9. 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+{"group_id": "math_034", "origin": "verified_reference", "answer": "$$\n6.11 \\sqrt{3} \\text {. }\n$$\n\nFor the equilateral $\\triangle ABC$ with a height of $12 \\times \\frac{\\sqrt{3}}{2}=6 \\sqrt{3}$, then\n$$\n\\begin{array}{l}\nP D+P E+P F=6 \\sqrt{3} . \\\\\n\\text { Also, } P D: P E: P F=1: 2: 3, \\text { so, } \\\\\nP D=\\sqrt{3}, P E=2 \\sqrt{3}, P F=3 \\sqrt{3} .\n\\end{array}\n$$\n\nAs shown in Figure 7, draw\n$P S / / B C$, intersecting $A B$ at $S$.\nSince $\\angle F S P=\\angle B$ $=60^{\\circ}$, therefore,\n$$\n\\begin{array}{l}\nF S=P F \\cot 60^{\\circ}=3, \\\\\nP S=6, \\\\\nB D=P S+P D \\cot 60^{\\circ}=7 .\n\\end{array}\n$$\n\nThus, $S_{\\text {quadrilateral BDPF }}=S_{\\triangle P S}+S_{\\text {trapezoid BDPS }}$\n$$\n=\\frac{1}{2} \\times 3 \\times 3 \\sqrt{3}+\\frac{6+7}{2} \\times \\sqrt{3}=11 \\sqrt{3} .\n$$", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 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+{"group_id": "math_035", "origin": "budgie", "answer": "The sequence starts from 1 and increases by 3 each time.\n\nHere is an example of the sequence:\n1; 4; 7; 10; 13; 16; 19; 22; 25; 28; 31; $$$\n\nThe $21^{\\varrho}$ term of the sequence is 28.\n\nFinal answer:\n#### 28", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 1780, 314, 260, 1885, 49161, 49160, 21990, 76, 102, 2110, 16147, 24362, 2115, 282, 260, 5972, 198, 198, 9212, 198, 49160, 8544, 216, 49161, 27, 49162, 8544, 216, 49163, 27, 49164, 27, 49165, 8544, 216, 49166, 27, 49167, 27, 49168, 27, 49160, 49159, 8544, 216, 49160, 49160, 27, 49160, 49161, 27, 49160, 49162, 27, 49160, 49163, 27, 49160, 49164, 8544, 3814, 400, 1565, 9148, 198, 9212, 49154, 49158, 198, 504, 5972, 5909, 429, 216, 49160, 284, 4495, 411, 216, 49162, 971, 655, 30, 198, 198, 4590, 314, 354, 1183, 282, 260, 5972, 42, 198, 49160, 43, 216, 49163, 43, 216, 49166, 43, 216, 49160, 49159, 43, 216, 49160, 49162, 43, 216, 49160, 49165, 43, 216, 49160, 49168, 43, 216, 49161, 49161, 43, 216, 49161, 49164, 43, 216, 49161, 49167, 43, 216, 49162, 49160, 43, 1885, 9212, 198, 198, 504, 1885, 49161, 49160, 21990, 76, 102, 2110, 16147, 24362, 2115, 282, 260, 5972, 314, 216, 49161, 49167, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49161, 49167, 49154], "prefix_len": 96, "old_logprobs": [-0.7064253091812134, -2.1128363609313965, -1.4904935359954834, -2.196011543273926, -0.5672488212585449, -0.11023297905921936, -0.39213189482688904, -1.5538829565048218, -0.09093422442674637, -0.10729856789112091, -0.6193104982376099, -0.20341934263706207, -0.2752462923526764, -0.5899540185928345, -1.2256301641464233, -0.08695174008607864, -4.820014953613281, -1.2590214014053345, -3.3636257648468018, -0.6942742466926575, -0.5064844489097595, -0.5588923096656799, -0.23836806416511536, -0.31788310408592224, -0.28988686203956604, -1.5490009784698486, -2.7020957469940186, -0.0972699224948883, -2.8533570766448975, -1.0686885118484497, -0.013619476929306984, -0.0490531399846077, -0.20197397470474243, -0.01366428006440401, -0.010436601005494595, -0.03175294026732445, -0.03866344690322876, -0.04938848316669464, -0.004479849245399237, -0.007804496213793755, -0.04377371817827225, -0.053258076310157776, -0.004611690063029528, -0.0012055517872795463, -0.06390839070081711, -0.08849840611219406, -0.012343090958893299, -0.0019602624233812094, -0.10365193337202072, -0.06747964769601822, -0.0005241450853645802, -0.006068968679755926, -0.033547792583703995, -0.06464362144470215, -0.0006235085893422365, -0.009080896154046059, -0.16216537356376648, -0.13239508867263794, -0.0006744970451109111, -0.003028926905244589, -0.02452835813164711, -0.09168770164251328, -0.011613735929131508, -0.009847158566117287, -0.12532299757003784, -4.928700923919678, -2.4139151573181152, -1.0522016286849976, -0.18397942185401917, -0.8604916930198669, -0.7772613167762756, -0.04756128042936325, -0.019902529194951057, -0.021333742886781693, -0.0025873063132166862, -0.00509487884119153, -0.00650052772834897, -0.08309584856033325, -0.016221093013882637, -0.02589593641459942, -0.7313510775566101, -0.579312801361084, -0.01413747388869524, -0.10231301188468933, -0.2035445123910904, -0.5125430226325989, -1.1856642961502075, -0.15004833042621613, -0.11035607010126114, -0.08339738100767136, -0.05282045528292656, -0.00033885458833537996, -0.0005536930402740836, -0.022069105878472328, -9.762764238985255e-05, -0.026370462030172348, -0.000598251644987613, -0.0035168009344488382, -0.0006611545104533434]}
+{"group_id": "math_035", "origin": "budgie", "answer": "The pattern in the sequence is an arithmetic sequence with first term $1$ and common difference $2$.\n\nThe $n^{\\varrho}$ term is given by the formula\n\n$$\n\\frac{n(a_1 + a_n)}{2}\n$$\n\nwhere $n$ is the term number, $a_1$ is the first term, and $a_n$ is the $n^{\\varrho}$ term.\n\nFor the $n^{\\varrho}$ term, $n$ is 15.\n\nSubstitute the values into the formula:\n\n$$\n\\frac{15(1 + 21)}{2} = \\frac{15(22)}{2} = 15 \\times 11 = \\boxed{165}\n$$", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 1780, 314, 260, 1885, 49161, 49160, 21990, 76, 102, 2110, 16147, 24362, 2115, 282, 260, 5972, 198, 198, 9212, 198, 49160, 8544, 216, 49161, 27, 49162, 8544, 216, 49163, 27, 49164, 27, 49165, 8544, 216, 49166, 27, 49167, 27, 49168, 27, 49160, 49159, 8544, 216, 49160, 49160, 27, 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+{"group_id": "math_035", "origin": "verified_reference", "answer": "Observe that the $21^{\\text{st}}$ term is the sum of 21 consecutive numbers. Taking the first term of each term, that is, 1, 2, 4, 7, 11, $16, \\ldots$, we have that\n\n$$\n\\begin{aligned}\n2 & =1+1 \\\\\n4 & =2+1+1 \\\\\n7 & =3+2+1+1 \\\\\n11 & =4+3+2+1+1 \\\\\n16 & =5+4+3+2+1+1\n\\end{aligned}\n$$\n\nThus, the first term of the $21^{\\text{st}}$ term is\n\n$$\n20+19+\\cdots+3+2+1+1=\\frac{20 \\times 21}{2}+1=211\n$$\n\nand the $21^{\\text{st}}$ term is\n\n$$\n211+212+\\cdots+230+231=\\frac{(211+231) \\times 21}{2}=221 \\times 21=4641\n$$", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 1780, 314, 260, 1885, 49161, 49160, 21990, 76, 102, 2110, 16147, 24362, 2115, 282, 260, 5972, 198, 198, 9212, 198, 49160, 8544, 216, 49161, 27, 49162, 8544, 216, 49163, 27, 49164, 27, 49165, 8544, 216, 49166, 27, 49167, 27, 49168, 27, 49160, 49159, 8544, 216, 49160, 49160, 27, 49160, 49161, 27, 49160, 49162, 27, 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+{"group_id": "math_036", "origin": "budgie", "answer": "The original sum of these numbers is:\n\\[ 1 + 2 + 3 + \\ldots + 2009 \\]\n\nA couple of the numbers occurring twice are written, with the remainder $2009 - (1 + 2 + 3 + \\ldots + 2009)$. Calculating the sum of this sequence, we have:\n\\[ 2009 \\times 13 - 1 = 26527 \\]\n\nSo the sum of the remaining two numbers is $2009 \\times 13 - 26527$ and their remainder is 26527.\n\nTherefore, the remainder when the sum of the erased numbers is divided by 13 is:\n\\[ 2009 \\times 13 - 26527 + 26527 = 2009 \\times 13 = 26527 \\]\n\nHence, the third number is $26527 - 2009 = \\boxed{24520}$.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 3280, 253, 4411, 28, 260, 2966, 429, 216, 49160, 288, 216, 49161, 49159, 49159, 49168, 359, 2855, 30, 330, 6075, 282, 601, 359, 40446, 284, 2489, 282, 601, 28, 335, 260, 4411, 314, 2855, 260, 17867, 282, 260, 2028, 282, 260, 40446, 2966, 6066, 411, 216, 49160, 49162, 30, 2550, 253, 6075, 282, 18388, 282, 451, 2812, 1817, 28, 805, 216, 49162, 2966, 359, 2049, 28, 282, 527, 827, 359, 216, 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+{"group_id": "math_036", "origin": "budgie", "answer": "Let's denote the sum of the numbers on the board before erasing and after erasing as S. By the problem's conditions, S = 1 + 2 + 3 + ... + 2009.\n\nThe first thing to do is to find the sum S. We know that S = 1(1 + 2 + 3 + ... + 2009) = 1(2009)(2010)/2 = 2010*2010 = 4,040,100.\n\nNow, let's subtract the sum of the numbers that are erased from this sum to find the sum of the numbers that remain.\n\nS = 4,040,100 - 9,099 = 4,030,011\n\nNow, we divide this sum by 13 to find the sum of the numbers that remain.\n\nS = 4,030,011/13 = 3,230,000\n\nThe sum of the numbers on the board (S) is 3,230,000.\n\nThe numbers on the board are: 3,230,000, 4,030,001, 4030,000.\n\nThe third number is 4030,000.\n\nThe final answer is \\boxed{4030000}.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 3280, 253, 4411, 28, 260, 2966, 429, 216, 49160, 288, 216, 49161, 49159, 49159, 49168, 359, 2855, 30, 330, 6075, 282, 601, 359, 40446, 284, 2489, 282, 601, 28, 335, 260, 4411, 314, 2855, 260, 17867, 282, 260, 2028, 282, 260, 40446, 2966, 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+{"group_id": "math_036", "origin": "verified_reference", "answer": "1. **Define the invariant:**\n   Let \\( I \\) be the remainder when the sum of all the integers on the board is divided by 13. Initially, the numbers from 1 to 2009 are written on the board. The sum of these numbers is:\n   \\[\n   S = \\sum_{k=1}^{2009} k = \\frac{2009 \\times 2010}{2} = 2009 \\times 1005\n   \\]\n   We need to find \\( S \\mod 13 \\).\n\n2. **Calculate \\( 2009 \\mod 13 \\) and \\( 1005 \\mod 13 \\):**\n   \\[\n   2009 \\div 13 = 154 \\quad \\text{remainder} \\quad 7 \\quad \\Rightarrow \\quad 2009 \\equiv 7 \\pmod{13}\n   \\]\n   \\[\n   1005 \\div 13 = 77 \\quad \\text{remainder} \\quad 4 \\quad \\Rightarrow \\quad 1005 \\equiv 4 \\pmod{13}\n   \\]\n\n3. **Calculate \\( S \\mod 13 \\):**\n   \\[\n   S = 2009 \\times 1005 \\equiv 7 \\times 4 = 28 \\equiv 2 \\pmod{13}\n   \\]\n   Therefore, \\( I = 2 \\).\n\n4. **Determine the third number \\( a \\):**\n   After several operations, only three numbers are left on the board: 9, 999, and \\( a \\). The sum of these three numbers modulo 13 must be equal to the invariant \\( I \\):\n   \\[\n   a + 9 + 999 \\equiv 2 \\pmod{13}\n   \\]\n\n5. **Calculate \\( 999 \\mod 13 \\):**\n   \\[\n   999 \\div 13 = 76 \\quad \\text{remainder} \\quad 11 \\quad \\Rightarrow \\quad 999 \\equiv 11 \\pmod{13}\n   \\]\n\n6. **Set up the equation:**\n   \\[\n   a + 9 + 11 \\equiv 2 \\pmod{13}\n   \\]\n   Simplify the equation:\n   \\[\n   a + 20 \\equiv 2 \\pmod{13}\n   \\]\n   Subtract 20 from both sides:\n   \\[\n   a \\equiv 2 - 20 \\pmod{13}\n   \\]\n   \\[\n   a \\equiv -18 \\pmod{13}\n   \\]\n   Since \\(-18 \\equiv -18 + 26 \\equiv 8 \\pmod{13}\\), we have:\n   \\[\n   a \\equiv 8 \\pmod{13}\n   \\]\n\n7. **Find the possible values of \\( a \\) between 1 and 2009:**\n   The possible values of \\( a \\) are of the form:\n   \\[\n   a = 8 + 13k \\quad \\text{for integer} \\quad k\n   \\]\n   We need \\( 1 \\leq a \\leq 2009 \\):\n   \\[\n   1 \\leq 8 + 13k \\leq 2009\n   \\]\n   \\[\n   -7 \\leq 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+{"group_id": "math_037", "origin": "budgie", "answer": "The last four digits of $2^{1999}$ are $1000000", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 2464, 30, 7985, 260, 1596, 1876, 18638, 282, 1885, 49161, 21990, 49160, 49168, 49168, 49168, 24362, 30, 198, 198, 8241, 12837, 260, 1694, 2120, 618, 2321, 28, 5361, 1446, 260, 2798, 1694, 506, 1761, 9102, 284, 4624, 28, 284, 3124, 260, 8483, 966, 3319, 30, 49154, 49158, 198, 504, 1596, 1876, 18638, 282, 1885, 49161, 21990, 49160, 49168, 49168, 49168, 24362, 359, 1885, 49160, 49159, 49159, 49159, 49159, 49159, 49159, 49154], "prefix_len": 75, "old_logprobs": [-1.6261858940124512, -0.7485172748565674, -0.016881341114640236, -0.002212578197941184, -0.04400874301791191, -0.2066299319267273, -0.029543381184339523, -0.3452100455760956, -0.022605936974287033, -0.003916214685887098, -0.0010151476599276066, -0.0018397325184196234, -0.026995036751031876, -0.2991894483566284, -1.1147418022155762, -1.8266735076904297, -1.1103037595748901, -1.7767319679260254, -0.21661123633384705, -0.7164345383644104, -0.1503499150276184, -0.14830179512500763, -14.781360626220703]}
+{"group_id": "math_037", "origin": "budgie", "answer": "I should not Areas have any new numerical value.", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 2464, 30, 7985, 260, 1596, 1876, 18638, 282, 1885, 49161, 21990, 49160, 49168, 49168, 49168, 24362, 30, 198, 198, 8241, 12837, 260, 1694, 2120, 618, 2321, 28, 5361, 1446, 260, 2798, 1694, 506, 1761, 9102, 284, 4624, 28, 284, 3124, 260, 8483, 966, 3319, 30, 49154, 49158, 198, 57, 868, 441, 20284, 457, 750, 725, 12141, 1685, 30, 49154], "prefix_len": 75, "old_logprobs": [-0.945827066898346, -2.6717734336853027, -1.6540647745132446, -10.62779426574707, -4.251129150390625, -2.1618282794952393, -3.0434930324554443, -6.8759355545043945, -1.8187967538833618, -1.4505869150161743, -0.7305818200111389]}
+{"group_id": "math_037", "origin": "verified_reference", "answer": "$$\n\\varphi(625)=\\varphi\\left(5^{4}\\right)=5^{4} \\times\\left(1-\\frac{1}{5}\\right)=500 \\text {. }\n$$\n\nSince $(2,625)=1$, by Euler's theorem, we get $2^{500} \\equiv 1(\\bmod 625)$.\n$$\n\\begin{array}{l}\n\\therefore 2^{2000} \\equiv 1(\\bmod 625) . \\\\\n\\therefore \\text { Let } 2^{2000}=625 m+1(m \\in \\mathbf{N}) .\n\\end{array}\n$$\n\nAlso, $2^{5} \\mid 2^{2000}$,\nso let $2^{2000}=2^{5} \\cdot n(n \\in \\mathbf{N})$.\nFrom (1) and (2), we get $625 m+1=2^{5} \\cdot n$.\nSolving this indeterminate equation, we get\n$$\n\\left\\{\\begin{array}{l}\nm=32 k+15, \\\\\nn=625 k+293\n\\end{array}\\right.\n$$\n$k \\in \\mathbf{Z}$.\nSubstituting $n=625 k+293$ into (2), we get\n$$\n\\begin{array}{l}\n2^{2000}=32(625 k+293)=20000 k+9376 . \\\\\n\\therefore 2^{1999}=10000 k+4688 .\n\\end{array}\n$$\n\nTherefore, the last four digits of $2^{1999}$ are 4688.", "reward": 1.0, "advantage": 1.4142115623759233, "ids": 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+{"group_id": "math_038", "origin": "budgie", "answer": "First, let's simplify the equations $x^{5}+x+1=0$ and $x+\\sqrt[5]{x}+1=0$.\n\nFor the first equation, recognize it as a binomial expansion:\n$$(x+1)^{5} = x^{5}+5\\cdot x^{4}+10\\cdot x^{3}+5\\cdot x^{2}+x = 0$$\n\nFor the second equation, recognize it as a root-sum equation:\n$$x+\\sqrt[5]{x}+1 = 0$$\n\nLet $y = x + \\sqrt[5]{x}$. Substitute $\\sqrt[5]{x}$ with $y-1$:\n$$y = x + (y-1)$$\n\nThis equation is a linear equation in terms of $y$. We can solve for $y$ by subtracting $x$ from both sides:\n$$y - x = 1$$\n\nNow we have a linear equation in $y$:\n$$y - x = 1$$\n\nWe can solve this equation for $x$ by using the Extended Euclidean Algorithm.\n\nFirst, express $1$ as a linear combination of $x$ and $\\sqrt[5]{x}$:\n$$1 = 1 - x - \\sqrt[5]{x}$$\n\nNow, apply the Extended Euclidean Algorithm to find the coefficients $a$ and $b$ such that:\n$$a = -1, b = 1$$\n\nNext, express $-1$ as a linear combination of $x$ and $\\sqrt[5]{x}$:\n$$-1 = -1 - x - \\sqrt[5]{x}$$\n\nNow, apply the Extended Euclidean Algorithm to find the coefficients", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 365, 17872, 429, 260, 216, 49160, 49162, 373, 476, 42519, 23278, 18, 6484, 269, 780, 29435, 25, 378, 1345, 5190, 282, 260, 9773, 1885, 104, 21990, 49164, 109, 27, 104, 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+{"group_id": "math_038", "origin": "budgie", "answer": "The problem states that the roots of both equations are $\\alpha, \\beta$. We can use Vieta's formulas to solve this problem.\n\nFor the first equation, $x^{5}+x+1=0$, the sum of the roots is $-\\frac{1}{2}$. Therefore, we have\n$$\\alpha + \\beta = -\\frac{1}{2}.$$\n\nFor the second equation, $x+\\sqrt[5]{x}+1=0$, the product of the roots is $-\\frac{1}{x}$. We can write this as\n$$(\\alpha + \\beta) \\cdot (\\alpha + \\beta) = -\\frac{1}{x}.$$\n\nSince $\\alpha + \\beta = -\\frac{1}{2}$ from the given options, we have\n$$-\\frac{1}{2} \\cdot (\\alpha + \\beta) = -\\frac{1}{x}.$$\n\nSubstituting the value of $\\alpha + \\beta$ from our previous result, we get\n$$-\\frac{1}{2} \\cdot (-\\frac{1}{2}) = -\\frac{1}{x}.$$\n\nSolving for $x$, we find that\n$$-\\frac{1}{4} = -\\frac{1}{x},$$\nwhich implies that\n$$x = 4.$$\n\nTherefore, the correct answer is 4, so we have\n$$\\boxed{4}$$\n\nFinal answer:\n#### -1", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 365, 17872, 429, 260, 216, 49160, 49162, 373, 476, 42519, 23278, 18, 6484, 269, 780, 29435, 25, 378, 1345, 5190, 282, 260, 9773, 1885, 104, 21990, 49164, 109, 27, 104, 27, 49160, 45, 49159, 20, 284, 1885, 104, 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+{"group_id": "math_038", "origin": "verified_reference", "answer": "Solution: Choose A. Reason: Consider the function $f(x)=x^{5}+x+1$, which is an increasing function on $\\mathbf{R}$,\nwhen $x^{5}+x+1=0$,\nwe get $x=\\sqrt[5]{-x-1}$, substituting $-x-1$ for $x$ yields $-x-1=\\sqrt[5]{x}$.\nFrom (2), we have $(-x-1)^{5}=x$, which is also $(-x-1)^{5}+(-x-1)+1=0$.\nLet $\\alpha$ be a root of equation (1), i.e., $f(\\alpha)=0$. Let $\\beta$ be a root of equation (2), i.e., $-1-\\beta=\\sqrt[5]{\\beta}$, thus from (3) we know that $f(-1-\\beta)=0$.\nTherefore, we have $f(\\alpha)=f(-1-\\beta)$, which means $\\alpha+\\beta=-1$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 365, 17872, 429, 260, 216, 49160, 49162, 373, 476, 42519, 23278, 18, 6484, 269, 780, 29435, 25, 378, 1345, 5190, 282, 260, 9773, 1885, 104, 21990, 49164, 109, 27, 104, 27, 49160, 45, 49159, 20, 284, 1885, 104, 42639, 16166, 75, 49164, 36939, 104, 109, 27, 49160, 45, 49159, 20, 359, 19573, 13433, 28, 3814, 17565, 20, 7827, 28, 965, 19573, 13433, 42639, 17565, 20, 16473, 365, 15165, 198, 49, 30, 731, 49160, 198, 50, 30, 1885, 46240, 18169, 107, 49160, 15009, 49161, 24362, 198, 51, 30, 19573, 18169, 107, 49160, 15009, 49161, 24362, 198, 52, 30, 216, 49160, 49154, 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17565, 24814, 16166, 75, 49164, 77, 26754, 17565, 24362, 28, 3310, 429, 365, 49162, 25, 392, 699, 338, 1885, 86, 10242, 49160, 46240, 17565, 36637, 49159, 28422, 198, 17308, 28, 392, 457, 1885, 86, 19407, 13433, 36637, 86, 10242, 49160, 46240, 17565, 25, 26265, 527, 1530, 19573, 13433, 42639, 17565, 15196, 49160, 28422, 49154], "prefix_len": 132, "old_logprobs": [-4.239291191101074, -0.03602408617734909, -10.380029678344727, -6.006158351898193, -1.6323580741882324, -14.995430946350098, -2.6167168617248535, -8.304129600524902, -0.6309458613395691, -2.6258506774902344, -0.3151669204235077, -0.20736399292945862, -0.04648115113377571, -0.08773563802242279, -0.490862637758255, -0.10086862742900848, -0.4489029049873352, -0.22785815596580505, -0.06976636499166489, -0.08413946628570557, -0.030051346868276596, -0.2565946578979492, -0.003767061745747924, -1.4398553371429443, -3.245041847229004, -0.49924278259277344, -1.2620635032653809, -1.9664454460144043, -0.013746591284871101, 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+{"group_id": "math_040", "origin": "budgie", "answer": "The problem requires us to find how many steps bring Henry's list of the first 1000 positive integers to an empty state. To do this, let's break down the process step by step.\n\nStep 1:\nWe start with a list of the first 1000 positive integers.\n\nStep 2:\nWe start erasing any nonpositive integers or integers with a repeated digit.\n\nStep 3:\nFor the remaining integers, we decrease them by 1.\n\nStep 4:\nAt the end, we have a list of the first 1000 integers.\n\nNow, let's count the number of steps required.\n\nThe final answer: 1000", "reward": -1, "advantage": -1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 27803, 5909, 351, 253, 1398, 282, 260, 808, 216, 49160, 49159, 49159, 49159, 2731, 23313, 28, 284, 16790, 253, 3086, 282, 3301, 335, 260, 1398, 30, 1814, 971, 1833, 28, 384, 2812, 1049, 750, 1603, 16185, 23313, 355, 750, 23313, 338, 457, 253, 6514, 11403, 28, 284, 965, 12191, 3117, 281, 260, 1398, 411, 216, 49160, 30, 1073, 800, 3301, 1072, 357, 1188, 327, 6661, 506, 1398, 288, 325, 7964, 47, 198, 198, 75, 89, 77, 2954, 11312, 411, 7937, 11583, 75, 31, 89, 77, 49154, 49158, 198, 504, 1732, 3073, 468, 288, 1042, 638, 800, 3301, 2635, 6661, 506, 1398, 282, 260, 808, 216, 49160, 49159, 49159, 49159, 2731, 23313, 288, 354, 7964, 1215, 30, 1626, 536, 451, 28, 1303, 506, 2373, 1187, 260, 980, 1833, 411, 1833, 30, 198, 198, 3512, 216, 49160, 42, 198, 1882, 1120, 351, 253, 1398, 282, 260, 808, 216, 49160, 49159, 49159, 49159, 2731, 23313, 30, 198, 198, 3512, 216, 49161, 42, 198, 1882, 1120, 2812, 1817, 750, 1603, 16185, 23313, 355, 23313, 351, 253, 6514, 11403, 30, 198, 198, 3512, 216, 49162, 42, 198, 2193, 260, 6219, 23313, 28, 392, 6302, 601, 411, 216, 49160, 30, 198, 198, 3512, 216, 49163, 42, 198, 3473, 260, 1112, 28, 392, 457, 253, 1398, 282, 260, 808, 216, 49160, 49159, 49159, 49159, 23313, 30, 198, 198, 2696, 28, 1303, 506, 985, 260, 1230, 282, 3301, 2309, 30, 198, 198, 504, 3403, 2988, 42, 216, 49160, 49159, 49159, 49159, 49154], "prefix_len": 115, "old_logprobs": [-1.871724009513855, -2.574619770050049, -2.096071243286133, -0.7273120880126953, -0.0002602000313345343, -0.7705178260803223, -2.7802019119262695, -0.04593171179294586, -0.20295001566410065, -5.081482887268066, -1.7981959581375122, -0.033860255032777786, -0.11285962164402008, -0.6855083703994751, -0.9442072510719299, -0.003649243852123618, -0.011937606148421764, -0.0024352199397981167, -0.00721787940710783, -0.01309144590049982, -0.00012385078298393637, -0.008104770444333553, -0.0010724276071414351, -0.39646559953689575, -1.7101092338562012, -0.04261443763971329, -0.6259003281593323, -0.21008506417274475, -3.8725905418395996, -1.3950518369674683, -0.023674197494983673, -0.02188263274729252, -3.4927611351013184, -0.01255536638200283, -1.1237075328826904, -0.18404120206832886, -0.4941917955875397, -0.5180196762084961, -0.5200074911117554, -0.16161076724529266, -0.00020346954988781363, -0.4072398245334625, -0.3694777190685272, -0.0006252956227399409, 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+{"group_id": "math_040", "origin": "budgie", "answer": "This is a classic example of the Cauchy-Ri plague algorithm. The steps are as follows:\n\n1.  Start with the list of 1000 positive integers.\n2.  Initialize an empty list.\n3. aced the first integer by erasing nonpositive integers or integers that have a repeated digit, and then reducing the list by 1.\n4.  Continue this process until the list becomes empty.\n\nThe algorithm runs in O(N^2) time, where N is the number of integers. Since the list grows extremely quickly, it will eventually become empty.\n\nI see that the example given shows how the algorithm works. For 2, the list is [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], which takes 129 steps.\n\nI should also note that for a list of 1000 numbers, the number of steps required is log2(1000) = 11, since 2^10 = 1024, and 1025 is the next power of two after 1024.\n\nI think the answer is 11 steps.\n\nFinal answer:\n#### 11", "reward": 1.0, "advantage": 0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 27803, 5909, 351, 253, 1398, 282, 260, 808, 216, 49160, 49159, 49159, 49159, 2731, 23313, 28, 284, 16790, 253, 3086, 282, 3301, 335, 260, 1398, 30, 1814, 971, 1833, 28, 384, 2812, 1049, 750, 1603, 16185, 23313, 355, 750, 23313, 338, 457, 253, 6514, 11403, 28, 284, 965, 12191, 3117, 281, 260, 1398, 411, 216, 49160, 30, 1073, 800, 3301, 1072, 357, 1188, 327, 6661, 506, 1398, 288, 325, 7964, 47, 198, 198, 75, 89, 77, 2954, 11312, 411, 7937, 11583, 75, 31, 89, 77, 49154, 49158, 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+{"group_id": "math_040", "origin": "verified_reference", "answer": "1. **Initial List and Conditions**:\n   - Henry starts with the list of the first 1000 positive integers: \\( \\{1, 2, 3, \\ldots, 1000\\} \\).\n   - At each step, he erases any nonpositive integers or any integers that have a repeated digit.\n   - After erasing, he decreases every remaining integer in the list by 1.\n\n2. **Understanding the Erasure Condition**:\n   - Nonpositive integers are erased immediately.\n   - Integers with repeated digits are also erased. For example, numbers like 11, 22, 101, etc., will be erased.\n\n3. **Effect of Decreasing by 1**:\n   - After each step, every integer in the list is decreased by 1. This means that the list will eventually contain nonpositive integers, which will be erased.\n\n4. **Key Observation**:\n   - The critical observation is to determine the maximum number of steps it takes for any integer to be erased.\n   - Consider the number 10. It will be decreased to 9, 8, 7, ..., 1, 0, and then -1. This process takes exactly 11 steps.\n\n5. **Erasure of Numbers with Repeated Digits**:\n   - Numbers like 11, 22, 33, ..., 99, 101, 111, etc., will be erased as soon as they appear in the list.\n   - For example, 11 will be erased in the first step, 22 in the second step, and so on.\n\n6. **General Case**:\n   - For any number \\( n \\) in the list, if it does not have repeated digits, it will be decreased step by step until it becomes nonpositive.\n   - The maximum number of steps required for any number to become nonpositive is determined by the number 10, which takes 11 steps.\n\n7. **Conclusion**:\n   - Since the number 10 takes exactly 11 steps to be erased, and all other numbers will be erased in fewer or equal steps, the entire list will be empty after 11 steps.\n\n\\[\n\\boxed{11}\n\\]", "reward": 1.0, "advantage": 0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 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+{"group_id": "math_041", "origin": "budgie", "answer": "The sum of the other two sides is 27, so the third side is 27 - 2x = 4, where x is the length of the side opposite the given side. Then, we have 27 - 2x = 4. Solving for x, we get x = 27/4.\n\nSince the inradius of the triangle is 4, which is the radius of the inscribed circle, the angle opposite the given side is equal to the angle opposite the radius. Therefore, the angle opposite the given side is equal to 360 degrees minus 180 degrees, or 180 degrees.\n\nThus, the answer is \\boxed{180}.", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49161, 30, 49160, 49167, 49159, 30, 330, 2103, 282, 260, 14973, 314, 4037, 288, 216, 49160, 49164, 28, 260, 2028, 282, 260, 550, 827, 5882, 314, 216, 49161, 49166, 30, 7985, 260, 39027, 282, 260, 7256, 7011, 260, 1836, 2103, 28, 585, 260, 13661, 282, 260, 30036, 7842, 281, 260, 14973, 314, 216, 49163, 30, 49154, 49158, 198, 504, 2028, 282, 260, 550, 827, 5882, 314, 216, 49161, 49166, 28, 588, 260, 3464, 2103, 314, 216, 49161, 49166, 731, 216, 49161, 104, 446, 216, 49163, 28, 837, 1792, 314, 260, 3519, 282, 260, 2103, 7011, 260, 1836, 2103, 30, 3875, 28, 392, 457, 216, 49161, 49166, 731, 216, 49161, 104, 446, 216, 49163, 30, 30238, 327, 1792, 28, 392, 820, 1792, 446, 216, 49161, 49166, 31, 49163, 30, 198, 198, 7230, 260, 281, 24440, 282, 260, 14973, 314, 216, 49163, 28, 527, 314, 260, 13661, 282, 260, 30036, 7842, 28, 260, 7256, 7011, 260, 1836, 2103, 314, 4037, 288, 260, 7256, 7011, 260, 13661, 30, 4882, 28, 260, 7256, 7011, 260, 1836, 2103, 314, 4037, 288, 216, 49162, 49165, 49159, 5027, 18253, 216, 49160, 49167, 49159, 5027, 28, 355, 216, 49160, 49167, 49159, 5027, 30, 198, 198, 14344, 28, 260, 2988, 314, 3814, 4810, 277, 107, 49160, 49167, 49159, 19181, 49154], "prefix_len": 85, "old_logprobs": [-1.949613094329834, -2.0315499305725098, -0.004266683477908373, -0.10915522277355194, -0.37602463364601135, -0.004523046314716339, -0.0024132197722792625, -0.4790869355201721, -0.12486499547958374, -0.10333548486232758, -0.10346878319978714, -0.7809114456176758, -0.9782068729400635, -0.9665250778198242, -1.3077925443649292, -0.01130437757819891, -0.5259026288986206, -0.09725499898195267, -0.17042315006256104, -0.02695419266819954, -0.28588464856147766, -0.08853746950626373, -0.7653873562812805, -3.4850168228149414, -1.0174169540405273, -0.07181113958358765, -3.8292784690856934, -1.6823101043701172, -0.9089520573616028, -0.00744272256270051, -0.05472160503268242, -0.025187214836478233, -0.5085362195968628, -0.050171539187431335, -0.18647298216819763, -0.9104083776473999, -0.6326968669891357, -0.14116470515727997, -1.6305606365203857, -0.2927303910255432, -0.1373077630996704, -3.223015069961548, -1.0170705318450928, -1.6682024002075195, -0.5599475502967834, -0.40137749910354614, -0.5418233871459961, -0.5164341330528259, -0.027581999078392982, -0.11155867576599121, -0.251260370016098, -0.05483660474419594, -0.10986039787530899, -0.065475232899189, -0.17217475175857544, -1.9649536609649658, -0.6427127718925476, -0.3693077564239502, -0.004739242140203714, -0.16412858664989471, -0.003243544837459922, -0.11420800536870956, -0.3675920367240906, -0.00998159870505333, -0.06703906506299973, -0.9075446128845215, -0.3002467751502991, -0.1638481169939041, -1.1923787593841553, -1.259132742881775, -0.8282867074012756, -0.06619960069656372, -2.9121999740600586, -0.5211148858070374, -3.985921859741211, -0.0005031750188209116, -0.5637041330337524, -0.6469448804855347, -0.026321114972233772, -0.019012944772839546, -0.37021782994270325, -0.05217503756284714, -0.022799383848905563, -3.8542938232421875, -0.09375808387994766, -1.4037925004959106, -0.7145781517028809, -0.03172718733549118, -0.025734923779964447, -0.10922896862030029, -0.000654602306894958, -0.03831886500120163, -0.4246523976325989, -0.6731109023094177, -0.024490319192409515, -0.20948360860347748, -0.6368272304534912, -0.0022104373201727867, -0.15884679555892944, -1.0959442853927612, -0.009781762026250362, -0.4484667181968689, -0.18613003194332123, -0.24014000594615936, -0.04385975003242493, -1.5531961917877197, -1.3021754026412964, -0.8861821293830872, -0.0012335318606346846, -0.16567693650722504, -0.6611570715904236, -0.10597717016935349, -0.0326773077249527, -0.9089555740356445, -0.0013174673076719046, -0.0449293777346611, -1.9763059616088867, -0.005495559424161911, -0.950429379940033, -2.480767250061035, -0.43628206849098206, -0.7508238554000854, -1.0170297622680664, -1.2447285652160645, -0.843635082244873, -1.22129225730896, -0.9981544017791748, -0.0029252853710204363, -0.020219286903738976, -0.3108171820640564, -0.9591777920722961, -0.048967666923999786, -0.023342112079262733, -0.001001571537926793, -0.00042012447374872863, -0.005242410581558943, -0.1029992550611496, -0.14862918853759766, -0.0013684204313904047, -2.4572300910949707, -0.0026750515680760145, -0.055143050849437714, -1.6075286865234375, -0.0012157914461567998, -1.2671606540679932, -0.01009477861225605, -8.702239938429557e-06, -0.005043996497988701, -0.05562093108892441, -0.0434034988284111, -0.0028055852744728327, -0.07963540405035019, -0.3330913484096527]}
+{"group_id": "math_041", "origin": "budgie", "answer": "12.180. A side of the triangle is equal to 15. The sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4.\n\nFinal answer: 4", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49161, 30, 49160, 49167, 49159, 30, 330, 2103, 282, 260, 14973, 314, 4037, 288, 216, 49160, 49164, 28, 260, 2028, 282, 260, 550, 827, 5882, 314, 216, 49161, 49166, 30, 7985, 260, 39027, 282, 260, 7256, 7011, 260, 1836, 2103, 28, 585, 260, 13661, 282, 260, 30036, 7842, 281, 260, 14973, 314, 216, 49163, 30, 49154, 49158, 198, 49160, 49161, 30, 49160, 49167, 49159, 30, 330, 2103, 282, 260, 14973, 314, 4037, 288, 216, 49160, 49164, 30, 378, 2028, 282, 260, 550, 827, 5882, 314, 216, 49161, 49166, 30, 7985, 260, 39027, 282, 260, 7256, 7011, 260, 1836, 2103, 28, 585, 260, 13661, 282, 260, 30036, 7842, 281, 260, 14973, 314, 216, 49163, 30, 198, 198, 29288, 2988, 42, 216, 49163, 49154], "prefix_len": 85, "old_logprobs": [-1.7625274658203125, -0.2852133810520172, -0.019678249955177307, -0.0632067546248436, -0.08933103084564209, -0.024749109521508217, -0.02094881422817707, -0.8523209095001221, -0.03433908149600029, -0.03505378216505051, -0.003901965217664838, -0.001149232150055468, -0.009642454795539379, -0.18531294167041779, -7.402622577501461e-05, -0.005557325202971697, -0.0015445933677256107, -0.0020831567235291004, -2.364210844039917, -0.09893722087144852, -0.09756212681531906, -0.00016556799528189003, -0.004048842005431652, -0.03579879552125931, -0.0016195764765143394, -0.00031537804170511663, -0.01855059526860714, -0.010237799026072025, -0.005612330511212349, -0.007803786545991898, -0.020496468991041183, -0.27346596121788025, -0.0033279056660830975, -0.05561517924070358, -0.0010408704401925206, -0.002529280725866556, -0.006299519911408424, -0.0021621440537273884, -0.056628718972206116, -0.013809612952172756, -0.0007113072206266224, -0.17834442853927612, -0.004774479195475578, -0.0027096483390778303, -0.003655538894236088, -0.00011002412065863609, -0.00020644917094614357, -0.0018772137118503451, -0.00020275443966966122, -0.04476592689752579, -0.00016735584358684719, -0.00011288482346571982, -0.00021026308240834624, -0.003204094711691141, -0.0007233387441374362, -0.0015987483784556389, -0.28829389810562134, -0.012479785829782486, -0.052587151527404785, -0.0007202415145002306, -0.0007626485894434154, -2.3861351013183594, -0.6596547365188599, -0.023943325504660606]}
+{"group_id": "math_041", "origin": "verified_reference", "answer": "## Solution.\n\nLet $a$ be the given side of the triangle, $a=15$, $b$ and $c$ be the other two sides, $b+c=27$, $c=27-b$. For definiteness, we will assume that $b>c$. The semiperimeter of the triangle $p=\\frac{a+b+c}{2}=21$, its area $S=p r=84$.\n\nBy Heron's formula $S=\\sqrt{p(p-a)(p-b)(p-c)}$, thus\n\n$$\n\\begin{aligned}\n& \\sqrt{21 \\cdot(21-15)(21-b)(21-27+b)}=84 \\Leftrightarrow \\sqrt{21 \\cdot 6(21-b)(b-6)}=84 \\Leftrightarrow \\\\\n& \\Leftrightarrow(21-b)(b-6)=56 \\Leftrightarrow b^{2}-27 b+182=0, \\\\\n& b_{1}=14, b_{2}=13 . \\\\\n& \\text { We have } b=14, c=13 \\text {. By the cosine rule }\n\\end{aligned}\n$$\n\n$$\n\\cos \\alpha=\\frac{b^{2}+c^{2}-a^{2}}{2 b c}=\\frac{5}{13}\n$$\n\nAnswer: $\\frac{5}{13}$.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49161, 30, 49160, 49167, 49159, 30, 330, 2103, 282, 260, 14973, 314, 4037, 288, 216, 49160, 49164, 28, 260, 2028, 282, 260, 550, 827, 5882, 314, 216, 49161, 49166, 30, 7985, 260, 39027, 282, 260, 7256, 7011, 260, 1836, 2103, 28, 585, 260, 13661, 282, 260, 30036, 7842, 281, 260, 14973, 314, 216, 49163, 30, 49154, 49158, 198, 712, 19857, 30, 198, 198, 4239, 1885, 81, 20, 325, 260, 1836, 2103, 282, 260, 14973, 28, 1885, 81, 45, 49160, 49164, 26265, 1885, 82, 20, 284, 1885, 83, 20, 325, 260, 550, 827, 5882, 28, 1885, 82, 27, 83, 45, 49161, 49166, 26265, 1885, 83, 45, 49161, 49166, 29, 82, 28422, 1068, 39416, 8051, 28, 392, 523, 7635, 338, 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+{"group_id": "math_046", "origin": "budgie", "answer": "This problem involves the properties of the circumcircle and incircle of a triangle, as well as the midpoint of an arc.\n\nThe key to solving this problem is to apply relevant theorems and theorems related to the circumcircle and the incircle.\n\nSince $O$ and $I$ are the centers of the circumcircle and incircle and $M$ is the midpoint of the arc $AC$, we can use the theorem of the triangle being equal to the sum of the lengths of any two sides to find the length of $AC$.\n\nFor the triangle $ABC$, we have $AB = 15$, $BC = 7$, and $MI = MO$. Applying the theorem of the triangle being equal to the sum of the lengths of any two sides, we can find the length of $AC$.\n\nThe theorem states that in any triangle, the sum of the lengths of the two smaller sides is equal to the length of the largest side.\n\nSo, for triangle $ABC$, we have $AB + BC = Introduce.\n\nApplying the theorem to triangle $ABC$, we get $15 + 7 = 22 = AC$.\n\nTherefore, the length of $AC$ is $22$ units.\n\nFinal answer:\n#### 22", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 24862, 1885, 63, 20, 284, 1885, 57, 20, 359, 260, 7292, 282, 260, 4584, 29733, 284, 1266, 337, 1668, 282, 14973, 1885, 35542, 26265, 284, 1885, 61, 20, 314, 260, 3921, 3154, 282, 260, 13210, 1885, 3144, 20, 282, 260, 4584, 29733, 365, 1766, 5085, 1885, 50, 20, 595, 657, 314, 1343, 338, 1885, 5431, 45, 49160, 49164, 28, 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+{"group_id": "math_046", "origin": "budgie", "answer": "The key to this problem lies in the fact that $M$ is the midpoint of the arc $AC$ of the circumcircle not containing $B$. This is known as the inscribed angle theorem.\n\nGiven that $AB=15$ and $BC=7$, we can find the radius of the circumcircle using the formula $R = \\frac{abc}{4 \\Delta}$, where $a$, $b$, and $c$ are the sides of the triangle and $\\Delta$ is the area of the triangle.\n\nSubstituting the given values for $a$, $b$, and $c$, we get $R = \\frac{15 \\times 7}{4 \\times \\sqrt{15^2 - 7^2}} = \\frac{105}{4 \\times 2\\sqrt{109}}$.\n\nNext, we can find the area of the triangle using Heron's formula. First, we need to find the semi-perimeter $s$ of the triangle:\n\n$s = \\frac{a + b + c}{2} = \\frac{15 + 7 + AC}{2} = \\frac{22 + AC}{2}$.\n\nNow, we can express the area of the triangle in terms of $AC$ using Heron's formula:\n\n$Area = \\sqrt{s(s - a)(s - b)(s - c)} = \\sqrt{\\frac{22 + AC}{2} \\left(\\frac{22 + AC}{2} - 15\\right) \\left(\\frac{22 + AC}{2} - 7\\right) \\left(\\frac{22 + AC}{2} - AC\\right)}}$.\n\nSimplify the expression to get the area", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 24862, 1885, 63, 20, 284, 1885, 57, 20, 359, 260, 7292, 282, 260, 4584, 29733, 284, 1266, 337, 1668, 282, 14973, 1885, 35542, 26265, 284, 1885, 61, 20, 314, 260, 3921, 3154, 282, 260, 13210, 1885, 3144, 20, 282, 260, 4584, 29733, 365, 1766, 5085, 1885, 50, 20, 595, 657, 314, 1343, 338, 1885, 5431, 45, 49160, 49164, 28, 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+{"group_id": "math_046", "origin": "verified_reference", "answer": "Answer: $A C=13$.\n\nSolution. (Fig. 3). First, we prove that $M I=M A$ (trident lemma).\n\nIndeed, the external angle $A I M$ of triangle $A I B$ is equal to the sum of angles $B A I$ and $A B I$, and since $A I$ and $B I$ are angle bisectors, $\\angle A I M=\\frac{1}{2} \\angle A+\\frac{1}{2} \\angle B$. Angle $I A M$ is equal to the sum of angles $I A C$ and $C A M$. But $\\angle I A C=\\frac{1}{2} \\angle A$, and $\\angle C A M=\\angle C B M=\\frac{1}{2} \\angle B$ as inscribed angles.\n\nFrom this, it follows that $\\angle I A M=\\frac{1}{2} \\angle A+\\frac{1}{2} \\angle B=\\angle A I M$, and therefore, triangle $A M I$ is isosceles, $M I=M A$. By the condition $M O=$ $M I$, so by the trident lemma $A O=M O=M I=M A$. This means that triangle $A O M$ is equilateral and $\\angle A O M=60^{\\circ}$. Since the central angle $A O M$ is twice the inscribed angle $A B M$, we have $\\frac{1}{2} \\angle B=30^{\\circ}$, that is, $\\angle B=60^{\\circ}$.\n\nBy the cosine theorem, $A C^{2}=A B^{2}+B C^{2}-2 \\cdot A B \\cdot B C \\cos 60^{\\circ}=$ $13^{2}$.\n\nCriteria. Answer without explanation - 0 points. Proved that angle $B$ is $60^{\\circ}-5$ points. Points are not deducted for the absence of proof of the trident lemma. Full solution - 7 points.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_7789e0097aa3cdbc6e57g-2.jpg?height=540&width=467&top_left_y=1779&top_left_x=1485)\n\nFig. 3", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 24862, 1885, 63, 20, 284, 1885, 57, 20, 359, 260, 7292, 282, 260, 4584, 29733, 284, 1266, 337, 1668, 282, 14973, 1885, 35542, 26265, 284, 1885, 61, 20, 314, 260, 3921, 3154, 282, 260, 13210, 1885, 3144, 20, 282, 260, 4584, 29733, 365, 1766, 5085, 1885, 50, 20, 595, 657, 314, 1343, 338, 1885, 5431, 45, 49160, 49164, 28, 7636, 45, 49166, 26265, 284, 1885, 11155, 45, 12850, 28422, 7985, 1885, 3144, 28422, 49154, 49158, 198, 21350, 42, 1885, 49, 340, 45, 49160, 49162, 28422, 198, 198, 37500, 30, 365, 10462, 30, 216, 49162, 595, 3508, 28, 392, 6824, 338, 1885, 61, 339, 45, 61, 330, 20, 365, 8489, 880, 20417, 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+{"group_id": "math_047", "origin": "budgie", "answer": "The key concept in this problem is the concept of inverse functions.\n\nLet $f(x) = \\frac{a-x}{x-a-1}$, then $y = f(f^{-1}(x))$.\n\nWe know that the inverse function $f^{-1}(x)$ is symmetric about the point $(-1,4)$.\n\nThe point $(-1,4)$ is the midpoint between the point $(a-1,1)$ and the point $(x,y)$ on the graph of $y = f(f^{-1}(x))$.\n\nSince $(a-1,1)$ is on the graph of $y = f(f^{-1}(x))$, we have $f(a-1) = 1$.\n\nNow, let's find the inverse function $f^{-1}(x)$. We know that the inverse function $f^{-1}(x)$ is symmetric about the point $(-1,4)$.\n\nThe point $(-1,4)$ is the midpoint between the point $(a-1,1)$ and the point $(x,-1)$.\n\nSince $(-1,4)$ is the midpoint between the point $(a-1,1)$ and the point $(x,-1)$, we have $f^{-1}(x) = -1$.\n\nSubstituting $a-1$ for $x$ in the equation $f^{-1}(x) = -1$, we get $f^{-1}(a-1) = -1$.\n\nNow, consider the inverse function $f^{-1}(x)$. 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+{"group_id": "math_047", "origin": "budgie", "answer": "The inverse function $f^{-1}(x)$ satisfies the condition:\n\\[ f^{-1}(f(x)) = x \\]\n\nSince the graph of the inverse function is symmetric about the point $(-1, 4)$, we have:\n\\[ f(-1 + x) = f(x) \\]\n\nLet $x = 0$ in the equation above:\n\\[ f(0) = f(0 + y) + f(0 - y) = f(0) + f(y) = 4 \\]\n\nTherefore, $y = 0$.\n\nHence, $f^{-1}(0) = 0$.\n\nSince the function $y = x^{-1}$ is an inverse function of $y = 2x - 1$, the point $0$ corresponds to the point $(-1, 4)$.\n\nSince $f(-1 + x) = f(x)$, it follows that:\n\\[ f(x) = f(0) = 2(0) - 1 = -1 \\]\n\nHence, the value of $a$ is:\n\\[ a = \\boxed{0} \\]\n\nFinal answer:\n#### 0", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 10887, 260, 1517, 1885, 105, 24814, 18169, 107, 81, 29, 104, 15009, 104, 29, 81, 29, 49160, 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+{"group_id": "math_047", "origin": "verified_reference", "answer": "2.3.-\n\nFrom the problem, we know that the graph of the function $y=\\frac{a-x}{x-a-1}$ is centrally symmetric about the point $(4,-1)$.\n$\\because y=\\frac{a-x}{x-a-1}=-1-\\frac{1}{x-(a+1)}$, we have $(y+1)[x-(a+1)]=-1$,\n$\\therefore$ the graph of the function is a hyperbola with its center at $(a+1,-1)$.\nAlso, $\\because$ the graph of the hyperbola is centrally symmetric about its center,\n$\\therefore a+1=4$, which means $a=3$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 10887, 260, 1517, 1885, 105, 24814, 18169, 107, 81, 29, 104, 15009, 104, 29, 81, 29, 49160, 24362, 28, 260, 3670, 282, 624, 20508, 1517, 314, 31458, 563, 260, 1225, 1885, 10242, 49160, 28, 49163, 25, 28422, 3875, 260, 1685, 282, 260, 1345, 1230, 1885, 81, 20, 314, 19573, 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+{"group_id": "math_048", "origin": "budgie", "answer": "Solution:\n\n1. Since $|\\overrightarrow{O A}|=|\\overrightarrow{O B}|=24$, then $|\\overrightarrow{O A}|^2=4\\overrightarrow{O B}^2=256$,\nwhich means $256 \\times 24=256\\times4\\overrightarrow{O B}^2 \\Rightarrow \\overrightarrow{O B}^2=256$,\nwhich means $\\overrightarrow{O B}=(1 it, 0 )$.\n\n2. $t \\overrightarrow{A B}-\\overrightarrow{A O}=t (|\\overrightarrow{O B}| - 24)=t (12-24)=-12t$, $|\\overrightarrow{B O}-(1-t) \\overrightarrow{A B}=(\\overrightarrow{B O}+\\overrightarrow{A B})-(1-t) \\overrightarrow{A B}=(\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{A B})$\n\n3. Therefore, $|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{A B}\\right|=|t-1|+|t^2+(4-t) \\overrightarrow{A B}|=|t-1|+|(t+4)(1-t)|=\\begin{cases} 2(t+1), & 0\\leqslant t\\leqsl", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 10887, 338, 19573, 1400, 2771, 5281, 107, 63, 330, 109, 3814, 456, 96, 3814, 1400, 2771, 5281, 107, 63, 389, 24362, 28, 284, 1885, 43509, 1400, 2771, 5281, 107, 63, 330, 109, 108, 45, 43509, 1400, 2771, 5281, 107, 63, 389, 109, 108, 45, 49161, 49163, 28422, 1094, 1885, 100, 3814, 254, 75, 49159, 28, 49160, 77, 26265, 965, 198, 9212, 198, 108, 100, 3814, 1400, 2771, 5281, 107, 49, 389, 39949, 76, 1400, 2771, 5281, 107, 49, 492, 109, 108, 42639, 8842, 43509, 18169, 107, 49164, 15009, 49160, 49161, 109, 3814, 1400, 2771, 5281, 107, 50, 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+{"group_id": "math_048", "origin": "budgie", "answer": "(A)\n\nThe dot product of the vectors is 0, so\n$$\n\\overrightarrow{O A} \\cdot \\overrightarrow{O B} = 0.\n$$\n\nIf $t \\in [0, 1]$, then\n$$\n|t \\overrightarrow{A B} - \\overrightarrow{A O}| = |t| \\cdot 24 - (\\overrightarrow{O A} \\cdot \\overrightarrow{O B}) = 24t - 0.\n$$\n\nTherefore,\n$$\n|t \\overrightarrow{A B} - \\overrightarrow{A O}| + |\\frac{5}{12} \\overrightarrow{B O} - (1-t) \\overrightarrow{B A}| = 24t + \\frac{5}{12}|\\overrightarrow{A B}| + |t|\\cdot 24 - |\\overrightarrow{A B}| \\cdot \\frac{5}{12} = 24t + \\frac{5}{12} \\cdot 24 + 48t - 24 = 24 \\sqrt{2} \\cdot t + 24.\n$$\n\nThus, the minimum value is 24 \\sqrt{2} \\cdot t + 24,\n$$\n\nHence, the answer is:\n$$\n\\boxed{(C) \\ 24 \\sqrt{2}}.\n$$", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 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B.\n\nConstruct a square $O A C B$, connect the diagonal $A B$, and let $D$ and $E$ be points on the diagonal $A B$ and the side $O B$ respectively, such that\n$$\n\\begin{array}{l}\nt \\overrightarrow{A B}-\\overrightarrow{A O}=\\overrightarrow{O D}, \\\\\n\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}=\\overrightarrow{E D}, \\\\\nE B=10, O D=D C . \\\\\n\\text { Therefore }|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right| \\\\\n=|\\overrightarrow{E D}|+|\\overrightarrow{D C}| \\leqslant|\\overrightarrow{E C}|=26 .\n\\end{array}\n$$", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 10887, 338, 19573, 1400, 2771, 5281, 107, 63, 330, 109, 3814, 456, 96, 3814, 1400, 2771, 5281, 107, 63, 389, 24362, 28, 284, 1885, 43509, 1400, 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+{"group_id": "math_049", "origin": "budgie", "answer": "1. We start by examining the general term of the series:\n\n$$\n\\frac{n}{(n+2)^{2} \\cdot (n+1)^{2}}\n$$\n\nLet's denote this as \\( a_n = \\frac{n}{(n+2)^{2} \\cdot (n+1)^{2}} \\).\n\n2. We can derive a recursive relationship for \\( a_n \\) by manipulating the terms of the series. We can rewrite the terms in the denominator as follows:\n\n$$\na_n = \\frac{n}{(n+2)^{2} \\cdot (n+1)^{2}} = \\frac{1}{A_{n-1} \\cdot A_{n+1}} + \\frac{1}{A_{n-2} \\cdot A_{n+2}} + \\cdots + \\frac{1}{A_2 \\cdot A_1} + \\cdots\n$$\n\nwhere \\( A_{n-k} = A_k + A_{k+1} + \\cdots \\) and \\( A_k = A_{n+k} - A_{n+k-k} = A_{n+k-k} + A_{n+k-k+1} + \\cdots \\).\n\n3. Denote the numerator of \\( a_n \\) by \\( B_k = n - k \\). We can express \\( B_k \\) in terms of \\( A_{n+k-k} \\). After some manipulations, we obtain:\n\n$$\nA_{n+k-k} = A_k + A_{k+1} + \\cdots = A_{A_k} + A_{", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49167, 30, 22983, 260, 1685, 282, 260, 2028, 198, 9212, 198, 76, 18169, 107, 49162, 15009, 49160, 21990, 49161, 109, 3814, 41591, 216, 49161, 21990, 49161, 17859, 42639, 18169, 107, 49164, 15009, 49161, 21990, 49161, 109, 3814, 41591, 216, 49162, 21990, 49161, 17859, 42639, 18169, 107, 49166, 15009, 49162, 21990, 49161, 109, 3814, 41591, 216, 49163, 21990, 49161, 17859, 42639, 13133, 1565, 42639, 18169, 107, 49161, 49168, 15009, 49160, 49163, 21990, 49161, 109, 3814, 41591, 216, 49160, 49164, 21990, 49161, 17859, 1673, 198, 9212, 49154, 49158, 198, 49160, 30, 1046, 1120, 411, 5893, 260, 2210, 2115, 282, 260, 3086, 42, 198, 198, 9212, 198, 76, 18169, 107, 94, 15009, 24, 94, 27, 49161, 25, 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+{"group_id": "math_049", "origin": "budgie", "answer": "1. The given sum can be simplified by first writing each fraction with a denominator of the form $n^2$:\n\n    $$\n    \\frac{3}{1^2 \\cdot 2^2} + \\frac{5}{2^2 \\cdot 3^2} + \\frac{7}{3^2 \\cdot 4^2} + \\cdots + \\frac{29}{14^2 \\cdot 15^2}.\n    $$\n\n2. Notice that the denominator of the first fraction in this series is a square of a prime number, $2^{2}$. This means that when we write each term of the series with a denominator of the form $n^2$, we are effectively adding more factors of 2 to the denominator.\n\n3. The denominator of the first fraction is $2^{2}$, which is a power of 2. Each subsequent denominator is twice the previous denominator, suggesting a factor of 2 in the numerators of the fractions.\n\n4. Each denominator can be expressed as a product of prime powers. The denominator of the first fraction is $2^{2}$, and the denominator of the second fraction is $2^{4}$. Therefore, the denominator of the entire series can be written as:\n\n    $$\n    n^2 \\cdot (2^{2})^{n-1} = n^2 \\cdot 2^{2n - 2}.\n    $$\n\n5. The numerator of the numerator in the given series is the sum of the first 29 prime numbers. The sum of the first $n$ prime numbers can be calculated using the formula for the sum of a series:\n\n    $$\n    S_{n} = \\frac{n(n+1)}{2}.\n    $$\n\n6. Substitute the value of $n$ into this formula to", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49167, 30, 22983, 260, 1685, 282, 260, 2028, 198, 9212, 198, 76, 18169, 107, 49162, 15009, 49160, 21990, 49161, 109, 3814, 41591, 216, 49161, 21990, 49161, 17859, 42639, 18169, 107, 49164, 15009, 49161, 21990, 49161, 109, 3814, 41591, 216, 49162, 21990, 49161, 17859, 42639, 18169, 107, 49166, 15009, 49162, 21990, 49161, 109, 3814, 41591, 216, 49163, 21990, 49161, 17859, 42639, 13133, 1565, 42639, 18169, 107, 49161, 49168, 15009, 49160, 49163, 21990, 49161, 109, 3814, 41591, 216, 49160, 49164, 21990, 49161, 17859, 1673, 198, 9212, 49154, 49158, 198, 49160, 30, 378, 1836, 2028, 416, 325, 20004, 411, 808, 2385, 971, 10236, 351, 253, 30842, 282, 260, 910, 1885, 94, 78, 49161, 20, 42, 1004, 17410, 472, 3814, 18169, 107, 49162, 15009, 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+{"group_id": "math_049", "origin": "verified_reference", "answer": "Solution: $224 / 225$ The sum telescopes as\n$$\n\\left(\\frac{1}{1^{2}}-\\frac{1}{2^{2}}\\right)+\\left(\\frac{1}{2^{2}}-\\frac{1}{3^{2}}\\right)+\\cdots+\\left(\\frac{1}{14^{2}}-\\frac{1}{15^{2}}\\right)=\\frac{1}{1^{2}}-\\frac{1}{15^{2}}=\\frac{224}{225} .\n$$", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49167, 30, 22983, 260, 1685, 282, 260, 2028, 198, 9212, 198, 76, 18169, 107, 49162, 15009, 49160, 21990, 49161, 109, 3814, 41591, 216, 49161, 21990, 49161, 17859, 42639, 18169, 107, 49164, 15009, 49161, 21990, 49161, 109, 3814, 41591, 216, 49162, 21990, 49161, 17859, 42639, 18169, 107, 49166, 15009, 49162, 21990, 49161, 109, 3814, 41591, 216, 49163, 21990, 49161, 17859, 42639, 13133, 1565, 42639, 18169, 107, 49161, 49168, 15009, 49160, 49163, 21990, 49161, 109, 3814, 41591, 216, 49160, 49164, 21990, 49161, 17859, 1673, 198, 9212, 49154, 49158, 198, 37500, 42, 1885, 49161, 49161, 49163, 2272, 216, 49161, 49161, 49164, 20, 378, 2028, 20368, 347, 198, 9212, 198, 76, 8842, 19407, 18169, 107, 49160, 15009, 49160, 21990, 49161, 17859, 46240, 18169, 107, 49160, 15009, 49161, 21990, 49161, 109, 17243, 2771, 18434, 76, 8842, 19407, 18169, 107, 49160, 15009, 49161, 21990, 49161, 17859, 46240, 18169, 107, 49160, 15009, 49162, 21990, 49161, 109, 17243, 2771, 18434, 76, 13133, 1565, 42639, 8842, 19407, 18169, 107, 49160, 15009, 49160, 49163, 21990, 49161, 17859, 46240, 18169, 107, 49160, 15009, 49160, 49164, 21990, 49161, 109, 17243, 2771, 25, 24814, 18169, 107, 49160, 15009, 49160, 21990, 49161, 17859, 46240, 18169, 107, 49160, 15009, 49160, 49164, 21990, 49161, 17859, 24814, 18169, 107, 49161, 49161, 49163, 15009, 49161, 49161, 49164, 109, 1673, 198, 9212, 49154], "prefix_len": 113, "old_logprobs": [-5.1915998458862305, -0.019102534279227257, -5.553123950958252, -2.125361442565918, -5.40551233291626, -3.449836492538452, -7.3291850090026855, -0.4743936359882355, -1.5833905935287476, -3.135874032974243, -0.48246580362319946, -2.4723854064941406, -8.017727851867676, -1.737647294998169, -5.529193878173828, -2.635599136352539, -2.3292083740234375, -0.44871193170547485, -0.19060561060905457, -0.07887586951255798, -2.3604836463928223, -0.6513047218322754, -0.01119145192205906, -0.023722510784864426, -2.8608155250549316, -0.10079328715801239, -1.5001734495162964, -1.4697316884994507, -0.022787265479564667, -2.7171618938446045, -2.240118980407715, -0.0026760026812553406, -0.00015186110977083445, -0.4244289994239807, -0.008682949468493462, -0.33442962169647217, -0.006431594956666231, -0.0005519058904610574, -0.3135577440261841, -0.0015730400336906314, -0.005822009406983852, -0.40552812814712524, -0.02727394551038742, -0.016379306092858315, -0.002074948512017727, -0.00012492353562265635, -0.00011538793478393927, -0.07464474439620972, -0.0017492959741503, -0.03797278553247452, -0.00252714054659009, -0.0003784178989008069, -0.009179767221212387, -0.03505849838256836, -7.652943895664066e-05, -5.602820692729438e-06, -0.001073975581675768, -3.325883881188929e-05, -0.0029667671769857407, -0.0001234931987710297, -8.4638240878121e-06, -0.0005467826849780977, -0.00015770144818816334, -3.194758028257638e-05, -0.012015936896204948, -0.07742787152528763, -2.651477336883545, -6.425174069590867e-05, -0.028445225208997726, -0.12810136377811432, -0.0021033799275755882, -9.48860906646587e-05, -4.410646579344757e-05, -0.02582705020904541, -0.006598825100809336, -0.060691602528095245, -0.17326772212982178, -0.002560077467933297, -4.1126360883936286e-05, -0.4333839416503906, -0.014012172818183899, -0.00011002412065863609, -8.344646857949556e-07, -0.001255200942978263, -0.001292585046030581, -0.005583405494689941, -0.0017851145239546895, -0.00012730741582345217, -3.075552376685664e-05, -0.0008425738196820021, -5.3165931603871286e-05, -2.1576648578047752e-05, -0.38326728343963623, -2.8214783668518066, -0.1723429411649704, -0.02174791693687439, -0.2526610791683197, -0.3228552043437958, -0.4851118326187134, -2.851935863494873, -0.007014174945652485, -1.2016106843948364, -0.26967477798461914, -0.017337316647171974, -0.004412913229316473, -0.044778693467378616, -0.02190922386944294, -0.9553692936897278, -0.18843165040016174, -0.0056563084945082664, -6.55629628454335e-05, -0.11438079923391342, -0.4433906674385071, -0.20652201771736145, -0.0021064728498458862, -1.8581511974334717, -0.8837118148803711, -1.2482788562774658, -0.8169817924499512, -0.5968276262283325, -0.026662427932024002, -0.017207611352205276, -1.1365848779678345, -0.4566560387611389, -0.08287828415632248, -0.0026537699159234762, -1.7867131233215332]}
+{"group_id": "math_050", "origin": "budgie", "answer": "#### Part (a)\n1. Let the length of the wire be a positive number $a$ such that $a > 0$.\n2. The lengths of the pieces are $a$ and $a-a=0$.\n3. By the triangle inequality, we have the following inequalities:\n\\[\n|a| < |0-(a-0)| = a\n\\]\n4. This simplifies to $|a| < a$, which means $a < 0$ as well.\n5. Given $a > 0$, the only possible value for $a$ is $a = 1$.\n\n#### Part (b)\n1. The sides of the squares are $1$ and $2$. Let the length of the side of each square be $x$.\n2. The sum of the areas of the squares is given by:\n\\[\nx^2 + (x-1)^2 = 4\n\\]\n3. Expanding and simplifying, we get:\n\\[\nx^2 + x^2 - 2x + 1 = 4 \\implies 2x^2 - 2x + 1 = 4\n\\]\n4. Rearranging, we have:\n\\[\n2x^2 - 2x - 3 = 0\n\\]\n5. Dividing the entire equation by $2$, we get:\n\\[\nx^2 - x - \\frac{3}{2} = 0\n\\]\n6. The solutions to this quadratic equation are given by:\n\\[\nx = \\frac{1 \\pm \\sqrt{1 + 6}}{2} = \\frac{1 \\pm \\sqrt{7}}{2}\n\\]\nHence, the possible values", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 17411, 95, 553, 253, 6192, 338, 314, 1885, 49160, 49159, 3814, 8898, 5167, 107, 110, 93, 24362, 986, 30, 909, 2022, 253, 2304, 418, 253, 1225, 335, 260, 6192, 28, 13725, 827, 4443, 30, 1963, 3800, 553, 253, 3519, 282, 1885, 104, 20, 284, 260, 550, 553, 253, 3519, 282, 1885, 49160, 49159, 29, 104, 20, 347, 3057, 281, 260, 4110, 2441, 42, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49159, 49160, 79, 84, 49159, 49161, 83, 49161, 49166, 49164, 49164, 344, 49162, 49162, 49166, 49162, 82, 1018, 49159, 49167, 454, 29, 49159, 49164, 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+{"group_id": "math_050", "origin": "budgie", "answer": "#### Part (a)\n\n1. **Identify the given information**: Augusto has a wire of length \\(10\\) meters and he cuts it into two pieces. One piece has a length of \\(x\\) meters and the other has a length of \\(10-x\\) meters.\n2. **Set up the equation**: The total length of the wire is the sum of the lengths of the two pieces, so we write the equation:\n   \\[\n   x + (10 - x) = 10\n   \\]\n3. **Solve the equation**:\n   \\[\n   x + 10 - x = 10 \\\\\n   10 = 10\n   \\]\n4. **Conclusion**: The equation holds true, so the length of the sides of both pieces is:\n   \\[\n   \\boxed{10 \\text{ meters each}}\n   \\]\n\n#### Part (b)\n\n1. **Express the given length**: \n   \\[\n   x = 10 - x - 10\n   \\]\n2. **Solve the equation**: \n   \\[\n   x = 10 - (10 - x) - 10 \\\\\n   10 = 10 - 10 + x - x \\\\\n   10 = 10 - 10 \\\\\n   10 = 10\n   \\]\n3. **Final answer**: The length of each piece is:\n   \\[\n   \\boxed{10\\ \\text{meters each}}\n   \\]\n\n#### Part (c)\n\n1. **Consider the total length of the wire**: We need to find the smallest number of pieces (or the sum of the lengths of the pieces) that can be cut from the 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+{"group_id": "math_050", "origin": "verified_reference", "answer": "Solution\n\nIn this problem, all lengths are given in meters and areas in square meters.\n\na) A piece of rope has length $x$ and another piece of rope has length $10-x$. Since a square has four sides of equal length, one square will have a side length of $\\frac{x}{4}$ and the other square will have a side length of $\\frac{10-x}{4}$.\n\nThe area of a square with side length $\\ell$ is $\\ell^{2}$. Therefore, one square will have an area of $\\left(\\frac{x}{4}\\right)^{2}=\\frac{x^{2}}{16}$, while the other square will have an area of $\\left(\\frac{10-x}{4}\\right)^{2}=\\frac{100-20 x+x^{2}}{16}$.\n\nb) Let $S(x)$ be the sum of the areas of the two squares. From the previous part, we have\n\n$$\nS(x)=\\frac{x^{2}}{16}+\\frac{100-20 x+x^{2}}{16}=\\frac{100-20 x+2 x^{2}}{16}=\\frac{1}{8} x^{2}-\\frac{5}{4} x+\\frac{25}{4}\n$$\n\nwhich is a quadratic function. The minimum of a function of the form\n\n$$\nf(x)=a x^{2}+b x+c\n$$\n\nwith $a>0$ is achieved at $x=\\frac{-b}{2 a}$. Thus, the minimum area will be achieved if\n\n$$\nx=-\\frac{\\left(-\\frac{5}{4}\\right)}{2 \\frac{1}{8}}=5\n$$\n\nIn other words, if the rope is cut exactly in the middle!\n\nc) From the previous part, we know that to minimize the sum of the areas, it is necessary to cut the rope exactly in the middle. Well, we claim that to minimize the area with nine cuts (i.e., creating ten squares), it is necessary that all pieces of rope be equal. To show this, consider the following argument: if two of the ten pieces of rope were different, it would be possible to reduce the area by cutting the pieces of rope so that these two were equal (we are using the previous part). Therefore, any two pieces of rope must be equal. 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+{"group_id": "math_051", "origin": "budgie", "answer": "To find $f(2013)$, we can express $2013$ in terms of $1$ and $2$ using the given equation:\n\\[ f(2013) = f(1) + f(2) - 2 \\times 1 \\times 2013 \\]\n\nUsing the given equation, we can substitute $x = 1$ and $y = 2$ to get:\n\\[ f(1) = f(1) + f(2) - 4026 \\]\n\nRearrange the equation to isolate $f(1)$:\n\\[ f(1) - f(1) = f(2) - 4026 \\Rightarrow 0 = f(2) - 4026 \\Rightarrow f(2) = 4026 \\]\n\nNow, we can find $f(2) + f(2) - 2 \\times 2013$:\n\\[ f(2) + f(2) - 2 \\times 2013 = 4026 + 4026 - 2 \\times 2013 = 4026 + 4026 - 2 \\times 2013 = 8052 - 2 \\times 2013 = 8052 - 4026 = \\boxed{4026} \\]\nFinal answer:\n#### 4026", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 49160, 7985, 1885, 86, 24, 49161, 49159, 49160, 49162, 25, 26265, 585, 327, 750, 1345, 1885, 104, 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+{"group_id": "math_051", "origin": "budgie", "answer": "To find $f(2013)$, we can use the given equation $f(x-y)=f(x)+f(y)-2xy$ for any real $x$ and $y$, and then substitute $x=2013$ and $y=1$ to find $f(2013)$.\n\nWe know $x-y=2013$ and $x+y=2014$. Using the given equation, we can express $f(2014)$ in terms of $f(2013)$.\n\nSubstitute $x=2013$ and $y=1$ into the given equation:\n\\[ f(2014) = f(2013) + f(1) - 2(2013)(1) \\]\n\nNow, substitute $x=2014$ and $y=1$ into the given equation:\n\\[ f(2014) = f(2013) + f(1) - 2(2013) \\]\n\nSubstitute $x=2013$ and $y=1$ into the equation:\n\\[ f(2013) = f(2013) + f(1) - 2(2013) - 4028 \\]\n\nRearrange the equation to isolate $f(2013)$:\n\\[ f(2013) - f(2013) = f(2013) + f(1) - 2(2013) - 4028 - 2013 \\]\n\n\\[ f(2", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 49160, 7985, 1885, 86, 24, 49161, 49159, 49160, 49162, 25, 26265, 585, 327, 750, 1345, 1885, 104, 20, 284, 1885, 105, 20, 260, 7897, 6650, 198, 198, 9212, 198, 86, 24, 104, 29, 105, 36637, 86, 24, 104, 18434, 86, 24, 105, 14400, 49161, 8860, 198, 9212, 49154, 49158, 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+{"group_id": "math_051", "origin": "verified_reference", "answer": "Solution. Substitute $x=y=0$. We get $f(0)=2 f(0)+0$, from which we obtain that $f(0)=0$. Substitute $x=y$. We get $0=f(0)=f(x)+f(x)-2 x^{2}$. Hence, $f(x)=x^{2}$.\n\nAnswer: 4052169. (C)\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_997c53c06135e2de9c16g-02.jpg?height=67&width=1157&top_left_y=1003&top_left_x=498)", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 49160, 7985, 1885, 86, 24, 49161, 49159, 49160, 49162, 25, 26265, 585, 327, 750, 1345, 1885, 104, 20, 284, 1885, 105, 20, 260, 7897, 6650, 198, 198, 9212, 198, 86, 24, 104, 29, 105, 36637, 86, 24, 104, 18434, 86, 24, 105, 14400, 49161, 8860, 198, 9212, 49154, 49158, 198, 37500, 30, 18714, 2881, 1885, 104, 45, 105, 45, 49159, 28422, 1046, 820, 1885, 86, 24, 49159, 36637, 49161, 275, 24, 49159, 18434, 49159, 26265, 429, 527, 392, 3478, 338, 1885, 86, 24, 49159, 36637, 49159, 28422, 18714, 2881, 1885, 104, 45, 105, 28422, 1046, 820, 1885, 49159, 45, 86, 24, 49159, 36637, 86, 24, 104, 18434, 86, 24, 104, 14400, 49161, 1792, 21990, 49161, 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+{"group_id": "math_052", "origin": "budgie", "answer": "First, we need to find the smallest base $b$ such that $N$ is a fourth power. We know that a fourth power is a number that can be expressed as $N = x^4$ for some integer $x$.\n\nSince $N = 777 = x^4$, the base $b$ must be a fourth power. We can find the smallest such $b$ by trying different values of $b$.\n\nLet's start by trying $b = 4$.\n\nFor $b = 4$, we have $N = x^4 = 777$. We can see that $x = 7$ is a solution, since $7^4 = 2401$. However, we need to check if $x = 7$ is also a solution to the congruence $x^4 \\equiv 21 \\pmod{1000}$.\n\nBy testing values of $x$, we can see that $x = 7$ satisfies the congruence.\n\nTherefore, the smallest base $b$ is:\n\\[ \\boxed{4} \\]", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 365, 49168, 373, 6805, 28440, 29435, 25, 1885, 62, 20, 314, 354, 15328, 28, 624, 6133, 281, 3159, 1885, 82, 20, 314, 216, 49166, 49166, 49166, 30, 7985, 260, 13341, 2731, 15328, 1885, 82, 20, 715, 338, 1885, 62, 20, 314, 253, 8629, 1149, 282, 354, 15328, 281, 17948, 20431, 30, 49154, 49158, 198, 5345, 28, 392, 737, 288, 1042, 260, 13341, 3159, 1885, 82, 20, 715, 338, 1885, 62, 20, 314, 253, 8629, 1149, 30, 1046, 699, 338, 253, 8629, 1149, 314, 253, 1230, 338, 416, 325, 6803, 347, 1885, 62, 446, 1792, 78, 49163, 20, 327, 634, 15328, 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+{"group_id": "math_052", "origin": "budgie", "answer": "We start with $b = 3$ because 0 is not a fourth power of 2.\n\nStart with $N = 777$, a three-digit number in base 3.\n\\[ 777_{10} = 1 7 7 7_{10} \\]\n\nConvert $N = 777_{10}$ to decimal:\n\\[ 777_{10} = 777_{3} = 342_{3} = 7^4_{10} = 7^4 \\times 3^0 = 7^4 \\times 1 = 7^3 = 343_{3} = 213_{2} = 3^3_{10} = 3^3 \\times 4^1 = 3^3 \\times 4 = 34_{10} \\]\n\nThe smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation is $\\boxed{3}$ because $N = 342_{3} = 7^4 \\times 3^0 = 7^4 \\times 1 = 7^3$ is a fourth power of an integer in decimal notation.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 365, 49168, 373, 6805, 28440, 29435, 25, 1885, 62, 20, 314, 354, 15328, 28, 624, 6133, 281, 3159, 1885, 82, 20, 314, 216, 49166, 49166, 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+{"group_id": "math_052", "origin": "verified_reference", "answer": "2. This problem is equivalent to finding the smallest positive integer $b$, such that the equation $7 b^{2}+7 b+7=x^{4}$, (1) has an integer solution for $x$. Since 7 is a prime number, it follows from equation (1) that 7 is a divisor of $x$. Therefore, let $x=7 k$, then equation (1) becomes $b^{2}+b+1=7^{3} k^{4}$. The smallest $b$ occurs when $k$ is at its minimum. Taking $k=1$, we then have $b^{2}+b+1=343, b^{2}+b-342=0$, which is $(b-18)(b+19)=0$, yielding the positive integer solution $b=18$. Thus, we have $(777)_{18}=\\left(7^{4}\\right)_{10}$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49161, 30, 365, 49168, 373, 6805, 28440, 29435, 25, 1885, 62, 20, 314, 354, 15328, 28, 624, 6133, 281, 3159, 1885, 82, 20, 314, 216, 49166, 49166, 49166, 30, 7985, 260, 13341, 2731, 15328, 1885, 82, 20, 715, 338, 1885, 62, 20, 314, 253, 8629, 1149, 282, 354, 15328, 281, 17948, 20431, 30, 49154, 49158, 198, 49161, 30, 669, 1732, 314, 7492, 288, 4212, 260, 13341, 2731, 15328, 1885, 82, 26265, 715, 338, 260, 8345, 1885, 49166, 278, 21990, 49161, 109, 27, 49166, 278, 27, 49166, 45, 104, 21990, 49163, 24362, 28, 365, 49160, 25, 553, 354, 15328, 3564, 327, 1885, 104, 28422, 4311, 216, 49166, 314, 253, 9552, 1230, 28, 357, 5558, 429, 8345, 365, 49160, 25, 338, 216, 49166, 314, 253, 3044, 40031, 282, 1885, 104, 28422, 4882, 28, 1303, 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+{"group_id": "math_053", "origin": "budgie", "answer": "The problem is about finding the volume of the intersection of two tetrahedra created by reflecting one of them through its center.", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49168, 30, 10887, 314, 253, 2649, 29889, 39513, 282, 4981, 216, 49160, 1673, 1046, 3478, 253, 1796, 2649, 29889, 39513, 411, 10329, 260, 1836, 582, 738, 624, 3712, 30, 1812, 314, 260, 4981, 282, 480, 10930, 47, 49154, 49158, 198, 504, 1732, 314, 563, 4212, 260, 4981, 282, 260, 10930, 282, 827, 29889, 770, 317, 2312, 411, 10329, 582, 282, 601, 738, 624, 3712, 30, 49154], "prefix_len": 66, "old_logprobs": [-1.550062894821167, -2.989487886428833, -2.296311378479004, -1.1432123184204102, -1.5341554880142212, -0.01694592460989952, -0.05713459849357605, -0.043754320591688156, -0.2523476779460907, -0.12494485080242157, -0.21934205293655396, -0.05688011273741722, -1.6691546440124512, -0.0011337526375427842, -0.5466839075088501, -3.511587619781494, -0.021064963191747665, -0.24207036197185516, -0.5093414187431335, -3.076242208480835, -1.1807098388671875, -0.055522371083498, -0.15194553136825562, -0.1955566555261612, -0.10770324617624283, -4.276659965515137]}
+{"group_id": "math_053", "origin": "budgie", "answer": "A regular tetrahedron with a volume of 1 is a regular tetrahedron depending on its edge length.\n\nThe volume of a regular tetrahedron is given by the formula:\n\n$$V = \\frac{\\sqrt{2}}{12} \\times (\\text{edge length})^3$$\n\nThe second tetrahedron is half of the first one because each edge of the first tetrahedron is now reflected through its center to form the second one.\n\nSo, the volume of the intersection is twice the volume of the second tetrahedron.\n\nHence, the answer is: $\\boxed{2\\sqrt{2}}$", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49168, 30, 10887, 314, 253, 2649, 29889, 39513, 282, 4981, 216, 49160, 1673, 1046, 3478, 253, 1796, 2649, 29889, 39513, 411, 10329, 260, 1836, 582, 738, 624, 3712, 30, 1812, 314, 260, 4981, 282, 480, 10930, 47, 49154, 49158, 198, 49, 2649, 29889, 39513, 351, 253, 4981, 282, 216, 49160, 314, 253, 2649, 29889, 39513, 4509, 335, 624, 5595, 3519, 30, 198, 198, 504, 4981, 282, 253, 2649, 29889, 39513, 314, 1836, 411, 260, 7961, 42, 198, 198, 9212, 70, 446, 3814, 18169, 26754, 16166, 107, 49161, 109, 15009, 49160, 49161, 109, 3814, 14602, 365, 76, 2692, 107, 7277, 3519, 8148, 78, 49162, 9212, 198, 198, 504, 1796, 29889, 39513, 314, 2745, 282, 260, 808, 582, 975, 971, 5595, 282, 260, 808, 29889, 39513, 314, 1209, 9807, 738, 624, 3712, 288, 910, 260, 1796, 582, 30, 198, 198, 2931, 28, 260, 4981, 282, 260, 10930, 314, 6757, 260, 4981, 282, 260, 1796, 29889, 39513, 30, 198, 198, 35391, 28, 260, 2988, 314, 42, 19573, 4810, 277, 107, 49161, 76, 16166, 107, 49161, 17859, 20, 49154], "prefix_len": 66, "old_logprobs": [-1.5268267393112183, -0.011417053639888763, -0.00980360060930252, -0.00029845553217455745, -2.7570624351501465, -1.0321513414382935, -0.15390600264072418, -0.004066057503223419, -0.02138613723218441, -0.0038498349022120237, -1.2024527788162231, -0.840359628200531, -0.5977230668067932, -0.44333934783935547, -0.0028651398606598377, -9.927957534790039, -0.042961087077856064, -0.8258200287818909, -0.12453719973564148, -0.027798019349575043, -0.4635339081287384, -1.286529541015625, -0.11656678467988968, -1.477001667022705, -1.5239051580429077, -0.08915635943412781, -0.19664758443832397, -0.03069235570728779, -0.007639005314558744, -0.00030322244856506586, -1.0974417924880981, -0.2368212193250656, -0.003986864350736141, -0.5041430592536926, -0.0019431296968832612, -1.1992218494415283, -0.18282870948314667, -0.32322821021080017, -0.44389188289642334, -0.11373955756425858, -0.4446525573730469, -0.009152363054454327, -0.03306155279278755, -0.20578810572624207, -0.005824616644531488, -0.01808229461312294, -0.0621364526450634, -0.020250244066119194, -0.017571140080690384, -0.05218465253710747, -0.011430253274738789, -0.05380994826555252, -1.9663575887680054, -0.38633692264556885, -0.4257713854312897, -0.16556571424007416, -0.02194071188569069, -0.0004593271005433053, -0.11788401007652283, -0.011235067620873451, -0.07252804934978485, -0.027683116495609283, -0.006976648699492216, -0.0062084193341434, -0.024529054760932922, -0.0015543533954769373, -1.4745618104934692, -0.8177616596221924, -0.47106412053108215, -0.000666277133859694, -0.5157479643821716, -6.993849754333496, -0.43704384565353394, -0.47260117530822754, -1.2254003286361694, -0.4145038425922394, -2.0234856605529785, -3.0246806144714355, -0.7801727056503296, -0.37492579221725464, -0.04081806540489197, -1.105315089225769, -0.2565179169178009, -0.00014506718434859067, -0.08136652410030365, -5.064205646514893, -1.2104003429412842, -0.1668986827135086, -0.23409920930862427, -0.034970443695783615, -1.6681430339813232, -0.3721422851085663, -0.4846370816230774, -0.1862044483423233, -0.7824249267578125, -0.09138736128807068, -0.3177088797092438, -0.001972516765818, -1.7063922882080078, -0.1416255533695221, -0.14741481840610504, -0.29360267519950867, -0.015308385714888573, -0.07909992337226868, -1.4486943483352661, -1.2485182285308838, -1.3840548992156982, -0.016923420131206512, -0.007106622215360403, -0.00045015214709565043, -0.06763752549886703, -1.3885653018951416, -0.026263637468218803, -0.00015317220822907984, -0.9156184196472168, -0.0627988800406456, -0.003077772678807378, -3.076115131378174, -0.06328597664833069, -0.08068615943193436, -0.2403603196144104, -0.003192092990502715, -0.29312893748283386, -3.428654193878174, -0.002806892851367593, -3.576278118089249e-07, -0.07123958319425583, -0.24469593167304993, -1.7872461080551147, -0.41067156195640564, -0.01043978612869978, -0.019031191244721413, -0.01818426139652729, -0.2570699155330658, -0.6266476511955261]}
+{"group_id": "math_053", "origin": "verified_reference", "answer": "Solution: $1 / 2$\nImagine placing the tetrahedron $A B C D$ flat on a table with vertex $A$ at the top. By vectors or otherwise, we see that the center is $3 / 4$ of the way from $A$ to the bottom face, so the reflection of this face lies in a horizontal plane halfway between $A$ and $B C D$. In particular, it cuts off the smaller tetrahedron obtained by scaling the original tetrahedron by a factor of $1 / 2$ about $A$. Similarly, the reflections of the other three faces cut off tetrahedra obtained by scaling $A B C D$ by $1 / 2$ about $B, C$, and $D$. On the other hand, the octahedral piece remaining remaining after we remove these four smaller tetrahedra is in the intersection of $A B C D$ with its reflection, since the reflection sends this piece to itself. So the answer we seek is just the volume of this piece, which is\n$$\n\\begin{array}{l}\n\\text { (volume of } A B C D)-4 \\cdot(\\text { volume of } A B C D \\text { scaled by a factor of } 1 / 2) \\\\\n=1-4(1 / 2)^{3}=1 / 2\n\\end{array}\n$$", "reward": 1.0, "advantage": 1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49168, 30, 10887, 314, 253, 2649, 29889, 39513, 282, 4981, 216, 49160, 1673, 1046, 3478, 253, 1796, 2649, 29889, 39513, 411, 10329, 260, 1836, 582, 738, 624, 3712, 30, 1812, 314, 260, 4981, 282, 480, 10930, 47, 49154, 49158, 198, 37500, 42, 1885, 49160, 2272, 216, 49161, 20, 198, 4792, 10482, 260, 29889, 39513, 1885, 49, 389, 340, 422, 20, 5562, 335, 253, 3252, 351, 24121, 1885, 49, 20, 418, 260, 1466, 30, 1428, 14799, 355, 5214, 28, 392, 963, 338, 260, 3712, 314, 1885, 49162, 2272, 216, 49163, 20, 282, 260, 970, 429, 1885, 49, 20, 288, 260, 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+{"group_id": "math_054", "origin": "budgie", "answer": "1.  **Assumptions and Setup**: \n   We have ten natural numbers on a circle, which are evenly marked. We place these numbers 1 through 10 in a single circle. We rotate the line passing through the center of the circle, which divides the numbers into two groups. Each group must then be divided by the line passing through the center, resulting in five distinct outcomes. For each arrangement, we need to find the maximum possible value of $K$.\n\n2. **Key Insight**:\n   The line passing through the center divides the circle into two groups. This means the line must intersect the numbers 1 through 10. The maximum possible value for $K$ occurs when the line is tangent to the circle.\n\n3. **Constructing the Arrangements**:\n   To achieve this, we can construct a different line through the center that does not intersect the numbers 1 through 10. For example, we can construct a line passing through the numbers 2 through 17,prime to 19, and 11 through 13,prime to 19. This line does not intersect the numbers 1 through 10 because it passes through prime to prime.\n\n4. **Final Arrangement**:\n   Using the above lines and configurations, we can visualize that the maximum possible value for $K$ is 1.\n\n5. **Considering Other Differences**:\n   We need to consider other differences, such as 2, 3, and 4. These differences are not useful in this context, as they would not help in maximizing $K$.\n\n6. **Constructing the Maximum Value**:\n   By carefully placing the numbers 1 through 10 on the circle, we can see that the maximum possible value for $K$ is 1.\n\n7. **Conclusion**:\n   The maximum value of $K$", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49162, 30, 365, 49160, 49164, 2876, 25, 1032, 3057, 281, 260, 4110, 28, 3772, 2876, 359, 15760, 6319, 335, 260, 27436, 282, 253, 7842, 30, 9373, 260, 3772, 1782, 2966, 1885, 49160, 3814, 8045, 216, 49160, 49159, 20, 335, 623, 3772, 2876, 30, 16744, 368, 253, 1761, 7685, 738, 260, 3712, 282, 260, 7842, 30, 1550, 260, 1761, 1072, 441, 1301, 738, 750, 6319, 2876, 28, 357, 20751, 260, 2966, 216, 49160, 288, 216, 49160, 49159, 618, 827, 2119, 30, 1068, 971, 11955, 28, 665, 359, 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+{"group_id": "math_054", "origin": "budgie", "answer": "11. This problem can be approached by considering the symmetry of the circle and the division of numbers. To maximize the value of $K$, we should consider the case where $K$ is minimized. The smallest product is achieved when the two groups are as close to equal as possible, and since there are five ways to do this, we can get a value of $K$ as close as possible to $10$, which is $K=\\boxed{5}$.\n\nFinal answer:\n#### 5", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49162, 30, 365, 49160, 49164, 2876, 25, 1032, 3057, 281, 260, 4110, 28, 3772, 2876, 359, 15760, 6319, 335, 260, 27436, 282, 253, 7842, 30, 9373, 260, 3772, 1782, 2966, 1885, 49160, 3814, 8045, 216, 49160, 49159, 20, 335, 623, 3772, 2876, 30, 16744, 368, 253, 1761, 7685, 738, 260, 3712, 282, 260, 7842, 30, 1550, 260, 1761, 1072, 441, 1301, 738, 750, 6319, 2876, 28, 357, 20751, 260, 2966, 216, 49160, 288, 216, 49160, 49159, 618, 827, 2119, 30, 1068, 971, 11955, 28, 665, 359, 2531, 1853, 288, 10335, 260, 2966, 347, 260, 1761, 35290, 30, 1068, 971, 7573, 28, 665, 314, 253, 1406, 282, 260, 24053, 282, 260, 827, 2119, 28, 284, 260, 13341, 282, 623, 2531, 2320, 314, 31712, 347, 1885, 59, 28422, 1812, 314, 260, 5428, 1685, 282, 1885, 59, 20, 1486, 511, 1636, 11775, 47, 49154, 49158, 198, 49160, 49160, 30, 669, 1732, 416, 325, 12055, 411, 6361, 260, 19536, 282, 260, 7842, 284, 260, 7573, 282, 2966, 30, 1626, 12726, 260, 1685, 282, 1885, 59, 26265, 392, 868, 1771, 260, 1671, 837, 1885, 59, 20, 314, 32451, 30, 378, 13341, 1406, 314, 6551, 645, 260, 827, 2119, 359, 347, 2800, 288, 4037, 347, 1636, 28, 284, 1675, 665, 359, 2531, 1853, 288, 536, 451, 28, 392, 416, 820, 253, 1685, 282, 1885, 59, 20, 347, 2800, 347, 1636, 288, 1885, 49160, 49159, 26265, 527, 314, 1885, 59, 24814, 4810, 277, 107, 49164, 24362, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49164, 49154], "prefix_len": 170, "old_logprobs": [-1.5749647617340088, -3.4995548725128174, -0.28815290331840515, -3.8406312465667725, -0.7434225678443909, -1.2087918519973755, -0.02243751659989357, -2.4861018657684326, -0.26891428232192993, -0.9247164130210876, -0.267414927482605, -3.4229257106781006, -0.7260379195213318, -0.09930482506752014, -1.1229243278503418, -0.33573028445243835, -0.22513626515865326, -1.8179881572723389, -0.3387903571128845, -0.476768434047699, -1.2044992446899414, -3.8428139686584473, -0.14608033001422882, -1.1503896713256836, -1.436244010925293, -0.015381169505417347, -0.11323980987071991, -0.008275032043457031, -0.0262267105281353, -0.20125699043273926, -1.2034345865249634, -1.6562790870666504, -0.6755614876747131, -3.5350499153137207, -0.16434980928897858, -1.5821049213409424, -0.42188894748687744, -0.13570307195186615, -0.03889119252562523, -1.9493823051452637, -0.23667500913143158, -2.7459139823913574, -1.2212700843811035, -1.0729178190231323, -1.2163156270980835, -0.7272261381149292, -0.06481156498193741, -0.7654646635055542, -2.175027370452881, -0.19105783104896545, -0.16946664452552795, -0.4311183989048004, -0.9449246525764465, -1.5917807817459106, -0.681716799736023, -0.008523158729076385, -0.0006898645660839975, -1.5318175554275513, -1.4919495582580566, -3.8361823558807373, -1.2708584070205688, -0.03544183075428009, -0.8166951537132263, -1.4285683631896973, -0.009360946714878082, -0.750688374042511, -0.07259467989206314, -0.0989997386932373, -0.919661819934845, -0.32716667652130127, -3.288177490234375, -0.7691940069198608, -1.017737865447998, -0.3865899443626404, -0.03841855749487877, -0.04886118695139885, -0.3217540681362152, -2.147792100906372, -0.5073598623275757, -1.8048293590545654, -0.39137381315231323, -0.06476619839668274, -1.067868947982788, -0.7331997752189636, -0.6485348343849182, -4.665074825286865, -0.5055636763572693, -0.08790509402751923, -0.7769777774810791, -1.8317192792892456, -3.6683976650238037, -0.15780994296073914, -3.576278118089249e-07, -0.008889504708349705, -1.3543790578842163, -0.5914703607559204, -0.0419258177280426, -1.9717118740081787, -0.19022062420845032, -0.06876706331968307, -0.0003675738989841193, -0.0006724718841724098, -0.007033114787191153, -0.002886179368942976, -0.011172355152666569, -0.0012494861148297787, -0.0001679517881711945]}
+{"group_id": "math_054", "origin": "verified_reference", "answer": "【Analysis】No matter how these 10 numbers are placed, their sum remains unchanged, $1+2+3+4+5+6+7+8+9+10=55$, divided into two groups, the closer the sums of the two groups, the larger the required product will be.\n\n【Solution】\n$$\n\\begin{array}{l}\n1+2+3+4+5+6+7+8+9+10=55 \\\\\n55=27+28\n\\end{array}\n$$\n\nTherefore, as long as the 10 numbers are divided into two parts with sums of 27 and 28, the product at this time is $27 \\times 28=756$. Answer: In all the arrangements, $K$ is maximized at 756.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49162, 30, 365, 49160, 49164, 2876, 25, 1032, 3057, 281, 260, 4110, 28, 3772, 2876, 359, 15760, 6319, 335, 260, 27436, 282, 253, 7842, 30, 9373, 260, 3772, 1782, 2966, 1885, 49160, 3814, 8045, 216, 49160, 49159, 20, 335, 623, 3772, 2876, 30, 16744, 368, 253, 1761, 7685, 738, 260, 3712, 282, 260, 7842, 30, 1550, 260, 1761, 1072, 441, 1301, 738, 750, 6319, 2876, 28, 357, 20751, 260, 2966, 216, 49160, 288, 216, 49160, 49159, 618, 827, 2119, 30, 1068, 971, 11955, 28, 665, 359, 2531, 1853, 288, 10335, 260, 2966, 347, 260, 1761, 35290, 30, 1068, 971, 7573, 28, 665, 314, 253, 1406, 282, 260, 24053, 282, 260, 827, 2119, 28, 284, 260, 13341, 282, 623, 2531, 2320, 314, 31712, 347, 1885, 59, 28422, 1812, 314, 260, 5428, 1685, 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+{"group_id": "math_056", "origin": "budgie", "answer": "To find the maximum number of red dragons that could have been at the table, we can consider the alternating pattern of truth and lie. Since each dragon has three heads, each dragon is either follows the truth or lies.\n\nIf we start with a truth-teller, the first position is a truth-teller, the second position is a truth-teller, the third position is a truth-teller, and so on. This pattern continues for all 530 dragons.\n\nThe maximum number of red dragons can be found by starting with the maximum possible number of truth-tellers and then alternating between truth-tellers and liars.\n\nSince each dragon has three heads, the maximum possible number of truth-tellers is 100, and the maximum possible number of liars is 100.\n\nWe start with the maximum number of truth-tellers (100), and then alternate between truth-tellers and liars. This pattern continues for all 530 dragons.\n\nTherefore, the maximum number of red dragons that could have been at the table is 100.\n\nThe answer is:\n\\[ \\boxed{100} \\]", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 33348, 216, 49165, 30, 49167, 30, 533, 260, 1798, 282, 28846, 7409, 28, 665, 2330, 2382, 28, 2654, 28, 284, 4461, 26225, 30, 3768, 17422, 553, 1296, 8648, 28, 971, 282, 527, 1811, 6200, 260, 4569, 355, 1811, 5721, 30, 4790, 28, 971, 17422, 553, 418, 2124, 582, 1680, 338, 6200, 260, 4569, 30, 1963, 1194, 28, 216, 49164, 49162, 49159, 26225, 2643, 1130, 253, 5075, 3252, 28, 284, 971, 282, 601, 1137, 42, 198, 198, 29, 216, 49160, 302, 1680, 42, 619, 2068, 957, 2049, 314, 253, 2654, 17422, 1184, 198, 29, 216, 49161, 268, 1680, 42, 619, 2068, 957, 1048, 314, 253, 4461, 17422, 1184, 198, 29, 216, 49162, 7212, 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+{"group_id": "math_056", "origin": "budgie", "answer": "To determine the maximum number of red dragons, we need to analyze the statements made by the dragons and their relationship to each other. If 530 dragons have statements that contradict the truth/ jogit of their heads, then no red dragon could have a head that tells the truth.\n\nLet's denote the number of red dragons as $R$ and the number of green dragons as $G$. Since each red dragon has exactly three heads, we can denote the number of heads as $3R$ and the number of heads as $3G$.\n\nNow, consider the statements made by the dragons. If the head that tells the truth tells the truth about the head that tells the truth about another dragon, then the head that tells the truth about the head with the truth about the head that tells the truth about another dragon must be green.\n\nThis leads to a contradiction, as there would be no head that tells the truth about another dragon, and it would not be possible for 530 dragons to add up to 530 (the number of heads in a round table).\n\nHowever, if the head that tells the truth about another dragon tells the truth about the head with the truth about another dragon, then the head that tells the truth about the head with the truth about the head that tells the truth about another dragon must be blue.\n\nTherefore, the maximum number of red dragons that could have been at the table is $R = 3G = 3(3R) = 9G$.\n\nSo, the answer is:\n\\[ \\boxed{9} \\]\nFinal answer:\n#### 9", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 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+{"group_id": "math_056", "origin": "verified_reference", "answer": "Answer: 176.\n\nSolution. Consider an arbitrary red dragon. To the right of this dragon, at least one head must tell the truth. Note that the 1st and 3rd heads cannot tell the truth (since there is a red dragon to the left), so the 2nd head must tell the truth, and to the right of this dragon, there must be a blue dragon. Now consider the dragon to the left of the chosen red dragon. By similar reasoning, we understand that only the 1st head can tell the truth, and to the left of this dragon, there must be a green one.\n\nFrom the previous reasoning, we get that for any red dragon, one position to the right there is definitely a blue dragon, and one position to the left - a green one. For each red dragon, we combine these three dragons into one group. Note that any dragon can be in at most one such group. Then we get that there are no more than $530 / 3$ < 177, i.e., no more than 176 such groups. Therefore, there are no more than 176 red dragons.\n\nNow let's provide an example. Arrange 176 red dragons in a circle, and in 175 gaps between them, place a green and a blue dragon: ...RGRB..., and in one gap, place two blue and two green dragons: ...RBGBGR.... It is easy to check that all conditions are met: each dragon has at least one head that will tell the truth, and there are exactly 530 dragons in total.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_8af0c885427e3e323cf9g-17.jpg?height=492&width=525&top_left_y=852&top_left_x=464)\n\n## 7th grade", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 33348, 216, 49165, 30, 49167, 30, 533, 260, 1798, 282, 28846, 7409, 28, 665, 2330, 2382, 28, 2654, 28, 284, 4461, 26225, 30, 3768, 17422, 553, 1296, 8648, 28, 971, 282, 527, 1811, 6200, 260, 4569, 355, 1811, 5721, 30, 4790, 28, 971, 17422, 553, 418, 2124, 582, 1680, 338, 6200, 260, 4569, 30, 1963, 1194, 28, 216, 49164, 49162, 49159, 26225, 2643, 1130, 253, 5075, 3252, 28, 284, 971, 282, 601, 1137, 42, 198, 198, 29, 216, 49160, 302, 1680, 42, 619, 2068, 957, 2049, 314, 253, 2654, 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+{"group_id": "math_059", "origin": "budgie", "answer": "This problem is about using the Law of Sines and the Law of Cosines to find the values of trigonometric functions. The goal is to find the value of $(y + z \\cos A)(z + x \\cos B)(x + y \\cos C) + (y \\cos A + z)(z \\cos B + x)(x \\cos C + y)$.\n\nTo begin with, we can rewrite the expression in a more convenient form. We know that $\\sin C = \\sin(180^\\circ - A - B) = \\sin(A + B)$, and similarly for the other angles.\n\nNow, we can apply the Law of Sines:\n$$y + z \\cos A = \\sin(A + B) + z \\cos A$$\n$$z + x \\cos B = \\sin(A + B) + x \\cos B$$\n$$x + y \\cos C = \\sin(A + B) + y \\cos C$$\n\nNow, we can substitute these expressions into the given expression:\n$$(y + z \\cos A)(z + x \\cos B)(x + y \\cos C) + (y \\cos A + z)(z \\cos B + x)(x \\cos C + y) = (\\sin(A + B) + z \\cos A)(\\sin(A + B) + x \\cos B)(\\sin(A + B) + y \\cos C) + (\\sin(A + B) + x \\cos A)(\\sin(A + B) + y \\cos C)$$\n\nSince $\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B$ and $\\sin(A + B) = \\sin A \\cos B - \\", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 533, 19573, 22128, 6935, 17638, 26265, 357, 314, 1343, 338, 1885, 104, 3814, 15495, 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+{"group_id": "math_059", "origin": "verified_reference", "answer": "In $\\triangle A B C$, $\\sin C=\\sin (A+B)=\\sin A \\cos B+\\cos A \\sin B$. Substituting the known conditions, we get $x \\sin A+y \\sin B+z(\\sin A \\cos B+\\cos A \\sin B)=0$, which simplifies to $\\sin A(x+z \\cos B)=-\\sin B(y+z \\cos A)$.\nSimilarly, $\\sin B(y+x \\cos C)=-\\sin C(z+x \\cos B)$, $\\sin C(z+y \\cos A)=-\\sin A(x+y \\cos C)$,\nMultiplying the three equations, we have:\n$(x+z \\cos B)(y+x \\cos C)(z+y \\cos A)=-(y+z \\cos A)(z+x \\cos B)(x+y \\cos C)$, which simplifies to $(y+z \\cos A)(z+x \\cos B)(x+y \\cos C)+(y \\cos A+z)(z \\cos B+x)(x \\cos C+y)=0$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 533, 19573, 22128, 6935, 17638, 26265, 357, 314, 1343, 338, 1885, 104, 3814, 15495, 330, 1232, 329, 3814, 15495, 389, 1232, 1892, 3814, 15495, 340, 446, 216, 49159, 28422, 7985, 260, 1685, 282, 46408, 105, 1232, 1892, 3814, 12374, 330, 14502, 106, 1232, 1792, 3814, 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+{"group_id": "math_061", "origin": "budgie", "answer": "If $|z|=|w|=|z-w|$, then $z=w$ or $z=w$ or $z=-w$. Since $z=w$ is the only possible solution, $\\left(\\frac{z}{w}\\right)^{99}=\\left(\\frac{w}{w}\\right)^{99}=1$.\n\nThis means that $\\left(\\frac{z}{w}\\right)^{99}=\\boxed{1}$.\n\nFinal answer:\n#### 1", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 22188, 389, 29, 49161, 30, 49160, 30, 365, 49161, 49159, 2876, 25, 1068, 1784, 2966, 1885, 106, 28, 267, 20, 715, 338, 1885, 108, 106, 108, 45, 108, 103, 108, 45, 108, 106, 29, 103, 108, 26265, 7529, 19573, 8842, 19407, 18169, 107, 106, 15009, 103, 17243, 2771, 25, 21990, 49168, 49168, 24362, 30, 49154, 49158, 198, 1937, 1885, 108, 106, 108, 45, 108, 103, 108, 45, 108, 106, 29, 103, 108, 26265, 965, 1885, 106, 45, 103, 20, 355, 1885, 106, 45, 103, 20, 355, 1885, 106, 15196, 103, 28422, 4311, 1885, 106, 45, 103, 20, 314, 260, 805, 1636, 3564, 28, 19573, 8842, 19407, 18169, 107, 106, 15009, 103, 17243, 2771, 25, 21990, 49168, 49168, 109, 24814, 8842, 19407, 18169, 107, 103, 15009, 103, 17243, 2771, 25, 21990, 49168, 49168, 109, 45, 49160, 28422, 198, 198, 1348, 1530, 338, 19573, 8842, 19407, 18169, 107, 106, 15009, 103, 17243, 2771, 25, 21990, 49168, 49168, 109, 24814, 4810, 277, 107, 49160, 24362, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49154], "prefix_len": 84, "old_logprobs": [-5.4676618576049805, -0.08267625421285629, -0.48951417207717896, -0.00837528333067894, -0.035947270691394806, -0.20950999855995178, -0.026094090193510056, -0.00300860358402133, -0.008101341314613819, -0.34878528118133545, -0.04217327758669853, -0.0012685356196016073, -0.005173746962100267, -4.7444173105759546e-05, -0.0038459161296486855, -0.10646867007017136, -0.15687593817710876, -0.5347756743431091, -0.2634028196334839, -1.780600666999817, -0.3890330195426941, -0.5366333723068237, -0.28852781653404236, -0.012083896435797215, -0.366604208946228, -0.4687512516975403, -0.10306990891695023, -0.6842280030250549, -0.27897170186042786, -0.01272981520742178, -0.1433679759502411, -2.5191280841827393, -0.00403613829985261, -0.17723913490772247, -2.315380811691284, -0.2344483584165573, -0.9342364072799683, -1.621399998664856, -0.03618483245372772, -0.3087265193462372, -1.9577724933624268, -1.188499093055725, -0.07433547079563141, -3.0158865451812744, -0.9246641397476196, -0.08035124838352203, -2.374654769897461, -0.2017061710357666, -0.028968919068574905, -0.00010954733443213627, -0.00017093151109293103, -0.0034398215357214212, -0.0003064401389565319, -0.000497336674015969, -0.00011860620725201443, -1.7881377516459906e-06, -0.005675748456269503, -0.0008301864145323634, -0.0032932115718722343, -0.00014995403762441128, -0.051952898502349854, -0.661983072757721, -0.04980333894491196, -0.15507034957408905, -0.03550821915268898, -0.004921702668070793, -0.5237862467765808, -0.0027215369045734406, -0.36131808161735535, -0.0005858612130396068, -5.686121585313231e-05, -0.00038985759601928294, -7.772143726469949e-05, -0.019822711125016212, -0.014727095142006874, -0.03479096665978432, -0.34551504254341125, -0.011373092420399189, -0.06380147486925125, -0.17162667214870453, -0.022371772676706314, -4.3474931716918945, -1.2452268600463867, -0.2738761305809021, -0.5261099934577942, -0.27741482853889465, -0.005650144536048174, -0.00035315475543029606, -0.0005619138828478754, -0.018781563267111778, -0.0038870032876729965, -0.000734178873244673, -0.000714404450263828, -3.4570634852570947e-06, -0.0006388768670149148, -0.0004365683125797659, -0.0013140147784724832, -0.00016842853801790625, -0.03923202306032181, -1.2012253999710083, -0.30498284101486206, -6.079655122448457e-06, -0.006577863823622465, -0.004407928790897131, -0.020266132429242134, -0.2134394347667694, -0.7960283160209656, -0.17046286165714264, -0.08317252993583679, -0.0023445994593203068, -0.0013304438907653093, -0.022198647260665894, -0.007149232551455498, -0.005019799806177616, -2.6464111215318553e-05, -0.000532842765096575]}
+{"group_id": "math_061", "origin": "budgie", "answer": "To approach this problem, we need to understand that if $r$ is a complex number such that $|z|=|w|=|z-w|$, then $z$ is a geometric progression with common ratio $r$.\n\nSince $r$ is a complex number, it can be written as $|r| = c$, where $c$ is a real number.\n\nTherefore, we can write:\n\\[ z = c \\left(\\frac{z}{w}\\right) = c \\left(\\frac{c}{w}\\right) = c^2 \\left(\\frac{1}{w}\\right) = c^2 \\left(\\frac{|z|}{|w|}\\right) = c^2 \\left(\\frac{|z-w|}{|w|}\\right) \\]\n\nSince $|z|=|w|=|z-w|$, we can simplify this to:\n\\[ z = c^2 \\left(\\frac{|z-w|}{|w|}\\right) = c^2 \\left(\\frac{|z|}{|w|}\\right) = c^2 \\left(\\frac{|z-w|}{|w|}\\right) = c^2 \\left(\\frac{|z-w|}{|w|}\\right) = c^2 \\left(\\frac{|z-w|}{|w|} \\times \\frac{1}{\\left|z-w|\\right|}\\right) = c^2 \\left(\\frac{|w|}{|z-w|}\\right) = c^2 \\left(\\frac{|w|}{|z-w|}\\right) = c^2 \\left(\\frac{1}{|z-w|}\\right)", "reward": -1, "advantage": -0.707106031187343, "ids": 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+{"group_id": "math_061", "origin": "verified_reference", "answer": "First solution. Let $u=\\frac{z}{w}=x+y i$. 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+{"group_id": "math_062", "origin": "budgie", "answer": "To perform the task of \"erasing\" link segments, we must firstaaaaa deal with the remaining 11 horizontal line segments and then deal with the remaining 11 vertical line segments, which is the least.\n\nThere are $11 + 11 = 22$ line segments on the $10 \\times 10$ grid paper.\n\nThe first step is to erase a certain number of link segments, $x$.\n\nThe remaining link segments are $22 - x$.\n\nThe remaining link segments are $22 - x - x = 22 - 2x$.\n\nThe problem requires that at most 3 link segments remain at each node.\n\nThus, we need to eliminate $3(22 - 2x)$ link segments.\n\n$3(22 - 2x) \\leq 22$.\n\n$66 - 6x \\leq 22$.\n\n$6x \\leq 44$.\n\n$x \\leq 6$.\n\nThus, $x$ has to be less than or equal to 6.\n\nSo, to make sure that at most 3 link segments remain at each node, we need to erase $3(22 - 2x) \\leq 22$ link segments.\n\nTherefore, we need to erase at least 6 link segments.\n\nThe answer is \\boxed{6}.", "reward": -1, 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+{"group_id": "math_062", "origin": "budgie", "answer": "To determine the minimum number of link segments that need to be erased, we can first think of placing the nodes in a grid where each node is connected to its neighbors, and we want to create a grid of nodes with the maximum number of link segments between any two nodes.\n\nImagine a grid where each node is connected to its neighbors, and we want to create a structure with the maximum number of link segments between any two nodes. To achieve this, we can start by connecting each node to its immediate neighbors, then connect each neighbor to its neighbors, and so on.\n\nHowever, this approach is not efficient because we want to maximize the number of link segments between any two nodes with each node being connected to its neighbors. If we attempt to do this, we will be creating a grid of nodes with a lot of link segments between each node.\n\nTo make it easier, we can think of this problem as having a \" simplify\" or \"level\" in the grid. This means that each node is connected to its immediate neighbors, and we can have a \" exceeds\" or \"root\" in the grid.\n\nThe simplest way to think about this is to have a \" net\" structure where each node is connected to its neighbors and each connection is a direct edge between two nodes. In this case, we can have a \" binocular\" in the grid with a \"step\" or \"net\" that each node is connected to its neighbors, and each connection is a direct edge between two nodes.\n\nHowever, if we want to maximize the number of link segments between any two nodes, we can try to build the grid in a \"([[\" or \"level\") shape. This means that each node is connected to its immediate neighbors and the neighbors of its neighbors, and we can have a \"level\" shape in the grid.\n\nTo achieve this, we can start", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49163, 30, 49168, 1985, 253, 1885, 49160, 49159, 3814, 14602, 216, 49160, 49159, 20, 6983, 1941, 28, 665, 359, 216, 49160, 49160, 10691, 6983, 3204, 284, 216, 49160, 49160, 8681, 6983, 3204, 30, 330, 1761, 9134, 8377, 827, 11358, 8661, 335, 260, 1142, 5173, 1761, 314, 1217, 253, 476, 8371, 9134, 2227, 1073, 800, 2795, 12523, 737, 288, 325, 40446, 418, 2124, 28, 588, 338, 418, 768, 216, 49162, 2795, 12523, 2612, 418, 971, 5994, 47, 49154, 49158, 198, 2068, 3346, 260, 5869, 1230, 282, 2795, 12523, 338, 737, 288, 325, 40446, 28, 392, 416, 808, 1510, 282, 10482, 260, 8661, 281, 253, 6983, 837, 971, 5994, 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+{"group_id": "math_062", "origin": "verified_reference", "answer": "[Solution] Removing the dotted links in the right figure satisfies the requirements of the problem, and there are 41 dotted links in the figure, so the minimum value required does not exceed 41.\n\nOn the other hand, there are 81 internal nodes on the grid paper, each with 4 links. Therefore, at least one link must be erased from each node to meet the requirement. Since each link connects two nodes, at least 41 links must be erased.\nIn summary, the minimum number of links that need to be erased is 41.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49163, 30, 49168, 1985, 253, 1885, 49160, 49159, 3814, 14602, 216, 49160, 49159, 20, 6983, 1941, 28, 665, 359, 216, 49160, 49160, 10691, 6983, 3204, 284, 216, 49160, 49160, 8681, 6983, 3204, 30, 330, 1761, 9134, 8377, 827, 11358, 8661, 335, 260, 1142, 5173, 1761, 314, 1217, 253, 476, 8371, 9134, 2227, 1073, 800, 2795, 12523, 737, 288, 325, 40446, 418, 2124, 28, 588, 338, 418, 768, 216, 49162, 2795, 12523, 2612, 418, 971, 5994, 47, 49154, 49158, 198, 75, 37500, 77, 40646, 260, 33697, 5823, 281, 260, 1048, 4110, 38475, 260, 4292, 282, 260, 1732, 28, 284, 665, 359, 216, 49163, 49160, 33697, 5823, 281, 260, 4110, 28, 588, 260, 5869, 1685, 2309, 1072, 441, 8769, 216, 49163, 49160, 30, 198, 198, 3280, 260, 550, 1369, 28, 665, 359, 216, 49167, 49160, 4611, 8661, 335, 260, 6983, 1941, 28, 971, 351, 216, 49163, 5823, 30, 4882, 28, 418, 2124, 582, 2795, 1251, 325, 40446, 429, 971, 5994, 288, 2220, 260, 9702, 30, 4311, 971, 2795, 13766, 827, 8661, 28, 418, 2124, 216, 49163, 49160, 5823, 1251, 325, 40446, 30, 198, 788, 7127, 28, 260, 5869, 1230, 282, 5823, 338, 737, 288, 325, 40446, 314, 216, 49163, 49160, 30, 49154], "prefix_len": 105, "old_logprobs": [-7.449690818786621, -1.9770647287368774, -0.15920168161392212, -10.69961166381836, -1.789900779724121, -11.838637351989746, -3.979261875152588, -3.605294942855835, -1.1912921667099, -7.498869895935059, -3.8084609508514404, -5.740559101104736, -0.28958895802497864, -3.902332067489624, -1.1528631448745728, -0.09817113727331161, -0.1340741068124771, -2.633474349975586, -1.7155736684799194, -2.7289886474609375, -0.5453047752380371, -1.146714210510254, -2.2426087856292725, -4.827389717102051, -12.58397102355957, -0.026423754170536995, -1.617440104484558, -0.4235057532787323, -1.399945616722107, -2.12799334526062, -1.3228168487548828, -1.0529439449310303, -3.0631728172302246, -6.422821044921875, -4.625144958496094, -8.781652450561523, -0.033182527869939804, -1.4572054147720337, -0.11708926409482956, -0.21313565969467163, -0.18142126500606537, -0.10972218215465546, -0.35778218507766724, -0.06791090220212936, -6.157463550567627, -0.9485287070274353, -1.8810614347457886, -0.007239655591547489, -0.0018123644404113293, -3.1792166233062744, -0.4323405921459198, -0.5759453773498535, -4.433464050292969, -2.7335517406463623, -7.434441566467285, -4.061356067657471, -3.5231809616088867, -0.3463122248649597, -1.1522432565689087, -2.9703993797302246, -0.43633532524108887, -5.184387683868408, -0.9019814729690552, -0.7719457149505615, -1.3656389713287354, -1.8687137365341187, -1.834261178970337, -1.1006048917770386, -0.0026856327895075083, -2.429999828338623, -0.29096072912216187, -2.4851255416870117, -0.6696619391441345, -1.6720387935638428, -0.14165224134922028, -2.0822014808654785, -3.248870611190796, -0.282360315322876, -0.9901083111763, -1.4724669456481934, -4.510658264160156, -0.02598026767373085, -0.5497615933418274, -0.5901196002960205, -4.733688831329346, -2.063830852508545, -1.370210886001587, -1.9551098346710205, -0.11643362790346146, -0.6598911285400391, -0.18370525538921356, -2.090676784515381, -0.17032140493392944, -0.9332923889160156, -1.4725711345672607, -1.9542450904846191, -1.1973481178283691, -0.9059697389602661, -0.03581397607922554, -0.04219784960150719, -0.7868223190307617, -0.10581129044294357, -8.253219604492188, -2.2577691078186035, -0.024062616750597954, -0.7008510231971741, -0.5332680344581604, -0.8551396131515503, -0.08624454587697983, -0.698243260383606, -1.0326505899429321, -0.33920007944107056, -0.001156733720563352, -0.000621840707026422, -0.037371110171079636, -0.22377364337444305, -0.4029815196990967, -0.16946452856063843, -0.0882166400551796, -0.10297312587499619, -4.6557207107543945]}
+{"group_id": "math_064", "origin": "budgie", "answer": "Given that $a^n - 2014$ has $2014$ digits, we need to find the smallest natural number $k$ such that $a$ cannot also be a $k$-digit number.\n\nIf $a$ is a $k$-digit number, then it can be written as $a = \\overline{abcd...}$, where $a, b, c, d,..., k$ are its digits.\n\nSubstituting $a = \\overline{abcd...}$ into the equation, we get:\n\n$$\\overline{abcd...}^n - 2014 = \\overline{abcd...}$$\n\nRearranging the equation, we have:\n\n$$\\overline{abcd...}^n = 2014\\overline{abcd...}$$\n\nTaking the logarithm of both sides, we get:\n\n$$n \\log \\overline{abcd...} = \\log 2014 \\overline{abcd...}$$\n\nNow, we need to find the smallest natural number $k$ such that $a$ cannot also be a $k$-digit number.\n\nUsing the definition of $a$ and $n$, we can write $n = 2014 - 2014 \\log \\overline{a}$.\n\nSubstituting this into the equation, we get:\n\n$$2014 - 2014 \\log \\overline{a} = \\log 2014$$\n\nSolving for $\\log \\overline{a}$, we get:\n\n$$\\log \\overline{a} = 4$$\n\nSince $a$ is", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 2959, 1885, 81, 20, 284, 1885, 94, 20, 325, 1782, 2966, 28, 284, 357, 314, 1343, 338, 1885, 81, 21990, 94, 39949, 49161, 49159, 49160, 49163, 20, 314, 253, 1230, 351, 1885, 49161, 49159, 49160, 49163, 20, 18638, 30, 7985, 260, 13341, 1782, 1230, 1885, 91, 20, 715, 338, 1885, 81, 20, 2526, 325, 253, 1885, 91, 20, 29, 23141, 1230, 30, 49154, 49158, 198, 15423, 338, 1885, 81, 78, 94, 731, 216, 49161, 49159, 49160, 49163, 20, 553, 1885, 49161, 49159, 49160, 49163, 20, 18638, 28, 392, 737, 288, 1042, 260, 13341, 1782, 1230, 1885, 91, 20, 715, 338, 1885, 81, 20, 2526, 597, 325, 253, 1885, 91, 20, 29, 23141, 1230, 30, 198, 198, 1937, 1885, 81, 20, 314, 253, 1885, 91, 20, 29, 23141, 1230, 28, 965, 357, 416, 325, 2855, 347, 1885, 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+{"group_id": "math_064", "origin": "budgie", "answer": "2014 is a multiple of 4, so\n$$(a^k - 2014) \\equiv 0 \\pmod{4}$$\n\nThe possible values of $a$ are 1, 2, 3, 4, and 5.\n\nIf $a = 5$, then $5^k - 2014 = 0 \\Rightarrow 5^k = 2014 \\Rightarrow k = 4$, which is not a $k$-digit number.\n\nIf $a = 4$, then $4^k - 2014 = 0 \\Rightarrow 4^k = 2014 \\Rightarrow k = 5$, which is a $k$-digit number.\n\nIf $a = 3$, then $3^k - 2014 = 0 \\Rightarrow 3^k = 2014 \\Rightarrow k = 6$, which is a $k$-digit number.\n\nIf $a = 2$, then $2^k - 2014 = 0 \\Rightarrow 2^k = 2014 \\Rightarrow k = 127$, which is not a $k$-digit number.\n\nIf $a = 1$, then $1^k - 2014 = 0 \\Rightarrow 1^k = 2014 \\Rightarrow k = 402$, which is not a $k$-digit number.\n\nIf $a = 0$, then $0^k - 2014 = 0 \\Rightarrow 0^k = 2014 \\Rightarrow k = 40", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 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+{"group_id": "math_064", "origin": "verified_reference", "answer": "Solution. Let $a$ be a $k$-digit number, then $10^{k-1} \\leq a2013$, i.e., $k>45$.\n\nWe will check for $k$ starting from 46, whether the condition of the absence of an integer in the specified interval is met. We will find that\n\n(for $k=46$) $2013 / 462013$ );\n\n(for $k=47$) $2013 / 472013$ );\n\n(for $k=48$) $2013 / 482013$ );\n\nbut (for $k=49$) $2013 / 49 > 41$ (since $\\quad 41 \\cdot 49 = 45^2 - 1^2 = 2025 - 16 < 2013$).\n\nThus, the interval $[2013 / 49, 2013 / 48)$ does not contain any integers, and for $k=49$, there are no suitable $n$.\n\n## Answer: $\\mathbf{k}=49$.\n\n## Criteria.\n\nThe problem is complex, so don't be stingy with points for reasonable thoughts. Any answer in the vicinity of $\\sqrt{2014}$, which is preceded by something reasonable, is worth points. 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+{"group_id": "math_065", "origin": "budgie", "answer": "To find the minimum sum of all real numbers written in $11$ edges, we can analyze the structure of the graph described.\n\nSince each edge is connecting two disks, we can model each edge as a pair of disks. The value of the real number in each disk is the absolute difference between the two real numbers written in the two disks that the edge connects.\n\nWe can represent the graph with edges as follows:\n\n-(2^1, 0) - 2^0, -2^0, -2^1, -2^1, -2^2 - 2^1, -2^1, -2^2 - 2^2, ...\n\nThe $x$-coordinate of each vertex is the sum of the squares of the distances from the vertices to the other two vertices. The $y$-coordinate of each vertex is the square of the difference between the two real numbers in the two disks that the edge connects.\n\nNotice that the values in the $y$-coordinate can be represented as a function of the distance in the $x$-direction. This function can be represented as:\n\n$$f(x, y) = |x - y|^2$$\n\nWe can see that the function $f(x, y)$ is a piecewise linear function, and the minimum value occurs when $x = y$. For this to happen, we must have $x = 0$ and $y = 1$, which would make $f(x, y) = 0$.\n\nTo find the minimum value of $f(x, y)$, we can take the derivative of $f(x, y)$ with respect to $x$ and set it equal to zero:\n\n$$f'(x, y) = 2|x - y", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 665, 325, 253, 4110, 351, 1885, 49168, 20, 26113, 284, 1885, 49160, 49160, 20, 8938, 28, 347, 3057, 2441, 30, 198, 198, 1882, 523, 2965, 253, 1345, 1230, 281, 971, 284, 897, 10599, 30, 3875, 28, 327, 971, 5595, 28, 392, 523, 2965, 260, 5222, 282, 260, 3193, 826, 260, 827, 1345, 2966, 2855, 281, 260, 827, 26113, 338, 260, 5595, 13766, 30, 1046, 1251, 2965, 1885, 49159, 20, 281, 10599, 1885, 49, 26265, 284, 1885, 49160, 20, 281, 10599, 1885, 57, 28422, 7985, 260, 5869, 2028, 282, 511, 1345, 2966, 2855, 281, 1885, 49160, 49160, 20, 8938, 30, 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+{"group_id": "math_065", "origin": "budgie", "answer": "The equation that defines the distances between the real numbers in the disks is the sum of the radii of the two disks. This can be written as $r_1 + r_2 = 9$.\n\nWe can also write this equation in terms of the radii of the two disks as $r_1 + r_2 = 9$ and $r_1 - r_2 = 9$.\n\nSolving these two equations, we find that $r_1 = 3$ and $r_2 = 6$.\n\nNow, we can find the sum of all real numbers written in the edges of the figure. This is equal to $11(r_1 + r_2) = 11(3 + 6) = 49$.\n\nTherefore, the minimum sum of all real numbers written in the edges is $\\boxed{49}$.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 665, 325, 253, 4110, 351, 1885, 49168, 20, 26113, 284, 1885, 49160, 49160, 20, 8938, 28, 347, 3057, 2441, 30, 198, 198, 1882, 523, 2965, 253, 1345, 1230, 281, 971, 284, 897, 10599, 30, 3875, 28, 327, 971, 5595, 28, 392, 523, 2965, 260, 5222, 282, 260, 3193, 826, 260, 827, 1345, 2966, 2855, 281, 260, 827, 26113, 338, 260, 5595, 13766, 30, 1046, 1251, 2965, 1885, 49159, 20, 281, 10599, 1885, 49, 26265, 284, 1885, 49160, 20, 281, 10599, 1885, 57, 28422, 7985, 260, 5869, 2028, 282, 511, 1345, 2966, 2855, 281, 1885, 49160, 49160, 20, 8938, 30, 49154, 49158, 198, 504, 8345, 338, 11754, 260, 9510, 826, 260, 1345, 2966, 281, 260, 26113, 314, 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+{"group_id": "math_065", "origin": "verified_reference", "answer": "1. **Model the problem as an electrical circuit:**\n   - We are given a graph with 9 disks (nodes) and 11 edges.\n   - We need to assign real numbers to each disk such that the sum of the squares of the differences between the numbers on the disks connected by edges is minimized.\n   - We are required to set the number on disk \\(A\\) to \\(0\\) and on disk \\(I\\) to \\(1\\).\n\n2. **Interpret the problem using electrical circuit theory:**\n   - Consider each disk as a node in an electrical circuit.\n   - Assign a voltage to each node, with disk \\(A\\) at \\(0V\\) and disk \\(I\\) at \\(1V\\).\n   - Each edge represents a resistor of \\(1 \\Omega\\).\n\n3. **Apply the concept of minimizing power in the circuit:**\n   - The goal is to minimize the sum of the squares of the differences, which is equivalent to minimizing the total power dissipated in the circuit.\n   - The power dissipated in a resistor \\(R\\) with voltage difference \\(V\\) is given by \\(P = \\frac{V^2}{R}\\).\n   - For a resistor of \\(1 \\Omega\\), the power dissipated is \\(P = V^2\\).\n\n4. **Calculate the equivalent resistance of the circuit:**\n   - The total resistance between nodes \\(A\\) and \\(I\\) is given as \\(\\frac{33}{20} \\Omega\\).\n   - This is derived from the specific configuration of the graph and the resistances.\n\n5. **Determine the total power in the circuit:**\n   - The total voltage across the circuit is \\(1V\\) (from \\(A\\) to \\(I\\)).\n   - The total power dissipated in the circuit is given by:\n     \\[\n     P = \\frac{V^2}{R_{\\text{eq}}} = \\frac{1^2}{\\frac{33}{20}} = \\frac{20}{33}\n     \\]\n\n6. **Conclusion:**\n   - The minimum sum of the squares of the differences between the numbers on the disks connected by edges is \\(\\frac{20}{33}\\).\n\nThe final answer is \\(\\boxed{\\frac{20}{33}}\\).", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 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+{"group_id": "math_066", "origin": "budgie", "answer": "We know that $A$ and $B$ are two vertices of $\\mathcal{P}$ on the circumcircle of $\\mathcal{Y}$. Let the other arc between $A$ and $B$ on $\\mathcal{Y}$ have $2^k+1$ vertices.\n\nLet $A$ be $P_1$ and $B$ be $P_2$. By the inscribed angle theorem, we know that $2\\sin(\\angle(AP_1, AP_2)) = 2\\sin(\\angle(BP_1, BP_2)$ and $2\\sin(\\angle(AP_1, BP_2)) = 2\\sin(\\angle(AP_2, BP_1)$.\n\nSince $A$ and $B$ are on the circumcircle of $\\mathcal{Y}$, we know that $AP_1 = BP_2$.\n\nTherefore, $\\angle(AP_1, AP_2) = \\angle(AP_1, BP_2) = \\angle(BP_1, BP_2) = \\angle(P_1B_2, P_2B_1) = \\angle(P_2A_2, P_1A_1) = \\angle(A_1P_2, A_2P_1) = \\angle(P_1A_1, PB_2) = \\angle(P_2A_2, PB_1)$.\n\nSince $\\angle(AP_1, BP_2) = \\angle(P_2A_2, PB_1)$, we know that $AP_2 = BP_1$.\n\nSince $\\angle(AP_1, AP_2) = \\angle(AP_2, BP_1) =", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 2122, 314, 253, 2649, 1885, 49160, 49166, 20, 29, 13539, 19573, 8898, 5020, 107, 64, 24362, 284, 624, 4584, 29733, 19573, 8898, 5020, 107, 73, 24362, 335, 260, 8303, 30, 216, 198, 504, 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+{"group_id": "math_066", "origin": "budgie", "answer": "This problem is about counting the number of colorings of a regular 17-gon with a circumcircle of modulus 2.\n\nFirst, we need to find the number of colorings. To do this, we can use the following recurrence relation:\n- The first vertex can be colored in red, blue, or green.\n- The second vertex can be colored in two choices: red or blue.\n- The third side of the polygon can be colored in three choices.\n- The fourth side of the polygon can be colored in two choices.\n- We continue this pattern until we reach the last vertex.\n\nUsing this recurrence relation, we can find the number of colorings for a regular 17-gon.\n\nLet's denote the number of colors as $c_n$. We can find $c_{17}$, the number of colorings for a regular 17-gon:\n\n$$c_{17} = 2c_{16} + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8$$\n\nNow, we can calculate the value of $c_{17}$.\n\n$$c_{17} = 2(2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8)$$\n\nTo find the value of $c_{17}$, we need to calculate the sum of the squares of the first 20 positive integers.\n\n$$c_{17} = (2^1 + 2^2 + 2^3 +... + 2^20) + (2", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 2122, 314, 253, 2649, 1885, 49160, 49166, 20, 29, 13539, 19573, 8898, 5020, 107, 64, 24362, 284, 624, 4584, 29733, 19573, 8898, 5020, 107, 73, 24362, 335, 260, 8303, 30, 216, 198, 504, 24796, 282, 19573, 8898, 5020, 107, 64, 24362, 359, 23692, 281, 715, 253, 970, 338, 1885, 49, 28, 50, 3814, 254, 3814, 8898, 5020, 107, 64, 24362, 359, 282, 673, 199, 6202, 7302, 28, 585, 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+{"group_id": "math_066", "origin": "verified_reference", "answer": "1. **Identify the relevant values of \\(2^k + 1\\):**\n   Since the minor arc \\(AB\\) has at most 9 vertices on it, the relevant values of \\(2^k + 1\\) are \\(3, 5,\\) and \\(9\\). This is because:\n   \\[\n   2^1 + 1 = 3, \\quad 2^2 + 1 = 5, \\quad 2^3 + 1 = 9\n   \\]\n\n2. **Enumerate the vertices:**\n   Enumerate the vertices of the 17-gon as \\(V_0, V_1, \\ldots, V_{16}\\).\n\n3. **Determine the conditions for different colors:**\n   We have that \\(V_i\\) and \\(V_j\\) are of different colors if \\(i - j \\equiv \\pm 2, \\pm 4, \\pm 8 \\pmod{17}\\). This is derived from the relevant values of \\(2^k + 1\\).\n\n4. **Establish a lower bound for the number of colors:**\n   Since \\(V_0, V_2, V_4\\) must have pairwise distinct colors, we have \\(n \\geq 3\\).\n\n5. **Check if 4 colors suffice:**\n   If there are 4 colors \\(A, B, C,\\) and \\(D\\), we can try to color the vertices such that the condition is satisfied. One possible coloring is:\n   \\[\n   \\{ V_0, V_1, \\cdots, V_{16} \\} = \\{ A, D, B, B, C, C, D, D, B, B, C, A, D, D, B, B, C \\}\n   \\]\n   This coloring ensures that no two vertices \\(V_i\\) and \\(V_j\\) with \\(i - j \\equiv \\pm 2, \\pm 4, \\pm 8 \\pmod{17}\\) have the same color. Therefore, \\(n \\leq 4\\).\n\n6. **Prove that 3 colors are insufficient:**\n   Assume there exists a solution with \\(n = 3\\). Without loss of generality, let \\(V_0 = A\\) and \\(V_2 = B\\). Then:\n   \\[\n   V_4 = C, \\quad V_6 = A, \\quad V_8 = B, \\quad V_{10} = C, \\quad V_{12} = A, \\quad V_{14} = B, \\quad V_{16} = C\n   \\]\n   Continuing this pattern, we find that \\(V_{13} = A\\), which contradicts the assumption that \\(n = 3\\) is a solution because \\(V_0\\) and \\(V_{13}\\) would be the same color, violating the condition for different colors.\n\nTherefore, the least number of colors required is \\(4\\).\n\nThe final answer is \\(\\boxed{4}\\).", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 2122, 314, 253, 2649, 1885, 49160, 49166, 20, 29, 13539, 19573, 8898, 5020, 107, 64, 24362, 284, 624, 4584, 29733, 19573, 8898, 5020, 107, 73, 24362, 335, 260, 8303, 30, 216, 198, 504, 24796, 282, 19573, 8898, 5020, 107, 64, 24362, 359, 23692, 281, 715, 253, 970, 338, 1885, 49, 28, 50, 3814, 254, 3814, 8898, 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+{"group_id": "math_068", "origin": "budgie", "answer": "(a) To find the measure of angle $A X C$, we can use the properties of inscribed quadrilaterals.\n\nSince $\\angle B = 32^{\\circ}$, we know that $\\angle B X D = 180^{\\circ} - 32^{\\circ} - 52^{\\circ} = 96^{\\circ}$.\n\nSimilarly, since $\\angle X D C = 52^{\\circ}$, we know that $\\angle X D C = 180^{\\circ} - 52^{\\circ} - 96^{\\circ} = 44^{\\circ}$.\n\nNow, we can use the fact that the sum of the angles in a quadrilateral is $360^{\\circ}$ to find the measure of $\\angle A X C$.\n\n$$\\angle A X C = x + \\angle X D C + \\angle B D C = x + 44^{\\circ} + 96^{\\circ}$$\n\nTo find the value of $x$, we can plug in $x = 44^{\\circ}$ and $x = 96^{\\circ}$ into the equation.\n\n$$\\angle A X C = 44^{\\circ} + 96^{\\circ} = 140^{\\circ}$$\n\nTherefore, the angle $A X C$ measures $\\boxed{140^{\\circ}}$.\n\n(b) To find the measure of angle $A C B$, we can use the properties of inscribed quadrilaterals.\n\nWe know that the sum of the angles in a quadrilateral is $360^{\\circ}$.\n\n$$\\angle A C B + \\angle", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 18584, 403, 216, 49168, 30, 49164, 30, 49161, 30, 334, 1677, 1885, 49, 422, 20, 284, 1885, 52, 340, 20, 282, 260, 30036, 20378, 13293, 1885, 49, 389, 340, 422, 20, 359, 4037, 30, 330, 1225, 1885, 72, 20, 314, 6319, 335, 2103, 1885, 50, 340, 20, 715, 338, 1885, 49, 389, 446, 389, 2273, 28422, 657, 314, 1343, 338, 19573, 6935, 389, 446, 216, 49162, 49161, 21990, 76, 21386, 24362, 284, 19573, 6935, 2273, 422, 340, 446, 216, 49164, 49161, 21990, 76, 21386, 24362, 30, 198, 24, 81, 25, 365, 49160, 1225, 25, 1073, 800, 5027, 1072, 260, 7256, 1885, 49, 2273, 340, 20, 2621, 47, 198, 198, 24, 82, 25, 365, 49162, 2876, 25, 1073, 800, 5027, 1072, 260, 7256, 1885, 49, 340, 389, 20, 2621, 47, 198, 198, 21653, 5257, 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+{"group_id": "math_068", "origin": "budgie", "answer": "(a) To find the measure of angle $A D C$, we can use the fact that the sum of the interior angles of a triangle is $180^\\circ$:\n\n$$A D C + A B = \\angle A B C + \\angle A B D$$\n\nSince $\\angle A B D = 180^\\circ - \\angle B = 180^\\circ - 32^\\circ = 148^\\circ$. Substituting this into the equation above, we get:\n\n$$A D C + 148^\\circ = 180^\\circ$$\n\nRearranging the equation, we get:\n\n$$A D C = 180^\\circ - 148^\\circ$$\n\n$$A D C = 32^\\circ$$\n\n(b) To find the measure of angle $A C B$, we can use the fact that the sum of the interior angles of a triangle is $180^\\circ$:\n\n$$A C B + A C B = 180^\\circ$$\n\nSince $A C B$ and $A C B$ are congruent triangles, we can equate the two expressions:\n\n$$A C B = 180^\\circ - A C B$$\n\nSubtracting $A C B$ from both sides, we get:\n\n$$0 = 180^\\circ - A C B - A C B$$\n\nRearranging the equation, we get:\n\n$$A C B = 180^\\circ$$\n\n(c) To find the measure of angle $C B D$, we can use the fact that the sum of the interior angles of a triangle is $180^\\circ$:\n\n$$", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 18584, 403, 216, 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+{"group_id": "math_068", "origin": "verified_reference", "answer": "Answer: (a) $106^{\\circ}$. (b) $48^{\\circ}$.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 18584, 403, 216, 49168, 30, 49164, 30, 49161, 30, 334, 1677, 1885, 49, 422, 20, 284, 1885, 52, 340, 20, 282, 260, 30036, 20378, 13293, 1885, 49, 389, 340, 422, 20, 359, 4037, 30, 330, 1225, 1885, 72, 20, 314, 6319, 335, 2103, 1885, 50, 340, 20, 715, 338, 1885, 49, 389, 446, 389, 2273, 28422, 657, 314, 1343, 338, 19573, 6935, 389, 446, 216, 49162, 49161, 21990, 76, 21386, 24362, 284, 19573, 6935, 2273, 422, 340, 446, 216, 49164, 49161, 21990, 76, 21386, 24362, 30, 198, 24, 81, 25, 365, 49160, 1225, 25, 1073, 800, 5027, 1072, 260, 7256, 1885, 49, 2273, 340, 20, 2621, 47, 198, 198, 24, 82, 25, 365, 49162, 2876, 25, 1073, 800, 5027, 1072, 260, 7256, 1885, 49, 340, 389, 20, 2621, 47, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49159, 49165, 79, 49160, 86, 49163, 49168, 49167, 49160, 49164, 49160, 49168, 49163, 49166, 49168, 49164, 49162, 49161, 16373, 83, 49160, 87, 29, 49160, 49162, 30, 13655, 47, 11551, 45, 49162, 49160, 49160, 22, 6930, 45, 49166, 49159, 49168, 22, 4961, 79, 8842, 79, 105, 45, 49161, 49161, 49159, 22, 4961, 79, 8842, 79, 104, 45, 49162, 49166, 49161, 25, 49154, 49158, 198, 21350, 42, 365, 81, 25, 1885, 49160, 49159, 49165, 21990, 76, 21386, 24362, 30, 365, 82, 25, 1885, 49163, 49167, 21990, 76, 21386, 24362, 30, 49154], "prefix_len": 244, "old_logprobs": [-4.0428147315979, -0.08647264540195465, -0.3793182373046875, -1.4903967380523682, -0.0379696860909462, -3.6785571575164795, -1.2445402145385742, -2.1928887367248535, -4.509901523590088, -0.9986282587051392, -0.0014760324265807867, -0.0010352734243497252, -0.08891186118125916, -1.569225549697876, -4.69000768661499, -2.6808485984802246, -0.004261935595422983, -0.9301959276199341, -3.1230106353759766, -2.5131139755249023, -0.39433079957962036, -0.0006468587671406567, -0.00030870441696606576, -0.029331324622035027, -0.12441731244325638, -4.077394962310791]}
+{"group_id": "math_069", "origin": "budgie", "answer": "For any non-isosceles trapezoid, the sum of the lengths of the two parallel sides equals the length of the base. Since the non-parallel sides are of different lengths, they can form two right triangles with the height as the hypotenuse. The other leg of each right triangle is half the difference between the two parallel sides, so each right triangle has sides 20 cm and 13 cm. Since it's an isosceles trapezoid, it follows that 20 = 2x and 13 = x, where x is the height of the trapezoid. Therefore, x = 10 cm. The height of the trapezoid is 10 cm.\n\nFinal answer:\n#### 10", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49159, 30, 49159, 49161, 49163, 30, 378, 15900, 282, 260, 7623, 5882, 282, 260, 635, 31999, 1633, 359, 216, 49161, 49164, 284, 216, 49163, 5973, 28, 284, 260, 15900, 282, 260, 1603, 29, 29769, 5882, 359, 216, 49161, 49159, 284, 216, 49160, 49162, 5973, 30, 7985, 260, 4614, 282, 260, 635, 31999, 1633, 30, 49154, 49158, 198, 2193, 750, 1603, 29, 271, 395, 5906, 266, 635, 31999, 1633, 28, 260, 2028, 282, 260, 15900, 282, 260, 827, 7623, 5882, 16473, 260, 3519, 282, 260, 3159, 30, 4311, 260, 1603, 29, 29769, 5882, 359, 282, 896, 15900, 28, 502, 416, 910, 827, 1048, 22694, 351, 260, 4614, 347, 260, 16392, 264, 1726, 30, 378, 550, 1475, 282, 971, 1048, 14973, 314, 2745, 260, 3193, 826, 260, 827, 7623, 5882, 28, 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+{"group_id": "math_069", "origin": "budgie", "answer": "In an isosceles trapezoid, the non-parallel sides are not necessarily equal, so we cannot directly use the Pythagorean theorem. However, we can use the fact that the sum of the lengths of the two parallel sides and one of the non-parallel sides is equal to the length of the other side.\n\nLet's denote the height of the trapezoid as h cm. The sum of the lengths of the parallel sides is 25 + 4 = 29 cm. Therefore, we can write:\n\nh + 25 + 20 = 29\n\nSimplify and solve for h:\n\nh + 45 = 29\nh = 29 - 45\nh = -16 cm\n\nThis is incorrect since the height cannot be negative.\n\nLet's try again. The sum of the lengths of the two parallel sides is 25 + 4 = 29 cm. The missing side has a length of h cm. Therefore, we can write:\n\n25 + 4 = 4 + h\n29 = 4 + h\nh = 29 - 4\nh = 25 cm\n\nThe height of the trapezoid is 25 cm.\n\nFinal answer:\n#### 25", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49159, 30, 49159, 49161, 49163, 30, 378, 15900, 282, 260, 7623, 5882, 282, 260, 635, 31999, 1633, 359, 216, 49161, 49164, 284, 216, 49163, 5973, 28, 284, 260, 15900, 282, 260, 1603, 29, 29769, 5882, 359, 216, 49161, 49159, 284, 216, 49160, 49162, 5973, 30, 7985, 260, 4614, 282, 260, 635, 31999, 1633, 30, 49154, 49158, 198, 788, 354, 314, 395, 5906, 266, 635, 31999, 1633, 28, 260, 1603, 29, 29769, 5882, 359, 441, 6834, 4037, 28, 588, 392, 2526, 3319, 722, 260, 47997, 22688, 30, 1423, 28, 392, 416, 722, 260, 1027, 338, 260, 2028, 282, 260, 15900, 282, 260, 827, 7623, 5882, 284, 582, 282, 260, 1603, 29, 29769, 5882, 314, 4037, 288, 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+{"group_id": "math_069", "origin": "verified_reference", "answer": "## Solution.\n\nGiven $B C=4 \\text{ cm}, A D=25 \\text{ cm}, A B=20 \\text{ cm}, C D=13 \\text{ cm}$ (Fig. 10.24). Draw $B E \\perp A D$ and $C F \\perp A D$. Let $B E=C F=h, A E=x$, $F D=y$. Then from $\\triangle A B E$ and $\\triangle C F D$ we find $h^{2}=20^{2}-x^{2}=13^{2}-y^{2}$. Considering that $y=25-4-x=21-x$, we have $20^{2}-x^{2}=13^{2}-(21-x)^{2}$ or $42 x=672$, from which $x=16$ (cm). Therefore, $h=\\sqrt{20^{2}-16^{2}}=12$ (cm).\n\nAnswer: 12 cm.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49159, 30, 49159, 49161, 49163, 30, 378, 15900, 282, 260, 7623, 5882, 282, 260, 635, 31999, 1633, 359, 216, 49161, 49164, 284, 216, 49163, 5973, 28, 284, 260, 15900, 282, 260, 1603, 29, 29769, 5882, 359, 216, 49161, 49159, 284, 216, 49160, 49162, 5973, 30, 7985, 260, 4614, 282, 260, 635, 31999, 1633, 30, 49154, 49158, 198, 712, 19857, 30, 198, 198, 15423, 1885, 50, 340, 45, 49163, 3814, 2692, 107, 5973, 6663, 330, 422, 45, 49161, 49164, 3814, 2692, 107, 5973, 6663, 330, 389, 45, 49161, 49159, 3814, 2692, 107, 5973, 6663, 340, 422, 45, 49160, 49162, 3814, 2692, 107, 5973, 24362, 365, 10462, 30, 216, 49160, 49159, 30, 49161, 49163, 595, 10602, 1885, 50, 414, 3814, 456, 96, 330, 422, 20, 284, 1885, 51, 426, 3814, 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+{"group_id": "math_070", "origin": "budgie", "answer": "This problem can be solved by finding the GCD of all possible permutations of the digits of the number 202120222023. The GCD is 3.", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49161, 30, 7985, 260, 452, 6998, 282, 511, 2966, 6238, 411, 511, 1636, 7205, 33301, 282, 260, 18638, 282, 260, 1230, 216, 49161, 49159, 49161, 49160, 49161, 49159, 49161, 49161, 49161, 49159, 49161, 49162, 198, 198, 37500, 30, 1428, 260, 3044, 271, 1968, 3938, 28, 511, 623, 2966, 359, 46522, 411, 216, 49168, 365, 1195, 2028, 282, 260, 18638, 314, 216, 49160, 49167, 595, 330, 5499, 2619, 288, 6824, 338, 665, 359, 787, 550, 2966, 314, 338, 260, 3193, 826, 750, 827, 715, 2966, 314, 597, 46522, 411, 260, 452, 6998, 30, 1068, 1183, 28, 216, 49161, 49161, 49161, 49161, 49161, 49161, 49160, 49159, 49159, 49159, 49162, 49161, 731, 216, 49161, 49161, 49161, 49161, 49161, 49161, 49160, 49159, 49159, 49159, 49161, 49162, 446, 216, 49168, 314, 46522, 411, 216, 49168, 284, 2526, 325, 2852, 670, 216, 49168, 30, 49154, 49158, 198, 1348, 1732, 416, 325, 14168, 411, 4212, 260, 452, 6998, 282, 511, 1636, 7205, 33301, 282, 260, 18638, 282, 260, 1230, 216, 49161, 49159, 49161, 49160, 49161, 49159, 49161, 49161, 49161, 49159, 49161, 49162, 30, 378, 452, 6998, 314, 216, 49162, 30, 49154], "prefix_len": 169, "old_logprobs": [-4.162250518798828, -0.5279743671417236, -1.8504750728607178, -0.01975691132247448, -0.08087814599275589, -0.2180236428976059, -2.7862915992736816, -0.043194133788347244, -1.1433933973312378, -0.0005972985527478158, -0.05813749134540558, -0.34744739532470703, -1.531062364578247, -0.06045081466436386, -0.0023694555275142193, -0.035987865179777145, -0.04790462553501129, -0.025946466252207756, -0.10412748903036118, -0.5967187881469727, -0.03968536853790283, -0.03482941910624504, -0.01022906694561243, -0.027027178555727005, -0.008818373084068298, -0.0039245267398655415, -0.007647404447197914, -0.0012988949893042445, -0.0004002247005701065, -0.0005734706646762788, -0.0031329141929745674, -0.006006049923598766, -0.0002696150622796267, -0.0004693360242526978, -0.27413374185562134, -2.191563129425049, -1.7300163507461548, -0.000604565953835845, -1.2303577661514282, -0.5960536003112793, -2.2327699661254883, -0.49995723366737366, -1.7536964416503906]}
+{"group_id": "math_070", "origin": "budgie", "answer": "The number 202120222023 has 11 digits, all of which are 2's, and 11 + 11 = 22 digits are 2's. Therefore, the GCD of all 22 numbers obtained by all possible permutations of the digits of 202120222023 is 22.\n\nFinal answer: $\\boxed{22}$", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49161, 30, 7985, 260, 452, 6998, 282, 511, 2966, 6238, 411, 511, 1636, 7205, 33301, 282, 260, 18638, 282, 260, 1230, 216, 49161, 49159, 49161, 49160, 49161, 49159, 49161, 49161, 49161, 49159, 49161, 49162, 198, 198, 37500, 30, 1428, 260, 3044, 271, 1968, 3938, 28, 511, 623, 2966, 359, 46522, 411, 216, 49168, 365, 1195, 2028, 282, 260, 18638, 314, 216, 49160, 49167, 595, 330, 5499, 2619, 288, 6824, 338, 665, 359, 787, 550, 2966, 314, 338, 260, 3193, 826, 750, 827, 715, 2966, 314, 597, 46522, 411, 260, 452, 6998, 30, 1068, 1183, 28, 216, 49161, 49161, 49161, 49161, 49161, 49161, 49160, 49159, 49159, 49159, 49162, 49161, 731, 216, 49161, 49161, 49161, 49161, 49161, 49161, 49160, 49159, 49159, 49159, 49161, 49162, 446, 216, 49168, 314, 46522, 411, 216, 49168, 284, 2526, 325, 2852, 670, 216, 49168, 30, 49154, 49158, 198, 504, 1230, 216, 49161, 49159, 49161, 49160, 49161, 49159, 49161, 49161, 49161, 49159, 49161, 49162, 553, 216, 49160, 49160, 18638, 28, 511, 282, 527, 359, 216, 49161, 506, 28, 284, 216, 49160, 49160, 1232, 216, 49160, 49160, 446, 216, 49161, 49161, 18638, 359, 216, 49161, 506, 30, 4882, 28, 260, 452, 6998, 282, 511, 216, 49161, 49161, 2966, 6238, 411, 511, 1636, 7205, 33301, 282, 260, 18638, 282, 216, 49161, 49159, 49161, 49160, 49161, 49159, 49161, 49161, 49161, 49159, 49161, 49162, 314, 216, 49161, 49161, 30, 198, 198, 29288, 2988, 42, 19573, 4810, 277, 107, 49161, 49161, 24362, 49154], "prefix_len": 169, "old_logprobs": [-1.2751632928848267, -1.8553056716918945, -0.5552741885185242, -0.03442765399813652, -0.44634881615638733, -0.009287605062127113, -0.017534945160150528, -0.017008626833558083, -0.0024380742106586695, -0.000590865034610033, -0.0007270314963534474, -0.004284369759261608, -0.013349791057407856, -0.002029741881415248, -0.004078167490661144, -0.6864653825759888, -0.620512068271637, -1.1484708786010742, -1.4627463817596436, -0.05690128728747368, -1.044712781906128, -2.544477701187134, -0.408455491065979, -0.016646651551127434, -0.06941532343626022, -1.3538349866867065, -0.17690707743167877, -0.9989707469940186, -1.1439831256866455, -1.0088485479354858, -1.4579106569290161, -0.7175869941711426, -1.3473387956619263, -3.9890966415405273, -0.015657702460885048, -0.3239790201187134, -0.5485996603965759, -0.2915966212749481, -0.011740515939891338, -0.0016959820641204715, -0.00032085992279462516, -0.37679311633110046, -0.8103131055831909, -0.33686041831970215, -0.3993087410926819, -0.04962616413831711, -0.23915044963359833, -1.35410475730896, -0.01473191101104021, -0.674564778804779, -0.8888201117515564, -0.000618624035269022, -0.1752222776412964, -0.2119254469871521, -3.2273378372192383, -0.05504140257835388, -0.06209511682391167, -0.6232460737228394, -0.8941295146942139, -0.07227402180433273, -0.27634313702583313, -0.41369861364364624, -0.05416978523135185, -0.0005273620481602848, -0.035330332815647125, -0.3823673725128174, -0.09567642956972122, -0.24226021766662598, -1.0454427003860474, -0.0029779395554214716, -0.006200718693435192, -0.0007712250226177275, -0.0005383234238252044, -0.005497930571436882, -0.0005443998379632831, -0.00037174468161538243, -0.0006104036583565176, -0.00283815641887486, -0.014971042983233929, -0.00013064485392533243, -0.0005133026279509068, -0.04219807684421539, -0.2716982662677765, -0.28270667791366577, -0.45719096064567566, -0.3799542486667633, -0.671518862247467, -0.017886998131871223, -0.3838563859462738, -0.00035279724397696555, -0.0012175773736089468, -3.311633586883545, -0.00011359999916749075, 0.0, -0.0009581027552485466, -0.0022381513845175505, -0.002102547325193882, -0.0013781820889562368, -0.11235304176807404]}
+{"group_id": "math_070", "origin": "verified_reference", "answer": "Answer: 9.\n\n## Criteria\n\n7 points - complete solution;\n\n3 points - GCD found, but not proven that there can be no other divisors.", "reward": 1.0, "advantage": 1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49161, 30, 7985, 260, 452, 6998, 282, 511, 2966, 6238, 411, 511, 1636, 7205, 33301, 282, 260, 18638, 282, 260, 1230, 216, 49161, 49159, 49161, 49160, 49161, 49159, 49161, 49161, 49161, 49159, 49161, 49162, 198, 198, 37500, 30, 1428, 260, 3044, 271, 1968, 3938, 28, 511, 623, 2966, 359, 46522, 411, 216, 49168, 365, 1195, 2028, 282, 260, 18638, 314, 216, 49160, 49167, 595, 330, 5499, 2619, 288, 6824, 338, 665, 359, 787, 550, 2966, 314, 338, 260, 3193, 826, 750, 827, 715, 2966, 314, 597, 46522, 411, 260, 452, 6998, 30, 1068, 1183, 28, 216, 49161, 49161, 49161, 49161, 49161, 49161, 49160, 49159, 49159, 49159, 49162, 49161, 731, 216, 49161, 49161, 49161, 49161, 49161, 49161, 49160, 49159, 49159, 49159, 49161, 49162, 446, 216, 49168, 314, 46522, 411, 216, 49168, 284, 2526, 325, 2852, 670, 216, 49168, 30, 49154, 49158, 198, 21350, 42, 216, 49168, 30, 198, 198, 712, 38042, 1116, 49166, 2876, 731, 2994, 3564, 43, 1116, 49162, 2876, 731, 452, 6998, 983, 28, 564, 441, 8220, 338, 665, 416, 325, 787, 550, 3044, 40520, 30, 49154], "prefix_len": 169, "old_logprobs": [-4.287250518798828, -0.026273159310221672, -0.6010149717330933, -0.7073802351951599, -3.0711803436279297, -1.5149413347244263, -0.044312141835689545, -7.475040912628174, -13.60683536529541, -4.241678237915039, -1.9305158853530884, -10.840970039367676, -8.009834289550781, -10.719900131225586, -5.60141658782959, -8.295293807983398, -3.8393115997314453, -4.912070274353027, -1.5972609519958496, -0.25642016530036926, -7.8958845138549805, -0.02761678770184517, -5.656700134277344, -3.1824111938476562, -4.286386013031006, -2.403550624847412, -3.8519606590270996, -5.302414417266846, -1.7391934394836426, -4.413415908813477, -0.34524405002593994, -2.577296257019043, -0.11411569267511368, -5.766369819641113, -0.04870814457535744, -0.8330802321434021, -3.127284049987793]}
+{"group_id": "math_071", "origin": "budgie", "answer": "There are 5 sums because $a_1+a_2$, $a_1+a_3$, $a_1+a_4$, $a_1+a_5$, and $a_2+a_3$ are the 5 sums. Since there are 5 numbers, and they are distinct, there is only 1 number left. Therefore, $m=1$.\n\nThe answer is:\n\\boxed{1}", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 1885, 81, 79, 49160, 28, 81, 79, 49161, 28, 81, 79, 49162, 28, 81, 79, 49163, 28, 81, 79, 49164, 20, 325, 4073, 1345, 2966, 30, 6689, 511, 24053, 282, 260, 910, 1885, 81, 79, 89, 1232, 253, 79, 90, 20, 837, 1885, 89, 28, 90, 3814, 254, 3814, 107, 49160, 28, 49161, 28, 49162, 28, 49163, 28, 49164, 76, 24362, 284, 1885, 89, 3814, 662, 97, 544, 28422, 2959, 1885, 93, 20, 325, 260, 1230, 282, 4073, 2966, 1486, 623, 24053, 30, 1812, 314, 260, 13341, 1636, 1685, 282, 1885, 93, 20, 47, 49154, 49158, 198, 2122, 359, 216, 49164, 24053, 975, 1885, 81, 79, 49160, 27, 81, 79, 49161, 26265, 1885, 81, 79, 49160, 27, 81, 79, 49162, 26265, 1885, 81, 79, 49160, 27, 81, 79, 49163, 26265, 1885, 81, 79, 49160, 27, 81, 79, 49164, 26265, 284, 1885, 81, 79, 49161, 27, 81, 79, 49162, 20, 359, 260, 216, 49164, 24053, 30, 4311, 665, 359, 216, 49164, 2966, 28, 284, 502, 359, 4073, 28, 665, 314, 805, 216, 49160, 1230, 2049, 30, 4882, 28, 1885, 93, 45, 49160, 28422, 198, 198, 504, 2988, 314, 42, 198, 76, 4810, 277, 107, 49160, 109, 49154], "prefix_len": 125, "old_logprobs": [-3.113186836242676, -0.08462987095117569, -0.8381704092025757, -0.3669227361679077, -1.8881096839904785, -4.855652332305908, -1.939371109008789, -0.9162423014640808, -0.024545922875404358, -0.2490803748369217, -1.7291144132614136, -0.0017780937487259507, -0.00026806574896909297, -0.0056962547823786736, -2.4100089073181152, -0.04025645926594734, -0.0006226746481843293, -0.0015499495202675462, -0.5376858711242676, -0.001544950413517654, -0.00017021637177094817, -7.593343616463244e-05, -0.12526148557662964, -0.009346893057227135, -0.05758395418524742, -0.0007675323868170381, -0.0006849803612567484, -0.08255431801080704, -0.00020716428116429597, -4.470248313737102e-05, -0.00014399446081370115, -0.001765955938026309, -0.00817914493381977, -0.03243233263492584, -0.0004015354788862169, -0.0007577646756544709, -0.03406386449933052, -7.60526381782256e-05, -3.802703940891661e-05, -0.0002843929105438292, -6.305972783593461e-05, -0.07334715127944946, -0.7788305282592773, -0.08500967174768448, -0.002091722097247839, -0.0006211258587427437, -0.007950800471007824, -0.0026016931515187025, -0.00034517052699811757, -0.0007416835287585855, -0.005346167832612991, -1.6080540418624878, -0.6820364594459534, -1.810835838317871, -2.636697769165039, -0.1660851240158081, -0.6825323104858398, -0.5372816920280457, -3.2197835445404053, -2.2614634037017822, -0.08498426526784897, -0.2480018585920334, -0.2158496379852295, -1.7172815799713135, -0.5309000611305237, -2.6364529132843018, -2.6850428581237793, -0.4874305725097656, -0.48716241121292114, -0.07813294976949692, -2.3537583351135254, -2.1851730346679688, -0.38934797048568726, -0.9084161520004272, -0.0012442474253475666, -0.6950001120567322, -0.2627752423286438, -1.3450559377670288, -1.0926463603973389, -0.031344275921583176, -0.7898772358894348, -0.01613534986972809, -0.468288779258728, -0.1359553337097168, -0.2142956256866455, -0.1638282835483551, -0.06685689091682434, -0.5006710290908813, -0.3590526878833771, -0.0006498370785266161, -0.2250765860080719, -0.037441615015268326, -0.09413974732160568, -1.374876856803894, -1.597391747054644e-05, -0.0004068977141287178, -0.0053018201142549515, -0.052733272314071655, -0.12261684983968735]}
+{"group_id": "math_071", "origin": "budgie", "answer": "We can see that for each sum that can be formed, there are four distinct values of $a_i + a_j$. The sums that cannot be formed are $m = 5^5 - 2^5 = 1313$, $m = 4^5 - 3^5 = 2643$, and $m = 3^5 - 2^5 = 19$.\n\nTherefore, the smallest possible value of $m$ is $19$.\n\nThe answer is:\n\\[ \\boxed{19} \\]\nFinal answer:\n#### 19", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 1885, 81, 79, 49160, 28, 81, 79, 49161, 28, 81, 79, 49162, 28, 81, 79, 49163, 28, 81, 79, 49164, 20, 325, 4073, 1345, 2966, 30, 6689, 511, 24053, 282, 260, 910, 1885, 81, 79, 89, 1232, 253, 79, 90, 20, 837, 1885, 89, 28, 90, 3814, 254, 3814, 107, 49160, 28, 49161, 28, 49162, 28, 49163, 28, 49164, 76, 24362, 284, 1885, 89, 3814, 662, 97, 544, 28422, 2959, 1885, 93, 20, 325, 260, 1230, 282, 4073, 2966, 1486, 623, 24053, 30, 1812, 314, 260, 13341, 1636, 1685, 282, 1885, 93, 20, 47, 49154, 49158, 198, 1882, 416, 963, 338, 327, 971, 2028, 338, 416, 325, 4331, 28, 665, 359, 1876, 4073, 2396, 282, 1885, 81, 79, 89, 1232, 253, 79, 90, 28422, 378, 24053, 338, 2526, 325, 4331, 359, 1885, 93, 446, 216, 49164, 78, 49164, 731, 216, 49161, 78, 49164, 446, 216, 49160, 49162, 49160, 49162, 26265, 1885, 93, 446, 216, 49163, 78, 49164, 731, 216, 49162, 78, 49164, 446, 216, 49161, 49165, 49163, 49162, 26265, 284, 1885, 93, 446, 216, 49162, 78, 49164, 731, 216, 49161, 78, 49164, 446, 216, 49160, 49168, 28422, 198, 198, 17308, 28, 260, 13341, 1636, 1685, 282, 1885, 93, 20, 314, 1885, 49160, 49168, 28422, 198, 198, 504, 2988, 314, 42, 198, 76, 75, 3814, 4810, 277, 107, 49160, 49168, 109, 3814, 77, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49168, 49154], "prefix_len": 125, "old_logprobs": [-2.132789134979248, -0.7716965675354004, -3.110560417175293, -0.08593370020389557, -1.987987756729126, -1.212799072265625, -1.5401246547698975, -5.645573616027832, -2.176344156265259, -0.009319375269114971, -0.32650235295295715, -0.608310341835022, -0.6677852869033813, -0.8955617547035217, -3.0503954887390137, -3.1945087909698486, -2.2192115783691406, -1.2678277492523193, -0.269671767950058, -0.0447160005569458, -0.016970185562968254, -0.08405803889036179, -0.1586689054965973, -0.0003182381624355912, -0.0004320403386373073, -0.0002474478678777814, -0.5418604016304016, -4.506051063537598, -2.9172840118408203, -0.6948451995849609, -2.217109441757202, -0.07299104332923889, -0.005205648019909859, -0.200049489736557, -1.6507158279418945, -3.159205913543701, -0.4549645483493805, -0.08387389779090881, -2.1496431827545166, -5.5863938331604, -1.2474703788757324, -0.7381654381752014, -0.08777265250682831, -2.287095546722412, -1.9305610656738281, -0.03940016031265259, -0.6807305812835693, -0.01818905957043171, -1.0829941034317017, -2.0865182876586914, -0.5093056559562683, -2.0061280727386475, -2.0728724002838135, -0.2748419940471649, -0.0501222163438797, -0.009834646247327328, -0.013796914368867874, -1.8849923610687256, -0.012428924441337585, -0.042379654943943024, -0.01973750814795494, -0.0021955687552690506, -0.18279553949832916, -0.0005368936690501869, -0.0004861365014221519, -0.03184312954545021, -0.0017273995326831937, -1.7331116199493408, -1.6780314445495605, -1.2822973728179932, -1.815400242805481, -0.17102473974227905, -0.7035549879074097, -0.019885700196027756, -0.001892800792120397, -0.00562560698017478, -0.005717589985579252, -0.6212297677993774, -0.0024126251228153706, -0.00047565114800818264, -0.00812794640660286, -0.001073380233719945, -0.32874810695648193, -0.0022280411794781685, -0.00567491864785552, -0.03639727830886841, -0.00326980440877378, -0.5756306052207947, -1.5491572618484497, -2.159425735473633, -1.0111730098724365, -0.0787346214056015, -1.1382776498794556, -0.003087517572566867, -0.16770271956920624, -0.1627691239118576, -0.0944795086979866, -0.038083530962467194, -0.0018857807153835893, -0.011094783432781696, -0.0009495278936810791, -0.0073648616671562195, -0.005330871790647507, -1.074412226676941, -0.3970789611339569, -1.2125120162963867, -0.5068919658660889, -0.030533038079738617, -0.06074623391032219, -0.5647497177124023, -0.3607828617095947, -0.0003499372396618128, -0.13594096899032593, -0.03238224238157272, -0.15164881944656372, -0.42200228571891785, -0.008243818767368793, -8.332382276421413e-05, -1.1920928244535389e-07, -4.184158387943171e-05, -0.003632140113040805, -0.0011609011562541127, -9.07141511561349e-05, -0.00021979777375236154, -0.00038818930624984205, -1.065896987915039, -0.315780907869339, -0.004167087841778994, -0.004666865803301334, -0.00954858772456646, -0.02962358482182026, -0.0031815171241760254, -7.629103492945433e-05, -3.528532761265524e-05, -8.225102646974847e-05]}
+{"group_id": "math_071", "origin": "verified_reference", "answer": "1. **Define the problem and the sums:**\n   We are given five distinct real numbers \\(a_1, a_2, a_3, a_4, a_5\\). We need to consider all possible sums of the form \\(a_i + a_j\\) where \\(i \\neq j\\). There are \\(\\binom{5}{2} = 10\\) such sums since each pair \\((i, j)\\) is considered once.\n\n2. **Calculate the number of distinct sums:**\n   We need to find the smallest possible number of distinct sums, denoted as \\(m\\).\n\n3. **Example with specific values:**\n   Let's try the values \\(a_1 = 0, a_2 = 1, a_3 = 2, a_4 = 3, a_5 = 4\\). The sums are:\n   \\[\n   \\begin{aligned}\n   &0+1 = 1, \\\\\n   &0+2 = 2, \\\\\n   &0+3 = 3, \\\\\n   &0+4 = 4, \\\\\n   &1+2 = 3, \\\\\n   &1+3 = 4, \\\\\n   &1+4 = 5, \\\\\n   &2+3 = 5, \\\\\n   &2+4 = 6, \\\\\n   &3+4 = 7.\n   \\end{aligned}\n   \\]\n   The distinct sums are \\{1, 2, 3, 4, 5, 6, 7\\}, so \\(m = 7\\).\n\n4. **Check if \\(m = 6\\) is possible:**\n   Suppose \\(a_1 = 0\\) (without loss of generality, since adding a constant to each \\(a_i\\) does not change the number of distinct sums). Let the remaining numbers be \\(a, b, c, d\\) in increasing order. The sums are:\n   \\[\n   \\begin{aligned}\n   &0 + a = a, \\\\\n   &0 + b = b, \\\\\n   &0 + c = c, \\\\\n   &0 + d = d, \\\\\n   &a + b, \\\\\n   &a + c, \\\\\n   &a + d, \\\\\n   &b + c, \\\\\n   &b + d, \\\\\n   &c + d.\n   \\end{aligned}\n   \\]\n   For \\(m = 6\\), we need some sums to be equal. Assume \\(a + b = d\\). Then the sums are:\n   \\[\n   \\{a, b, c, d, a + c, a + d, b + c, b + d, c + d\\}.\n   \\]\n   Since \\(a + b = d\\), we have:\n   \\[\n   \\{a, b, c, d, a + c, d + a, b + c, b + d, c + d\\}.\n   \\]\n   We need to check if we can have only 6 distinct sums. If \\(a + c = b + d\\), then \\(a + c = b + (a + b) = a + 2b\\), so \\(c = 2b\\). This leads to contradictions as we need all numbers to be distinct.\n\n5. **Conclusion:**\n   After trying different configurations and ensuring all numbers are distinct, we find that the smallest possible value of \\(m\\) is indeed 7.\n\nThe final answer is \\(\\boxed{7}\\).", "reward": 1.0, "advantage": 1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 1885, 81, 79, 49160, 28, 81, 79, 49161, 28, 81, 79, 49162, 28, 81, 79, 49163, 28, 81, 79, 49164, 20, 325, 4073, 1345, 2966, 30, 6689, 511, 24053, 282, 260, 910, 1885, 81, 79, 89, 1232, 253, 79, 90, 20, 837, 1885, 89, 28, 90, 3814, 254, 3814, 107, 49160, 28, 49161, 28, 49162, 28, 49163, 28, 49164, 76, 24362, 284, 1885, 89, 3814, 662, 97, 544, 28422, 2959, 1885, 93, 20, 325, 260, 1230, 282, 4073, 2966, 1486, 623, 24053, 30, 1812, 314, 260, 13341, 1636, 1685, 282, 1885, 93, 20, 47, 49154, 49158, 198, 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+{"group_id": "math_072", "origin": "budgie", "answer": "This is a problem of similarity transformation.\n\nSince $B$ is the midpoint of the side, we know $BC=BP\\cdot \\frac{1}{2}$\n\nThis means that $\\frac{BC}{BC}=\\frac{BP}{2}$ so $BP=2$\n\nTherefore, $A_1B_1=2\\cdot AB$, so $AB^2=4$\n\nTherefore, the area of triangle $ABC_1$ is $S_{ABC_1}=S_{ABC} \\cdot AB = 5\\cdot 2= \\boxed{10}$\n\nFinal answer:\n#### 10", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 10887, 253, 32857, 20378, 13293, 1885, 5431, 6998, 20, 351, 1557, 1885, 64, 28422, 1046, 8290, 2103, 1885, 5431, 20, 3070, 1885, 50, 20, 288, 1885, 49, 79, 49160, 20, 715, 338, 19573, 1400, 1311, 107, 5431, 109, 24814, 1400, 1311, 107, 10833, 79, 49160, 24362, 28, 965, 1885, 6190, 20, 3070, 1885, 51, 20, 288, 1885, 50, 79, 49160, 20, 715, 338, 19573, 1400, 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+{"group_id": "math_072", "origin": "budgie", "answer": "Let $x$ be the area of quadrilateral $A_1B_1C_1D_1$.\n\nWe can express $x$ in terms of $P$ and the area of quadrilateral $ABCD$:\n\n$P = \\frac{1}{4}AB \\cdot BC \\cdot CD \\cdot DA$\n\nSince $AB = DC$, $BC = CD$, $BC = \\frac{1}{2}BC$, $CD = \\frac{1}{2}CD$, $DA = DC$, we have:\n\n$x = \\frac{1}{4}(AB)(BC)(CD)(DA)$\n\nSince $AB = DC$ and $BC = CD$, we have:\n\n$x = \\frac{1}{4}(AB)(BC)(CD) \\cdot BC = \\frac{1}{4}(AB)(BC) \\cdot CD \\cdot BC$\n\nSince $BC = \\frac{1}{2}BC$, we have:\n\n$x = \\frac{1}{4}(AB)(BC) \\cdot CD \\cdot BC = \\frac{1}{4}(AB)(AC) \\cdot CD \\cdot BC$\n\nSince $AB = DC$ and $AC = AD$, we have:\n\n$x = \\frac{1}{4}(DC)(AC) \\cdot CD \\cdot AC = \\frac{1}{4}(DC)(AD) \\cdot CD \\cdot AD$\n\nSince $DC = DC$, we have:\n\n$x = \\frac{1}{4}(DC)(AD) \\cdot DC = \\frac{1}{4}(DC)(AD) \\cdot AD = \\frac{1}{4}(AD)(DC) \\cdot AD$\n\nSince $DA = DA$, we have:", "reward": -1, "advantage": -0.9805795218461241, "ids": 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+{"group_id": "math_072", "origin": "verified_reference", "answer": "Solution. We connect $A$ with $C_{1}$, $B$ with $D_{1}$, $C$ with $A_{1}$, and $D$ with $B_{1}$. Then,\n\n$$\n\\begin{aligned}\n& P_{\\triangle A B C}=P_{I}=P_{I I} \\\\\n& P_{\\triangle B C D}=P_{I I I}=P_{I V} \\\\\n& P_{\\triangle C D A}=P_{V}=P_{V I} \\\\\n& P_{\\triangle D A B}=P_{V I I}=P_{I I I}\n\\end{aligned}\n$$\n\nwhere\n\n$$\n\\begin{array}{ll}\nP_{I}=P_{\\Delta A_{1} C}, & P_{I I}=P_{\\Delta C A_{1} B_{1}} \\\\\nP_{I I I}=P_{\\Delta D C B_{1}}, & P_{I V}=P_{\\Delta C_{1} D B_{1}} \\\\\nP_{V}=P_{\\Delta C_{1} D A}, & P_{V I}=P_{\\Delta C_{1} D_{1} A} \\\\\nP_{I I I}=P_{\\Delta A D_{1} B}, & P_{V I I I}=P_{\\Delta B D_{1} A_{1}}\n\\end{array}\n$$\n\n![](https://cdn.mathpix.com/cropped/2024_06_05_ea01241e897ed9d0948dg-02.jpg?height=507&width=641&top_left_y=708&top_left_x=838)\n\nWe have:\n\n$$\n\\begin{aligned}\nP_{\\Delta A B C}+P_{\\Delta B C D}+P_{\\Delta C D A}+P_{\\Delta D A B} & =P_{I}+P_{I I I}+P_{V}+P_{I I}= \\\\\n& =P_{I I}+P_{I V}+P_{V I}+P_{V I I I}=2 P \\\\\nP_{1}=P+\\left(P_{I}+P_{I I}+P_{I I I}+P_{I V}+P_{V}+P_{I I}\\right. & \\left.+P_{V I I}+P_{I I 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+{"group_id": "math_073", "origin": "budgie", "answer": "To find $f(x)$, we need to substitute $f(x) = 4x^3 - \\frac{5}{x^7} - 8 \\sqrt[6]{x^5} + 10$ into the given equation.\n\nLet's denote $g(x) = f(x) - 10$.\n\nNow, let's find $g(x)$ by substituting $f(x)$ into $g(x) = g(x-1) + 10$.\n\nWe have:\n\n$$g(x) = g(x-1) + 10$$\n\nNow, we can substitute $g(x)$ in terms of $f(x)$ into the given equation $f(x) = f(x-1) - 10$.\n\nLet's denote $x = g(x)$ and $x = x-g(x)$. Then, we have:\n\n$$g(x) = g(x-1) + 10$$\n\nSince $x = g(x)$, we have:\n\n$$x = g(x-1) + 10$$\n\nNow, we can substitute $g(x-1)$ for $x$ in the equation $g(x) = g(x-1) + 10$ to get:\n\n$$g(x) - 10 = g(x-1)$$\n\nNow, we can solve this equation for $g(x)$.\n\nNotice that $g(x) = g(x-1)$ for all $x$. Let's set $x = 0$ in the equation to find $g(0)$:\n\n$$g(0) = g(0-1", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49165, 30, 49166, 30, 1885, 86, 24, 104, 36637, 49163, 1792, 21990, 49162, 39949, 76, 18169, 107, 49164, 15009, 104, 21990, 49166, 17859, 29, 49167, 3814, 16166, 75, 49165, 36939, 104, 21990, 49164, 17859, 27, 49160, 49159, 28422, 1116, 49165, 30, 49166, 30, 1885, 86, 24, 104, 36637, 49163, 1792, 21990, 49162, 39949, 76, 18169, 107, 49164, 15009, 104, 21990, 49166, 17859, 29, 49167, 3814, 16166, 75, 49165, 36939, 104, 21990, 49164, 17859, 27, 49160, 49159, 28422, 49154, 49158, 198, 2068, 1042, 1885, 86, 24, 104, 25, 26265, 392, 737, 288, 13649, 1885, 86, 24, 104, 25, 446, 216, 49163, 104, 78, 49162, 731, 3814, 18169, 107, 49164, 15009, 104, 78, 49166, 109, 731, 216, 49167, 3814, 16166, 75, 49165, 36939, 104, 78, 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+{"group_id": "math_073", "origin": "budgie", "answer": "The given expression is $f(x) = 4 x^{3} - 5x^{-7} - 8 x^{5} \\sqrt[6]{x^{5}} + 10$.\n\nTo simplify $f(x)$, we can substitute the fractional exponents into the equation.\n\n\\[ f(x) = 4 x^{3} - 5 x^{-7} - 8 x^{5} \\sqrt[6]{x^{5}} + 10 \\]\n\nNotice that we can factor out $x^{-15}$ from the terms involving $x^{5}$.\n\n\\[ f(x) = x^{-15} (4 x^{3} - 5 x^{-7}) + 10 \\]\n\nNow, we can see that this is a hyperbolic equation.\n\nThe general solution is $f(x) = A(x - \\lambda)^{1/2}$, where $\\lambda = \\pm 1$.\n\nSubstituting $\\lambda = 1$ into the equation, we get $f(x) = A(x - 1)^{1/2}$.\n\nTo find the value of $A$, we can use the initial condition $f(0) = 1$.\n\n\\[ f(0) = 1 \\Rightarrow A \\cdot 0^{1/2} = 1 \\Rightarrow A = 1 \\]\n\nSo, the function $f(x)$ can be written as $f(x) = 1(x - 1)^{1/2}$.\n\nThe final answer is: $\\boxed{(x - 1)^{1/2}}$", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 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+{"group_id": "math_074", "origin": "budgie", "answer": "Let's analyze the given inequalities and eliminate the variables $a, b, c, d$ step by step.\n\n1. $a<2b$\n2. $b<3c$\n3. $c<4d$\n\nWe can rewrite these inequalities as follows:\n\n* $a < 2b$\n* $b < 3c$\n* $c < 4d$\n\nNow, let's move to the product of these inequalities:\n\n$$a\\cdot b\\cdot c\\cdot d > (2b)(3c)(4d)$$\n\nNotice that $2b \\cdot 3c \\cdot 4d = 24abc$ is the product of the fourth powers of the three numbers, so:\n\n$$a\\cdot b\\cdot c\\cdot d > 24abc$$\n\nWe can rewrite this as:\n\n$$a < 24abc / 24 = abc$$\n\nSince the product of the three numbers is less than the product of their fifth powers, we can write:\n\n$$abc < (abc)^5$$\n\nTaking the fifth root of both sides:\n\n$$abc < abc$$\n\nThis inequality is always true.\n\nHowever, we know that the product of the numbers is greater than 130000. Since the product of the numbers is less than 130000, we can choose the values of $a, b, c, d$ such that the product of these three numbers is greater than 130000.\n\nLet's choose $a = 130000 - (3c)^5$ and $b = 130000 - 3c^5$.\n\nThen, we", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 2959, 1885, 81, 28, 278, 28, 265, 28, 287, 20, 325, 23313, 28, 284, 1885, 81, 44, 49161, 82, 28, 278, 44, 49162, 83, 28, 265, 44, 49163, 84, 28422, 10887, 1885, 84, 44, 49160, 49159, 49159, 26265, 965, 260, 5428, 1636, 1685, 282, 1885, 81, 20, 314, 365, 15165, 198, 24, 49, 25, 216, 49161, 49162, 49165, 49166, 198, 24, 50, 25, 216, 49161, 49162, 49166, 49164, 198, 24, 51, 25, 216, 49161, 49162, 49168, 49160, 198, 24, 52, 25, 216, 49161, 49162, 49168, 49168, 49154, 49158, 198, 4239, 506, 6524, 260, 1836, 21544, 284, 8534, 260, 6031, 1885, 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+{"group_id": "math_074", "origin": "budgie", "answer": "Considering the inequality $a<2b, b<3c, c<4d$, we can write the inequalities for each variable:\n\n$$a<2b, b<3c, c<4d$$\n\nSince $d<100$, we know that $0<d<100$.\n\nFor $c$, the inequality becomes $c<4d$.\n\nFor $b$, the second inequality becomes $b<3c$.\n\nFor $a$, the first inequality becomes $a<2b$.\n\nGiven that $a<2b, b<3c, c<4d$, we can simplify these inequalities to find the range of possible values for $a$.\n\nSince $0<d<100$, $d$ can take values from 0 to 99.\n\nFor $c$, the lowest inequality is $c<4d$. Since $d<100$, this means $c<4*100=400$.\n\nFor $b$, the second inequality becomes $b<3c$. This implies that $b<3*400=1200$.\n\nFor $a$, the first inequality becomes $a<2b$. Given that $0<d<100$, this means $a<2*400=800$.\n\nTo maximize $a$, we need to minimize $b$ and $c$ while keeping $a<800$. The smallest possible value for $a$ is $800$.\n\nTherefore, the maximum possible value of $a$ is 800.\n\nAnswer:\n#### 800", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 2959, 1885, 81, 28, 278, 28, 265, 28, 287, 20, 325, 23313, 28, 284, 1885, 81, 44, 49161, 82, 28, 278, 44, 49162, 83, 28, 265, 44, 49163, 84, 28422, 10887, 1885, 84, 44, 49160, 49159, 49159, 26265, 965, 260, 5428, 1636, 1685, 282, 1885, 81, 20, 314, 365, 15165, 198, 24, 49, 25, 216, 49161, 49162, 49165, 49166, 198, 24, 50, 25, 216, 49161, 49162, 49166, 49164, 198, 24, 51, 25, 216, 49161, 49162, 49168, 49160, 198, 24, 52, 25, 216, 49161, 49162, 49168, 49168, 49154, 49158, 198, 39539, 260, 9724, 1885, 81, 44, 49161, 82, 28, 278, 44, 49162, 83, 28, 265, 44, 49163, 84, 26265, 392, 416, 2965, 260, 21544, 327, 971, 5811, 42, 198, 198, 9212, 81, 44, 49161, 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+{"group_id": "math_074", "origin": "verified_reference", "answer": "$-.1 .(\\mathrm{A})$.\nThen, if $d=99$, we have $c<4 \\times 99=395$. Taking $c=395$, we get $b<3 \\times 395=1185$. Taking $b=1184$, we get $a<2368$. Therefore, the greatest integer for $a$ is 2367.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 2959, 1885, 81, 28, 278, 28, 265, 28, 287, 20, 325, 23313, 28, 284, 1885, 81, 44, 49161, 82, 28, 278, 44, 49162, 83, 28, 265, 44, 49163, 84, 28422, 10887, 1885, 84, 44, 49160, 49159, 49159, 26265, 965, 260, 5428, 1636, 1685, 282, 1885, 81, 20, 314, 365, 15165, 198, 24, 49, 25, 216, 49161, 49162, 49165, 49166, 198, 24, 50, 25, 216, 49161, 49162, 49166, 49164, 198, 24, 51, 25, 216, 49161, 49162, 49168, 49160, 198, 24, 52, 25, 216, 49161, 49162, 49168, 49168, 49154, 49158, 198, 20, 43526, 49160, 1673, 19407, 8898, 5167, 107, 49, 8148, 28422, 198, 10039, 28, 585, 1885, 84, 45, 49168, 49168, 26265, 392, 457, 1885, 83, 44, 49163, 3814, 14602, 216, 49168, 49168, 45, 49162, 49168, 49164, 28422, 14985, 1885, 83, 45, 49162, 49168, 49164, 26265, 392, 820, 1885, 82, 44, 49162, 3814, 14602, 216, 49162, 49168, 49164, 45, 49160, 49160, 49167, 49164, 28422, 14985, 1885, 82, 45, 49160, 49160, 49167, 49163, 26265, 392, 820, 1885, 81, 44, 49161, 49162, 49165, 49167, 28422, 4882, 28, 260, 6250, 15328, 327, 1885, 81, 20, 314, 216, 49161, 49162, 49165, 49166, 30, 49154], "prefix_len": 117, "old_logprobs": [-6.425846099853516, -16.364072799682617, -2.287951946258545, -8.431763648986816, -8.98987102508545, -5.425016403198242, -0.5020139813423157, -0.0022081772331148386, -4.566595554351807, -2.083723783493042, -2.1881768703460693, -0.9943889379501343, -8.527412414550781, -1.5335103273391724, -4.344038963317871, -0.3912464678287506, -1.526413917541504, -2.8187415599823, -3.6707684993743896, -0.1920979917049408, -1.0199874639511108, -2.6167168617248535, -0.4177727699279785, -0.23391488194465637, -2.704787015914917, -2.0005509853363037, -1.717228889465332, -1.7626054286956787, -0.09881271421909332, -0.1463354527950287, -0.0624006912112236, -0.000740968796890229, -0.8555609583854675, -0.17192263901233673, -0.013412484899163246, -16.31250762939453, -0.8968100547790527, -6.658372402191162, -0.5423813462257385, -1.5255768299102783, -0.2701251804828644, -0.5963861346244812, -0.18563984334468842, -0.5782619118690491, -0.2764997184276581, -0.48854631185531616, -0.8883193731307983, -0.1381218135356903, -2.384387254714966, -0.23763535916805267, -0.5992197394371033, -0.4430578649044037, -0.0012672259472310543, -0.009162403643131256, -0.02776312083005905, -0.0010917185572907329, -0.00099323526956141, -0.14730484783649445, -0.007869075983762741, -0.024035966023802757, -0.09253354370594025, -0.0010805251076817513, -0.3366527259349823, -1.168932557106018, -0.005158448126167059, -0.0667290985584259, -0.01810126006603241, -0.014152283780276775, -0.0058921691961586475, -0.00818600319325924, -4.64556884765625, -0.18131357431411743, -0.05474394932389259, -0.9498416185379028, -0.03900183364748955, -0.21976259350776672, -0.11610379815101624, -0.3498428761959076, -4.435967445373535, -3.406467914581299, -1.9928441047668457, -0.5053212642669678, -2.519426107406616, -0.029143076390028, -0.2695752680301666, -5.4710001945495605, -0.695104718208313, -5.377488613128662, -0.2628493905067444, -0.04393276572227478, -0.03003700263798237, -0.10434316843748093, -0.3839997947216034, -0.02400093711912632, -0.0019070786656811833, -0.5576317310333252, -0.4087328612804413, -0.13717380166053772, -5.950602054595947]}
+{"group_id": "math_075", "origin": "budgie", "answer": "32. Let $f(x)=16 \\cdot 5^{2 x-1}-2 \\cdot 5^{x-1}-0.048$. When $x=0$, $f(0)=0$.\n\nAnswer:\n#### 0", "reward": 1.0, "advantage": 0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49161, 49163, 49161, 30, 19573, 8842, 24, 49160, 49165, 3814, 41591, 216, 49164, 21990, 49161, 1792, 29, 49160, 39949, 49161, 3814, 41591, 216, 49164, 21990, 104, 29, 49160, 39949, 49159, 30, 49159, 49163, 49167, 76, 2771, 25, 3814, 92, 87, 3814, 8842, 24, 104, 21990, 49162, 109, 27, 49161, 1792, 27, 49160, 76, 2771, 36637, 49159, 28422, 49154, 49158, 198, 49162, 49161, 30, 2959, 1885, 86, 24, 104, 36637, 49160, 49165, 3814, 41591, 216, 49164, 21990, 49161, 1792, 29, 49160, 39949, 49161, 3814, 41591, 216, 49164, 21990, 104, 29, 49160, 39949, 49159, 30, 49159, 49163, 49167, 28422, 1550, 1885, 104, 45, 49159, 26265, 1885, 86, 24, 49159, 36637, 49159, 28422, 198, 198, 21350, 42, 198, 1229, 216, 49159, 49154], "prefix_len": 88, "old_logprobs": [-3.4670050144195557, -2.1015844345092773, -1.198379635810852, -3.9908227920532227, -0.728891134262085, -2.5934014320373535, -0.05405392125248909, -0.06436410546302795, -0.3244601786136627, -1.6596819162368774, -0.024508466944098473, -1.2133287191390991, -0.07153483480215073, -0.0489482581615448, -0.006597996223717928, -0.010280866175889969, -0.05378735437989235, -1.0601204633712769, -0.03440750017762184, -0.0028884378261864185, -0.08698343485593796, -0.019465375691652298, -0.014873212203383446, -0.000764792668633163, -0.0017982041463255882, -0.00010775939153973013, -0.0003587556129787117, -0.004680866841226816, -0.001304609584622085, -1.2874520507466514e-05, -0.009680116549134254, -0.06938262283802032, -0.00025829317746683955, -0.01409986149519682, -0.0011800715001299977, -0.00047803416964598, -1.0689234733581543, -4.980095863342285, -0.08800455182790756, -0.3799773156642914, -0.6907307505607605, -1.1965309381484985, -0.4314729571342468, -0.18463991582393646, -0.009664060547947884, -0.04461681842803955, -0.5420650839805603, -0.2076900601387024, -0.36814284324645996, -0.45088624954223633, -0.6046979427337646, -0.10241943597793579, -4.209035396575928, -0.043070707470178604, -0.2714500427246094, -0.1879296451807022, -0.10618666559457779, -0.18912285566329956, -0.026781858876347542]}
+{"group_id": "math_075", "origin": "budgie", "answer": "1024.\n\nFinal answer:\n#### 1024", "reward": -0.5, "advantage": -1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49161, 49163, 49161, 30, 19573, 8842, 24, 49160, 49165, 3814, 41591, 216, 49164, 21990, 49161, 1792, 29, 49160, 39949, 49161, 3814, 41591, 216, 49164, 21990, 104, 29, 49160, 39949, 49159, 30, 49159, 49163, 49167, 76, 2771, 25, 3814, 92, 87, 3814, 8842, 24, 104, 21990, 49162, 109, 27, 49161, 1792, 27, 49160, 76, 2771, 36637, 49159, 28422, 49154, 49158, 198, 49160, 49159, 49161, 49163, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49159, 49161, 49163, 49154], "prefix_len": 88, "old_logprobs": [-1.2940069437026978, -1.8934171199798584, -1.748826503753662, -0.637182354927063, -0.7875847220420837, -2.3630528450012207, -0.038016509264707565, -0.4355299174785614, -0.002022841712459922, -0.002695143921300769, -0.16291376948356628, -0.005939097609370947, -0.038427162915468216, -0.24012304842472076, -0.0016104121459648013, -0.001712642959319055, -0.0010882653295993805, -0.005151806864887476]}
+{"group_id": "math_075", "origin": "verified_reference", "answer": "Solution.\n\nDomain: $x^{3}+2 x+1>0$.\n\nFrom the condition $16 \\cdot 5^{2 x-1}-2^{x-1}-0.048=0$ or $\\lg \\left(x^{3}+2 x+1\\right)=0$. Rewrite the first equation as\n\n$\\frac{16}{5} \\cdot 5^{2 x}-\\frac{2}{5} \\cdot 5^{x}-0.048=0 \\Leftrightarrow 16 \\cdot 5^{2 x}-2 \\cdot 5^{x}-0.24=0$.\n\nSolving this equation as a quadratic in terms of $5^{x}$, we get $5^{x}=-\\frac{3}{40}$ (no solutions), or $5^{x}=5^{-1} \\Leftrightarrow x_{1}=-1$ (does not satisfy the domain). From the second equation, we have\n\n$x^{3}+2 x+1=1 \\Leftrightarrow x^{3}+2 x=0 \\Leftrightarrow x\\left(x^{2}+2\\right)=0, x_{3}=0, x^{2}+2 \\neq 0$.\n\nAnswer: 0.", "reward": 1.0, "advantage": 0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49161, 49163, 49161, 30, 19573, 8842, 24, 49160, 49165, 3814, 41591, 216, 49164, 21990, 49161, 1792, 29, 49160, 39949, 49161, 3814, 41591, 216, 49164, 21990, 104, 29, 49160, 39949, 49159, 30, 49159, 49163, 49167, 76, 2771, 25, 3814, 92, 87, 3814, 8842, 24, 104, 21990, 49162, 109, 27, 49161, 1792, 27, 49160, 76, 2771, 36637, 49159, 28422, 49154, 49158, 198, 37500, 30, 198, 198, 34804, 42, 1885, 104, 21990, 49162, 109, 27, 49161, 1792, 27, 49160, 46, 49159, 28422, 198, 198, 5212, 260, 2619, 1885, 49160, 49165, 3814, 41591, 216, 49164, 21990, 49161, 1792, 29, 49160, 39949, 49161, 21990, 104, 29, 49160, 39949, 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+{"group_id": "math_077", "origin": "budgie", "answer": "First, we notice that the given equation is always true regardless of the values of $x$ and $y$. However, the question asks us to find the minimum value of the function $u = y^{2} - 2y + x^{2} + 4x$ when the inequality $2x + y \\geqslant 1$ holds. To do this, we can rewrite the given equation as follows:\n\n$$u = (y^{2} - 2y + x^{2} + 4x) + (2y - 2x) = (y + x)^{2} - x^{2}$$\n\nNow, let's use the Cauchy-Schwarz inequality to find the minimum value of $u$:\n\n$$(y + x)^{2} \\leq 1 + 1 = 2$$\n\nTherefore, the minimum value of $u$ is:\n\n$$u = (y + x)^{2} - x^{2} = 2 - x^{2}$$\n\nThe minimum value of the function $u$ is 2 - x².\n\nFinal answer:\n#### 2", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 2464, 30, 365, 49161, 49164, 2876, 25, 1094, 1885, 49161, 1792, 27, 105, 3814, 361, 97, 7483, 403, 216, 49160, 28422, 8061, 288, 1042, 260, 5869, 1685, 282, 260, 1517, 1885, 101, 45, 105, 21990, 49161, 39949, 49161, 329, 20, 1885, 27, 104, 21990, 49161, 109, 27, 49163, 1792, 28422, 49154, 49158, 198, 5345, 28, 392, 4887, 338, 260, 1836, 8345, 314, 1811, 2629, 7142, 282, 260, 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+{"group_id": "math_077", "origin": "budgie", "answer": "25. (25 points)\n\n**Analysis**\n\nThis problem can be solved by completing the square for both the $y$-terms and the $x$-terms.\n\nFirst, rewrite the function to express it in vertex form, then complete the square for the $y$-terms.\n\n**Step-by-step Solution**\n\nThe function $y^2 - 2y$ can be rewritten as $(y - 1)^2 - 1$.\n\nThe function $x^2 + 4x$ can be rewritten as $(x + 2)^2 - 4$.\n\nNow, we complete the square for the $y$-terms:\n\n$$u = (y - 1)^2 - 1$$\n\n$$u = (y + (1) - 1)^2 - 1$$\n\n$$u = (y + 1 - 1)^2 - 1$$\n\n$$u = (y + 1 - 1)^2 - 1$$\n\n$$u = (y + 1 - 1)^2 - 1$$\n\n$$u = (y + 1 - 1 - 0.5)^2 - 1$$\n\n$$u = (y - 0.5)^2 - 1$$\n\n$$u = (y - 0.5)^2 - 1$$\n\nNow, we write the function in vertex form by completing the square:\n\n$$u = (y + 0.5)^2 - 1$$\n\n$$u = (y - 0.5)^2 - 1$$\n\nThe given inequality $2x + y \\geqslant 1$ can be written as:\n\n$$2x + y -", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 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+{"group_id": "math_077", "origin": "verified_reference", "answer": "$\\therefore$ From $u=y^{2}-2 y+x^{2}+4 x$ completing the square, we get\n$$\n(x+2)^{2}+(y-1)^{2}=u+5 \\text {. }\n$$\n\nSince $(x+2)^{2}+(y-1)^{2}$ can be regarded as the square of the distance from point $P(x, y)$ to the fixed point $(-2,1)$. The constraint $2 x+y \\geqslant 1$ indicates that point $P$ is in the region $G$ which is the right upper area of the line $l: 2 x+y=1$. Therefore, the problem is transformed into finding the shortest distance from point $A(-2,1)$ to the region $G$.\nFrom the right figure, it can be seen that the distance from point $A$ to\n\nline $I$ is the minimum value of the distance from point $A$ to any point in region $G$.\n$$\n\\begin{aligned}\nd & =\\frac{|2 \\cdot(-2)+1-1|}{\\sqrt{2^{2}+1^{2}}} \\\\\n& =\\frac{4}{\\sqrt{5}}, \\\\\n& \\therefore d^{2}=\\frac{16}{5} \\text {, i.e., } u+5=\\frac{16}{5}\n\\end{aligned}\n$$\n$\\therefore u=-\\frac{9}{5}$. 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+{"group_id": "math_078", "origin": "budgie", "answer": "To solve this problem, we need to find the maximum number of red pencils (80 kopecks) and blue pencils (50 kopecks) that can be bought, while ensuring that the number of red pencils is as few as possible, but the number of blue pencils is as large as possible.\n\nSince the total cost of a red pencil is 27 kopecks, a blue pencil costs 23 kopecks, we will first find the maximum number of red pencils that can be bought with the given budget. This can be done by dividing the total budget by the cost of a red pencil.\n\n$$Maximum\\, red\\, pencils = Total\\, budget / Cost\\, of\\, a\\, red\\, pencil$$\n\nNow, plug in the given values for the total budget and cost of a red pencil to calculate the maximum number of red pencils.\n\n$$Maximum\\, red\\, pencils = 40 / 27$$\n\n$$Maximum\\, red\\, pencils = 1\\, (\\text{exactly 1 red pencil})$$\n\nSince the number of red pencils should be as few as possible, we should use the smallest possible value for the number of red pencils. Therefore, we should consider the case where the number of red pencils is equal to the number of red pencils.\n\n$$Maximum\\, red\\, pencils = 1\\, (\\text{one red pencil})$$\n\nNow that we have the maximum number of red pencils, we can calculate the number of blue pencils that can be bought. The cost of a blue pencil is 23 kopecks, so we can find the maximum number of blue pencils by dividing the total budget by the cost of a blue pencil.\n\n$$Maximum\\, blue\\, pencils = Total\\,", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 49163, 49159, 330, 2382, 15779, 3499, 216, 49161, 49166, 501, 371, 41907, 28, 253, 4461, 582, 731, 216, 49161, 49162, 501, 371, 41907, 30, 2838, 540, 670, 216, 49168, 6460, 735, 216, 49163, 49159, 501, 371, 41907, 416, 325, 4957, 335, 12217, 25487, 30, 657, 314, 2371, 288, 7131, 260, 5428, 1636, 2719, 1230, 282, 2382, 284, 4461, 25487, 30, 1814, 260, 1142, 655, 28, 260, 1230, 282, 2382, 25487, 868, 325, 347, 1443, 347, 1636, 28, 564, 260, 1230, 282, 4461, 25487, 868, 441, 9094, 429, 260, 1230, 282, 2382, 25487, 411, 540, 670, 216, 49160, 49159, 30, 1073, 800, 2382, 284, 638, 800, 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+{"group_id": "math_078", "origin": "budgie", "answer": "The problem can be solved by how many red pencils and blue pencils should be purchased under the conditions of the problem.\n\nFirst, we should consider the total cost. The cost of a red pencil is 27 kopecks and the cost of a blue one is 23 kopecks.\n\nThe number of red pencils purchased is max(27, 23) = 49.\n\nSince the number of red pencils should be as few as possible, the number of blue pencils purchased is 27 - 10 = 17.\n\nTherefore, under the given conditions, we should purchase 49 red pencils and 17 blue pencils.\n\nFinal answer:\n#### 49", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 49163, 49159, 330, 2382, 15779, 3499, 216, 49161, 49166, 501, 371, 41907, 28, 253, 4461, 582, 731, 216, 49161, 49162, 501, 371, 41907, 30, 2838, 540, 670, 216, 49168, 6460, 735, 216, 49163, 49159, 501, 371, 41907, 416, 325, 4957, 335, 12217, 25487, 30, 657, 314, 2371, 288, 7131, 260, 5428, 1636, 2719, 1230, 282, 2382, 284, 4461, 25487, 30, 1814, 260, 1142, 655, 28, 260, 1230, 282, 2382, 25487, 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+{"group_id": "math_078", "origin": "verified_reference", "answer": "3.40 Let $x$ red and $y$ blue pencils be bought. According to the conditions, $27 x+23 y \\leqslant 940$ and $y-x \\leqslant 10$. Let's construct the lines\n\n![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-108.jpg?height=597&width=625&top_left_y=113&top_left_x=287)\n\n\\[\n\\begin{aligned}\n& 27 x+23 y=940 \\\\\n& y-x=10\n\\end{aligned}\n\\]\n\nFrom Fig. 3.8, it is clear that these lines intersect at point $A$, the coordinates of which satisfy equations (1) and (2), and at this point, the maximum possible sum $x+y$ is achieved. Solving the system (1), (2) and considering that the numbers $x$ and $y$ are natural, we get $x=14, y=24$.\n\nAnswer: 14 red and 24 blue pencils.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 30, 49163, 49159, 330, 2382, 15779, 3499, 216, 49161, 49166, 501, 371, 41907, 28, 253, 4461, 582, 731, 216, 49161, 49162, 501, 371, 41907, 30, 2838, 540, 670, 216, 49168, 6460, 735, 216, 49163, 49159, 501, 371, 41907, 416, 325, 4957, 335, 12217, 25487, 30, 657, 314, 2371, 288, 7131, 260, 5428, 1636, 2719, 1230, 282, 2382, 284, 4461, 25487, 30, 1814, 260, 1142, 655, 28, 260, 1230, 282, 2382, 25487, 868, 325, 347, 1443, 347, 1636, 28, 564, 260, 1230, 282, 4461, 25487, 868, 441, 9094, 429, 260, 1230, 282, 2382, 25487, 411, 540, 670, 216, 49160, 49159, 30, 1073, 800, 2382, 284, 638, 800, 4461, 25487, 868, 325, 10610, 656, 260, 1836, 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+{"group_id": "math_079", "origin": "budgie", "answer": "And, we have $x^{2} y z=1$.\n\nThe solution to the equation is $x=\\frac{1}{\\frac{1}{y}+1+\\frac{1}{z}}=\\frac{y z+1}{y+z}.$\n\nSince $y+z \\leqslant 2$, $y+z \\leqslant 2$ is not possible, but we may assume that $y+z=\\frac{1}{2}$.\n\nSo, $x=\\frac{y z+1}{\\frac{1}{2}}=\\boxed{4y z}.$", "reward": -1, "advantage": -1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 15423, 1885, 49159, 3814, 290, 97, 7483, 403, 1792, 28, 329, 28, 1892, 3814, 290, 97, 7483, 403, 216, 49160, 26265, 5482, 260, 8345, 42, 198, 9212, 76, 18169, 107, 104, 15009, 49160, 27, 105, 27, 106, 1792, 109, 42639, 18169, 107, 105, 15009, 49160, 27, 106, 27, 104, 329, 109, 42639, 18169, 107, 106, 15009, 49160, 27, 104, 27, 105, 1892, 109, 24814, 18169, 107, 49162, 15009, 104, 27, 105, 27, 106, 109, 1673, 9212, 49154, 49158, 198, 3528, 28, 392, 457, 1885, 104, 21990, 49161, 109, 329, 1892, 45, 49160, 28422, 198, 198, 504, 3564, 288, 260, 8345, 314, 1885, 104, 24814, 18169, 107, 49160, 41322, 18169, 107, 49160, 15009, 105, 109, 27, 49160, 42639, 18169, 107, 49160, 15009, 106, 17859, 24814, 18169, 107, 105, 1892, 27, 49160, 15009, 105, 27, 106, 19181, 20, 198, 198, 7230, 1885, 105, 27, 106, 3814, 290, 97, 7483, 403, 216, 49161, 26265, 1885, 105, 27, 106, 3814, 290, 97, 7483, 403, 216, 49161, 20, 314, 441, 1636, 28, 564, 392, 654, 7635, 338, 1885, 105, 27, 106, 24814, 18169, 107, 49160, 15009, 49161, 24362, 30, 198, 198, 2931, 28, 1885, 104, 24814, 18169, 107, 105, 1892, 27, 49160, 41322, 18169, 107, 49160, 15009, 49161, 17859, 24814, 4810, 277, 107, 49163, 105, 1892, 19181, 20, 49154], "prefix_len": 106, "old_logprobs": [-8.75673770904541, -3.4261460304260254, -3.5458288192749023, -1.56813383102417, -2.2374930381774902, -0.8862387537956238, -5.024696350097656, -0.13346561789512634, -0.28033146262168884, -2.6508588790893555, -1.348811388015747, -1.567143201828003, -0.9688009023666382, -1.445141077041626, -0.6645085215568542, -0.14603088796138763, -2.736523389816284, -3.135577917098999, -1.5819978713989258, -0.25776031613349915, -0.6321702003479004, -0.11931177228689194, -2.076040267944336, -0.23133166134357452, -2.4882450103759766, -0.5926836729049683, -0.028871523216366768, -0.07371044158935547, -0.8454666137695312, -2.22076416015625, -0.006205813027918339, -0.4527757167816162, -0.12459518760442734, -0.6619689464569092, -1.3236027956008911, -1.4375725984573364, -0.8798022866249084, -1.126152515411377, -0.00921650230884552, -0.002704060636460781, -0.071990966796875, -0.12959828972816467, -0.06494183093309402, -0.05719922110438347, -1.6074366569519043, -0.11729633063077927, -0.021861523389816284, -0.8066228628158569, -2.24392032623291, -3.0859782695770264, -0.4857226014137268, -0.08395413309335709, -0.19864605367183685, -0.15390589833259583, -0.07328566908836365, -3.3746237754821777, -0.19239552319049835, -0.2791374623775482, -0.02929798513650894, -2.509556293487549, -0.18853649497032166, -1.2912299633026123, -2.2996022701263428, -0.005353045184165239, -0.6999714374542236, -0.9143136739730835, -0.05695263668894768, -0.09390829503536224, -5.757642793469131e-05, -0.23085282742977142, -1.0138683319091797, -0.9492963552474976, -2.5884132385253906, -0.15097570419311523, -1.6720058917999268, -0.025201397016644478, -0.42641326785087585, -0.4316529631614685, -0.00020680672605521977, -0.008617476560175419, -0.00019596086349338293, -0.5309330821037292, -0.37634778022766113, -1.3125964403152466, -1.7722691297531128, -2.890915870666504, -0.28401246666908264, -0.4846906065940857, -5.123191833496094, -1.8105454444885254, -3.891129493713379, -0.6775973439216614, -0.5891727209091187, -0.10989126563072205, -0.06442781537771225, -0.1410648077726364, -0.0010850501712411642, -3.7402753829956055, -0.0922781229019165, -0.002067572670057416, -0.25736671686172485, -0.08732399344444275, -0.34467795491218567, -0.20383769273757935, -0.459760457277298, -0.18379949033260345, -0.01901528425514698, -3.987239360809326, -0.08736921846866608, -1.1129399538040161, -0.6243656873703003, -0.1474321037530899, -0.05326802283525467, -0.030246825888752937, -1.907172679901123, -0.1035759449005127, -0.19997677206993103, -0.009429795667529106, -1.0584288835525513, -0.01390296220779419, -0.0010993395699188113, -0.11074105650186539, -0.006311484612524509, -0.32092806696891785, -0.41086775064468384, -0.2852543592453003, -0.4323851466178894, -6.794906312279636e-06, -0.18848219513893127, -1.3418103456497192, -0.5676193237304688, -0.7976429462432861, -2.1800999641418457, -0.0012915135594084859, -0.5573234558105469]}
+{"group_id": "math_079", "origin": "budgie", "answer": "Since $0 \\leqslant x, y, z \\leqslant 1$, we have $0 \\leqslant x, y, z < 1,$ and $\\frac{x}{1+y+z x} < \\frac{3}{x+y+z}.$  Therefore,\n$$\\frac{x}{1+y+z x} + \\frac{y}{1+z+x y} + \\frac{z}{1+x+y z} > 0,$$so\n$$\\frac{x}{1+y+z x} + \\frac{y}{1+z+x y} + \\frac{z}{1+x+y z} < -\\frac{3}{x+y+z}.$$This contradicts the original equation, so there is no solution.\n\nFinal answer:\n#### 1", "reward": 0.5, "advantage": 0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 15423, 1885, 49159, 3814, 290, 97, 7483, 403, 1792, 28, 329, 28, 1892, 3814, 290, 97, 7483, 403, 216, 49160, 26265, 5482, 260, 8345, 42, 198, 9212, 76, 18169, 107, 104, 15009, 49160, 27, 105, 27, 106, 1792, 109, 42639, 18169, 107, 105, 15009, 49160, 27, 106, 27, 104, 329, 109, 42639, 18169, 107, 106, 15009, 49160, 27, 104, 27, 105, 1892, 109, 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+{"group_id": "math_079", "origin": "verified_reference", "answer": "3. It is not difficult to prove: $\\frac{x}{1+y+z x} \\leqslant \\frac{1}{x+y+z}$, $\\frac{y}{1+z+x y} \\leqslant \\frac{1}{x+y+z}$, $\\frac{z}{1+x+y z} \\leqslant \\frac{1}{x+y+z}$. Therefore, if the equality holds, it is easy to get $x=y=z=1$.\n\nNote: This problem can also be solved as follows: First prove $\\frac{x}{1+y+z x} \\leqslant \\frac{x}{x+y+z}$, etc., so we can get $x+y+z \\geqslant 3$, hence $x=y=z=1$.", "reward": 1.0, "advantage": 0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 15423, 1885, 49159, 3814, 290, 97, 7483, 403, 1792, 28, 329, 28, 1892, 3814, 290, 97, 7483, 403, 216, 49160, 26265, 5482, 260, 8345, 42, 198, 9212, 76, 18169, 107, 104, 15009, 49160, 27, 105, 27, 106, 1792, 109, 42639, 18169, 107, 105, 15009, 49160, 27, 106, 27, 104, 329, 109, 42639, 18169, 107, 106, 15009, 49160, 27, 104, 27, 105, 1892, 109, 24814, 18169, 107, 49162, 15009, 104, 27, 105, 27, 106, 109, 1673, 9212, 49154, 49158, 198, 49162, 30, 657, 314, 441, 1990, 288, 6824, 42, 19573, 18169, 107, 104, 15009, 49160, 27, 105, 27, 106, 1792, 109, 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+{"group_id": "math_080", "origin": "budgie", "answer": "We know that $A_1A_2$ and $A_3A_4$ intersect at an acute angle of $\\theta$. This means that $\\tan \\theta > 0$. \n\nGiven the coordinates of $A_1$, $A_2$, $A_3$, and $A_4$, we can write the equations of the lines as follows:\n\n$$y = a_1^2x, \\, y = a_2^2x, \\, y = a_3^2x, \\, y = a_4^2x$$\n\nSolving for $x$ in each line equation, we get:\n\n$$x = \\frac{a_1^2}{a_1^2 - a_1^2a_1}, \\, x = \\frac{a_2^2}{a_2^2 - a_2^2a_2}, \\, x = \\frac{a_3^2}{a_3^2 - a_3^2a_3}, \\, x = \\frac{a_4^2}{a_4^2 - a_4^2a_4}$$\n\nSimplifying each equation, we get:\n\n$$x = \\frac{a_1^2}{a_1^2 - a_1^2a_1}, \\, x = \\frac{a_2^2}{a_2^2 - a_2^2a_2}, \\, x = \\frac{a_3^2}{a_3^2 - a_3^2a_3}, \\, x = \\frac{a_4^2}{a_4^2 - a_4^2a_4}$$\n\nSimplifying further", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 1885, 81, 79, 49160, 28, 253, 79, 49161, 28, 253, 79, 49162, 28, 253, 79, 49163, 20, 325, 23313, 351, 4073, 8435, 2396, 30, 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+{"group_id": "math_080", "origin": "budgie", "answer": "To find the maximum possible value for $\\tan \\theta$, we need to consider the properties of the intersection point $A_3$. Since $A_3A_4$ is a vertical line, its slope is undefined. Therefore, the slope of $A_3A_4$ is undefined.\n\nHowever, we can still find the maximum possible value of $\\tan \\theta$. If we take the derivative of $A_3A_4$ with respect to $\\theta$ and set it to zero, we get $A_3A_4 = A_3A_3$. This means that the slope of $A_3A_4$ is undefined.\n\nSince the slope of $A_1A_2$ is infinite, the slope of $A_1A_2$ is also infinite. Therefore, there is no maximum possible value for $\\tan \\theta$.\n\nHowever, we can still find the maximum possible value of $\\tan \\theta$. Since $A_1A_2$ is a perpendicular line, the maximum possible value of $\\tan \\theta$ is also infinite.\n\nBut we can still find the maximum possible value of $\\tan \\theta$. Since $A_2$ is a point on the $y$-axis, we can see that $\\tan \\theta \\ge 0$.\n\nTherefore, the maximum possible value of $\\tan \\theta$ is infinite.\n\nHowever, we can still find the maximum possible value of $\\tan \\theta$. Since $A_1$ is a point on the $x$-axis, we can see that $\\tan \\theta$ is approximately $\\tan \\theta \\approx 99.246497$.\n\nTherefore, the maximum possible value of $\\tan \\theta$ is approximately $99.24649", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 1885, 81, 79, 49160, 28, 253, 79, 49161, 28, 253, 79, 49162, 28, 253, 79, 49163, 20, 325, 23313, 351, 4073, 8435, 2396, 30, 533, 260, 13033, 8303, 28, 1303, 1885, 49, 79, 49160, 8615, 81, 79, 49160, 28, 81, 79, 49160, 78, 49161, 25, 26265, 1885, 49, 79, 49161, 8615, 81, 79, 49161, 28, 81, 79, 49161, 78, 49161, 25, 26265, 1885, 49, 79, 49162, 8615, 81, 79, 49162, 28, 81, 79, 49162, 78, 49161, 38224, 284, 1885, 49, 79, 49163, 8615, 81, 79, 49163, 28, 81, 79, 49163, 78, 49161, 25, 28422, 45132, 338, 3204, 1885, 49, 79, 49160, 49, 79, 49161, 20, 284, 1885, 49, 79, 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+{"group_id": "math_080", "origin": "verified_reference", "answer": "1. **Given Information and Setup:**\n   - We have integers \\(a_1, a_2, a_3, a_4\\) with distinct absolute values.\n   - Points in the coordinate plane are \\(A_1 = (a_1, a_1^2)\\), \\(A_2 = (a_2, a_2^2)\\), \\(A_3 = (a_3, a_3^2)\\), and \\(A_4 = (a_4, a_4^2)\\).\n   - Lines \\(A_1A_2\\) and \\(A_3A_4\\) intersect on the \\(y\\)-axis at an acute angle \\(\\theta\\).\n   - We need to find the maximum possible value for \\(\\tan \\theta\\) expressed as \\(\\frac{m}{n}\\) for relatively prime positive integers \\(m\\) and \\(n\\), and then find \\(100m + n\\).\n\n2. **Slope Calculation:**\n   - The slope of line \\(A_1A_2\\) is:\n     \\[\n     m_1 = \\frac{a_2^2 - a_1^2}{a_2 - a_1} = a_2 + a_1\n     \\]\n   - The slope of line \\(A_3A_4\\) is:\n     \\[\n     m_2 = \\frac{a_4^2 - a_3^2}{a_4 - a_3} = a_4 + a_3\n     \\]\n\n3. **Intersection on the \\(y\\)-axis:**\n   - Since the lines intersect on the \\(y\\)-axis, their \\(x\\)-coordinates at the intersection point are zero. Thus, the \\(y\\)-intercepts of the lines are:\n     \\[\n     c_1 = a_1^2 \\quad \\text{and} \\quad c_2 = a_3^2\n     \\]\n\n4. **Angle Between Lines:**\n   - The tangent of the angle \\(\\theta\\) between the two lines is given by:\n     \\[\n     \\tan \\theta = \\left| \\frac{m_1 - m_2}{1 + m_1 m_2} \\right| = \\left| \\frac{(a_1 + a_2) - (a_3 + a_4)}{1 + (a_1 + a_2)(a_3 + a_4)} \\right|\n     \\]\n\n5. **Maximizing \\(\\tan \\theta\\):**\n   - Let \\(x = a_1 + a_2\\) and \\(y = a_3 + a_4\\). We need to maximize:\n     \\[\n     f(x, y) = \\left| \\frac{x - y}{1 + xy} \\right|\n     \\]\n\n6. **Case Analysis:**\n   - **Case 1: \\(x, y > 0\\)**\n     - Here, \\(x, y \\geq 1\\) since they are integers. In this case, \\(|f(x, y)| < 1\\), which cannot exceed our current maximum.\n   - **Case 2: \\(x > 0, y < 0\\)**\n     - The numerator is positive, and the denominator is negative, so \\(|f(x, y)|\\) cannot be our maximum.\n   - **Case 3: \\(x < 0, y > 0\\)**\n     - Let \\(x = -u\\) and \\(y = v\\). We want to maximize:\n       \\[\n       \\frac{u + v}{uv - 1}\n       \\]\n     - We need to check when:\n       \\[\n       \\frac{u + v}{uv - 1} > \\frac{3}{2} \\implies 3(uv - 1) < 2(u + v) \\implies 3uv - 3 < 2u + 2v \\implies 3uv - 2u - 2v < 3\n       \\]\n     - If \\(u, v > 1\\), then it's impossible because \\(3uv - 2u - 2v \\geq 3\\) is false. Thus, we can let \\(v = 1\\) and need \\(u < 5\\).\n\n7. **Checking Specific Values:**\n   - For \\(u = 4\\) and \\(v = 1\\):\n     - We have \\(a_1 = -6, a_2 = 2, a_3 = 4, a_4 = -3\\).\n     - This gives:\n       \\[\n       \\tan \\theta = \\left| \\frac{(-6 + 2) - (4 - 3)}{1 + (-6 + 2)(4 - 3)} \\right| = \\left| \\frac{-4 - 1}{1 - 4} \\right| = \\left| \\frac{-5}{-3} \\right| = \\frac{5}{3}\n       \\]\n\nThe final answer is \\(\\boxed{503}\\).", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 4239, 1885, 81, 79, 49160, 28, 253, 79, 49161, 28, 253, 79, 49162, 28, 253, 79, 49163, 20, 325, 23313, 351, 4073, 8435, 2396, 30, 533, 260, 13033, 8303, 28, 1303, 1885, 49, 79, 49160, 8615, 81, 79, 49160, 28, 81, 79, 49160, 78, 49161, 25, 26265, 1885, 49, 79, 49161, 8615, 81, 79, 49161, 28, 81, 79, 49161, 78, 49161, 25, 26265, 1885, 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+{"group_id": "math_081", "origin": "budgie", "answer": "The two wheels rotate at a combined speed of $2 \\times 30 \\times 60 = 3600$ revolutions per minute.\n\nThe two wheels make 3600 revolutions per minute, so the smaller wheel takes 3600/30 = $\\boxed{120}$ seconds to rotate three times as fast as the larger wheel.", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 365, 49160, 49159, 2876, 25, 5713, 14296, 19087, 28, 11450, 770, 351, 971, 550, 28, 1130, 6450, 16418, 7685, 738, 260, 7292, 282, 14296, 1885, 49, 20, 284, 1885, 50, 28422, 378, 47317, 282, 260, 14296, 9094, 411, 253, 4186, 282, 1296, 30, 378, 3764, 8656, 2022, 216, 49162, 49159, 29742, 567, 8427, 30, 22983, 638, 800, 7345, 260, 3227, 8656, 21768, 335, 582, 5245, 47, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49159, 49165, 79, 49162, 49160, 83, 49162, 49162, 49166, 49165, 85, 49167, 84, 49163, 49166, 49165, 49165, 49164, 15659, 86, 49168, 49159, 87, 29, 49159, 49163, 30, 13655, 47, 11551, 45, 49163, 49159, 49162, 22, 6930, 45, 49163, 49167, 49159, 22, 4961, 79, 8842, 79, 105, 45, 49168, 49164, 49167, 22, 4961, 79, 8842, 79, 104, 45, 49167, 49165, 49167, 25, 49154, 49158, 198, 504, 827, 14296, 19087, 418, 253, 5276, 3547, 282, 1885, 49161, 3814, 14602, 216, 49162, 49159, 3814, 14602, 216, 49165, 49159, 446, 216, 49162, 49165, 49159, 49159, 20, 29742, 567, 8427, 30, 198, 198, 504, 827, 14296, 919, 216, 49162, 49165, 49159, 49159, 29742, 567, 8427, 28, 588, 260, 3764, 8656, 2933, 216, 49162, 49165, 49159, 49159, 31, 49162, 49159, 446, 19573, 4810, 277, 107, 49160, 49161, 49159, 24362, 7345, 288, 19087, 1296, 1711, 347, 3538, 347, 260, 3227, 8656, 30, 49154], "prefix_len": 183, "old_logprobs": [-1.4609476327896118, -2.627300262451172, -0.014013583771884441, -0.7484188079833984, -2.2794382572174072, -0.9261645078659058, -0.4807943105697632, -0.9631984829902649, -0.07189545035362244, -1.0450222492218018, -0.3002413809299469, -0.9950499534606934, -0.12939459085464478, -0.029764996841549873, -0.6436401605606079, -0.06925348937511444, -1.7881590127944946, -0.36630573868751526, -0.0868588462471962, -0.7296174764633179, -0.08340813219547272, -0.23776327073574066, -0.02078898809850216, -0.002308919792994857, -0.0031475310679525137, -2.13382354559144e-05, -0.011841481551527977, -0.024782951921224594, -0.7868180871009827, -0.009084794670343399, -0.011082522571086884, -0.028470484539866447, -0.5486528277397156, -0.09471310675144196, -1.1696217060089111, -3.588625907897949, -0.007486857008188963, -1.1202483177185059, -1.0956332683563232, -0.11845885962247849, -0.8795124888420105, -0.00281461956910789, -0.00984562374651432, -0.24169279634952545, -0.7204097509384155, -0.04912475869059563, -0.9118077754974365, -0.4916650354862213, -1.0592422485351562, -1.772507667541504, -0.01157072652131319, -1.8825381994247437, -0.41431236267089844, -0.0917191356420517, -0.013622770085930824, -0.0008238735608756542, -0.0006006343755871058, -0.11846309155225754, -0.09389016777276993, -0.45568305253982544, -0.2004159837961197, -3.185530424118042, -0.01651287078857422, -1.9073468138230965e-06, -0.020648052915930748, -0.019722197204828262, -0.00022754464589525014, -0.1470804363489151, -0.0052775111980736256, -1.1125140190124512, -0.4562505781650543, -3.4451723098754883, -3.5588934421539307, -0.018925100564956665, -1.0304131507873535, -0.12462970614433289, -0.5154629349708557, -0.05884100869297981, -0.03766952455043793, -0.036643240600824356, -0.15740782022476196, -0.40046897530555725]}
+{"group_id": "math_081", "origin": "budgie", "answer": "First, we need to find the distance the larger wheel travels in one revolution. Since the circumference of the wheel is the same for both wheels, we can find the distance traveled by the larger wheel in one revolution by multiplying the circumference of the larger wheel by the number of revolutions it makes in a minute.\n\nThe formula for the circumference of a circle is C = 2πr, where r is the radius of the circle.\n\nFor the larger wheel, the circumference in one minute is 2π(30/60) = 2π.\n\nSince the smaller wheel makes 30 revolutions per minute, the distance traveled by the smaller wheel in one minute is 2π(30/30) = 2π.\n\nTherefore, the larger wheel travels at a distance of 30*2π = 60π miles in one minute.\n\nNow, we can find the time the larger wheel spends on one revolution by dividing the distance traveled by the number of revolutions it makes in one minute by the number of revolutions it makes in a minute.\n\nTime = Distance / Number of pmid\nTime = (60π / 30) / 30\nTime = (6π / 30) minutes\nTime = (3/5) minutes\n\nSince we want the answer in seconds, we convert the time to seconds by multiplying by 60.\n\nTime = (3/5) * 60 seconds\nTime = 36 seconds\n\nSo, the larger wheel spends 36 seconds on one revolution.\n#### 36", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 365, 49160, 49159, 2876, 25, 5713, 14296, 19087, 28, 11450, 770, 351, 971, 550, 28, 1130, 6450, 16418, 7685, 738, 260, 7292, 282, 14296, 1885, 49, 20, 284, 1885, 50, 28422, 378, 47317, 282, 260, 14296, 9094, 411, 253, 4186, 282, 1296, 30, 378, 3764, 8656, 2022, 216, 49162, 49159, 29742, 567, 8427, 30, 22983, 638, 800, 7345, 260, 3227, 8656, 21768, 335, 582, 5245, 47, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49159, 49165, 79, 49162, 49160, 83, 49162, 49162, 49166, 49165, 85, 49167, 84, 49163, 49166, 49165, 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+{"group_id": "math_081", "origin": "verified_reference", "answer": "Answer: $6 s$\n\nSolution. The speeds of points lying on the edge of the wheels are the same.\n\n(2 points)\n\nThe distances traveled by these points differ by three times.\n\nThe small wheel spends on one revolution: $t_{\\text {small }}=\\frac{1 \\text { min }}{30 \\text { revolutions }}=\\frac{60 s}{30 \\text { revolutions }}=2 s$. (3 points)\n\nTherefore, the large wheel spends on one revolution:\n\n$t_{\\text {large }}=3 t_{\\text {small }}=3 \\cdot 2=6 s$.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 365, 49160, 49159, 2876, 25, 5713, 14296, 19087, 28, 11450, 770, 351, 971, 550, 28, 1130, 6450, 16418, 7685, 738, 260, 7292, 282, 14296, 1885, 49, 20, 284, 1885, 50, 28422, 378, 47317, 282, 260, 14296, 9094, 411, 253, 4186, 282, 1296, 30, 378, 3764, 8656, 2022, 216, 49162, 49159, 29742, 567, 8427, 30, 22983, 638, 800, 7345, 260, 3227, 8656, 21768, 335, 582, 5245, 47, 198, 198, 21653, 5257, 4687, 1742, 13133, 94, 30, 8898, 18449, 30, 824, 31, 22340, 4194, 31, 49161, 49159, 49161, 49163, 79, 49159, 49164, 79, 49159, 49165, 79, 49162, 49160, 83, 49162, 49162, 49166, 49165, 85, 49167, 84, 49163, 49166, 49165, 49165, 49164, 15659, 86, 49168, 49159, 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+{"group_id": "math_084", "origin": "budgie", "answer": "First, note that if $n!$ divided by $2$ does not have an integer factorial, it is impossible to factor it into the product of two factorials. Therefore, $n$ must be an even number or a multiple of $4$. There are $4$ possibilities: $4, 6, 8,...$ for $n$ values. If $n$ is a multiple of $4$, then $k = 4k$ and $l = 4l$, which implies $k \\cdot l = 4k \\cdot 4l$ does not have a factorial factorial. If $n = 8$ for the first case, then $k = 2$ and $l = 8$ for the second case, so there is only one solution. If $n = 6$, then $k = 3$ and $l = 6$ for the third case, so there is only one solution. If $n = 2$, then $k = 1$ and $l = 2$ for the fourth case, and only one solution. If $n = 1$, then $k = 1$ and $l = 1$ for the fifth case, and one solution. If $n = 0$, then $k = 0$ and $l = 0$ for the sixth case, and one solution. If $n = 5$, then $k = 1$ and $l = 5$ for the seventh case, and one solution. If $n = 3$, then $k = 1$ and $l = 3$ for the eighth and ninth cases, and one solution. If $n = 1$, then $k = 1$ and $l = 1$ for the tenth and eleventh cases, and one solution", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49160, 30, 43104, 260, 8345, 19573, 18169, 107, 94, 17, 15009, 49161, 109, 45, 91, 17, 27, 92, 17, 20, 281, 1782, 2966, 28, 837, 1885, 94, 42732, 49160, 3814, 41591, 216, 49161, 3814, 41591, 3814, 400, 1565, 304, 28422, 1094, 665, 359, 787, 3860, 28, 2965, 216, 49159, 43, 585, 665, 314, 582, 3564, 28, 2965, 1885, 94, 20, 43, 585, 665, 359, 2701, 3860, 28, 2965, 260, 2028, 282, 260, 2396, 282, 1885, 94, 20, 327, 511, 3860, 30, 37271, 338, 253, 3564, 314, 253, 3149, 12488, 46408, 94, 28, 501, 28, 303, 38224, 43, 585, 3860, 9094, 281, 418, 2124, 582, 5077, 28, 502, 359, 2500, 896, 30, 49154, 49158, 198, 5345, 28, 4109, 338, 585, 1885, 94, 17, 20, 6066, 411, 1885, 49161, 20, 1072, 441, 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+{"group_id": "math_084", "origin": "budgie", "answer": "To solve the equation $\\frac{n!}{2} = k! + l!$, start by noticing that it can be as simple as 1 factorial on both the left side and the right side.\n\nThe smallest $n$ for which this does not hold true is $n = 4$ because $4! = 24$.\n\nThus, the answer is $\\boxed{4}$.\n\nNote: If there are multiple solutions, then the sum of their values is not unique.", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49160, 30, 43104, 260, 8345, 19573, 18169, 107, 94, 17, 15009, 49161, 109, 45, 91, 17, 27, 92, 17, 20, 281, 1782, 2966, 28, 837, 1885, 94, 42732, 49160, 3814, 41591, 216, 49161, 3814, 41591, 3814, 400, 1565, 304, 28422, 1094, 665, 359, 787, 3860, 28, 2965, 216, 49159, 43, 585, 665, 314, 582, 3564, 28, 2965, 1885, 94, 20, 43, 585, 665, 359, 2701, 3860, 28, 2965, 260, 2028, 282, 260, 2396, 282, 1885, 94, 20, 327, 511, 3860, 30, 37271, 338, 253, 3564, 314, 253, 3149, 12488, 46408, 94, 28, 501, 28, 303, 38224, 43, 585, 3860, 9094, 281, 418, 2124, 582, 5077, 28, 502, 359, 2500, 896, 30, 49154, 49158, 198, 2068, 5482, 260, 8345, 19573, 18169, 107, 94, 17, 15009, 49161, 109, 446, 501, 17, 1232, 303, 17, 26265, 1120, 411, 30528, 338, 357, 416, 325, 347, 2232, 347, 216, 49160, 4186, 409, 335, 1062, 260, 2049, 2103, 284, 260, 1048, 2103, 30, 198, 198, 504, 13341, 1885, 94, 20, 327, 527, 451, 1072, 441, 2205, 2629, 314, 1885, 94, 446, 216, 49163, 20, 975, 1885, 49163, 17, 446, 216, 49161, 49163, 28422, 198, 198, 14344, 28, 260, 2988, 314, 19573, 4810, 277, 107, 49163, 24362, 30, 198, 198, 8928, 42, 1094, 665, 359, 2701, 3860, 28, 965, 260, 2028, 282, 480, 2396, 314, 441, 2116, 30, 49154], "prefix_len": 142, "old_logprobs": [-1.752201795578003, -0.23252420127391815, -0.38638609647750854, -0.02730526402592659, -0.24131788313388824, -0.0008070787298493087, -0.0002307625545654446, -0.006066480651497841, -0.0009224927052855492, -6.949660019017756e-05, -0.00018809456378221512, -0.00014006110723130405, -2.0225536823272705, -0.007939211092889309, -0.0014055621577426791, -0.04013221338391304, -0.0032390295527875423, -0.0026566232554614544, -0.6967863440513611, -4.457576751708984, -0.0786355584859848, -3.227722406387329, -0.166713148355484, -3.8902087211608887, -1.658910870552063, -0.019678249955177307, -10.357717514038086, -0.24977558851242065, -0.009714826010167599, -1.8815957307815552, -1.0306024551391602, -2.7271223068237305, -0.5626351833343506, -4.272486686706543, -1.320894718170166, -2.147026538848877, -0.3655475676059723, -0.7275599837303162, -0.709673285484314, -0.3043581247329712, -0.003196133067831397, -0.0016683719586580992, -0.73082035779953, -1.437998652458191, -0.08504339307546616, -3.029097318649292, -2.9758925437927246, -0.7527299523353577, -0.23528411984443665, -0.05562160536646843, -0.7382365465164185, -0.011888484470546246, -0.8811951875686646, -3.9085164070129395, -0.0400206632912159, -0.16688042879104614, -1.4710924625396729, -0.15427222847938538, -1.1851508617401123, -0.07550346106290817, -0.19464540481567383, -0.019566265866160393, -2.58644962310791, -1.847943902015686, -0.22161594033241272, -0.6745989322662354, -0.43096819519996643, -0.029274020344018936, -0.36712732911109924, -0.010231191292405128, -0.23011383414268494, -0.0006462631281465292, -2.101041078567505, -0.6142739653587341, -0.031150398775935173, -4.088268280029297, -0.01759737730026245, -0.8761716485023499, -1.8372199535369873, -0.017119256779551506, -2.0780539512634277, -0.0013398488517850637, -2.4437606043647975e-05, -0.0019221893744543195, -0.30396193265914917, -0.10821603983640671, -0.13172098994255066, -1.4083329439163208, -0.08927074074745178, -1.5658046007156372, -0.7187583446502686, -3.292543649673462, -1.179271936416626, -0.6118783354759216, -0.41102781891822815, -0.07096734642982483, -0.16836699843406677, -4.315783977508545, -1.1153658628463745, -0.8865266442298889, -0.09175002574920654, -2.0168392658233643, -0.15043504536151886, -0.4328908920288086, -2.6725282669067383, -1.33939528465271, -0.8214519023895264, -0.26096025109291077]}
+{"group_id": "math_084", "origin": "verified_reference", "answer": "Answer: 10 (all triples of solutions $(n, k, l):(3,1,2),(3,2,1),(4,3,3)$).\n\nSolution. Note that if $k4$, then $n!>4 \\cdot(n-1)!\\geqslant 2 \\cdot(k!+l!)$, so there are no such solutions. If $n=2$, we get $1=k!+l!$ - no solutions; if $n=3$, the equation $3=k!+l!$ has two solutions: $k=1, l=2$ and $k=2, l=1$; if $n=4$, the equation $12=k!+l!$ gives one more solution $k=l=3$.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49166, 30, 49160, 30, 43104, 260, 8345, 19573, 18169, 107, 94, 17, 15009, 49161, 109, 45, 91, 17, 27, 92, 17, 20, 281, 1782, 2966, 28, 837, 1885, 94, 42732, 49160, 3814, 41591, 216, 49161, 3814, 41591, 3814, 400, 1565, 304, 28422, 1094, 665, 359, 787, 3860, 28, 2965, 216, 49159, 43, 585, 665, 314, 582, 3564, 28, 2965, 1885, 94, 20, 43, 585, 665, 359, 2701, 3860, 28, 2965, 260, 2028, 282, 260, 2396, 282, 1885, 94, 20, 327, 511, 3860, 30, 37271, 338, 253, 3564, 314, 253, 3149, 12488, 46408, 94, 28, 501, 28, 303, 38224, 43, 585, 3860, 9094, 281, 418, 2124, 582, 5077, 28, 502, 359, 2500, 896, 30, 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+{"group_id": "math_085", "origin": "budgie", "answer": "To find the minimum value of $a+b+c$ given that $a$, $b$, and $c$ are all positive integers and that the parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A$ and $B$, we can proceed as follows:\n\nSince the parabola intersects the $x$-axis at $A$ and $B$, the equation of the parabola can be written as:\n\\[ y = a x^2 + b x + c = a(x^2 - 1) + b(x - 1) \\]\n\nWe know that the distance from $A$ to $B$ is less than 1, so:\n\\[ \\sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} < 1 \\Rightarrow (x_B - x_A)^2 + (y_B - y_A)^2 < 1 \\]\n\nSquaring both sides and expanding, we get:\n\\[ x_B^2 - 2x_Ax_B + x_A^2 + y_B^2 - 2y_Ay_B + y_A^2 + x_B^2 + y_B^2 - 2x_Ay_B + x_A^2 + y_A^2 < 1 \\]\n\nSimplifying, we get:\n\\[ 2x_B^2 - 2x_Ax_B + 2y_B^2 - 2y_Ay_B + 2x_A^2 + 2y_A^2 - 2x_A^2", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 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+{"group_id": "math_085", "origin": "budgie", "answer": "1. The equation of the parabola is $y = ax^2 + bx + c$.\n2. Since the parabola intersects the $x$-axis at points $A$ and $B$, we know that $A$ and $B$ have the same $y$-coordinate.\n3. The $y$-coordinates of $A$ and $B$ are the same, so we can write $y = k(ax^2 + bx + c)$, where $k$ is a constant.\n4. The roots of the quadratic equation $ax^2 + bx + c = 0$ are $x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$.\n5. Since $A$ and $B$ have the same $y$-coordinate, we have $k = k \\cdot \\frac{-b + \\sqrt{b^2 - 4ac}}{2a}$.\n6. Since the distances from $A$ and $B$ to the origin are both less than 1, we have $\\sqrt{(a - 0)^2 + b^2} < 1$.\n7. Squaring both sides of the equation, we get $(a - 0)^2 + b^2 < 1$.\n8. Expanding the left-hand side, we get $a^2 + b^2 < 1$.\n9. Since $a^2 + b^2 - 1 = 0$, we can apply the Cauchy-Schwarz inequality to get $\\sqrt{(a - 0)^2 + b^2} < 1$.\n10. 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+{"group_id": "math_085", "origin": "verified_reference", "answer": "Solution: Let $A\\left(x_{1}, 0\\right)$ and $B\\left(x_{2}, 0\\right)$, and $x_{1} < 0 < x_{2}$,  \nthen $x_{1} < 0, \\\\ \n\\therefore b > 2 \\sqrt{a c} . \\\\ \n\\text{Also, } \\because |O A| = |x_{1}| > 1$. Therefore, the parabola opens upwards, and when $x = -1$, $y > 0$, so $a(-1)^{2} + b(-1) + c > 0$, which means $b < a + c + 1$. Since $b > 2 \\sqrt{a c}$, we have $2 \\sqrt{a c} + 1 < a + c$,  \ni.e., $(\\sqrt{a} - \\sqrt{c})^{2} > 1$.  \nFrom equation (2), we get $\\sqrt{a} - \\sqrt{c} > 1$,  \n\ni.e., $\\sqrt{a} > \\sqrt{c} + 1$.  \nThus, $a > (\\sqrt{c} + 1)^{2} \\geq (\\sqrt{1} + 1)^{2} = 4$.  \nSolving, we get $a \\geq 5$.  \n$$\n\\begin{array}{l}\n\\text{Also, } \\because b > 2 \\sqrt{a c} \\geq 2 \\sqrt{5 \\times 1} > 4, \\\\\n\\therefore b \\geq 5 .\n\\end{array}\n$$\n\nWhen $a = 5, b = 5, c = 1$, the parabola $y = 5 x^{2} + 5 x + 1$ satisfies the given conditions.  \nTherefore, the minimum value of $a + b + c$ is 11.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49160, 49162, 10887, 338, 1885, 81, 26265, 1885, 82, 26265, 284, 1885, 83, 20, 359, 511, 2731, 23313, 28, 284, 260, 43099, 7866, 1885, 105, 45, 81, 1792, 21990, 49161, 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+{"group_id": "math_086", "origin": "budgie", "answer": "Let's consider the terms of the function $f(x)$:\n\n$$f(x) = \\left(x^{2}+4 x+5\\right)\\left(x^{2}+4 x+2\\right)+2 x^{2}+8 x+1$$\n\nWe can rewrite each term as $(x^{2}+4 x+2)(x^{2}+4 x+5)$.\n\nExpanding the product, we get:\n\n$$f(x) = (x^{2}+4 x+2)(x^{2}+4 x+5) = x^{4}+8 x^{3}+2 x^{2}+4 x^{3}+8 x^{2}+16 x+2 x^{2}+8 x+5$$\n\nCombining like terms, we get:\n\n$$f(x) = x^{4}+12 x^{3}+17 x^{2}+14 x+5$$\n\nNow, let's set $f(x)$ to a constant value to find the minimum value. We can do this by setting the expression $x^{4}+12 x^{3}+17 x^{2}+14 x+5 = k$, where $k$ is a constant.\n\nRearranging the equation, we get:\n\n$$x^{4}+12 x^{3}+17 x^{2}+14 x+5 - k = 0$$\n\nThis is a quartic equation in $x$, which can be difficult to solve. 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+{"group_id": "math_086", "origin": "budgie", "answer": "To solve this problem, we first need to find the minimum value of the function $f(x) = \\left(x^{2} + 4x + 5\\right) \\left(x^{2} + 4x + 2\\right) + 2x^{2} + 8x + 1$.\n\nLet's denote the first expression as $A$ and the second expression as $B$:\n\n$$A = x^{2} + 4x + 5$$\n$$B = x^{2} + 4x + 2$$\n\nNow, we can rewrite $A$ and $B$ as follows:\n\n$$A = x^{2} + 4x + 5 = (x^{2} + 4x + 2) + 2x^{2} + 4x + 3$$\n$$B = x^{2} + 4x + 2 = (x^{2} + 4x + 4) + 2x^{2} + 4x + 4 + 1$$\n\nSince $A$ and $B$ are equal, we can set the expressions equal to each other:\n\n$$(x^{2} + 4x + 2) + 2x^{2} + 4x + 3 = (x^{2} + 4x + 4) + 2x^{2} + 4x + 4 + 1$$\n\nSimplify the right side of the equation:\n\n$$\\Rightarrow x^{2} + 2x^{2} + 2x^{2} + 8x + 1 = 5x^{2} + 12x + 5$$\n\nRearrange the", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 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+{"group_id": "math_086", "origin": "verified_reference", "answer": "Let $u=x^{2}+4 x$, then\n$$\nf(x)=g(u)=(u+5)(u+2)+2 u+1=u^{2}+9 u+11 \\text {. }\n$$\n\nBy the properties of quadratic functions, when $u=-\\frac{9}{2}$, $g(u)$ reaches its minimum value, and when $u>-\\frac{9}{2}$, $g(u)$ is a strictly increasing function of $u$. Since $u=x^{2}+4 x=(x+2)^{2}-4$, we know that $u \\geqslant-4$, and when $x=-2$, $u=-4$. Therefore, when $x=-2$, $f(x)$ reaches its minimum value, which is -9.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 8412, 216, 49161, 2959, 1885, 104, 3814, 254, 3814, 33918, 107, 66, 24362, 28, 1576, 288, 1042, 260, 5869, 1685, 282, 260, 1517, 1885, 86, 24, 104, 25, 24814, 8842, 24, 104, 21990, 49161, 109, 27, 49163, 1792, 27, 49164, 76, 2771, 24492, 8842, 24, 104, 21990, 49161, 109, 27, 49163, 1792, 27, 49161, 76, 2771, 18434, 49161, 1792, 21990, 49161, 109, 27, 49167, 1792, 27, 49160, 28422, 49154, 49158, 198, 4239, 1885, 101, 45, 104, 21990, 49161, 109, 27, 49163, 1792, 26265, 965, 198, 9212, 198, 86, 24, 104, 36637, 87, 24, 101, 25, 8615, 101, 27, 49164, 14502, 101, 27, 49161, 18434, 49161, 326, 27, 49160, 45, 101, 21990, 49161, 109, 27, 49168, 326, 27, 49160, 49160, 3814, 2692, 1662, 30, 4029, 198, 9212, 198, 198, 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+{"group_id": "math_087", "origin": "budgie", "answer": "This problem mainly tests the properties of a focal length of a lens, which is a function of the angle between the incident ray and the plane of the lens.\n\nThe angle between the incident ray and the plane of the lens is related to the distance from the entry point to the lens through the lens's shape. The angle between the incident ray and the plane of the lens is essentially the angle between the rays and the plane of the lens. It is because the point is on the edge of the lens that the lightbars will form a right angle with the rays.\n\nSince the lightbars form a right angle with the rays, the focal length can be determined by the distance from the lightblue point to the lens. The distance from the lightblue point to the lens is the same as the distance from the lightblue point to the lens, which is half of the focal length of the lens.\n\nThe answer is: $\\boxed{10\\sqrt{2}}$ cm", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49167, 30, 365, 49160, 49164, 2876, 25, 330, 5607, 19156, 282, 1420, 8227, 335, 253, 5607, 4984, 2707, 7247, 418, 253, 4052, 282, 1885, 104, 45, 49160, 49159, 20, 5973, 429, 624, 10454, 3712, 30, 378, 7256, 826, 260, 9208, 19156, 284, 260, 8303, 282, 260, 7247, 19573, 13433, 45, 49163, 49164, 21990, 76, 21386, 24362, 28, 284, 260, 7256, 826, 260, 21112, 277, 19156, 284, 260, 8303, 282, 260, 7247, 19573, 17565, 45, 49162, 49159, 21990, 76, 21386, 24362, 30, 22983, 624, 17350, 3519, 30, 49154, 49158, 198, 1348, 1732, 5630, 3642, 260, 3849, 282, 253, 17350, 3519, 282, 253, 7247, 28, 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+{"group_id": "math_087", "origin": "budgie", "answer": "The focal length of the lens is given by the sum of the distances from the light source to the focal points.  Therefore, the focal length of the lens is $x+x=2x$.\n\nHence, the answer is:\n\n\\[ \\boxed{2x} \\]\n\nThe focal length of the lens is $2x$.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49167, 30, 365, 49160, 49164, 2876, 25, 330, 5607, 19156, 282, 1420, 8227, 335, 253, 5607, 4984, 2707, 7247, 418, 253, 4052, 282, 1885, 104, 45, 49160, 49159, 20, 5973, 429, 624, 10454, 3712, 30, 378, 7256, 826, 260, 9208, 19156, 284, 260, 8303, 282, 260, 7247, 19573, 13433, 45, 49163, 49164, 21990, 76, 21386, 24362, 28, 284, 260, 7256, 826, 260, 21112, 277, 19156, 284, 260, 8303, 282, 260, 7247, 19573, 17565, 45, 49162, 49159, 21990, 76, 21386, 24362, 30, 22983, 624, 17350, 3519, 30, 49154, 49158, 198, 504, 17350, 3519, 282, 260, 7247, 314, 1836, 411, 260, 2028, 282, 260, 9510, 429, 260, 1420, 2257, 288, 260, 17350, 2876, 30, 216, 4882, 28, 260, 17350, 3519, 282, 260, 7247, 314, 1885, 104, 27, 104, 45, 49161, 104, 28422, 198, 198, 35391, 28, 260, 2988, 314, 42, 198, 198, 76, 75, 3814, 4810, 277, 107, 49161, 104, 109, 3814, 77, 198, 198, 504, 17350, 3519, 282, 260, 7247, 314, 1885, 49161, 104, 28422, 49154], "prefix_len": 115, "old_logprobs": [-1.6423529386520386, -1.8036441802978516, -0.021930566057562828, -0.27133235335350037, -0.46621987223625183, -0.6992856860160828, -0.4222971796989441, -1.2089314460754395, -0.021433627232909203, -1.2431578636169434, -1.2867406606674194, -0.03392789885401726, -0.013428246602416039, -0.4706249237060547, -0.13301445543766022, -0.3747159242630005, -2.626441478729248, -0.043738119304180145, -0.11362124234437943, -0.18013767898082733, -3.027092456817627, -0.1778271645307541, -1.2479476928710938, -5.642955303192139, -2.841688394546509, -0.09926445782184601, -0.7446991205215454, -0.16762298345565796, -0.016065435484051704, -0.8568283915519714, -0.08398526161909103, -0.12953680753707886, -0.06648128479719162, -1.2334785461425781, -0.7478652000427246, -0.8828119039535522, -0.6608858108520508, -0.9784277081489563, -0.18139192461967468, -0.05453592911362648, -1.033318042755127, -1.1257203817367554, -0.12333303689956665, -3.191793441772461, -0.0310861486941576, -0.15096011757850647, -0.29360446333885193, -0.005857445299625397, -1.1482656002044678, -0.3377196490764618, -1.1740810871124268, -3.3703620433807373, -0.9121911525726318, -0.3129950165748596, -0.0019891730044037104, -4.768370445162873e-07, -0.0033521433360874653, -0.02178022637963295, -0.07638783007860184, -0.005004379898309708, -0.0009324969723820686, -0.0014873405452817678, -0.30634787678718567, -0.04693099856376648, -2.2015762329101562, -0.5450804829597473, -0.007443905808031559, -0.15827223658561707, -0.0460357740521431, -0.033400919288396835, -0.07599413394927979, -0.7017744779586792, -0.3742057979106903, -0.030160311609506607, -0.6276863813400269, -0.6498512029647827]}
+{"group_id": "math_087", "origin": "verified_reference", "answer": "Answer: $\\approx 13.7$ cm.\n\nSolution. Parallel rays intersect at the focus, so $F$ is the focus of the given lens.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_7a4bc87492c06fc34cd6g-06.jpg?height=480&width=302&top_left_y=1502&top_left_x=957)\n\nIn triangle $O A F$: angle $F A O=30^{\\circ}$, angle $O F A=15^{\\circ}$, angle $A O F=135^{\\circ}$.\n\nTherefore, $\\frac{O F}{\\sin 30^{\\circ}}=\\frac{A O}{\\sin 15^{\\circ}}$.\n\nWe obtain that the focal length:\n\n$F C=O F \\sin 45^{\\circ}=A O \\frac{\\sin 30^{\\circ}}{\\sin 15^{\\circ}} \\sin 45^{\\circ} \\approx 13.7 \\mathrm{~cm}$.\n\n## Multidisciplinary Engineering Olympiad \"Star\" in Natural Sciences\n\n10th grade\n\nFinal Round\n\n$2022-2023$\n\nVersion 2\n\nProblems, answers, and evaluation criteria", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49167, 30, 365, 49160, 49164, 2876, 25, 330, 5607, 19156, 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+{"group_id": "math_089", "origin": "budgie", "answer": "#### Part (a)\n1. Let's denote the number of performances sold by the performers as \\( S_{\\text{per former}} \\), the number of performances sold by theppromades as \\( S_{\\text{per new}} \\), and the number of performances by the audience as \\( S_{\\text{per new}} \\).\n2. According to the problem, the number of violin and possession is three times the number ofadders. This can be written as:\n   \\[\n   S_{\\text{per former}} = 3 \\times S_{\\text{per new}}\n   \\]\n3. From this, we get a relationship between the number of performances and the number of adders:\n   \\[\n   S_{\\text{per former}} = 3 \\times S_{\\text{per new}} = 3S_{\\text{per adder}}\n   \\]\n4. This relationship holds because the total number of performances is equal to the sum of the number of performances by the performers and the number of adders.\n5. Therefore, we can express the total number of performances as:\n   \\[\n   S_{\\text{per former}} + S_{\\text{per new}} = S_{\\text{per adder}}\n   \\]\n6. Substituting the expression for \\( S_{\\text{per adder}} \\) from the previous step, we get:\n   \\[\n   3S_{\\text{per old}} = S_{\\text{per new}}\n   \\]\n7. Hence, it follows that:\n   \\[\n   S_{\\text{per old}} = \\frac{S_{\\text{per new}}}{3}\n   \\]\n8. 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+{"group_id": "math_089", "origin": "budgie", "answer": "a) The number of program HBV is 1/4 of the total number of performance performances.\n\nLet x be the total number of performances. Then, we know that 1/4*x = 200.\n\nSolving for x, we find that x = 800.\n\nb) To find the number of normalized visitors, we need to find the number of visitors who visited the animal show. We can do this by dividing the number of visitors who bought the Conserv by three.\n\nx = (200/3*x) + (x/3)\n\nx = (800/3) + (800/9) + (x/3)\n\nNow, we can simplify this expression.\n\nx = (800/3) + (1667/9) + (x/3)\n\nx = (800/3) + (1667/9) + (819/9)\n\nx = (800 + 1667 + 819)/9\n\nx = (2556)/9\n\nx = 286\n\nSo, the number of normalized visitors is 286.\n\nFinal answer:\n#### 286", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 216, 49160, 49161, 49167, 30, 8948, 10203, 12920, 1885, 49160, 2272, 216, 49166, 49165, 20, 198, 198, 49, 38735, 4502, 216, 49161, 49159, 49159, 13171, 281, 260, 1596, 3645, 28, 511, 282, 527, 592, 3459, 578, 30, 378, 1230, 282, 14670, 281, 260, 38735, 16044, 314, 1296, 1711, 260, 8629, 599, 282, 260, 1230, 282, 13171, 1836, 30, 198, 198, 81, 25, 1073, 800, 1578, 40217, 592, 8057, 585, 582, 8629, 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+{"group_id": "math_089", "origin": "verified_reference", "answer": "From $(200: 4) \\cdot 3=150$ it follows that the circus has over 150 seats.\n\na) $(200 \\cdot 150): 4=7500$; 7500 program booklets were sold during the season.\n\nb) $(200 \\cdot 150): 2=15000$ and $30 \\cdot 15000=450000$; from the animal show, €450,000 was earned.", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 712, 216, 49160, 49161, 49167, 30, 8948, 10203, 12920, 1885, 49160, 2272, 216, 49166, 49165, 20, 198, 198, 49, 38735, 4502, 216, 49161, 49159, 49159, 13171, 281, 260, 1596, 3645, 28, 511, 282, 527, 592, 3459, 578, 30, 378, 1230, 282, 14670, 281, 260, 38735, 16044, 314, 1296, 1711, 260, 8629, 599, 282, 260, 1230, 282, 13171, 1836, 30, 198, 198, 81, 25, 1073, 800, 1578, 40217, 592, 8057, 585, 582, 8629, 282, 260, 7803, 10897, 253, 48797, 47, 198, 198, 82, 25, 1073, 800, 36559, 592, 21769, 5771, 429, 260, 11759, 12235, 327, 260, 3161, 1138, 585, 357, 436, 8797, 411, 2745, 282, 260, 7803, 284, 260, 11759, 13276, 436, 216, 49159, 30, 49162, 49159, 372, 47, 49154, 49158, 198, 5212, 46408, 49161, 49159, 49159, 42, 216, 49163, 25, 3814, 41591, 216, 49162, 45, 49160, 49164, 49159, 20, 357, 5558, 338, 260, 38735, 553, 690, 216, 49160, 49164, 49159, 14670, 30, 198, 198, 81, 25, 46408, 49161, 49159, 49159, 3814, 41591, 216, 49160, 49164, 49159, 727, 216, 49163, 45, 49166, 49164, 49159, 49159, 20, 43, 216, 49166, 49164, 49159, 49159, 1578, 1863, 6770, 592, 3459, 981, 260, 3645, 30, 198, 198, 82, 25, 46408, 49161, 49159, 49159, 3814, 41591, 216, 49160, 49164, 49159, 727, 216, 49161, 45, 49160, 49164, 49159, 49159, 49159, 20, 284, 1885, 49162, 49159, 3814, 41591, 216, 49160, 49164, 49159, 49159, 49159, 45, 49163, 49164, 49159, 49159, 49159, 49159, 20, 43, 429, 260, 3161, 1138, 28, 24927, 49163, 49164, 49159, 28, 49159, 49159, 49159, 436, 11420, 30, 49154], "prefix_len": 148, "old_logprobs": [-5.501042366027832, -8.446799278259277, -1.7892224788665771, -0.2276446670293808, -0.03379940614104271, -7.018219470977783, -1.21592116355896, -2.7752671241760254, -1.0411169528961182, -2.465818405151367, -2.898998498916626, -0.317328542470932, -1.1797640323638916, -2.7640480995178223, -1.6300458908081055, -2.0705618858337402, -0.02860696241259575, -0.5023309588432312, -7.442901134490967, -2.0864834785461426, -0.14977413415908813, -1.7550089359283447, -8.909077644348145, -3.310453414916992, -6.042674541473389, -0.2023615837097168, -0.6108866930007935, -0.4219259023666382, -0.026059595867991447, -3.002293586730957, -0.669857382774353, -0.30619552731513977, -0.3699648976325989, -2.59531831741333, -0.05388449877500534, -6.646946907043457, -1.0511711835861206, -0.01883525960147381, -0.005173391196876764, -3.8303399085998535, -2.2966740131378174, -0.11441162973642349, -1.278995156288147, -1.6159230470657349, -0.06959681212902069, -5.837584018707275, -0.6162245273590088, -2.1569061279296875, -1.7401385307312012, -3.6571054458618164, -2.397738218307495, -0.43141618371009827, -2.086458683013916, -1.4653723239898682, -4.603309631347656, -2.738957643508911, -0.7245733737945557, -0.11834109574556351, -0.05124225467443466, -0.024035267531871796, -11.320363998413086, -13.698543548583984, -3.758654832839966, -1.2459903955459595, -2.439926862716675, -4.68990421295166, -0.3229067325592041, -2.6251823902130127, -0.4582195580005646, -0.09894413501024246, -0.08927749842405319, -0.3696715533733368, -0.007363323122262955, -1.0422818660736084, -1.0685571432113647, -0.022764425724744797, -0.007480112835764885, -0.8188632130622864, -0.07470249384641647, -0.06314028054475784, -0.9951005578041077, -0.3773737847805023, -0.011652973480522633, -1.595185399055481, -0.1597268432378769, -2.3164823055267334, -1.5878303050994873, -1.6291325092315674, -1.3906404972076416, -0.012220979668200016, -0.055220890790224075, -0.8074251413345337, -0.439653605222702, -5.717396259307861, -1.8903231620788574, -1.8633973598480225, -0.799562394618988, -3.69037127494812, -0.32166391611099243, -0.09617637097835541, -0.8457388877868652, -0.5099091529846191, -0.014638286083936691, -1.6042590141296387, -0.30810344219207764, -0.5227669477462769, -0.5400387644767761, -0.03186887875199318, -0.058957893401384354, -0.017649265006184578, -0.00423071626573801, -0.22188161313533783, -0.6913185119628906, -1.9762012958526611, -5.063515663146973, -1.2091901302337646, -5.544290542602539, -0.053764987736940384, -1.7792181968688965, -13.154029846191406, -1.8135794401168823, -0.12425834685564041, -0.07622811943292618, -2.7730517387390137, -0.004688935354351997, -0.00010406429646536708, -0.0006721144891344011, -1.3766717910766602, -8.299947738647461, -0.8010165691375732, -4.170498371124268]}
+{"group_id": "math_090", "origin": "budgie", "answer": "To find the value of $2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)$, we need to simplify the expression inside the parentheses first.\n\n\\[ 2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right) = 2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\frac{1}{16}\\right)\\]\n\nNext, we need to find the sum inside the parentheses. To do this, we can add the fractions one by one and then find a common denominator.\n\n\\[ 1+(1+1+1+1+1) = 5 \\]\n\nSo, we have:\n\n\\[ 2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\frac{1}{16}\\right) = 2^{4}\\left(5\\right) = 2^{4}\\cdot 5 = 16\\cdot 2 = \\boxed{32} \\]\n\nFinal answer:\n#### 32", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 44208, 260, 1685, 282, 1885, 49161, 21990, 49163, 17243, 8842, 24, 49160, 42639, 18169, 107, 49160, 15009, 49161, 109, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49161, 17859, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49162, 17859, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49163, 109, 17243, 2771, 25, 28422, 49154, 49158, 198, 2068, 1042, 260, 1685, 282, 1885, 49161, 21990, 49163, 17243, 8842, 24, 49160, 42639, 18169, 107, 49160, 15009, 49161, 109, 42639, 18169, 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+{"group_id": "math_090", "origin": "budgie", "answer": "We can simplify the expression by grouping the terms into sets of four.\n\nLet's break down the expression step by step:\n\n\\begin{align*}\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right) &= 2^{4}\\cdot\\frac{1}{2^4}\\\\\n&= 2^4\\cdot\\frac{1}{16}\\\\\n&= 2^4\\cdot\\frac{1}{2^4}\\\\\n&= 2^4\\cdot\\frac{1}{2^4}\\\\\n&= 2^3\n\\end{align*}\n\nSo, the answer is $\\boxed{8}$.", "reward": -1, "advantage": -0.707106031187343, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 44208, 260, 1685, 282, 1885, 49161, 21990, 49163, 17243, 8842, 24, 49160, 42639, 18169, 107, 49160, 15009, 49161, 109, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49161, 17859, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49162, 17859, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49163, 109, 17243, 2771, 25, 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+{"group_id": "math_090", "origin": "verified_reference", "answer": "Solution 1\n\nMultiplying through, we obtain\n\n$$\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)=16+\\frac{16}{2}+\\frac{16}{4}+\\frac{16}{8}+\\frac{16}{16}=16+8+4+2+1=31\n$$\n\n## Solution 2\n\nUsing a common denominator inside the parentheses, we obtain\n\n$$\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)=16\\left(\\frac{16}{16}+\\frac{8}{16}+\\frac{4}{16}+\\frac{2}{16}+\\frac{1}{16}\\right)=16\\left(\\frac{31}{16}\\right)=31\n$$\n\nANSWER: 31", "reward": 1.0, "advantage": 1.4142120623746859, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 44208, 260, 1685, 282, 1885, 49161, 21990, 49163, 17243, 8842, 24, 49160, 42639, 18169, 107, 49160, 15009, 49161, 109, 42639, 18169, 107, 49160, 15009, 49161, 21990, 49161, 17859, 42639, 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+{"group_id": "math_091", "origin": "budgie", "answer": "5.  Let's consider a special case when $a=b=\\frac{1}{4}$.\n\n$$M=\\sqrt{1+a^{2}}+\\sqrt{1+2 a^{2}}$$\n\nSince $a=\\frac{1}{4}$, we have $a^{2}=1$.\n\nTherefore,\n$$M=\\sqrt{1+1}+\\sqrt{1+2 \\times 1}=\\sqrt{2}+\\sqrt{3}$$\n\nHence, $M=\\boxed{\\sqrt{2}+\\sqrt{3}}$.\n\nFinal answer:\n#### √2 + √3", "reward": -1, "advantage": -1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 10887, 2731, 1345, 2966, 1885, 81, 20, 284, 1885, 82, 20, 15765, 1885, 81, 27, 82, 45, 49160, 26265, 965, 1885, 61, 45, 20, 19573, 16166, 107, 49160, 27, 81, 21990, 49161, 17859, 42639, 16166, 107, 49160, 27, 49161, 278, 24362, 260, 15328, 599, 314, 49154, 49158, 198, 49164, 30, 216, 2959, 506, 1771, 253, 1767, 1671, 645, 1885, 81, 45, 82, 24814, 18169, 107, 49160, 15009, 49163, 24362, 30, 198, 198, 9212, 61, 24814, 16166, 107, 49160, 27, 81, 21990, 49161, 17859, 42639, 16166, 107, 49160, 27, 49161, 253, 21990, 49161, 17859, 9212, 198, 198, 7230, 1885, 81, 24814, 18169, 107, 49160, 15009, 49163, 24362, 28, 392, 457, 1885, 81, 21990, 49161, 109, 45, 49160, 28422, 198, 198, 17308, 28, 198, 9212, 61, 24814, 16166, 107, 49160, 27, 49160, 109, 42639, 16166, 107, 49160, 27, 49161, 3814, 14602, 216, 49160, 109, 24814, 16166, 107, 49161, 109, 42639, 16166, 107, 49162, 109, 9212, 198, 198, 35391, 28, 1885, 61, 24814, 4810, 277, 26754, 16166, 107, 49161, 109, 42639, 16166, 107, 49162, 17859, 28422, 198, 198, 29288, 2988, 42, 198, 1229, 19995, 244, 49161, 1232, 19995, 244, 49162, 49154], "prefix_len": 76, "old_logprobs": [-2.846531629562378, -0.03406040742993355, -4.166222095489502, -2.840928077697754, -1.116208553314209, -2.6269969940185547, -2.360008716583252, -2.644103765487671, -0.13075914978981018, -1.709085464477539, -0.08773389458656311, -0.1674066185951233, -0.45026957988739014, -0.4025063216686249, -1.9347126483917236, -0.021537235006690025, -0.016591312363743782, -0.004272262565791607, -0.003325529396533966, -2.3060858249664307, -0.01989259570837021, -0.32860100269317627, -2.343108654022217, -0.35041362047195435, -0.8137670755386353, -0.35823944211006165, -1.1011433601379395, -0.0266399085521698, -0.018920421600341797, -0.04576364532113075, -0.313053697347641, -0.9258660078048706, -0.06296231597661972, -0.00021419614495243877, -0.0037188436836004257, -0.025423860177397728, -0.0007682471186853945, -0.000458254711702466, -0.014812489040195942, -0.10120955854654312, -0.01616150699555874, -0.4749419093132019, -0.4295516908168793, -0.006863115355372429, -0.057760849595069885, -0.935051441192627, -0.027185779064893723, -0.03566214442253113, -1.4840718507766724, -0.1568746119737625, -0.1386883556842804, -1.4881163835525513, -0.004076742567121983, -0.0005445189890451729, -0.000426439888542518, -0.0002933310461230576, -0.023000596091151237, -0.07482971251010895, -0.2320021688938141, -0.432095468044281, -0.2007625550031662, -0.5616754293441772, -0.6401876211166382, -0.08956190943717957, -0.0032513870391994715, -0.005480265710502863, -0.22316892445087433, -0.08599911630153656, -1.439324140548706, -0.8431919813156128, -0.01872715912759304, -1.7364661693572998, -0.012653541751205921, -0.9566826820373535, -1.515427589416504, -0.06602709740400314, -0.05883089080452919, -0.08356757462024689, -0.015858128666877747, -0.0509844534099102, -0.1148604229092598, -0.6392412185668945, -0.08376426994800568, -0.025838203728199005, -0.0012767505832016468, -0.001587679609656334, -0.013515633530914783, -0.02587154321372509, -0.043960489332675934, -0.6111393570899963, -0.3930513262748718, -0.4097169041633606, -0.0012403184082359076, -0.007044715341180563, -0.8323615193367004, -0.13654327392578125, -0.014525854028761387, -0.01672789640724659, -0.005758363753557205, -0.29382699728012085, -0.0014293702552095056, -0.0010836211731657386, -0.002593608107417822, -0.004586414899677038, -0.011721429415047169, -0.007652727887034416, -0.01286739856004715, -1.8427907228469849, -0.0032561400439590216, -1.6527280807495117, -0.025943096727132797, -0.5494641065597534, -0.5537412166595459, -7.152555099310121e-07, -0.08994169533252716, -0.005313678178936243, -0.0016684910515323281, -0.10787931084632874, -0.002161073498427868, -0.26456332206726074, -0.000417741306591779, -5.411955135059543e-05, -0.00019453064305707812, -0.0007630059262737632, -0.19240330159664154, -0.6442713737487793, -0.11247054487466812, -0.19564655423164368, -0.0009012927766889334, -0.0019124329555779696, -0.11099318414926529, -0.003806845983490348, -3.2233340740203857, -0.05750877410173416, -0.03351528197526932, -0.14153139293193817, -0.006622391752898693, -0.0022782341111451387, -0.0006158839096315205, -0.028644032776355743]}
+{"group_id": "math_091", "origin": "budgie", "answer": "To find the integer part of $M$, we start by simplifying the expression inside the square root.\n\n\\begin{align*}\n\\sqrt{1+2 b} &= \\sqrt{1+2 b} \\\\\n&= \\sqrt{1+2 b} \\cdot \\frac{1}{\\sqrt{1+2 b} + 1} \\\\\n&= \\sqrt{1+2 b} \\\\\n&= \\frac{1+2 b}{\\sqrt{1+2 b} + 1}\n\\end{align*}\n\nThen, we rewrite the expression for $M$ as follows:\n\n\\begin{align*}\nM &= \\sqrt{1+2 b} \\\\\n&= \\frac{1+2 b}{\\sqrt{1+2 b} + 1} \\\\\n&= \\frac{1+2 b}{\\frac{1+2 b}{\\sqrt{1+2 b}+1}} \\\\\n&= \\frac{(1+2 b)^{3/2}}{\\frac{(1+2 b)^{3/2}}{\\sqrt{1+2 b}+1}} \\\\\n&= (1+2 b)^{3/2} \\cdot \\frac{\\sqrt{1+2 b}+1}{\\sqrt{1+2 b}+1} \\\\\n&= (1+2 b)^{3/2} \\\\cdot \\frac{\\sqrt{1+2 b}}{\\sqrt{1+2 b} + 1} \\\\\n&= (1+2 b)^{3/2} \\\\cdot \\frac{\\sqrt{1+2 b}}{\\sqrt{1+2 b} + 1} \\\\\n&= (1 + 2 b)^{3/2} \\\\", "reward": 0.5, "advantage": 0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 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+{"group_id": "math_091", "origin": "verified_reference", "answer": "5. 2\n\nTranslate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.", "reward": 1.0, "advantage": 0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49164, 30, 10887, 2731, 1345, 2966, 1885, 81, 20, 284, 1885, 82, 20, 15765, 1885, 81, 27, 82, 45, 49160, 26265, 965, 1885, 61, 45, 20, 19573, 16166, 107, 49160, 27, 81, 21990, 49161, 17859, 42639, 16166, 107, 49160, 27, 49161, 278, 24362, 260, 15328, 599, 314, 49154, 49158, 198, 49164, 30, 216, 49161, 198, 198, 8241, 12837, 260, 1694, 2120, 618, 2321, 28, 5361, 10756, 260, 2798, 1694, 506, 1761, 9102, 284, 4624, 28, 284, 3124, 260, 8483, 966, 3319, 30, 49154], "prefix_len": 76, "old_logprobs": [-2.881516218185425, -0.03053673729300499, -4.172260284423828, -3.2807745933532715, -4.813244819641113, -0.12185550481081009, -10.604605674743652, -1.077478051185608, -0.30144616961479187, -4.413824558258057, -5.857640266418457, -1.492982268333435, -0.5671310424804688, -1.210426688194275, -3.9845781326293945, -2.5507583618164062, -0.1242259219288826, -0.8155735731124878, -0.5026180744171143, -0.251913845539093, -2.529411792755127, -0.149778351187706, -0.8450430631637573, -1.4876989126205444, -0.2718856632709503, -0.1869080662727356, -3.0427935123443604, -0.18700121343135834, -1.4640156030654907, -0.5077035427093506, -1.4796782732009888, -0.9137620329856873, -0.8676198124885559]}
+{"group_id": "math_092", "origin": "budgie", "answer": "The simplest geometric reasoning follows from the problem statement: the vertices of a rectangle inscribed in a circle divide it into four arcs, and the distance from the midpoint of one of the larger arcs to the vertices of the rectangle is equal to the sum of the lengths of the arcs.\n\nFinal answer:\n#### 24", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 49164, 49166, 30, 378, 24796, 282, 253, 23710, 30036, 281, 253, 7842, 10335, 357, 618, 1876, 36260, 30, 7985, 260, 4052, 429, 260, 3921, 3154, 282, 582, 282, 260, 3227, 36260, 288, 260, 24796, 282, 260, 23710, 28, 585, 624, 5882, 359, 216, 49161, 49163, 284, 216, 49166, 5973, 30, 49154, 49158, 198, 504, 17519, 15932, 10115, 5558, 429, 260, 1732, 4814, 42, 260, 24796, 282, 253, 23710, 30036, 281, 253, 7842, 10335, 357, 618, 1876, 36260, 28, 284, 260, 4052, 429, 260, 3921, 3154, 282, 582, 282, 260, 3227, 36260, 288, 260, 24796, 282, 260, 23710, 314, 4037, 288, 260, 2028, 282, 260, 15900, 282, 260, 36260, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49161, 49163, 49154], "prefix_len": 81, "old_logprobs": [-0.9225724935531616, -8.11801815032959, -2.475101947784424, -2.399379253387451, -2.098318099975586, -0.10373091697692871, -0.10450682789087296, -3.714146137237549, -1.0484709739685059, -1.7804032564163208, -1.3354288339614868, -3.129392385482788, -0.26005586981773376, -0.1865401417016983, -0.016722973436117172, -0.06445910781621933, -0.0024624522775411606, -0.019344717264175415, -0.0011403016978874803, -0.10942763090133667, -0.01949448697268963, -0.0004804172203876078, -0.050700411200523376, -0.0034445736091583967, -1.9437302350997925, -0.7639788389205933, -0.3310529589653015, -1.5954161882400513, -0.02484295703470707, -0.012217800132930279, -0.032082244753837585, -0.0010390841634944081, -0.0017922543920576572, -0.01971670426428318, -0.004711597226560116, -0.1482938677072525, -0.019299348816275597, -0.0033154303673654795, -0.012198487296700478, -0.015538342297077179, -0.028347894549369812, -0.06154847517609596, -0.0026814716402441263, -0.0013087764382362366, -0.1445506364107132, -1.2931203842163086, -0.025262190029025078, -0.3255472481250763, -3.166843891143799, -0.021751299500465393, -0.09776949882507324, -1.4930616617202759, -0.0056566642597317696, -0.16373437643051147, -1.9984781742095947, -0.733363926410675, -0.35564538836479187, -0.008282124996185303, -0.18282464146614075, -0.0004895919119007885, -0.0008877150830812752, -0.16668610274791718, -0.0015323336701840162, -0.016827890649437904, -0.2285822331905365, -0.20685425400733948, -0.27670755982398987]}
+{"group_id": "math_092", "origin": "budgie", "answer": "To find the distance from the midpoint of one of the larger arcs to the vertices of the rectangle, we need to first understand the geometric relationship between the larger arcs and the rectangle.\n\nThe smaller arcs can be seen as parts of a circle. The larger arcs are the ones that make up the rectangle. These larger arcs, when considered as a whole, will form a full circle.\n\nThe distance from the midpoint of one of the larger arcs to the vertices of the rectangle can be found by considering the inscribed rectangle and the large circle. The rectangle is inscribed in the circle, and the vertices of the rectangle are the points where the circle touches the sides of the rectangle.\n\nSince the rectangle is inscribed in the circle, the diagonals of the rectangle are equal to the diameter of the circle. Let's denote the distance from the midpoint of one of the larger arcs to the vertices of the rectangle as d.\n\nThe diagonals of a rectangle bisect each other, so the distance from the midpoint of one of the larger arcs to the vertices of the rectangle can be found using the Pythagorean theorem in the right triangle formed by the radius of the circle (24 cm), half the length of the larger arc, and half the length of the smaller arc.\n\nThe distance d is equal to the length of the midpoint of the larger arc divided by 2 (since the two halves of the arc are equal in length).\n\n\\[ d = \\frac{24}{2} \\]\n\n\\[ d = 12 \\text{ cm} \\]\n\nTherefore, the distance from the midpoint of one of the larger arcs to the vertices of the rectangle is 12 cm.\n\n\\[ \\boxed{12} \\]", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 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+{"group_id": "math_092", "origin": "verified_reference", "answer": "1.57. Since $A C$ is the diameter of the circle (Fig. 1.44), then $R=0.5 \\sqrt{24^{2}+7^{2}}=12.5 \\quad$ (cm). In $\\triangle B O F$ we have $O F=$ $=\\sqrt{O B^{2}-B F^{2}}=\\sqrt{12.5^{2}-12^{2}}=3.5$ (cm); hence, $M F=12.5-3.5=9$ (cm), $M K=12.5+3.5=16$ (cm). From $\\triangle M B F$ and $\\triangle M A K$ we find the required distances: $\\quad M B=\\sqrt{M F^{2}+B F^{2}}=\\sqrt{9^{2}+12^{2}}=15(\\mathrm{~cm})$, $M A=\\sqrt{M K^{2}+K A^{2}}=\\sqrt{16^{2}+12^{2}}=20(\\mathrm{~cm})$.\n\nAnswer: 15 and 20 cm.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 30, 49164, 49166, 30, 378, 24796, 282, 253, 23710, 30036, 281, 253, 7842, 10335, 357, 618, 1876, 36260, 30, 7985, 260, 4052, 429, 260, 3921, 3154, 282, 582, 282, 260, 3227, 36260, 288, 260, 24796, 282, 260, 23710, 28, 585, 624, 5882, 359, 216, 49161, 49163, 284, 216, 49166, 5973, 30, 49154, 49158, 198, 49160, 30, 49164, 49166, 30, 4311, 1885, 49, 340, 20, 314, 260, 8805, 282, 260, 7842, 365, 10462, 30, 216, 49160, 30, 49163, 49163, 643, 965, 1885, 66, 45, 49159, 30, 49164, 3814, 16166, 107, 49161, 49163, 21990, 49161, 109, 27, 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+{"group_id": "math_093", "origin": "budgie", "answer": "15. The area of square $\\mathrm{ABCD}$ is 196 square centimeters. The square is divided into 4 smaller squares by dividing each side into two equal segments. Therefore, the side length of each smaller square is 12 cm.\n\nThe area of the shaded part is 1 small square cm.\n\nFinal answer:\n#### 1", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49164, 30, 1032, 3057, 281, 260, 4110, 28, 260, 1557, 282, 5222, 19573, 8898, 5167, 107, 5431, 6998, 24362, 314, 216, 49160, 49168, 49165, 5222, 25556, 28, 284, 357, 3593, 827, 11729, 20168, 3764, 16207, 30, 378, 3227, 282, 260, 827, 3764, 16207, 553, 354, 1557, 338, 314, 216, 49163, 1711, 260, 1557, 282, 260, 3764, 582, 28, 284, 260, 20168, 1557, 282, 260, 827, 16207, 314, 216, 49160, 5222, 46595, 30, 4882, 28, 260, 1557, 282, 260, 32416, 599, 314, 19573, 97, 32984, 20, 5222, 25556, 30, 49154, 49158, 198, 49160, 49164, 30, 378, 1557, 282, 5222, 19573, 8898, 5167, 107, 5431, 6998, 24362, 314, 216, 49160, 49168, 49165, 5222, 25556, 30, 378, 5222, 314, 6066, 618, 216, 49163, 3764, 16207, 411, 15213, 971, 2103, 618, 827, 4037, 12523, 30, 4882, 28, 260, 2103, 3519, 282, 971, 3764, 5222, 314, 216, 49160, 49161, 5973, 30, 198, 198, 504, 1557, 282, 260, 32416, 599, 314, 216, 49160, 1165, 5222, 5973, 30, 198, 198, 29288, 2988, 42, 198, 1229, 216, 49160, 49154], "prefix_len": 118, "old_logprobs": [-0.5755974650382996, -1.2770622968673706, -0.02278691530227661, -1.571797251701355, -0.85223388671875, -0.020022420212626457, -1.0354936122894287, -0.5070701241493225, -0.00289022084325552, -0.00023934361524879932, -1.8596476365928538e-05, -0.12913626432418823, -0.0025201248936355114, -0.004853609949350357, -0.0412442684173584, -0.20215484499931335, -0.009424482472240925, -0.012954599224030972, -0.0005149707430973649, -0.05195244774222374, -0.011320053599774837, -1.0292720794677734, -0.8696209788322449, -3.043081283569336, -2.2533950805664062, -0.5496581196784973, -0.09482023864984512, -1.236257553100586, -1.5183311700820923, -0.48693495988845825, -0.09079738706350327, -1.2180376052856445, -3.796616554260254, -1.563967227935791, -0.16696274280548096, -0.4388931095600128, -1.4668605327606201, -0.13274359703063965, -2.5360066890716553, -0.5679627656936646, -2.532691478729248, -0.011691974475979805, -0.29342275857925415, -0.8611952066421509, -0.06697005778551102, -0.019261227920651436, -0.5855500102043152, -0.17280147969722748, -0.006078921724110842, -0.01304708793759346, -0.43452200293540955, -0.5153100490570068, -1.5189309120178223, -0.9432300329208374, -0.14479860663414001, -0.99957275390625, -0.40196138620376587, -1.1500089168548584, -0.4706791043281555, -0.005474337842315435, -0.6900084018707275, -0.892620325088501, -0.19035573303699493, -0.17088738083839417, -1.4987080097198486, -0.4772089123725891, -5.316197872161865, -0.027098534628748894, -1.2322791814804077, -0.5750228762626648, -1.0874078273773193, -0.1757064014673233, -1.173536777496338, -0.0005969410995021462, -0.0020638848654925823, -0.046831920742988586, -0.000998951611109078, -0.060320522636175156, -0.005276918411254883, -0.031155481934547424]}
+{"group_id": "math_093", "origin": "budgie", "answer": "1. We need to find the area of the shaded part. Since the larger of the two smaller squares has an area of 4 times the area of the smaller one, and the overlapping area of the two squares is a square centimeter, the total shaded area is 4 cm^2.\n2. Now, we need to find the side length of the larger square. From the diagram, we can see that the total area of the larger square is 196 cm^2, which is equal to the side length squared. Let $s$ be the side length of the larger square, then:\n$$s^2 = 196$$\n\nSolving for $s$, we get:\n$$s = \\sqrt{196} = 14\\, cm$$\n\nNow that we have the side length of the larger square, we can find its area. We can do this by multiplying the side length by itself twice:\n$$s^2 = (s)(s) = (14) \\times (14) = 196\\, cm^2$$\n\nTherefore, the area of the shaded part is:\n$$\\boxed{196\\, cm^2}$$", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49164, 30, 1032, 3057, 281, 260, 4110, 28, 260, 1557, 282, 5222, 19573, 8898, 5167, 107, 5431, 6998, 24362, 314, 216, 49160, 49168, 49165, 5222, 25556, 28, 284, 357, 3593, 827, 11729, 20168, 3764, 16207, 30, 378, 3227, 282, 260, 827, 3764, 16207, 553, 354, 1557, 338, 314, 216, 49163, 1711, 260, 1557, 282, 260, 3764, 582, 28, 284, 260, 20168, 1557, 282, 260, 827, 16207, 314, 216, 49160, 5222, 46595, 30, 4882, 28, 260, 1557, 282, 260, 32416, 599, 314, 19573, 97, 32984, 20, 5222, 25556, 30, 49154, 49158, 198, 49160, 30, 1046, 737, 288, 1042, 260, 1557, 282, 260, 32416, 599, 30, 4311, 260, 3227, 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+{"group_id": "math_093", "origin": "verified_reference", "answer": "【Answer】 72\n【Analysis】Key point: Skillful area calculation\nThe area of the square is 196, so the side length is 14. The overlapping area is 1, so the side length is 1; the area of the larger square is 4 times that of the smaller square, so the side length of the larger square is twice that of the smaller square, and the sum of the side lengths of the larger and smaller squares is $14+1=15$.\nTherefore, the side length of the smaller square is $15 \\div 3=5$, and the side length of the larger square is $5 \\times 2=10$.\nThe area of the smaller rectangle is $(5-1) \\times(10-1)=36$, so the area of the two smaller rectangles is $36 \\times 2=72\\left(\\mathrm{~cm}^{2}\\right)$.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49160, 49164, 30, 1032, 3057, 281, 260, 4110, 28, 260, 1557, 282, 5222, 19573, 8898, 5167, 107, 5431, 6998, 24362, 314, 216, 49160, 49168, 49165, 5222, 25556, 28, 284, 357, 3593, 827, 11729, 20168, 3764, 16207, 30, 378, 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+{"group_id": "math_094", "origin": "budgie", "answer": "321. $\\sqrt{2x+5}+\\sqrt{x-1}=8$\n\nThe final answer is 321.", "reward": -0.5, "advantage": -0.7071057811879616, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 49161, 49160, 30, 19573, 16166, 107, 49161, 1792, 27, 49164, 109, 42639, 16166, 107, 104, 29, 49160, 109, 45, 49167, 28422, 198, 198, 8241, 12837, 260, 1694, 2120, 618, 2321, 28, 4825, 260, 2798, 1694, 506, 1761, 9102, 284, 4624, 28, 284, 3124, 260, 8483, 966, 3319, 30, 1116, 49162, 49161, 49160, 30, 19573, 16166, 107, 49161, 1792, 27, 49164, 109, 42639, 16166, 107, 104, 29, 49160, 109, 45, 49167, 28422, 49154, 49158, 198, 49162, 49161, 49160, 30, 19573, 16166, 107, 49161, 104, 27, 49164, 109, 42639, 16166, 107, 104, 29, 49160, 109, 45, 49167, 20, 198, 198, 504, 3403, 2988, 314, 216, 49162, 49161, 49160, 30, 49154], "prefix_len": 101, "old_logprobs": [-1.4745551347732544, -0.02883770316839218, -0.004770801402628422, -0.06657921522855759, -0.3931163549423218, -0.006997010204941034, -0.001413180842064321, -0.022863933816552162, -2.1348114013671875, -0.031304072588682175, -0.0021686863619834185, -0.02322261780500412, -0.07187259942293167, -0.0005793085438199341, -0.00027044929447583854, -0.002303448738530278, -0.0014575821114704013, -6.639736966462806e-05, -0.002167734783142805, -0.004769971128553152, -0.00416210200637579, -1.7389655113220215, -1.1067054271697998, -0.00705856503918767, -2.281137704849243, -1.993486762046814, -0.018200531601905823, -0.28281915187835693, -0.650902509689331, -0.9431203007698059, -0.043964024633169174, -0.007983086630702019, -0.7139750719070435, -0.09577511996030807]}
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+{"group_id": "math_094", "origin": "verified_reference", "answer": "Solution. Isolate one of the radicals and square both sides of the equation:\n\n$$\n(\\sqrt{2 x+5})^{2}=(8-\\sqrt{x-1})^{2} ; 2 x+5=64-16 \\sqrt{x-1}+(x-1)\n$$\n\nMove $16 \\sqrt{x-1}$ to the left side, and all other terms to the right side:\n\n$$\n16 \\sqrt{x-1}=64+x-1-2 x-5 ; 16 \\sqrt{x-1}=58-x\n$$\n\nSquare both sides of the equation again:\n\n$(16 \\sqrt{x-1})^{2}=(58-x)^{2} ; 256 x-256=3364-116 x+x^{2} ; x^{2}-372 x+3620=0$.\n\nSolve the resulting quadratic equation. Since $a=1, b=-372$, $c=3620, \\quad D=b^{2}-4 a c=(-372)^{2}-4 \\cdot 1 \\cdot 3620=138384-14480=123904$; $\\sqrt{D}=352$, then\n\n$$\nx_{1}=\\frac{-b-\\sqrt{D}}{2 a}=\\frac{372-352}{2}=10 ; \\quad x_{2}=\\frac{-b+\\sqrt{D}}{2 a}=\\frac{372+352}{2}=362\n$$\n\nCheck both values $x_{1}=10 ; x_{2}=362$.\n\nIf $x=10$, then $\\sqrt{2 \\cdot 10+5}+\\sqrt{10-1}=8 ; \\sqrt{25}+\\sqrt{9}=8 ; 5+3=8$; $8=8$\n\nIf $x=362$, then $\\sqrt{2 \\cdot 362+5}+\\sqrt{362-1}=8 ; \\sqrt{729}+\\sqrt{361}=8 ; 27+$ $+19 \\neq 8$; therefore, $x=362$ is an extraneous root.\n\nThe root of the given equation is $x=10$.", "reward": 1.0, "advantage": 1.4142115623759233, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 1116, 49162, 49161, 49160, 30, 19573, 16166, 107, 49161, 1792, 27, 49164, 109, 42639, 16166, 107, 104, 29, 49160, 109, 45, 49167, 28422, 198, 198, 8241, 12837, 260, 1694, 2120, 618, 2321, 28, 4825, 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+{"group_id": "math_095", "origin": "budgie", "answer": "1. **Define the Polynomial and its Properties:**\n\n   A polynomial with integer coefficients is given by:\n   \\[\n   P(x) = a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0\n   \\]\n   where \\(a_i\\) are integers.\n\n2. **Given Conditions:**\n\n   We know:\n   \\[\n   P(1) = 2019, \\quad P(2019) = 1, \\quad P(k) = k\n   \\]\n\n3. **Analyze the Polynomial at \\(x = \\frac{1}{k}\\):**\n\n   Since \\(P(1) = 2019\\), substituting \\(x = 1\\) gives:\n   \\[\n   2019 = a_4 + a_3 + a_2 + a_1 + a_0\n   \\]\n   Therefore:\n   \\[\n   2019 = a_4 + a_3 + a_2 + a_1 + a_0 \\implies a_4 + a_3 + a_2 + a_1 = 1\n   \\]\n\n4. **Analyze the Polynomial at \\(x = \\frac{2}{k}\\):**\n\n   Similarly, substituting \\(x = 2019\\) into the polynomial yields:\n   \\[\n   2019 = a_4(2019)^4 + a_3(2019)^3 + a_2(2019)^2 + a_1(2019) + a_0\n   \\]\n   Thus:\n   \\[\n   2019 = a_4(2019)^4 + a_", "reward": -1, "advantage": -0.9805795218461241, "ids": [49153, 49157, 198, 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+{"group_id": "math_095", "origin": "budgie", "answer": "1998", "reward": -0.5, "advantage": -0.39223180873844965, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49162, 30, 198, 198, 49, 23539, 1885, 64, 24, 104, 38224, 351, 15328, 20432, 553, 260, 3849, 198, 198, 9212, 198, 64, 24, 49160, 36637, 49161, 49159, 49160, 49168, 28, 3814, 32984, 377, 24, 49161, 49159, 49160, 49168, 36637, 49160, 28, 3814, 32984, 377, 24, 91, 36637, 91, 28, 198, 9212, 198, 198, 2942, 260, 1230, 1885, 91, 20, 314, 354, 15328, 30, 7985, 451, 1230, 1885, 91, 28422, 198, 198, 19, 49154, 49158, 198, 49160, 49168, 49168, 49167, 49154], "prefix_len": 105, "old_logprobs": [-1.392725944519043, -2.4470789432525635, -1.060302734375, -2.2288358211517334, -1.0910454988479614]}
+{"group_id": "math_095", "origin": "verified_reference", "answer": "# Solution.\n\nSince the polynomial $P(x)$ has integer coefficients, $P(a)-P(b)$ is divisible by $a-b$ for any integers $a$ and $b$.\n\nWe get that\n\n$$\n\\begin{gathered}\nP(k)-P(1)=(k-2019) \\text { is divisible by }(k-1), \\\\\nP(k)-P(2019)=(k-1) \\text { is divisible by }(k-2019) .\n\\end{gathered}\n$$\n\nThis can only be true if $|k-1|=|k-2019|$.\n\nThe solution to the obtained equation is $k=1010$.\n\nAnswer: $\\quad k=1010$.", "reward": 1.0, "advantage": 1.3728113305845737, "ids": [49153, 49157, 198, 16339, 307, 451, 6530, 1732, 5054, 30, 13843, 19484, 2545, 515, 10115, 284, 1830, 260, 3403, 2988, 4763, 418, 260, 1112, 30, 198, 198, 19, 15509, 216, 49162, 30, 198, 198, 49, 23539, 1885, 64, 24, 104, 38224, 351, 15328, 20432, 553, 260, 3849, 198, 198, 9212, 198, 64, 24, 49160, 36637, 49161, 49159, 49160, 49168, 28, 3814, 32984, 377, 24, 49161, 49159, 49160, 49168, 36637, 49160, 28, 3814, 32984, 377, 24, 91, 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diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_encoding_manifest.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_encoding_manifest.json
new file mode 100644
index 0000000000000000000000000000000000000000..e7323ffc7ac8516982e68c83c85d9b0b720feed7
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_encoding_manifest.json
@@ -0,0 +1,7 @@
+{
+  "groups": 72,
+  "encoded": 216,
+  "mean_tokens": 443.5925925925926,
+  "mean_response_tokens": 307.73148148148147,
+  "seconds": 3.2618279457092285
+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_rl_manifest.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_rl_manifest.json
new file mode 100644
index 0000000000000000000000000000000000000000..a670ea55eabd68a5f304b36241e859521ea47f91
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_rl_manifest.json
@@ -0,0 +1,13 @@
+{
+  "scanned": 341,
+  "symbolically_verified_available": 120,
+  "selected": 96,
+  "rejections": {
+    "verify": 213,
+    "length": 7
+  },
+  "benchmark_family_rows": 0,
+  "decontamination": "normalized exact + fixed-gate shared 12-token windows",
+  "source": "open-r1/OpenR1-Math-220k",
+  "verification": "math_verify(reference solution, reference answer)"
+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_rl_pool.jsonl b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_rl_pool.jsonl
new file mode 100644
index 0000000000000000000000000000000000000000..80c7e214a781c43248ef46335aaf3598ec4dea3a
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_rl_pool.jsonl
@@ -0,0 +1,96 @@
+{"problem": "Lines $l_1^{}$ and $l_2^{}$ both pass through the origin and make first-quadrant angles of $\\frac{\\pi}{70}$ and $\\frac{\\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l^{}_{}$, the transformation $R(l)^{}_{}$ produces another line as follows: $l^{}_{}$ is reflected in $l_1^{}$, and the resulting line is reflected in $l_2^{}$. Let $R^{(1)}(l)=R(l)^{}_{}$ and $R^{(n)}(l)^{}_{}=R\\left(R^{(n-1)}(l)\\right)$. Given that $l^{}_{}$ is the line $y=\\frac{19}{92}x^{}_{}$, find the smallest positive integer $m^{}_{}$ for which $R^{(m)}(l)=l^{}_{}$.", "solution": "Let $l$ be a line that makes an angle of $\\theta$ with the positive $x$-axis. Let $l'$ be the reflection of $l$ in $l_1$, and let $l''$ be the reflection of $l'$ in $l_2$. \nThe angle between $l$ and $l_1$ is $\\theta - \\frac{\\pi}{70}$, so the angle between $l_1$ and $l'$ must also be $\\theta - \\frac{\\pi}{70}$. Thus,  $l'$ makes an angle of $\\frac{\\pi}{70}-\\left(\\theta-\\frac{\\pi}{70}\\right) = \\frac{\\pi}{35}-\\theta$ with the positive $x$-axis.  \nSimilarly, since the angle between $l'$ and $l_2$ is $\\left(\\frac{\\pi}{35}-\\theta\\right)-\\frac{\\pi}{54}$, the angle between $l''$ and the positive $x$-axis is $\\frac{\\pi}{54}-\\left(\\left(\\frac{\\pi}{35}-\\theta\\right)-\\frac{\\pi}{54}\\right) = \\frac{\\pi}{27}-\\frac{\\pi}{35}+\\theta = \\frac{8\\pi}{945} + \\theta$. \nThus, $R(l)$ makes an $\\frac{8\\pi}{945} + \\theta$ angle with the positive $x$-axis. So $R^{(n)}(l)$ makes an $\\frac{8n\\pi}{945} + \\theta$ angle with the positive $x$-axis. \nTherefore, $R^{(m)}(l)=l$ iff $\\frac{8m\\pi}{945}$ is an integral multiple of $\\pi$. Thus, $8m \\equiv 0\\pmod{945}$. Since $\\gcd(8,945)=1$, $m \\equiv 0 \\pmod{945}$, so the smallest positive integer $m$ is $\\boxed{945}$.", "answer": "945", "source": "amc_aime", "problem_type": "Geometry", "uuid": "8791f647-68c1-5dec-b9f3-b235014b5116"}
+{"problem": "9. Given that $M N$ is a moving chord of the circumcircle of equilateral $\\triangle A B C$ with side length $2 \\sqrt{6}$, $M N=4$, and $P$ is a moving point on the sides of $\\triangle A B C$. Then the maximum value of $\\overrightarrow{M P} \\cdot \\overrightarrow{P N}$ is", "solution": "9.4.\n\nLet the circumcenter of $\\triangle ABC$ be $O$.\nIt is easy to find that the radius of $\\odot O$ is $r=2 \\sqrt{2}$.\nAlso, $MN=4$, so $\\triangle OMN$ is an isosceles right triangle, and\n$$\n\\begin{aligned}\n& \\overrightarrow{O M} \\cdot \\overrightarrow{O N}=0,|\\overrightarrow{O M}+\\overrightarrow{O N}|=4 . \\\\\n& \\text { Let }|\\overrightarrow{O P}|=x . \\\\\n& \\text { Then } \\sqrt{2} \\leqslant x \\leqslant 2 \\sqrt{2} \\text {, and } \\\\\n& \\overrightarrow{M P} \\cdot \\overrightarrow{P N}=(\\overrightarrow{O P}-\\overrightarrow{O M}) \\cdot(\\overrightarrow{O N}-\\overrightarrow{O P}) \\\\\n& =-\\overrightarrow{O P}{ }^{2}+(\\overrightarrow{O M}+\\overrightarrow{O N}) \\cdot \\overrightarrow{O P}-\\overrightarrow{O M} \\cdot \\overrightarrow{O N} \\\\\n= & -x^{2}+(\\overrightarrow{O M}+\\overrightarrow{O N}) \\cdot \\overrightarrow{O P} \\\\\n\\leqslant & -x^{2}+|\\overrightarrow{O M}+\\overrightarrow{O N}||\\overrightarrow{O P}| \\\\\n= & -x^{2}+4 x=4-(x-2)^{2} \\leqslant 4,\n\\end{aligned}\n$$\n\nThe equality holds if and only if $x=2$ and $\\overrightarrow{O P}$ is in the same direction as $\\overrightarrow{O M}+\\overrightarrow{O N}$.\nTherefore, the maximum value of $\\overrightarrow{M P} \\cdot \\overrightarrow{P N}$ is 4.", "answer": "4", "source": "cn_contest", "problem_type": "Geometry", "uuid": "fea23b7f-e22e-521d-be6d-4c469b94d965"}
+{"problem": "25. Anna, Bridgit and Carol run in a $100 \\mathrm{~m}$ race. When Anna finishes, Bridgit is $16 \\mathrm{~m}$ behind her and when Bridgit finishes, Carol is $25 \\mathrm{~m}$ behind her. The girls run at constant speeds throughout the race. How far behind was Carol when Anna finished?\nA $37 \\mathrm{~m}$\nB $41 \\mathrm{~m}$\nC $50 \\mathrm{~m}$\nD $55 \\mathrm{~m}$\nE $60 \\mathrm{~m}$", "solution": "25. A Carol finishes 25 metres behind Bridgit, so she travels 75 metres while Bridgit runs 100 metres. Therefore she runs 3 metres for every 4 metres Bridgit runs. When Anna finishes, Bridgit has run 84 metres, so that at that time Carol has run $\\frac{3}{4} \\times 84$ metres $=63$ metres. Hence Carol finishes $(100-63)$ metres $=37$ metres behind Anna.", "answer": "37\\mathrm{~}", "source": "olympiads", "problem_type": "Algebra", "uuid": "f7bf0e57-7e91-5810-bab6-d1e4ad0d4461"}
+{"problem": "4. The sequence $\\left\\{x_{n}\\right\\}$ is defined as follows:\n$$\nx_{1}=\\frac{2}{3}, x_{n+1}=\\frac{x_{n}}{2(2 n+1) x_{n}+1}\\left(n \\in \\mathbf{Z}_{+}\\right) \\text {. }\n$$\n\nThen $x_{1}+x_{2}+\\cdots+x_{2014}=$ $\\qquad$", "solution": "4. $\\frac{4028}{4029}$.\n\nBy definition, $\\frac{1}{x_{n+1}}=2(2 n+1)+\\frac{1}{x_{n}}$.\nLet $y_{n}=\\frac{1}{x_{n}}$.\nThus, $y_{1}=\\frac{3}{2}$, and $y_{n+1}-y_{n}=2(2 n+1)$.\nTherefore, $y_{n}=\\left(y_{n}-y_{n-1}\\right)+\\left(y_{n-1}-y_{n-2}\\right)+\\cdots+$ $\\left(y_{2}-y_{1}\\right)+y_{1}$\n$=2[(2 n-1)+(2 n-3)+\\cdots+3]+\\frac{3}{2}$\n$=2\\left(n^{2}-1\\right)+\\frac{3}{2}=\\frac{1}{2}(2 n+1)(2 n-1)$,\n$\\sum_{i=1}^{2014} x_{i}=\\sum_{i=1}^{2014} \\frac{1}{y_{i}}=\\sum_{i=1}^{2014} \\frac{2}{(2 i+1)(2 i-1)}$\n$=\\sum_{i=1}^{2014}\\left(\\frac{1}{2 i-1}-\\frac{1}{2 i+1}\\right)$\n$=1-\\frac{1}{4029}=\\frac{4028}{4029}$.", "answer": "\\frac{4028}{4029}", "source": "olympiads", "problem_type": "Algebra", "uuid": "8c80f4bc-5be2-599b-b225-3c48179b1052"}
+{"problem": "Example 6 The rules of a \"level-up game\" stipulate: On the $n$-th level, a die must be rolled $n$ times. If the sum of the points obtained from these $n$ rolls is greater than $2^{n}$, the level is considered passed. Questions:\n(1) What is the maximum number of levels a person can pass in this game?\n(2) What is the probability that he can pass the first three levels consecutively?\n(Note: A die is a uniform cube with points numbered $1,2,3,4,5,6$ on its faces. The number of points on the face that lands up after rolling the die is the result of the roll.)", "solution": "Solution: Since the dice is a uniform cube, the possibility of each number appearing after a roll is equal.\n(1) Because the maximum number that can appear on a single dice is 6, and $6 \\times 1 > 2^4, 6 \\times 5 < 2$.\nTherefore, when $n \\geqslant 5$, the sum of the numbers that appear after rolling the dice $n$ times cannot be greater than 2, which means this is an impossible event, and the probability of passing is 0.\nSo, the maximum number of consecutive levels that can be passed is 4.\n(2) Let event $A_{n}$ be \"failing the $n$-th level\", then the complementary event $\\overline{A_{n}}$ is \"passing the $n$-th level\". In the $n$-th level game, the total number of basic events is $6^n$.\nLevel 1: The number of basic events contained in event $A_{1}$ is 2 (i.e., the cases where the number is 1 or 2),\nSo the probability of passing this level is: $P\\left(\\overline{A_{1}}\\right)=1-P\\left(A_{1}\\right)=1-\\frac{2}{6}=\\frac{2}{3}$\nLevel 2: The number of basic events contained in event $A_{2}$ is the sum of the number of positive integer solutions to the equation $x+y=a$ when $a$ takes the values $2,3,4$. That is, $C_{1}^{3}+C_{3}^{1}+C_{3}^{1}=1+2+3=6$ (cases).\nSo the probability of passing this level is $P\\left(\\bar{A}_{2}\\right)=1-P\\left(A_{2}\\right)=1-\\frac{6}{6^{2}}=\\frac{5}{6}$\nLevel 3: The number of basic events contained in event $A_{3}$ is the sum of the number of positive integer solutions to the equation $x+y+z=a$ when $a$ takes the values $3,4,5,6,7,8$. That is, $C_{2}^{3}+C_{3}^{3}+C_{4}^{2}+C_{5}^{3}+C_{6}^{3}+C_{5}^{2}=1+3+6+10+15+21=56$ (cases).\nSo the probability of passing this level is: $P\\left(\\bar{A}_{3}\\right)=1-P\\left(A_{3}\\right)=1-\\frac{56}{6^{3}}=\\frac{20}{27}$\nTherefore, the probability of passing the first 3 levels consecutively is: $P\\left(\\bar{A}_{1}\\right) \\cdot P\\left(\\bar{A}_{2}\\right) \\cdot P\\left(\\bar{A}_{3}\\right)=\\frac{2}{3} \\times \\frac{5}{6} \\times \\frac{20}{27}=\\frac{100}{243}$", "answer": "\\frac{100}{243}", "source": "olympiads", "problem_type": "Combinatorics", "uuid": "96c9bf5f-cffe-505b-80c0-e679e10d2cf9"}
+{"problem": "6. (1999 American High School Mathematics Exam) A regular tetrahedron has two spheres, one inscribed and one circumscribed. Additionally, there are four smaller spheres between each face of the tetrahedron and the circumscribed sphere, each tangent to the center of the face. Point $P$ is a point inside the circumscribed sphere. The probability that $P$ falls within one of the five smaller spheres is closest to ( ).\nA. 0\nB. 0.1\nC. 0.2\nD. 0.3\nE. 0.4", "solution": "6. C. Reason: Let the edge length of the regular tetrahedron be \\( a \\). It is known that the radius of its inscribed sphere is \\( R_{1} = \\frac{\\sqrt{6}}{12} a \\), and the radius of its circumscribed sphere is \\( R_{2} = \\frac{\\sqrt{6}}{4} a \\).\n\nIf the radius of the sphere that is tangent to the circumscribed sphere and the faces of the tetrahedron is \\( r \\), then\n\\[\nr = \\frac{R_{2} - R_{1}}{2} = \\frac{\\sqrt{6}}{12} a \\text{.}\n\\]\n\nThus, the required probability is\n\\[\n\\frac{\\frac{4}{3} \\pi R_{1}^{3} + 4 \\cdot \\frac{4}{3} \\pi r^{3}}{\\frac{4}{3} \\pi R_{2}^{3}} = \\frac{5 \\left( \\frac{1}{12} \\right)^{3}}{\\left( \\frac{1}{4} \\right)^{3}} = \\frac{5}{27} \\approx 0.19.\n\\]", "answer": "0.19", "source": "olympiads", "problem_type": "Geometry", "uuid": "6539a719-572d-506d-9260-e0bd1489885e"}
+{"problem": "8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).\n\nTranslate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.", "solution": "8. Solution: $\\because 7 \\equiv-1(\\bmod 4), \\therefore 7^{7^{7}} \\equiv$ $-1(\\bmod 4)(98$ sevens$)$, let $7^{\\gamma^{\\prime 7}}=4 k+3(98$ sevens$), k \\in \\mathbf{Z}^{+}$, then $7^{7^{\\prime 7}}=7^{4 k+3}$ (99 sevens), $\\because 7^{4} \\equiv$ $1(\\bmod 100), \\therefore 7^{4 k} \\equiv 1(\\bmod 100), 7^{7^{\\prime 2}} \\equiv 7^{4 k+3}$ $\\equiv 7^{3} \\equiv 43(\\bmod 100)$. Let $7^{\\gamma^{7}}=7^{100 m+43}$ (100 sevens), $m \\in \\mathbf{Z}^{+}$.\n$$\n\\because 7^{4} \\equiv 2401(\\bmod 10000), \\therefore 7^{8} \\equiv 4801\n$$\n$(\\bmod 10000), 7^{16} \\equiv 9601(\\bmod 10000)$,\n$$\n7^{32} \\equiv 9201(\\bmod 10000), \\quad 7^{64} \\equiv 8401\n$$\n$(\\bmod 10000), 7^{100} \\equiv 7^{4} \\times 7^{32} \\times 7^{64} \\equiv 2401 \\times 9201$ $\\times 8401 \\equiv 1(\\bmod 10000), 7^{100 m} \\equiv 1(\\bmod 10000)$, $7^{100 m+43} \\equiv 7^{43} \\equiv 7^{3} \\times 7^{8} \\times 7^{32} \\equiv 2343$ $(\\bmod 10000), \\therefore 7^{\\gamma^{7}}(100$ sevens$) \\equiv 7^{100 m+43} \\equiv$ $2343(\\bmod 10000), 7^{7^{77}}(100$ sevens$)$ has the last four digits as 2343.", "answer": "2343", "source": "olympiads", "problem_type": "Number Theory", "uuid": "5c348b9a-119c-5e81-8cef-b2905b202dbd"}
+{"problem": "# Problem 5. (3 points)\n\nIn trapezoid $A B C D$, the lateral side $B C$ is equal to the diagonal $B D$. On the smaller arc $A B$ of the circumscribed circle of triangle $A B C$, a point $E$ is chosen such that $B C=B E$. Find the angle $\\angle A E D$.", "solution": "Answer: $90^{\\circ}$.\n\n## Solution:\n\nFrom the isosceles triangle $B C D$ and the parallelism of $A B$ and $C D$, we get $\\angle B C D = \\angle B D C = \\angle D B A = \\alpha$.\n\nLet the line $C D$ intersect the circumcircle of triangle $A B C$ at point $F$. Then $B C F A$ is an inscribed, i.e., isosceles, trapezoid, from which the arcs $B C, B E$, and $A F$ are equal. Hence, $\\angle B C E = \\angle A C F = \\beta$.\n\n$\\angle E B A = \\angle E C A = \\gamma$, since these angles subtend the same arc.\n\n$\\angle B C A = \\angle B C E + \\angle E C A = \\beta + \\gamma$.\n\n$\\angle E B D = \\angle E B A + \\angle D B A = \\gamma + \\alpha$. Therefore, in the isosceles triangle $E B D$, the equality $\\angle B D E = \\angle B E D = \\frac{180^{\\circ} - \\alpha - \\gamma}{2}$ holds.\n\nMoreover, $\\alpha = \\angle B C D = \\angle B C F = \\angle B C A + \\angle A C F = 2 \\beta + \\gamma$.\n\n$\\angle A E D = \\angle B E A - \\angle B E D = (180^{\\circ} - \\angle B C A) - \\frac{180^{\\circ} - \\alpha - \\gamma}{2} = 180^{\\circ} - \\beta - \\gamma - 90^{\\circ} + \\frac{2 \\beta + 2 \\gamma}{2} = 90^{\\circ}$", "answer": "90", "source": "olympiads", "problem_type": "Geometry", "uuid": "02f1d413-9526-5c96-a94d-ea128382c247"}
+{"problem": "Problem 6. (Option 1).\n\nIn the right triangle $ABC: \\angle ACB=90^{\\circ}, AC=6, BC=4$. On the line $BC$, a point $D (CD>BD)$ is marked such that $\\angle ADC=45^{\\circ}$. On the line $AD$, a point $E$ is marked such that the perimeter of triangle $CBE$ is the smallest possible. Then, on the line $DC$, a point $F$ is marked such that the perimeter of triangle $AFE$ is the smallest possible. Find $CF$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_3fdcc7ef9dfdd3dab8ecg-07.jpg?height=846&width=557&top_left_y=228&top_left_x=241)\n\n## Construction and Proof:\n\nFor convenience, construct a square ACDK (AD is its diagonal, since $\\angle ADC=45^{\\circ}$. Mark point $B_1$ on side $DK (B_1D=BD \\Rightarrow B_1B \\perp AD)$.\n\nDraw line $B_1C$, which intersects $AD$ at point $E$.\n\nThe perimeter of triangle $CBE$ is the smallest because among all possible points $E_1$ on line $AD$, the sum of the lengths of segments $B_1E + EC$ is the smallest ($B_1E + EC < B_1E_1 + E_1C$ - triangle inequality) and $B_1E = EB$. Similarly, mark point $A_1$ on side $AC$ ($A_1C = AC$). Draw line $A_1E$, which intersects $CD$ at point $F$.\n\nThe perimeter of triangle $AFE$ is the smallest because among all possible points $F_1$ on line $AD$, the sum of the lengths of segments $A_1F + EF$ is the smallest ($A_1F + EF < A_1F_1 + F_1E$ - triangle inequality) and $A_1F = FA$.", "solution": "# Solution:\n\n$C D=A C=6 \\Rightarrow B D=B_{1} D=6-4=2$. Since triangles $\\mathrm{ACE}$ and $\\mathrm{B}_{1} \\mathrm{ED}$\n\nare similar ( $\\angle A E C=\\angle B_{1} E D$ - vertical angles and $\\angle E A C=\\angle B_{1} D E=45^{\\circ}$ ), then\n\n$E C: B_{1} E=A C: D B_{1}=A E: E D=6: 2=3: 1$\n\nLet $E H \\perp D C \\Rightarrow$ right triangles ACD and EHD are similar ( $\\angle A C D-$ common), then $C D: H D=A C: E H=A D: E D=4: 1 \\Rightarrow E H=H D=1.5 \\Rightarrow C H=4.5$.\n\nRight triangles FHE and $\\mathrm{FCA}_{1}$ are similar $\\left(\\angle E F H=\\angle A_{1} F C\\right.$ vertical angles) $)$\n\nthen $C F: H F=A_{1} F: E F=A_{1} C: E H=6: 1.5=4: 1 ; C H=4.5 \\Rightarrow C F=0.8 \\cdot 4.5=3.6$\n\nAnswer: 3.6 .", "answer": "3.6", "source": "olympiads", "problem_type": "Geometry", "uuid": "090bc9bd-3980-510f-bf7d-7a0c75345340"}
+{"problem": "Problem 10.7. The graph of the function $f(x)=\\frac{1}{12} x^{2}+a x+b$ intersects the $O x$ axis at points $A$ and $C$, and the $O y$ axis at point $B$, as shown in the figure. It turned out that for the point $T$ with coordinates $(3 ; 3)$, the condition $T A=T B=T C$ is satisfied. Find $b$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-39.jpg?height=321&width=517&top_left_y=195&top_left_x=468)", "solution": "Answer: -6.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-39.jpg?height=359&width=614&top_left_y=600&top_left_x=420)\n\nFig. 11: to the solution of problem 10.7\n\nSolution. Let point $A$ have coordinates $\\left(x_{1} ; 0\\right)$, and point $C$ have coordinates $\\left(x_{2} ; 0\\right)$. From the condition, it is clear that $x_{1} < 0$ and $x_{2} > 0$. Since $x_{1}$ and $x_{2}$ are the roots of the quadratic trinomial $f(x)$, by Vieta's theorem, we have $x_{1} \\cdot x_{2} = 12b$, from which we get $b = \\frac{x_{1} \\cdot x_{2}}{12} < 0$.\n\nLet $H$ be the point with coordinates $(3 ; 0)$ (Fig. 11). Clearly, in the isosceles triangle $A T C$, the segment $T H$ is the height, and therefore it is also the median. Thus, $3 - x_{1} = A H = H C = x_{2} - 3$, from which we get $x_{1} = 6 - x_{2}$.\n\nLet $M$ be the point with coordinates $(0, 3)$. Since $T H = T M = 3$ and $T A = T B$, the right triangles $A T H$ and $B T M$ are equal by the leg and hypotenuse. Therefore, $3 - x_{1} = H A = M B = 3 - b$, that is, $x_{1} = b = \\frac{x_{1} \\cdot x_{2}}{12}$ (by Vieta's theorem), from which we find $x_{2} = 12$. Finally, $x_{1} = 6 - x_{2} = 6 - 12 = -6$ and $b = x_{1} = -6$.\n\nAnother solution. As in the previous solution, let the abscissas of points $A$ and $C$ be $x_{1}$ and $x_{2}$, respectively; we will also use the fact that point $B$ has coordinates $(0 ; b)$. Immediately, we understand that $O A = |x_{1}| = -x_{1}$, $O C = |x_{2}| = x_{2}$, and $O B = |b| = -b$.\n\nLet's find the second intersection of the circle with the y-axis, let this be point $D$ with coordinates $(0 ; d)$ (Fig. 12). The chords $A C$ and $B D$ of the circle intersect at the origin $O$; from the properties of the circle, we know that $O A \\cdot O C = O B \\cdot O D$. We get $-x_{1} \\cdot x_{2} = -b \\cdot d$, from which, replacing $x_{1} \\cdot x_{2}$ with $12b$ by Vieta's theorem, we get $d = 12$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-40.jpg?height=521&width=571&top_left_y=83&top_left_x=437)\n\nFig. 12: to the solution of problem 10.7\n\nIt remains to note that triangle $BTD$ is isosceles, and the midpoint of its base, point $M$, has coordinates $(0 ; 3)$. Reflecting point $D(0 ; 12)$ relative to it, we get $B(0 ; -6)$.", "answer": "-6", "source": "olympiads", "problem_type": "Algebra", "uuid": "4f14788f-fd45-586e-a5e9-c4653ba17885"}
+{"problem": "## 3. Math Puzzle $8 / 65$\n\nA crane is being transported to a construction site. To measure the length of the crane, Peter walks from the tip of the crane to the end and takes 15 steps (step length 80 $\\mathrm{cm}$).\n\nTo walk back from the end to the tip of the crane while the train is moving at a constant speed, he needs 75 steps at the same walking speed.\n\nHow long is the crane?", "solution": "With $v=$ speed, $s=$ distance, $t_{1}=$ time for 15 steps, $t_{2}=$ time for 75 steps, and $l=$ crane length, we have $l-v t_{1}=12 \\mathrm{~m}, l+v t_{2}=60 \\mathrm{~m}$.\n\nIt holds that $\\frac{t_{1}}{t_{2}}=\\frac{1}{5} \\rightarrow t_{1}=\\frac{1}{5} t_{2}$.\n\nThus, $l-v \\cdot \\frac{1}{5} t_{2}=12 \\mathrm{~m}$ and $l+v t_{2}=60 \\mathrm{~m}$. Multiplying the first equation by 5 and adding both equations yields $6 l=120 \\mathrm{~m}, l=20 \\mathrm{~m}$. The length of the crane is $20 \\mathrm{~m}$.", "answer": "20\\mathrm{~}", "source": "olympiads", "problem_type": "Logic and Puzzles", "uuid": "d350c85b-5988-5321-93ae-87b2dbeadca2"}
+{"problem": "## Zadatak B-1.2.\n\nNa slici su prikazani pravilni peterokut. $A B C D E$ i kvadrat $A B F G$. Odredite mjeru kuta $F A D$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_30_d455efeee432fadf0574g-01.jpg?height=234&width=257&top_left_y=1842&top_left_x=797)", "solution": "## Rješenje.\n\nZbroj mjera unutrašnjih kutova pravilnog peterokuta iznosi $(5-2) 180^{\\circ}=540^{\\circ}$ pa je mjera unutrašnjeg kuta pravilnog peterokuta jednaka $\\frac{540^{\\circ}}{5}=108^{\\circ}$.\n\nBudući da je trokut $A E D$ jednakokračan, $\\varangle D A E=\\frac{180^{\\circ}-108^{\\circ}}{2}=36^{\\circ}$\n\nKut $\\varangle B A F=45^{\\circ}$ jer je to kut između dijagonale i stranice kvadrata.\n\nTada je $\\varangle F A D=\\varangle B A E-\\varangle B A F-\\varangle D A E=108^{\\circ}-45^{\\circ}-36^{\\circ}=27^{\\circ}$.", "answer": "27", "source": "olympiads", "problem_type": "Geometry", "uuid": "95bcc8e7-a809-533d-8cbd-45e190531b31"}
+{"problem": "The area of the region in the $xy$-plane satisfying the inequality \\[\\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, \\, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1\\] is $k\\pi$, for some integer $k$. Find $k$.\n\n[i]Proposed by Michael Tang[/i]", "solution": "To find the area of the region in the $xy$-plane satisfying the inequality \n\\[\n\\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, \\, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1,\n\\]\nwe need to analyze the given inequality step by step.\n\n1. **Understanding the Inequality:**\n   The inequality involves two expressions:\n   \\[\n   \\max\\left(\\frac{x^2 + y^2}{4n^2}, 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1}\\right) \\le 1.\n   \\]\n   This means that for some integer $n$ in the range $1 \\le n \\le 10$, the maximum of the two expressions must be less than or equal to 1.\n\n2. **Breaking Down the Inequality:**\n   For the maximum of the two expressions to be less than or equal to 1, both expressions must individually be less than or equal to 1:\n   \\[\n   \\frac{x^2 + y^2}{4n^2} \\le 1 \\quad \\text{and} \\quad 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\le 1.\n   \\]\n\n3. **Simplifying Each Expression:**\n   - For the first expression:\n     \\[\n     \\frac{x^2 + y^2}{4n^2} \\le 1 \\implies x^2 + y^2 \\le 4n^2.\n     \\]\n   - For the second expression:\n     \\[\n     2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\le 1 \\implies \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\ge 1 \\implies x^2 + y^2 \\ge 4n^2 - 4n + 1.\n     \\]\n\n4. **Combining the Conditions:**\n   Therefore, for some $n$ in the range $1 \\le n \\le 10$, the point $(x, y)$ must satisfy:\n   \\[\n   4n^2 - 4n + 1 \\le x^2 + y^2 \\le 4n^2.\n   \\]\n   This describes a ring (annulus) in the $xy$-plane with inner radius $\\sqrt{4n^2 - 4n + 1}$ and outer radius $2n$.\n\n5. **Calculating the Area of Each Annulus:**\n   The area of an annulus with inner radius $r_1$ and outer radius $r_2$ is given by:\n   \\[\n   \\pi (r_2^2 - r_1^2).\n   \\]\n   For our specific case:\n   \\[\n   r_2 = 2n \\quad \\text{and} \\quad r_1 = \\sqrt{4n^2 - 4n + 1}.\n   \\]\n   Therefore, the area of the annulus is:\n   \\[\n   \\pi \\left((2n)^2 - (4n^2 - 4n + 1)\\right) = \\pi \\left(4n^2 - (4n^2 - 4n + 1)\\right) = \\pi (4n - 1).\n   \\]\n\n6. **Summing the Areas for $n$ from 1 to 10:**\n   We need to sum the areas of the annuli for $n = 1, 2, \\ldots, 10$:\n   \\[\n   \\sum_{n=1}^{10} \\pi (4n - 1) = \\pi \\sum_{n=1}^{10} (4n - 1).\n   \\]\n   The sum inside the parentheses is:\n   \\[\n   \\sum_{n=1}^{10} (4n - 1) = 4 \\sum_{n=1}^{10} n - \\sum_{n=1}^{10} 1 = 4 \\left(\\frac{10 \\cdot 11}{2}\\right) - 10 = 4 \\cdot 55 - 10 = 220 - 10 = 210.\n   \\]\n\n7. **Final Area:**\n   Therefore, the total area is:\n   \\[\n   210\\pi.\n   \\]\n\nThe final answer is $\\boxed{210}$.", "answer": "210", "source": "aops_forum", "problem_type": "Inequalities", "uuid": "36667977-8bb5-552e-8fcc-d328f2a797c0"}
+{"problem": "Four. (30 points) Given the sequence $\\left\\{a_{n}\\right\\}$ satisfies\n$$\na_{1}=\\frac{1}{2}, a_{n}=2 a_{n} a_{n+1}+3 a_{n+1}\\left(n \\in \\mathbf{N}_{+}\\right) \\text {. }\n$$\n(1) Find the general term formula of the sequence $\\left\\{a_{n}\\right\\}$;\n(2) If the sequence $\\left\\{b_{n}\\right\\}$ satisfies $b_{n}=1+\\frac{1}{a_{n}}\\left(n \\in \\mathbf{N}_{+}\\right)$, and for any positive integer $n(n \\geqslant 2)$, the inequality\n$$\n\\sum_{k=1}^{n} \\frac{1}{n+\\log _{3} b_{k}}>\\frac{m}{24}\n$$\n\nalways holds, find the maximum value of the integer $m$.", "solution": "(1) From\n$$\n\\begin{array}{l}\na_{n}=2 a_{n} a_{n+1}+3 a_{n+1} \\\\\n\\Rightarrow a_{n+1}=\\frac{a_{n}}{2 a_{n}+3} \\Rightarrow a_{n}>0 . \\\\\n\\text { Hence } \\frac{1}{a_{n+1}}-\\frac{3}{a_{n}}=2 \\Rightarrow \\frac{1}{a_{n+1}}+1=3\\left(\\frac{1}{a_{n}}+1\\right) .\n\\end{array}\n$$\n\nTherefore, $\\left\\{\\frac{1}{a_{n}}+1\\right\\}$ is a geometric sequence with the first term 3 and common ratio 3.\nThus, $\\frac{1}{a_{n}}+1=3^{n} \\Rightarrow a_{n}=\\frac{1}{3^{n}-1}\\left(n \\in \\mathbf{N}_{+}\\right)$.\n(2) From (1), we get $b_{n}=1+\\frac{1}{a_{n}}=3^{n}$.\nTherefore, $\\log _{3} b_{k}=k(k=1,2, \\cdots)$.\nHence, $\\sum_{k=1}^{n} \\frac{1}{n+\\log _{3} b_{k}}>\\frac{m}{24}$\n$\\Leftrightarrow \\frac{1}{n+1}+\\frac{1}{n+2}+\\cdots+\\frac{1}{n+n}>\\frac{m}{24}$.\nLet $f(n)=\\frac{1}{n+1}+\\frac{1}{n+2}+\\cdots+\\frac{1}{n+n}$. Then\n$$\n\\begin{array}{l}\nf(n+1)-f(n) \\\\\n=\\frac{1}{2 n+1}+\\frac{1}{2 n+2}-\\frac{1}{n+1} \\\\\n=\\frac{1}{2 n+1}-\\frac{1}{2 n+2}>0 .\n\\end{array}\n$$\n\nTherefore, $f(n)$ is monotonically increasing.\nHence, $f(n)_{\\min }=f(2)=\\frac{1}{3}+\\frac{1}{4}=\\frac{7}{12}$.\nThus, $\\frac{7}{12}>\\frac{m}{24} \\Rightarrow m<14$.\nTherefore, the maximum integer value of $m$ is 13.", "answer": "13", "source": "cn_contest", "problem_type": "Algebra", "uuid": "3929973c-a32c-5fed-884e-cdaf87e29bbc"}
+{"problem": "Example 7 How many positive integer factors does 20! have?", "solution": "Analyze writing 20! in its standard factorization form $n=\\beta_{1}^{a_{1}} \\beta_{2}^{a_{2}} \\cdots \\beta_{k}^{q_{k}}$, and then using $r(n)=\\left(\\alpha_{1}+1\\right)\\left(\\alpha_{2}+1\\right) \\cdots$ $\\left(\\alpha_{k}+1\\right)$ to calculate the result.\nSolution Since the prime numbers less than 20 are $2,3,5,7,11,13,17,19$, in the standard factorization of 20!, the highest power of 2 $=\\left[\\frac{20}{2}\\right]+\\left[\\frac{20}{4}\\right]+\\left[\\frac{20}{8}\\right]+\\left[\\frac{20}{16}\\right]=18$,\nthe highest power of 3 $=\\left[\\frac{20}{3}\\right]+\\left[\\frac{20}{9}\\right]=8$,\nthe highest power of 5 $=\\left[\\frac{20}{5}\\right]=4$,\nthe highest powers of $7,11,13,17,19$ are $2,1,1,1,1$ respectively.\nThus, the standard factorization of 20! is\n$$20!=2^{18} \\cdot 3^{8} \\cdot 5^{4} \\cdot 7^{2} \\cdot 11 \\cdot 13 \\cdot 17 \\cdot 19$$\n\nTherefore, the number of positive divisors $r(20!)$ of 20! is:\n$$\\begin{aligned}\nr(20!) & =(18+1)(8+1)(4+1)(2+1)(1+1)(1+1)(1+1)(1+1) \\\\\n& =19 \\cdot 9 \\cdot 5 \\cdot 3 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2 \\\\\n& =41040\n\\end{aligned}$$\n\nThus, the number of positive divisors of $20!$ is 41040.", "answer": "41040", "source": "number_theory", "problem_type": "Number Theory", "uuid": "38902e28-3cc8-5f40-a032-89bdea866710"}
+{"problem": "Francisco has 3 daughters: Alina, Valentina, and Civela. An interesting fact is that all three daughters were born on March 18. Today, March 18, 2014, is their birthday. Noting another curious fact, Francisco says:\n\n- Alina, your age is now double the age of Valentina.\n\na) Show that this could never have happened before and that, after the next birthday of Valentina, it will never happen again.\n\nThen, Alina, who was very clever, exclaimed:\n\n- Dad, exactly 10 years ago, the age of one of us three was double the age of another, and in 10 years, the same fact will happen again!\n\nb) It is known that the oldest of the daughters is over 30 years old. How old is Civela?", "solution": "Solution\n\na) It is clear that Alina is older than Valentina. Let's call the difference in age between Alina and Valentina $d$. Since they were born on the same day of the year, $d$ does not vary (note that this would not be true if they were born on different days, as in that case, the difference would change on the birthday of each one).\n\nIf Valentina's age today is $\\alpha$, then Alina's age must be $2 \\alpha$, so the age difference between the two is given by:\n\n$$\nd=2 \\alpha-\\alpha\n$$\n\nWe conclude that $d=\\alpha$, so Alina is $2 d$ years old and Valentina is $d$ years old.\n\nSuch a relationship between the ages of Alina and Valentina could not have happened before, as Valentina was younger than $d$ years old. It also will not happen again, as Valentina will be older than $d$ years.\n\nb) Let's call the age of the youngest daughter $A$, the age of the middle daughter $B$, and the age of the oldest daughter $C$. Let $x=B-A$ be the age difference between the two youngest, and $y=C-B$ be the age difference between the two oldest daughters. The age difference between the youngest and the oldest is therefore $C-A=x+y$. Since they were born on the same day of the year, these differences do not vary.\n\nIt is known that in each of the years 2004, 2014, and 2024, the age of one of the daughters was, is, or will be double the age of one of the others. For this, there are only 3 possibilities:\n\n(i) $B$ is double $A$.\n\n(ii) $C$ is double $B$.\n\n(iii) $C$ is double $A$.\n\nBy the reasoning used in part a), we know that each situation can only occur in one of the three years. Therefore, the three situations: (i), (ii), and (iii) must happen in the three years: 2004, 2014, and 2024, but we do not yet know in what order. Also, from part a), we know that:\n\n- When (i) happens, $B$ must be equal to $2 x$ and $A$ must be equal to $x$. Moreover, $C$ is equal to $B+y$, that is, $C$ must be equal to $2 x+y$.\n- When (ii) happens, $B=y, C=2 y$, and therefore, $A=y-x$.\n- When (iii) happens, $C=2(x+y), A=x+y$, and therefore, $B=2 x+y$.\n\nObserve that the ages of $A$ in cases (i), (ii), and (iii) are\n\n$$\nx, \\quad y-x \\quad \\text { and } \\quad x+y\n$$\n\nrespectively. Since $x<x+y$ and $y-x<x+y$, we conclude that $x+y$ is the largest of the three ages and, therefore, (iii) must happen in the year 2024. Only remains to see in which of the two years 2004 or 2014 each of the events in (i) and (ii) happens. Let's assume that (i) and (ii) happen in the years 2014 and 2004, respectively. Then, the ages of $A$ in the years 2004, 2014, and 2024 would be $y-x, x$, and $x+y$, respectively. Thus, we would have the equations\n\n$$\ny-x+10=x \\quad \\text { and } \\quad x+10=x+y\n$$\n\nSolving, we would have $x=10$ and $y=10$. But in that case, $C$ would be today equal to $2 x+y=30$. But the oldest daughter is more than 30 years old, as stated in the problem. We discard, then, this possibility.\n\nWe know then that (i) and (ii) happen in the years 2004 and 2014, respectively. Then, the ages of $A$ in the years 2004, 2014, and 2024 are $x, y-x$, and $x+y$, respectively. Therefore, we have that\n\n$$\nx+10=y-x \\quad \\text { and } \\quad y-x+10=x+y .\n$$\n\nSolving, we get $x=5$ and $y=20$. Finally, the ages of Francisco's daughters today are equal to\n\n$$\ny-x=15, \\quad y=20 \\quad \\text { and } \\quad 2 y=40\n$$\n\nSince Alina's age must be double Valentina's age, the ages of Alina and Valentina must be 40 and 20, respectively. Therefore, Civela is 15 years old today.", "answer": "15", "source": "olympiads", "problem_type": "Logic and Puzzles", "uuid": "9e1d477e-bfd3-5d98-a9ef-b7c0570221fb"}
+{"problem": "4. Let the sequence $\\left\\{a_{n}\\right\\}$ satisfy $a_{1}=0, a_{2}=1$, and for all $n \\geqslant 3, a_{n}$ is the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \\cdots, a_{n}$ that forms an arithmetic sequence. Find $a_{2014}$.", "solution": "4. First, prove a lemma using mathematical induction.\n\nLemma A non-negative integer appears in the sequence if and only if its ternary expansion contains only 0 and 1.\nProof It is obvious that the proposition holds for 0.\nAssume the proposition holds for all non-negative integers less than \\( N \\), and consider \\( N \\).\n\nIf the ternary expansion of \\( N \\) contains the digit 2, replace all occurrences of 2 with 0 to get the number \\( N_{0} \\); replace all occurrences of 2 with 1 to get the number \\( N_{1} \\). Thus, the ternary expansions of \\( N_{0} \\) and \\( N_{1} \\) do not contain the digit 2.\nBy the induction hypothesis, the numbers \\( N_{0} \\) and \\( N_{1} \\) are in the sequence.\nSince \\( N_{0} \\), \\( N_{1} \\), and \\( N \\) form an arithmetic sequence, \\( N \\) does not appear in the sequence. If the ternary expansion of \\( N \\) does not contain the digit 2, then we need to prove that \\( N \\) must appear in the sequence. If not, then there exist terms \\( N_{0} \\) and \\( N_{1} \\) in the sequence such that \\( N_{0} \\), \\( N_{1} \\), and \\( N \\) form an arithmetic sequence. Let the common difference be \\( d \\), and \\( 3^{k} \\| d \\). Then the ternary expansions of these three numbers have the same lowest \\( k-1 \\) digits, and the \\( k \\)-th digit is different for each pair, so one of the numbers must have a 2 in the \\( k \\)-th digit, which contradicts the assumption.\nReturning to the original problem.\nWe only need to find the 2014th non-negative integer whose ternary expansion does not contain the digit 2.\n\nNotice that under this restriction, the ternary carry-over method is equivalent to binary, so we just need to write 2013 (note \\( a_{1}=0 \\)) in binary \\((11111011101)_{2}\\) and then convert it to ternary \\( a_{2014}=88327 \\).", "answer": "88327", "source": "cn_contest", "problem_type": "Combinatorics", "uuid": "27ef0a0a-88ec-50b0-a32a-0ae07e230bed"}
+{"problem": "3. It is known that for some positive coprime numbers $m$ and $n$, the numbers $m+1947 n$ and $n+1947 m$ have a common prime divisor $d>9$. Find the smallest possible value of the number $d$ under these conditions.", "solution": "Answer: $d_{\\min }=139$.\n\nFor example,\n\n$$\n\\begin{aligned}\n& m=138, n=1 \\rightarrow 1947 m+n=1947 \\cdot 138+1=139 \\cdot 1933 \\\\\n& m+1947 n=138+1947=139 \\cdot 15\n\\end{aligned}\n$$", "answer": "139", "source": "olympiads", "problem_type": "Number Theory", "uuid": "381718c3-4fbf-5607-ac47-c87e50bb9dcf"}
+{"problem": "6. (10 points) As shown in the figure, the area of square $A B C D$ is 840 square centimeters, $A E=E B, B F=2 F C, D F$ intersects $E C$ at $G$. Then the area of quadrilateral $A E G D$ is square centimeters.", "solution": "510\n【Analysis】This figure easily leads one to think about finding the areas of triangles $BCE$, $CDF$, and $CGF$. The most challenging part is finding the area of $\\triangle CGF$. According to the given conditions, we should draw $GM \\perp BC$ with the intersection point at $M$. This forms two sets of similar triangles corresponding to $\\triangle BCE$ and $\\triangle CDF$. By using the property that the ratio of the areas of similar triangles is equal to the square of the similarity ratio, we can find the area of $\\triangle CGF$. Finally, based on the figure, we can calculate the area of the required shape.\n\n【Solution】Draw $GM \\perp BC$ with the intersection point at $M$,\n$$\n\\begin{array}{l}\n\\therefore \\triangle FMG \\sim \\triangle FCD \\Rightarrow FM: FC = MG: CD, \\\\\n\\because BF = 2FC \\Rightarrow BC = 3FC, \\\\\n\\therefore MG = 3FM, \\\\\n\\because \\triangle CGM: \\triangle CEB \\Rightarrow CM: CB = GM: BE, \\\\\nBC = 2BE, \\\\\n\\therefore GM = \\frac{1}{2} CM = 3FM \\Rightarrow CM = 6FM, \\\\\n\\therefore FM: FC = 1: 7, \\\\\nCM: CB = 2: 7, \\\\\nS \\triangle BCE = \\frac{1}{4} \\times \\square ABCD = 210, \\\\\nS \\triangle CGM = 4 \\div 49 \\times 210 = 840 \\div 49 = 120 \\div 7, \\\\\nS \\triangle CDF = S \\square ABCD \\div 6 = 140, \\\\\nS \\triangle MGF = 140 \\times 1 \\div 49 = 140 \\div 49 = 20 \\div 7, \\\\\nS \\triangle CGM + \\triangle MGF = 120 \\div 7 + 20 \\div 7 = 20, \\\\\n840 - 210 - 140 + 20 = 510 \\text{ (square centimeters). }\n\\end{array}\n$$\n\nTherefore, the area of quadrilateral $AEGD$ is 510 square centimeters.", "answer": "510", "source": "olympiads", "problem_type": "Geometry", "uuid": "2a224068-3c17-5d14-ad3a-426aec3cae2b"}
+{"problem": "1. In tetrahedron $ABCD$, let $AB=1$, $CD=\\sqrt{3}$, the distance between lines $AB$ and $CD$ is 2, and the angle between them is $\\frac{\\pi}{3}$. Then the volume of tetrahedron $ABCD$ is ( ).\n(2003 National High School League Question)\nA. $\\frac{\\sqrt{3}}{2}$\nB. $\\frac{1}{2}$\nC. $\\frac{1}{3}$\nD. $\\frac{\\sqrt{3}}{3}$", "solution": "1. Choose B. Reason: The area of the parallelogram formed by $\\overrightarrow{A B}$ and $\\overrightarrow{C D}$ is $|\\overrightarrow{A B}| \\cdot|\\overrightarrow{C D}| \\cdot \\sin 60^{\\circ}=\\frac{3}{2}$. If a line $C E \\mathbb{E}$ is drawn through point $C$ parallel to $A B$, then $S_{\\triangle C D E}=\\frac{1}{2} \\cdot \\frac{3}{2}=\\frac{3}{4}$, and the volume of the prism $A B F-E C D$ with $\\triangle C D E$ as the base and $B C$ as the side edge is 3 times that of $V_{A B C D}$, and $V_{A B F-E C D}=S_{\\triangle C D E} \\cdot 2=\\frac{3}{2}$, hence $V_{A B C D}=\\frac{1}{2}$.", "answer": "\\frac{1}{2}", "source": "olympiads", "problem_type": "Geometry", "uuid": "33b39e35-cec5-5faf-a661-1defa78a43a2"}
+{"problem": "3. (6 points) A construction company was building a tunnel. When $\\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \\%$ and reduced the working hours to $80 \\%$ of the original. As a result, it took a total of 185 days to complete the tunnel. If they had not used the new equipment and continued at the original speed, it would have taken $\\qquad$ days to complete the tunnel.", "solution": "3. (6 points) A construction company builds a tunnel. When $\\frac{1}{3}$ of the tunnel is completed at the original speed, new equipment is used, increasing the construction speed by $20 \\%$ and reducing the daily working hours to $80 \\%$ of the original. As a result, the tunnel is completed in 185 days. If the new equipment was not used and the original speed was maintained, it would take $\\qquad$ 180 days.\n\n【Solution】Solution: $\\left(1-\\frac{1}{3}\\right) \\div[(1+20 \\%) \\times 80 \\%]$\n$$\n\\begin{array}{l}\n=\\frac{2}{3} \\div[120 \\% \\times 80 \\%], \\\\\n=\\frac{2}{3} \\div \\frac{24}{25} \\\\\n=\\frac{25}{36} ; \\\\\n185 \\div\\left(\\frac{1}{3}+\\frac{25}{36}\\right) \\\\\n=185 \\div \\frac{37}{36}, \\\\\n=180 \\text { (days). }\n\\end{array}\n$$\n\nAnswer: If the original speed was maintained, it would take 180 days.\nTherefore, the answer is: 180.", "answer": "180", "source": "olympiads", "problem_type": "Algebra", "uuid": "1d57f4b3-c516-5522-91cc-a60dd91f6f66"}
+{"problem": "Exercise 8. The price (in euros) of a diamond corresponds to its mass (in grams) squared and then multiplied by 100. The price (in euros) of a crystal corresponds to three times its mass (in grams). Martin and Théodore unearth a treasure composed of precious stones which are either diamonds or crystals, and the total value is $5000000 €$. They cut each precious stone in half, and each takes one half of each stone. The total value of Martin's stones is $2000000 €$. In euros, what was the total initial value of the diamonds contained in the treasure?\n\nOnly a numerical answer is expected here.", "solution": "Solution to Exercise 8: Let $r$ be the number of stones that are diamonds, and let $d_{1}, \\ldots, d_{r}$ be the respective masses of the diamonds unearthed. We also denote $c$ as the mass of crystal unearthed. We aim to determine the value of $100\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)$.\n\nSince the total value of the precious stones unearthed is 5000000, we have\n\n$$\n100\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)+3 c=5000000\n$$\n\nSince Martin receives half of the total mass of diamonds and half of the total mass of crystal, and the value of Martin's share is 2000000, we also have\n\n$$\n100\\left[\\left(\\frac{d_{1}}{2}\\right)^{2}+\\ldots+\\left(\\frac{d_{r}}{2}\\right)^{2}\\right]+3 \\cdot \\frac{c}{2}=2000000\n$$\n\nBy subtracting twice (2) from (1), we get\n\n$$\n100\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)-2 \\times 25\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)+3 c-2 \\times 3 \\cdot \\frac{c}{2}=5000000-2 \\times 2000000\n$$\n\nwhich simplifies to $50\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)=1000000$. Therefore, $100\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)=2000000$, and the initial total value of the diamonds in the treasure is 2000000.\n\nGrader Comments: The exercise was well done by the majority. The mistakes made are mainly due to poor modeling of the problem statement. Several students simply wrote a numerically incorrect result and thus received no points for this first exercise, which was the simplest in the high school subject. Even if only a numerical answer is expected, it is recommended to write down the reasoning to earn some points.", "answer": "2000000", "source": "olympiads", "problem_type": "Algebra", "uuid": "d1ae3c87-5993-573c-891f-1453c3724013"}
+{"problem": "5. 5.1. Find the largest natural number in which all digits are different and any two adjacent digits differ by 6 or 7.", "solution": "Answer: 60718293.\n\n## Solution.\n\nWe will map each digit from 0 to 9 to a vertex in a graph and connect the vertices with an edge if the corresponding digits differ by 6 or 7.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_8a1d5b347fbc679ac0eeg-5.jpg?height=448&width=590&top_left_y=524&top_left_x=819)\n\nWe see that the vertices corresponding to the digits 4 and 5 are not connected to anything, so the maximum number that can be obtained will be 8 digits long. The vertices corresponding to the digits 3 and 6 can only be the first or last in the number. Since we need to find the maximum number, it is uniquely determined by moving along the edges of the graph and will be equal to 60718293.", "answer": "60718293", "source": "olympiads", "problem_type": "Number Theory", "uuid": "7edacb04-aea8-5a67-b4b4-4a1930dcb4df"}
+{"problem": "Example 12 Given $a, b, c, d \\in \\mathbf{R}^{+}$, try to find\n$f(a, b, c, d)=\\frac{a}{b+c+d}+\\frac{b}{c+d+a}+\\frac{c}{a+b+d}+\\frac{d}{a+b+c}$ the minimum value.", "solution": "Solution 1 Since directly eliminating the denominator is complicated, consider changing the problem.\nLet $b+c+d=A, c+d+a=B, d+a+b=C, a+b+c=D$.\nAdding these four equations, we get $a+b+c+d=\\frac{1}{3}(A+B+C+D)$.\nThus, we can find that $a=\\frac{1}{3}(B+C+D-2 A), b=\\frac{1}{3}(C+D+A-2 B)$,\n$$\nc=\\frac{1}{3}(D+A+B-2 C), d=\\frac{1}{3}(A+B+C-2 D).\n$$\n\nTherefore, the original expression $=\\frac{B+C+D-2 A}{3 A}+\\frac{C+D+A-2 B}{3 B}+\\frac{D+A+B-2 C}{3 C}+$\n$$\n\\frac{A+B+C-2 D}{3 D},\n$$\n\nwhich is $3 \\cdot f(a, b, c, d)+12$\n$$\n\\begin{array}{l}\n=\\frac{A+B+C+D}{A}+\\frac{A+B+C+D}{B}+\\frac{A+B+C+D}{C}+\\frac{A+B+C+D}{D} \\\\\n=(A+B+C+D) \\cdot\\left(\\frac{1}{A}+\\frac{1}{B}+\\frac{1}{C}+\\frac{1}{D}\\right) \\geqslant 16,\n\\end{array}\n$$\n\ni.e., $3 \\cdot f(a, b, c, d)+12 \\geqslant 16$, hence $f(a, b, c, d) \\geqslant \\frac{4}{3}$, where equality holds if and only if $A=$ $B=C=D$, i.e., $a=b=c=d$.\nTherefore, the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.\nSolution 2 Noting the weighted sum inequality, we have\n$$\n\\begin{aligned}\nf(a, b, c, d) & =\\frac{a^{2}}{a(b+c+d)}+\\frac{b^{2}}{b(a+c+d)}+\\frac{c^{2}}{c(a+b+d)}+\\frac{d^{2}}{d(a+b+c)} \\\\\n& \\geqslant \\frac{(a+b+c+d)^{2}}{2(a b+a c+a d+b c+b d+c d)} \\\\\n& =\\frac{(a+b+c+d)^{2}}{(a+b+c+d)^{2}-\\left(a^{2}+b^{2}+c^{2}+d^{2}\\right)}\n\\end{aligned}\n$$\n\nand $a^{2}+b^{2}+c^{2}+d^{2}=\\frac{a^{2}}{1}+\\frac{b^{2}}{1}+\\frac{c^{2}}{1}+\\frac{d^{2}}{1} \\geqslant \\frac{(a+b+c+d)^{2}}{4}$, thus\n$$\nf(a, b, c, d) \\geqslant \\frac{(a+b+c+d)^{2}}{(a+b+c+d)^{2}-\\frac{1}{4}(a+b+c+d)^{2}}=\\frac{4}{3} .\n$$\n\nwhere equality holds if and only if $a=b=c=d$, hence the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.\nSolution 3 Noting the Cauchy inequality or the weighted sum inequality, we have\n$$\n\\begin{aligned}\nf(a, b, c, d) & =\\frac{a+b+c+d}{b+c+d}+\\frac{a+b+c+d}{a+c+d}+\\frac{a+b+c+d}{a+b+d}+ \\\\\n& \\frac{a+b+c+d}{a+b+c}-4 \\\\\n& \\geqslant(a+b+c+d)\\left(\\frac{1}{b+c+d}+\\frac{1}{a+c+d}+\\frac{1}{a+b+d}+\\right. \\\\\n& \\left.\\frac{1}{a+b+c}\\right)-4 \\\\\n& \\geqslant(a+b+c+d) \\cdot \\frac{4^{2}}{3(a+b+c+d)}-4=\\frac{4}{3} .\n\\end{aligned}\n$$\n\nwhere equality holds if and only if $a=b=c=d$, hence the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.", "answer": "\\frac{4}{3}", "source": "olympiads", "problem_type": "Inequalities", "uuid": "10813f94-5443-58ca-9c8c-20d9f8a70b52"}
+{"problem": "1. (6 points) Today is January 31, 2015, welcome to the 2015 \"Spring Cup\" Final. The calculation result of the expression $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$ is", "solution": "【Solution】Solve: $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$\n$$\n\\begin{array}{l}\n=\\frac{\\left(\\frac{2015}{1}+\\frac{2015}{0.31}\\right) \\times 0.31}{(1+0.31) \\times 0.31} \\\\\n=\\frac{(1+0.31) \\times 2015}{(1+0.31) \\times 0.31} \\\\\n=6500 ;\n\\end{array}\n$$\n\nTherefore, the answer is: 6500.", "answer": "6500", "source": "olympiads", "problem_type": "Algebra", "uuid": "46eb6c3b-bc5d-5fc0-8913-2c4a06359b71"}
+{"problem": "Problem 4. A segment is divided by three points into four unequal parts. The distance between the midpoints of the inner parts is $6 \\mathrm{~cm}$, and the distance between the midpoints of the outer parts is $16 \\mathrm{~cm}$. What is the length of the given segment?", "solution": "Solution. Let the endpoints of the given segment be $A$ and $B$, and let the points $C, D, E$ divide it into four unequal parts. Let $M$ be the midpoint of\n\n![](https://cdn.mathpix.com/cropped/2024_06_05_dc4d34c0a9dd2a3ba572g-04.jpg?height=113&width=639&top_left_y=1155&top_left_x=570)\n\n$AC$, $N$ the midpoint of $CD$, $P$ the midpoint of $DE$, and $Q$ the midpoint of $EB$. Then, from the condition $\\overline{NP}=6 \\text{ cm}$ and $\\overline{MQ}=16 \\text{ cm}$ (see the diagram). Now, from $\\overline{NP}=6 \\text{ cm}$, we have that $\\overline{CD}+\\overline{DE}=2 \\cdot 6=12 \\text{ cm}$. From this and from $\\overline{MQ}=16 \\text{ cm}$, we get that $\\overline{MC}+\\overline{EQ}=16-12=4 \\text{ cm}$, and since $M$ is the midpoint of $AC$ and $Q$ is the midpoint of $EB$, it follows that $\\overline{AM}+\\overline{QB}=4 \\text{ cm}$. Finally, the length of the given segment is $\\overline{AB}=(\\overline{AM}+\\overline{QB})+\\overline{MQ}=4+16=20 \\text{ cm}$.", "answer": "20", "source": "olympiads", "problem_type": "Geometry", "uuid": "5c51b71b-267b-590e-be6a-eaf5657d85d3"}
+{"problem": "6. The set of all real number pairs $(x, y)$ that satisfy $\\left\\{\\begin{array}{l}4^{-x}+27^{-y}=\\frac{5}{6} \\\\ \\log _{27} y-\\log _{4} x \\geqslant \\frac{1}{6} \\text { is } \\\\ 27^{y}-4^{x} \\leqslant 1\\end{array}\\right.$ $\\qquad$ . $\\qquad$", "solution": "6. $\\left\\{\\left(\\frac{1}{2}, \\frac{1}{3}\\right)\\right\\}$ Detailed Explanation: Let $a=4^{x}, b=27^{y}$, then $\\frac{1}{a}+\\frac{1}{b}=\\frac{5}{6}, a>\\frac{6}{5}, b>\\frac{6}{5}$. From $\\frac{1}{a}+\\frac{1}{b}=\\frac{5}{6}$, we get $b=\\frac{6 a}{5 a-6}$, then $\\frac{6 a}{5 a-6}-a \\leqslant 1$, solving this yields $a \\geqslant 2$ or $a>\\frac{6}{5}$, so $a \\geqslant 2$. Similarly, we get $b \\leqslant 3$. Therefore, $x \\geqslant \\frac{1}{2}, y \\leqslant \\frac{1}{3} . \\log _{27} y-\\log _{4} x \\leqslant \\log _{27} \\frac{1}{3}-\\log _{4} \\frac{1}{2}=\\frac{1}{6}$. So $x=\\frac{1}{2}, y=\\frac{1}{3}$.", "answer": "(\\frac{1}{2},\\frac{1}{3})", "source": "olympiads", "problem_type": "Algebra", "uuid": "01b0b161-74b6-5dd2-958d-c4fa3a8e18a3"}
+{"problem": "4. (6 points) A number when divided by 3 leaves a remainder of 2, when divided by 4 leaves a remainder of 3, and when divided by 5 leaves a remainder of 4. This number is $\\qquad$", "solution": "【Analysis】Interpreting \"divided by 3 leaves a remainder of 2, divided by 4 leaves a remainder of 3, divided by 5 leaves a remainder of 4\" as being 1 less than when divided by 3, 1 less than when divided by 4, and 1 less than when divided by 5, means this number is at least 1 less than the least common multiple (LCM) of 3, 4, and 5. Since 3, 4, and 5 are pairwise coprime, the LCM of these three numbers is their product. Find the LCM of 3, 4, and 5, then subtract 1.\n\n【Solution】Solution: $3 \\times 4 \\times 5-1$,\n$$\n\\begin{array}{l}\n=60-1, \\\\\n=59 ;\n\\end{array}\n$$\n\nAnswer: The number is 59.\nTherefore, the answer is: 59.\n【Comment】This problem mainly examines the method of finding the least common multiple (LCM) when three numbers are pairwise coprime: If three numbers are pairwise coprime, their LCM is the product of these three numbers.", "answer": "59", "source": "olympiads", "problem_type": "Number Theory", "uuid": "7cfc118d-0022-572d-bacc-abb0a3b763a4"}
+{"problem": "A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all possible values of $n$.\n\nA rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all possible values of $n$.", "solution": "Answer: $n=324$.\n\nLet $a b=n$ and $c d=n+76$, where $a, b$ and $c, d$ are the numbers of squares in each direction for the partitioning of the rectangle into $n$ and $n+76$ squares, respectively. Then $\\frac{a}{c}=\\frac{b}{d}$, or $a d=b c$. Denote $u=\\operatorname{gcd}(a, c)$ and $v=\\operatorname{gcd}(b, d)$, then there exist positive integers $x$ and $y$ such that $\\operatorname{gcd}(x, y)=1, a=u x, c=u y$ and $b=v x, d=v y$. Hence we have\n\n$$\nc d-a b=u v\\left(y^{2}-x^{2}\\right)=u v(y-x)(y+x)=76=2^{2} \\cdot 19 .\n$$\n\nSince $y-x$ and $y+x$ are positive integers of the same parity and $\\operatorname{gcd}(x, y)=1$, we have $y-x=1$ and $y+x=19$ as the only possibility, yielding $y=10, x=9$ and $u v=4$. Finally we have $n=a b=x^{2} u v=324$.", "answer": "324", "source": "olympiads_ref", "problem_type": "Number Theory", "uuid": "NaN"}
+{"problem": "$[$ [Arithmetic. Mental calculation, etc.] $]$\n\nAuthors: Gaityerri G.A., Grierenko D.:\n\n2002 is a palindrome year, which means it reads the same backward as forward. The previous palindrome year was 11 years earlier (1991). What is the maximum number of non-palindrome years that can occur consecutively (between 1000 and 9999 years)?", "solution": "Let the current year be a palindrome of the form $\\overline{a b b a}$. When will the next such year occur? Consider two cases:\n\na) $b=9$ (year of the form $\\overline{a 99 a}, a<9)$. Then, 11 years later, another palindromic year will occur: $\\overline{(a+1) 00(a+1)}$. For example, the years 3993 and 4004.\n\nb) $b<9$. In this case, the next palindromic year will occur 110 years later: $\\overline{a(b+1)(b+1) a}$. For example, the years 9339 and 9449. Therefore, the maximum number of consecutive non-palindromic years is 109.\n\n## Answer\n\n109 years.", "answer": "109", "source": "olympiads", "problem_type": "Number Theory", "uuid": "effbcdc3-5088-5c32-8530-ff0fddfa69ea"}
+{"problem": "Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie? \n\n$ \\text{(A)}\\ 10\\qquad\\text{(B)}\\ 20\\qquad\\text{(C)}\\ 30\\qquad\\text{(D)}\\ 50\\qquad\\text{(E)}\\ 72 $", "solution": "1. First, we need to determine the number of students who prefer each type of pie. We know:\n   - 12 students prefer chocolate pie.\n   - 8 students prefer apple pie.\n   - 6 students prefer blueberry pie.\n\n2. Calculate the total number of students who prefer chocolate, apple, or blueberry pie:\n   \\[\n   12 + 8 + 6 = 26\n   \\]\n\n3. Subtract this number from the total number of students in the class to find the number of students who prefer either cherry or lemon pie:\n   \\[\n   36 - 26 = 10\n   \\]\n\n4. Since half of the remaining students prefer cherry pie and the other half prefer lemon pie, we divide the remaining students by 2:\n   \\[\n   \\frac{10}{2} = 5\n   \\]\n   Therefore, 5 students prefer cherry pie.\n\n5. To find the number of degrees Richelle should use for cherry pie in her pie graph, we need to calculate the proportion of students who prefer cherry pie and then convert this proportion to degrees. The proportion of students who prefer cherry pie is:\n   \\[\n   \\frac{5}{36}\n   \\]\n\n6. Multiply this proportion by the total number of degrees in a circle (360 degrees):\n   \\[\n   \\frac{5}{36} \\times 360 = 50\n   \\]\n\n7. Therefore, Richelle should use 50 degrees for cherry pie in her pie graph.\n\nThe final answer is $\\boxed{50}$.", "answer": "50", "source": "aops_forum", "problem_type": "Combinatorics", "uuid": "4f89da07-3f71-5edc-85a6-7460e959c360"}
+{"problem": "# Problem 6. (3 points)\n\nThree runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.\n\nAt some point, the first runner meets the second. After 15 minutes, the second runner meets the third for the first time. Another 25 minutes later, the third runner meets the first for the first time.\n\nHow many minutes does it take for one runner to complete the entire track?", "solution": "# Answer: 80\n\n## Solution:\n\nLet the first runner meet the second, then after $a$ minutes the second runner meets the third for the first time, and after another $b$ minutes the third runner meets the first for the first time.\n\nLet the first and second runners meet at point $A$, the second and third at point $B$, and the first and third at point $C$. Additionally, let the first runner be at point $D$ at the moment the second and third runners meet.\n\nThen the points are arranged on the circle in the following order: $D, A, B, C$, where the first runner ran from $D$ to $A$ in $a$ minutes, the second runner ran from $A$ to $B$ in $a$ minutes, the third runner ran from $B$ to $C$ in $b$ minutes, and the first runner ran from $D$ to $C$ in $b$ minutes. These four segments together form the entire track, so it is covered in $2a + 2b$ minutes.", "answer": "80", "source": "olympiads", "problem_type": "Logic and Puzzles", "uuid": "62791263-04f8-508f-9886-2dd0d0a262be"}
+{"problem": "2. A helicopter and an airplane flew towards each other at the same time. By the time they met, the helicopter had flown $100 \\mathrm{~km}$ less than the airplane, and the helicopter reached the airplane's takeoff location 3 hours after the meeting. The airplane reached the helicopter's takeoff location 1 hour and 20 minutes after the meeting. Determine the speeds of the helicopter and the airplane, and the distance between the takeoff locations.", "solution": "Solution. Let the helicopter fly $x \\mathrm{~km}$ to the meeting. Then the airplane flew $(x+100) \\mathrm{~km}$ to the meeting. The speed of the helicopter is $\\frac{x+100}{3} \\mathrm{~km} / \\mathrm{h}$, and the speed of the airplane is $\\frac{x}{1 \\frac{1}{3}} \\mathrm{~km} / \\mathrm{h}$. Thus, the time the helicopter flew from its takeoff point to the meeting is $x: \\frac{x+100}{3}=\\frac{3 x}{x+100}$ hours, and the time the airplane flew from its takeoff point to the meeting is $\\frac{1 \\frac{1}{3}(x+100)}{x}$ hours. They flew the same amount of time to the meeting, so\n\n$$\n\\frac{1 \\frac{1}{3}(x+100)}{x}=\\frac{3 x}{x+100}, \\text { i.e. }\\left(\\frac{x+100}{x}\\right)^{2}=\\frac{9}{4}\n$$\n\nFrom this, since $\\frac{9}{4}=\\left(\\frac{3}{2}\\right)^{2}=\\left(-\\frac{3}{2}\\right)^{2}$, we get\n\n$$\n\\frac{x+100}{x}=\\frac{3}{2} \\text { or } \\frac{x+100}{x}=-\\frac{3}{2} \\text {. }\n$$\n\nHowever, $\\frac{x+100}{x}>0$, so $\\frac{x+100}{x}=\\frac{3}{2}$, from which we get $x=200 \\mathrm{~km}$.\n\nThus, the helicopter flew $200 \\mathrm{~km}$, the airplane flew $300 \\mathrm{~km}$, the speed of the helicopter is $100 \\mathrm{~km} / \\mathrm{h}$, and the speed of the airplane is $150 \\mathrm{~km} / \\mathrm{h}$. The distance between the takeoff points is $500 \\mathrm{~km}$.", "answer": "500\\mathrm{~}", "source": "olympiads", "problem_type": "Algebra", "uuid": "3ed05b96-c2f2-5248-aa03-084302cdd1b5"}
+{"problem": "14. Let $k$ be a positive integer, such that for any positive numbers $a, b, c$ satisfying the inequality $k(a b+b c+c a)>5\\left(a^{2}+b^{2}\\right.$ $+c^{2}$), there must exist a triangle with side lengths $a, b, c$. Find the minimum value of $k$.", "solution": "14. $k(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right) \\geqslant 5(a b+a c+b c), k \\leqslant 5$ when, the inequality has no solution. When $k=6$, the inequality becomes $6(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right)$. Without loss of generality, assume $c=\\max (a, b, c)$, transform the inequality into $5 c^{2}-6(a+b) c+5\\left(a^{2}+b^{2}\\right)-6 a b0$, $5\\left(a^{2}+b^{2}\\right)-6 a b>0$, so the quadratic equation corresponding to the inequality has two positive roots $c_{1}, c_{2}$, the solution to the quadratic inequality in $c$ is $c \\in\\left(c_{1}, c_{2}\\right)$. If $c_{2} \\leqslant a+b$, then $c6$ when, $k(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right)$ has solutions, but they cannot form a triangle, so the only $k$ that satisfies the condition is $k=6$.", "answer": "6", "source": "olympiads", "problem_type": "Inequalities", "uuid": "562e031a-910d-5c5d-b73e-56e9bc99849c"}
+{"problem": "6. As shown in Figure 2, let $P$ be a point inside the equilateral $\\triangle ABC$ with side length 12. Draw perpendiculars from $P$ to the sides $BC$, $CA$, and $AB$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $PD: PE: PF = 1: 2: 3$. Then, the area of quadrilateral $BDPF$ is", "solution": "$$\n6.11 \\sqrt{3} \\text {. }\n$$\n\nFor the equilateral $\\triangle ABC$ with a height of $12 \\times \\frac{\\sqrt{3}}{2}=6 \\sqrt{3}$, then\n$$\n\\begin{array}{l}\nP D+P E+P F=6 \\sqrt{3} . \\\\\n\\text { Also, } P D: P E: P F=1: 2: 3, \\text { so, } \\\\\nP D=\\sqrt{3}, P E=2 \\sqrt{3}, P F=3 \\sqrt{3} .\n\\end{array}\n$$\n\nAs shown in Figure 7, draw\n$P S / / B C$, intersecting $A B$ at $S$.\nSince $\\angle F S P=\\angle B$ $=60^{\\circ}$, therefore,\n$$\n\\begin{array}{l}\nF S=P F \\cot 60^{\\circ}=3, \\\\\nP S=6, \\\\\nB D=P S+P D \\cot 60^{\\circ}=7 .\n\\end{array}\n$$\n\nThus, $S_{\\text {quadrilateral BDPF }}=S_{\\triangle P S}+S_{\\text {trapezoid BDPS }}$\n$$\n=\\frac{1}{2} \\times 3 \\times 3 \\sqrt{3}+\\frac{6+7}{2} \\times \\sqrt{3}=11 \\sqrt{3} .\n$$", "answer": "11 \\sqrt{3}", "source": "cn_contest", "problem_type": "Geometry", "uuid": "2a4c0374-d9ea-538e-8e0f-0811359d6cf6"}
+{"problem": "What is the $21^{\\varrho}$ term of the sequence\n\n$$\n1 ; 2+3 ; 4+5+6 ; 7+8+9+10 ; 11+12+13+14+15 ; \\ldots ?\n$$", "solution": "Observe that the $21^{\\text{st}}$ term is the sum of 21 consecutive numbers. Taking the first term of each term, that is, 1, 2, 4, 7, 11, $16, \\ldots$, we have that\n\n$$\n\\begin{aligned}\n2 & =1+1 \\\\\n4 & =2+1+1 \\\\\n7 & =3+2+1+1 \\\\\n11 & =4+3+2+1+1 \\\\\n16 & =5+4+3+2+1+1\n\\end{aligned}\n$$\n\nThus, the first term of the $21^{\\text{st}}$ term is\n\n$$\n20+19+\\cdots+3+2+1+1=\\frac{20 \\times 21}{2}+1=211\n$$\n\nand the $21^{\\text{st}}$ term is\n\n$$\n211+212+\\cdots+230+231=\\frac{(211+231) \\times 21}{2}=221 \\times 21=4641\n$$", "answer": "4641", "source": "olympiads", "problem_type": "Number Theory", "uuid": "ac33a322-ff2a-5a42-8d4e-dc84713e0a1f"}
+{"problem": "On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number.", "solution": "1. **Define the invariant:**\n   Let \\( I \\) be the remainder when the sum of all the integers on the board is divided by 13. Initially, the numbers from 1 to 2009 are written on the board. The sum of these numbers is:\n   \\[\n   S = \\sum_{k=1}^{2009} k = \\frac{2009 \\times 2010}{2} = 2009 \\times 1005\n   \\]\n   We need to find \\( S \\mod 13 \\).\n\n2. **Calculate \\( 2009 \\mod 13 \\) and \\( 1005 \\mod 13 \\):**\n   \\[\n   2009 \\div 13 = 154 \\quad \\text{remainder} \\quad 7 \\quad \\Rightarrow \\quad 2009 \\equiv 7 \\pmod{13}\n   \\]\n   \\[\n   1005 \\div 13 = 77 \\quad \\text{remainder} \\quad 4 \\quad \\Rightarrow \\quad 1005 \\equiv 4 \\pmod{13}\n   \\]\n\n3. **Calculate \\( S \\mod 13 \\):**\n   \\[\n   S = 2009 \\times 1005 \\equiv 7 \\times 4 = 28 \\equiv 2 \\pmod{13}\n   \\]\n   Therefore, \\( I = 2 \\).\n\n4. **Determine the third number \\( a \\):**\n   After several operations, only three numbers are left on the board: 9, 999, and \\( a \\). The sum of these three numbers modulo 13 must be equal to the invariant \\( I \\):\n   \\[\n   a + 9 + 999 \\equiv 2 \\pmod{13}\n   \\]\n\n5. **Calculate \\( 999 \\mod 13 \\):**\n   \\[\n   999 \\div 13 = 76 \\quad \\text{remainder} \\quad 11 \\quad \\Rightarrow \\quad 999 \\equiv 11 \\pmod{13}\n   \\]\n\n6. **Set up the equation:**\n   \\[\n   a + 9 + 11 \\equiv 2 \\pmod{13}\n   \\]\n   Simplify the equation:\n   \\[\n   a + 20 \\equiv 2 \\pmod{13}\n   \\]\n   Subtract 20 from both sides:\n   \\[\n   a \\equiv 2 - 20 \\pmod{13}\n   \\]\n   \\[\n   a \\equiv -18 \\pmod{13}\n   \\]\n   Since \\(-18 \\equiv -18 + 26 \\equiv 8 \\pmod{13}\\), we have:\n   \\[\n   a \\equiv 8 \\pmod{13}\n   \\]\n\n7. **Find the possible values of \\( a \\) between 1 and 2009:**\n   The possible values of \\( a \\) are of the form:\n   \\[\n   a = 8 + 13k \\quad \\text{for integer} \\quad k\n   \\]\n   We need \\( 1 \\leq a \\leq 2009 \\):\n   \\[\n   1 \\leq 8 + 13k \\leq 2009\n   \\]\n   \\[\n   -7 \\leq 13k \\leq 2001\n   \\]\n   \\[\n   0 \\leq k \\leq 154\n   \\]\n   Therefore, the possible values of \\( a \\) are:\n   \\[\n   a = 8, 21, 34, \\ldots, 2002\n   \\]\n\nThe final answer is \\( \\boxed{8} \\).", "answer": "8", "source": "aops_forum", "problem_type": "Number Theory", "uuid": "26ac9e7a-f8f3-5ef9-b051-f84dfb5129ce"}
+{"problem": "II. Find the last four digits of $2^{1999}$.\n\nTranslate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.", "solution": "$$\n\\varphi(625)=\\varphi\\left(5^{4}\\right)=5^{4} \\times\\left(1-\\frac{1}{5}\\right)=500 \\text {. }\n$$\n\nSince $(2,625)=1$, by Euler's theorem, we get $2^{500} \\equiv 1(\\bmod 625)$.\n$$\n\\begin{array}{l}\n\\therefore 2^{2000} \\equiv 1(\\bmod 625) . \\\\\n\\therefore \\text { Let } 2^{2000}=625 m+1(m \\in \\mathbf{N}) .\n\\end{array}\n$$\n\nAlso, $2^{5} \\mid 2^{2000}$,\nso let $2^{2000}=2^{5} \\cdot n(n \\in \\mathbf{N})$.\nFrom (1) and (2), we get $625 m+1=2^{5} \\cdot n$.\nSolving this indeterminate equation, we get\n$$\n\\left\\{\\begin{array}{l}\nm=32 k+15, \\\\\nn=625 k+293\n\\end{array}\\right.\n$$\n$k \\in \\mathbf{Z}$.\nSubstituting $n=625 k+293$ into (2), we get\n$$\n\\begin{array}{l}\n2^{2000}=32(625 k+293)=20000 k+9376 . \\\\\n\\therefore 2^{1999}=10000 k+4688 .\n\\end{array}\n$$\n\nTherefore, the last four digits of $2^{1999}$ are 4688.", "answer": "4688", "source": "cn_contest", "problem_type": "Number Theory", "uuid": "e89e22c1-c45e-57cb-ba19-292af932d553"}
+{"problem": "Example 1 (Question from the 13th \"Hope Cup\" Invitational Competition) The real roots of the equations $x^{5}+x+1=0$ and $x+\\sqrt[5]{x}+1=0$ are $\\alpha, \\beta$ respectively, then $\\alpha+\\beta$ equals ( ).\nA. -1\nB. $-\\frac{1}{2}$\nC. $\\frac{1}{2}$\nD. 1", "solution": "Solution: Choose A. Reason: Consider the function $f(x)=x^{5}+x+1$, which is an increasing function on $\\mathbf{R}$,\nwhen $x^{5}+x+1=0$,\nwe get $x=\\sqrt[5]{-x-1}$, substituting $-x-1$ for $x$ yields $-x-1=\\sqrt[5]{x}$.\nFrom (2), we have $(-x-1)^{5}=x$, which is also $(-x-1)^{5}+(-x-1)+1=0$.\nLet $\\alpha$ be a root of equation (1), i.e., $f(\\alpha)=0$. Let $\\beta$ be a root of equation (2), i.e., $-1-\\beta=\\sqrt[5]{\\beta}$, thus from (3) we know that $f(-1-\\beta)=0$.\nTherefore, we have $f(\\alpha)=f(-1-\\beta)$, which means $\\alpha+\\beta=-1$.", "answer": "-1", "source": "olympiads", "problem_type": "Algebra", "uuid": "1eb53f35-0255-526f-b139-9f8efdda61ec"}
+{"problem": "Example 13 Given $a+b+c=0, a^{3}+b^{3}+c^{3}$ $=0$. Find the value of $a^{15}+b^{15}+c^{15}$.", "solution": "Solution: From $a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)$ - $\\left(a^{2}+b^{2}+c^{2}-a b-b c-a c\\right)$, we have $a b c=0$. Therefore, at least one of $a, b, c$ is 0.\nAssume $c=0$, then $a, b$ are opposites,\n$$\n\\therefore a^{15}+b^{15}+c^{15}=0 \\text {. }\n$$", "answer": "0", "source": "cn_contest", "problem_type": "Algebra", "uuid": "7a8e22bb-4320-53a4-b25b-c2f1855f675d"}
+{"problem": "Henry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?\n\n[i]Proposed by Michael Ren[/i]", "solution": "1. **Initial List and Conditions**:\n   - Henry starts with the list of the first 1000 positive integers: \\( \\{1, 2, 3, \\ldots, 1000\\} \\).\n   - At each step, he erases any nonpositive integers or any integers that have a repeated digit.\n   - After erasing, he decreases every remaining integer in the list by 1.\n\n2. **Understanding the Erasure Condition**:\n   - Nonpositive integers are erased immediately.\n   - Integers with repeated digits are also erased. For example, numbers like 11, 22, 101, etc., will be erased.\n\n3. **Effect of Decreasing by 1**:\n   - After each step, every integer in the list is decreased by 1. This means that the list will eventually contain nonpositive integers, which will be erased.\n\n4. **Key Observation**:\n   - The critical observation is to determine the maximum number of steps it takes for any integer to be erased.\n   - Consider the number 10. It will be decreased to 9, 8, 7, ..., 1, 0, and then -1. This process takes exactly 11 steps.\n\n5. **Erasure of Numbers with Repeated Digits**:\n   - Numbers like 11, 22, 33, ..., 99, 101, 111, etc., will be erased as soon as they appear in the list.\n   - For example, 11 will be erased in the first step, 22 in the second step, and so on.\n\n6. **General Case**:\n   - For any number \\( n \\) in the list, if it does not have repeated digits, it will be decreased step by step until it becomes nonpositive.\n   - The maximum number of steps required for any number to become nonpositive is determined by the number 10, which takes 11 steps.\n\n7. **Conclusion**:\n   - Since the number 10 takes exactly 11 steps to be erased, and all other numbers will be erased in fewer or equal steps, the entire list will be empty after 11 steps.\n\n\\[\n\\boxed{11}\n\\]", "answer": "11", "source": "aops_forum", "problem_type": "Logic and Puzzles", "uuid": "6aa339e9-2248-53cf-9c07-634bb5e8d596"}
+{"problem": "12.180. A side of the triangle is equal to 15, the sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4.", "solution": "## Solution.\n\nLet $a$ be the given side of the triangle, $a=15$, $b$ and $c$ be the other two sides, $b+c=27$, $c=27-b$. For definiteness, we will assume that $b>c$. The semiperimeter of the triangle $p=\\frac{a+b+c}{2}=21$, its area $S=p r=84$.\n\nBy Heron's formula $S=\\sqrt{p(p-a)(p-b)(p-c)}$, thus\n\n$$\n\\begin{aligned}\n& \\sqrt{21 \\cdot(21-15)(21-b)(21-27+b)}=84 \\Leftrightarrow \\sqrt{21 \\cdot 6(21-b)(b-6)}=84 \\Leftrightarrow \\\\\n& \\Leftrightarrow(21-b)(b-6)=56 \\Leftrightarrow b^{2}-27 b+182=0, \\\\\n& b_{1}=14, b_{2}=13 . \\\\\n& \\text { We have } b=14, c=13 \\text {. By the cosine rule }\n\\end{aligned}\n$$\n\n$$\n\\cos \\alpha=\\frac{b^{2}+c^{2}-a^{2}}{2 b c}=\\frac{5}{13}\n$$\n\nAnswer: $\\frac{5}{13}$.", "answer": "\\frac{5}{13}", "source": "olympiads", "problem_type": "Geometry", "uuid": "d0246a87-5e79-5337-904b-c098b279a7bd"}
+{"problem": "The function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\\forall$ $x\\in \\mathbb{R}$ and $f(x)=x(x-1)$ for $\\forall$ $x\\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\\geq -\\frac{8}{9}$ is true for \n$\\forall$ $x\\in (-\\infty , m]$.", "solution": "1. **Given Conditions and Initial Analysis:**\n   - The function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfies \\( f(x+1) = 2f(x) \\) for all \\( x \\in \\mathbb{R} \\).\n   - For \\( x \\in (0,1] \\), \\( f(x) = x(x-1) \\).\n\n2. **Behavior of \\( f(x) \\) in \\( (0,1] \\):**\n   - We need to find the minimum and maximum values of \\( f(x) \\) in the interval \\( (0,1] \\).\n   - \\( f(x) = x(x-1) \\) is a quadratic function opening downwards.\n   - The vertex of the parabola \\( f(x) = x(x-1) \\) is at \\( x = \\frac{1}{2} \\).\n   - Evaluating at \\( x = \\frac{1}{2} \\):\n     \\[\n     f\\left(\\frac{1}{2}\\right) = \\frac{1}{2} \\left(\\frac{1}{2} - 1\\right) = \\frac{1}{2} \\left(-\\frac{1}{2}\\right) = -\\frac{1}{4}\n     \\]\n   - Evaluating at the endpoints \\( x = 0 \\) and \\( x = 1 \\):\n     \\[\n     f(0) = 0 \\quad \\text{and} \\quad f(1) = 0\n     \\]\n   - Therefore, \\( \\min_{x \\in (0,1]} f(x) = -\\frac{1}{4} \\) and \\( \\max_{x \\in (0,1]} f(x) = 0 \\).\n\n3. **General Form of \\( f(x) \\) for \\( x \\in (k, k+1] \\):**\n   - For \\( x \\in (k, k+1] \\), where \\( k \\in \\mathbb{Z} \\), we have:\n     \\[\n     f(x) = 2^k f(x-k)\n     \\]\n   - Since \\( f(x-k) = (x-k)(x-k-1) \\) for \\( x-k \\in (0,1] \\), we get:\n     \\[\n     f(x) = 2^k (x-k)(x-k-1)\n     \\]\n\n4. **Analyzing \\( f(x) \\) for \\( x \\in (1,2] \\):**\n   - For \\( x \\in (1,2] \\):\n     \\[\n     f(x) = 2 \\cdot (x-1)(x-2)\n     \\]\n   - The minimum value in this interval is at \\( x = \\frac{3}{2} \\):\n     \\[\n     f\\left(\\frac{3}{2}\\right) = 2 \\left(\\frac{3}{2} - 1\\right) \\left(\\frac{3}{2} - 2\\right) = 2 \\cdot \\frac{1}{2} \\cdot \\left(-\\frac{1}{2}\\right) = -\\frac{1}{2}\n     \\]\n   - Therefore, \\( f(x) \\geq -\\frac{1}{2} \\) for \\( x \\in (1,2] \\).\n\n5. **Analyzing \\( f(x) \\) for \\( x \\in (2,3] \\):**\n   - For \\( x \\in (2,3] \\):\n     \\[\n     f(x) = 4 \\cdot (x-2)(x-3)\n     \\]\n   - The minimum value in this interval is at \\( x = \\frac{5}{2} \\):\n     \\[\n     f\\left(\\frac{5}{2}\\right) = 4 \\left(\\frac{5}{2} - 2\\right) \\left(\\frac{5}{2} - 3\\right) = 4 \\cdot \\frac{1}{2} \\cdot \\left(-\\frac{1}{2}\\right) = -1\n     \\]\n   - Therefore, \\( f(x) \\in [-1, 0] \\) for \\( x \\in (2,3] \\).\n\n6. **Finding \\( m \\) such that \\( f(x) \\geq -\\frac{8}{9} \\):**\n   - We need to solve \\( 4(x-2)(x-3) = -\\frac{8}{9} \\):\n     \\[\n     4(x-2)(x-3) = -\\frac{8}{9}\n     \\]\n     \\[\n     36(x-2)(x-3) = -8\n     \\]\n     \\[\n     9(x-2)(x-3) = -2\n     \\]\n     \\[\n     9(x^2 - 5x + 6) = -2\n     \\]\n     \\[\n     9x^2 - 45x + 54 = -2\n     \\]\n     \\[\n     9x^2 - 45x + 56 = 0\n     \\]\n   - Solving this quadratic equation:\n     \\[\n     x = \\frac{45 \\pm \\sqrt{2025 - 2016}}{18} = \\frac{45 \\pm 3}{18}\n     \\]\n     \\[\n     x_1 = \\frac{48}{18} = \\frac{8}{3}, \\quad x_2 = \\frac{42}{18} = \\frac{7}{3}\n     \\]\n   - Therefore, \\( f(x) \\geq -\\frac{8}{9} \\) for \\( 2 < x \\leq \\frac{7}{3} \\).\n\n7. **Conclusion:**\n   - The greatest real number \\( m \\) for which \\( f(x) \\geq -\\frac{8}{9} \\) for all \\( x \\in (-\\infty, m] \\) is \\( m = \\frac{7}{3} \\).\n\nThe final answer is \\( \\boxed{\\frac{7}{3}} \\)", "answer": "\\frac{7}{3}", "source": "aops_forum", "problem_type": "Calculus", "uuid": "1617e146-e118-5933-a973-11c199667efe"}
+{"problem": "A3. The length of the aquarium is $50 \\mathrm{~cm}$, the width is $20 \\mathrm{~cm}$, and the height is $25 \\mathrm{~cm}$. How many $\\mathrm{cm}$ from the top edge of the aquarium will the water level be if we pour 19 liters of water into it?\n(A) $19 \\mathrm{~cm}$\n(B) $1,9 \\mathrm{~cm}$\n(C) $10,9 \\mathrm{~cm}$\n(D) $6 \\mathrm{~cm}$\n(E) $0,6 \\mathrm{~cm}$", "solution": "A3. We find that the volume of water is $19 \\mathrm{dm}^{3}$. We write the data into the formula for the volume of water ( $x$ is the height of the water $\\mathrm{v}$ in the aquarium) and get the equation $V=5 \\cdot 2 \\cdot x=19$. We calculate $x=1.9 \\mathrm{dm}$ $=19 \\mathrm{~cm}$. The water level will thus be $6 \\mathrm{~cm}$ lower than the top edge of the aquarium. The correct answer is D.", "answer": "6\\mathrm{~}", "source": "olympiads", "problem_type": "Geometry", "uuid": "75a3c347-b7d7-5064-ab17-c6895b558311"}
+{"problem": "4.206 There are two forces $f_{1}$ and $f_{2}$ acting on the origin $O$ of the coordinate axis,\n$$\\begin{array}{l}\n\\vec{f}_{1}=\\overrightarrow{O A}=\\sqrt{2}\\left(\\cos 45^{\\circ}+i \\sin 45^{\\circ}\\right) \\\\\n\\vec{f}_{2}=\\overrightarrow{O B}=2\\left[\\cos \\left(-30^{\\circ}\\right)+i \\sin \\left(-30^{\\circ}\\right)\\right]\n\\end{array}$$\n(1) Find the magnitude and direction of their resultant force;\n(2) Find the distance between points $A$ and $B$ (accurate to 0.1).", "solution": "[Solution]\n$$\\text { (1) Resultant force } \\begin{aligned}\n\\vec{f}= & \\overrightarrow{f_{1}}+\\overrightarrow{f_{2}} \\\\\n= & \\sqrt{2}\\left(\\cos 45^{\\circ}+i \\sin 45^{\\circ}\\right) \\\\\n& +2\\left[\\cos \\left(-30^{\\circ}\\right)+i \\sin \\left(-30^{\\circ}\\right)\\right] \\\\\n= & 1+i+\\sqrt{3}-i \\\\\n= & \\sqrt{3}+1 \\\\\n= & (\\sqrt{3}+1)\\left(\\cos 0^{\\circ}+i \\sin 0^{\\circ}\\right) .\n\\end{aligned}$$\n\nSo the magnitude of the resultant force is $\\sqrt{3}+1$, and its direction is the same as the $x$-axis.\n(2) Since $\\overrightarrow{O A}=1+i$, the Cartesian coordinates of point $A$ are $(1,1)$, and $\\overrightarrow{O B}=\\sqrt{3}-i$, so the Cartesian coordinates of point $B$ are $(\\sqrt{3},-1)$.\n$$\\text { Therefore } \\begin{aligned}\n|A B| & =\\sqrt{(\\sqrt{3}-1)^{2}+(-1-1)^{2}} \\\\\n& =\\sqrt{8-2 \\sqrt{3}} \\approx 2.1\n\\end{aligned}$$", "answer": "2.1", "source": "inequalities", "problem_type": "Algebra", "uuid": "e93a8cbd-336b-5aa7-9e39-b81d5037510d"}
+{"problem": "Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily  distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?", "solution": "1. **Restate the problem in a more manageable form:**\n   We need to prove that the number \\(1\\) can be represented as a sum of a finite number \\(n\\) of real numbers, each less than \\(1\\), and each containing only the digits \\(0\\) and \\(7\\) in their decimal representation. We also need to find the least possible value of \\(n\\).\n\n2. **Reformulate the problem using a known rational number:**\n   Consider the rational number \\(\\frac{1}{7} = 0.\\overline{142857}\\). We need to express \\(\\frac{1}{7}\\) as a sum of \\(n\\) real numbers, each less than \\(\\frac{1}{7}\\) and containing only the digits \\(0\\) and \\(1\\) in their decimal representation.\n\n3. **Assume \\(n < 10\\) and find the least value of \\(n\\):**\n   We assume that all the real numbers are positive and that \\(n < 10\\). The least value of \\(n\\) is found to be \\(8\\), for which we have the identity:\n   \\[\n   \\frac{1}{7} = 0.\\overline{1} + 0.\\overline{011111} + 0.\\overline{010111} + 0.\\overline{010111} + 0.\\overline{000111} + 0.\\overline{000101} + 0.\\overline{000101} + 0.\\overline{000100}.\n   \\]\n\n4. **Consider the case with both positive and negative real numbers:**\n   Suppose now that \\(\\frac{1}{7}\\) is the sum of \\(i > 0\\) positive real numbers and \\(j > 0\\) negative real numbers, with \\(i + j \\leq 7\\). Denote by \\(u = 0.u_1u_2\\ldots\\) (respectively by \\(-v = -0.v_1v_2\\ldots\\)) the sum of the positive (respectively negative) real numbers, so that \\(\\frac{1}{7} + v = u\\). Note that \\(u\\) and \\(v\\) are necessarily strictly less than \\(1\\).\n\n5. **Analyze the constraints on \\(u\\) and \\(v\\):**\n   Since \\(i, j < 7\\), it follows that \\(v_6 \\geq 3\\), so that \\(j \\geq 3\\) and \\(i \\leq 4\\). Then, we must have \\(v_5 \\geq 4\\), so that \\(j \\geq 4\\) and \\(i \\leq 3\\). In this case, we find \\(u_3 \\geq 3\\) and thus \\(i = 3\\). This finally implies that \\(v_2 \\geq 6\\), which is impossible since \\(j = 4\\).\n\n6. **Conclude the least value of \\(n\\):**\n   The least value of \\(n\\) is therefore \\(8\\).\n\nThe final answer is \\( \\boxed{ 8 } \\).", "answer": "8", "source": "aops_forum", "problem_type": "Number Theory", "uuid": "1cd82e3d-2e7a-5eca-9571-62f5669824c4"}
+{"problem": "5. Points $O$ and $I$ are the centers of the circumcircle and incircle of triangle $ABC$, and $M$ is the midpoint of the arc $AC$ of the circumcircle (not containing $B$). It is known that $AB=15, BC=7$, and $MI=MO$. Find $AC$.", "solution": "Answer: $A C=13$.\n\nSolution. (Fig. 3). First, we prove that $M I=M A$ (trident lemma).\n\nIndeed, the external angle $A I M$ of triangle $A I B$ is equal to the sum of angles $B A I$ and $A B I$, and since $A I$ and $B I$ are angle bisectors, $\\angle A I M=\\frac{1}{2} \\angle A+\\frac{1}{2} \\angle B$. Angle $I A M$ is equal to the sum of angles $I A C$ and $C A M$. But $\\angle I A C=\\frac{1}{2} \\angle A$, and $\\angle C A M=\\angle C B M=\\frac{1}{2} \\angle B$ as inscribed angles.\n\nFrom this, it follows that $\\angle I A M=\\frac{1}{2} \\angle A+\\frac{1}{2} \\angle B=\\angle A I M$, and therefore, triangle $A M I$ is isosceles, $M I=M A$. By the condition $M O=$ $M I$, so by the trident lemma $A O=M O=M I=M A$. This means that triangle $A O M$ is equilateral and $\\angle A O M=60^{\\circ}$. Since the central angle $A O M$ is twice the inscribed angle $A B M$, we have $\\frac{1}{2} \\angle B=30^{\\circ}$, that is, $\\angle B=60^{\\circ}$.\n\nBy the cosine theorem, $A C^{2}=A B^{2}+B C^{2}-2 \\cdot A B \\cdot B C \\cos 60^{\\circ}=$ $13^{2}$.\n\nCriteria. Answer without explanation - 0 points. Proved that angle $B$ is $60^{\\circ}-5$ points. Points are not deducted for the absence of proof of the trident lemma. Full solution - 7 points.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_7789e0097aa3cdbc6e57g-2.jpg?height=540&width=467&top_left_y=1779&top_left_x=1485)\n\nFig. 3", "answer": "13", "source": "olympiads", "problem_type": "Geometry", "uuid": "a4f66b73-0bb9-55c1-b2b7-58da34d0e16f"}
+{"problem": "2. Given the function $y=\\frac{a-x}{x-a-1}$, the graph of its inverse function is symmetric about the point $(-1,4)$. Then the value of the real number $a$ is $\\qquad$ .", "solution": "2.3.-\n\nFrom the problem, we know that the graph of the function $y=\\frac{a-x}{x-a-1}$ is centrally symmetric about the point $(4,-1)$.\n$\\because y=\\frac{a-x}{x-a-1}=-1-\\frac{1}{x-(a+1)}$, we have $(y+1)[x-(a+1)]=-1$,\n$\\therefore$ the graph of the function is a hyperbola with its center at $(a+1,-1)$.\nAlso, $\\because$ the graph of the hyperbola is centrally symmetric about its center,\n$\\therefore a+1=4$, which means $a=3$.", "answer": "3", "source": "cn_contest", "problem_type": "Algebra", "uuid": "ffccaa4e-8118-5206-b8c0-cbb5298378ed"}
+{"problem": "6. Given that $\\overrightarrow{O A} \\perp \\overrightarrow{O B}$, and $|\\overrightarrow{O A}|=|\\overrightarrow{O B}|=24$. If $t \\in[0,1]$, then\n$$\n|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right|\n$$\n\nthe minimum value is ( ).\n(A) $2 \\sqrt{193}$\n(B) 26\n(C) $24 \\sqrt{2}$\n(D) 24", "solution": "6. B.\n\nConstruct a square $O A C B$, connect the diagonal $A B$, and let $D$ and $E$ be points on the diagonal $A B$ and the side $O B$ respectively, such that\n$$\n\\begin{array}{l}\nt \\overrightarrow{A B}-\\overrightarrow{A O}=\\overrightarrow{O D}, \\\\\n\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}=\\overrightarrow{E D}, \\\\\nE B=10, O D=D C . \\\\\n\\text { Therefore }|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right| \\\\\n=|\\overrightarrow{E D}|+|\\overrightarrow{D C}| \\leqslant|\\overrightarrow{E C}|=26 .\n\\end{array}\n$$", "answer": "26", "source": "olympiads", "problem_type": "Geometry", "uuid": "09c374b3-283a-5d5c-b647-4b3336197400"}
+{"problem": "8. Determine the value of the sum\n$$\n\\frac{3}{1^{2} \\cdot 2^{2}}+\\frac{5}{2^{2} \\cdot 3^{2}}+\\frac{7}{3^{2} \\cdot 4^{2}}+\\cdots+\\frac{29}{14^{2} \\cdot 15^{2}} .\n$$", "solution": "Solution: $224 / 225$ The sum telescopes as\n$$\n\\left(\\frac{1}{1^{2}}-\\frac{1}{2^{2}}\\right)+\\left(\\frac{1}{2^{2}}-\\frac{1}{3^{2}}\\right)+\\cdots+\\left(\\frac{1}{14^{2}}-\\frac{1}{15^{2}}\\right)=\\frac{1}{1^{2}}-\\frac{1}{15^{2}}=\\frac{224}{225} .\n$$", "answer": "\\frac{224}{225}", "source": "olympiads", "problem_type": "Algebra", "uuid": "1f447ae2-6a9a-5160-b486-9a19b5aedace"}
+{"problem": "Augusto has a wire that is $10 \\mathrm{~m}$ long. He makes a cut at a point on the wire, obtaining two pieces. One piece has a length of $x$ and the other has a length of $10-x$ as shown in the figure below:\n\n![](https://cdn.mathpix.com/cropped/2024_05_01_d02c2755ad3373bde08ag-05.jpg?height=645&width=1166&top_left_y=568&top_left_x=527)\n\nAugusto uses the two pieces of wire to make two squares.\n\na) What is the length of the side of each of the squares? What is the area of each?\n\nb) What is the length of each of the two pieces of wire so that the sum of the areas of the squares obtained is minimized?\n\nc) Suppose Augusto cuts the wire into ten pieces and uses each one to make a square. What should be the size of each piece so that the sum of the areas of the squares obtained is minimized?", "solution": "Solution\n\nIn this problem, all lengths are given in meters and areas in square meters.\n\na) A piece of rope has length $x$ and another piece of rope has length $10-x$. Since a square has four sides of equal length, one square will have a side length of $\\frac{x}{4}$ and the other square will have a side length of $\\frac{10-x}{4}$.\n\nThe area of a square with side length $\\ell$ is $\\ell^{2}$. Therefore, one square will have an area of $\\left(\\frac{x}{4}\\right)^{2}=\\frac{x^{2}}{16}$, while the other square will have an area of $\\left(\\frac{10-x}{4}\\right)^{2}=\\frac{100-20 x+x^{2}}{16}$.\n\nb) Let $S(x)$ be the sum of the areas of the two squares. From the previous part, we have\n\n$$\nS(x)=\\frac{x^{2}}{16}+\\frac{100-20 x+x^{2}}{16}=\\frac{100-20 x+2 x^{2}}{16}=\\frac{1}{8} x^{2}-\\frac{5}{4} x+\\frac{25}{4}\n$$\n\nwhich is a quadratic function. The minimum of a function of the form\n\n$$\nf(x)=a x^{2}+b x+c\n$$\n\nwith $a>0$ is achieved at $x=\\frac{-b}{2 a}$. Thus, the minimum area will be achieved if\n\n$$\nx=-\\frac{\\left(-\\frac{5}{4}\\right)}{2 \\frac{1}{8}}=5\n$$\n\nIn other words, if the rope is cut exactly in the middle!\n\nc) From the previous part, we know that to minimize the sum of the areas, it is necessary to cut the rope exactly in the middle. Well, we claim that to minimize the area with nine cuts (i.e., creating ten squares), it is necessary that all pieces of rope be equal. To show this, consider the following argument: if two of the ten pieces of rope were different, it would be possible to reduce the area by cutting the pieces of rope so that these two were equal (we are using the previous part). Therefore, any two pieces of rope must be equal. Hence, all must be equal!", "answer": "5", "source": "olympiads", "problem_type": "Algebra", "uuid": "e8e60a66-4791-5625-bbfd-31ac1c2af0d6"}
+{"problem": "2.1 Find $f(2013)$, if for any real $x$ and $y$ the equality holds\n\n$$\nf(x-y)=f(x)+f(y)-2xy\n$$", "solution": "Solution. Substitute $x=y=0$. We get $f(0)=2 f(0)+0$, from which we obtain that $f(0)=0$. Substitute $x=y$. We get $0=f(0)=f(x)+f(x)-2 x^{2}$. Hence, $f(x)=x^{2}$.\n\nAnswer: 4052169. (C)\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_997c53c06135e2de9c16g-02.jpg?height=67&width=1157&top_left_y=1003&top_left_x=498)", "answer": "4052169", "source": "olympiads", "problem_type": "Algebra", "uuid": "8b03f2a1-15de-52a8-b2b5-24806ad8d00a"}
+{"problem": "2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation.", "solution": "2. This problem is equivalent to finding the smallest positive integer $b$, such that the equation $7 b^{2}+7 b+7=x^{4}$, (1) has an integer solution for $x$. Since 7 is a prime number, it follows from equation (1) that 7 is a divisor of $x$. Therefore, let $x=7 k$, then equation (1) becomes $b^{2}+b+1=7^{3} k^{4}$. The smallest $b$ occurs when $k$ is at its minimum. Taking $k=1$, we then have $b^{2}+b+1=343, b^{2}+b-342=0$, which is $(b-18)(b+19)=0$, yielding the positive integer solution $b=18$. Thus, we have $(777)_{18}=\\left(7^{4}\\right)_{10}$.", "answer": "18", "source": "olympiads", "problem_type": "Number Theory", "uuid": "39a6c760-dfa3-5201-aa87-105127951e16"}
+{"problem": "9. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?", "solution": "Solution: $1 / 2$\nImagine placing the tetrahedron $A B C D$ flat on a table with vertex $A$ at the top. By vectors or otherwise, we see that the center is $3 / 4$ of the way from $A$ to the bottom face, so the reflection of this face lies in a horizontal plane halfway between $A$ and $B C D$. In particular, it cuts off the smaller tetrahedron obtained by scaling the original tetrahedron by a factor of $1 / 2$ about $A$. Similarly, the reflections of the other three faces cut off tetrahedra obtained by scaling $A B C D$ by $1 / 2$ about $B, C$, and $D$. On the other hand, the octahedral piece remaining remaining after we remove these four smaller tetrahedra is in the intersection of $A B C D$ with its reflection, since the reflection sends this piece to itself. So the answer we seek is just the volume of this piece, which is\n$$\n\\begin{array}{l}\n\\text { (volume of } A B C D)-4 \\cdot(\\text { volume of } A B C D \\text { scaled by a factor of } 1 / 2) \\\\\n=1-4(1 / 2)^{3}=1 / 2\n\\end{array}\n$$", "answer": "\\frac{1}{2}", "source": "olympiads", "problem_type": "Geometry", "uuid": "6bb59c32-e03b-53f3-9abf-a23cdeb0c3a1"}
+{"problem": "13. (15 points) As shown in the figure, ten points are evenly marked on the circumference of a circle. Place the ten natural numbers $1 \\sim 10$ on these ten points. Rotate a line passing through the center of the circle. When the line does not pass through any marked points, it divides the numbers 1 to 10 into two groups. For each arrangement, there are five ways to divide the numbers as the line rotates. For each division, there is a product of the sums of the two groups, and the smallest of these five products is denoted as $K$. What is the maximum value of $K$ among all possible arrangements?", "solution": "【Analysis】No matter how these 10 numbers are placed, their sum remains unchanged, $1+2+3+4+5+6+7+8+9+10=55$, divided into two groups, the closer the sums of the two groups, the larger the required product will be.\n\n【Solution】\n$$\n\\begin{array}{l}\n1+2+3+4+5+6+7+8+9+10=55 \\\\\n55=27+28\n\\end{array}\n$$\n\nTherefore, as long as the 10 numbers are divided into two parts with sums of 27 and 28, the product at this time is $27 \\times 28=756$. Answer: In all the arrangements, $K$ is maximized at 756.", "answer": "756", "source": "olympiads", "problem_type": "Combinatorics", "uuid": "cbdaf71c-479e-5ef7-b26b-7b66cd85e7cf"}
+{"problem": "## 232. Math Puzzle $9 / 84$\n\nA motorcycle battery ( $6 \\mathrm{~V}$ ) is discharged through a lamp $6 \\mathrm{~V} / 0.5 \\mathrm{~W}$. The lamp burns for 48 hours.\n\nHow many Ah was the battery charged if we assume that the discharge current remains constant?", "solution": "The discharge current is $I=\\frac{0.5 \\mathrm{~W}}{6 \\mathrm{~V}}=\\frac{1}{12} \\mathrm{~A}$.\n\nIf we assume that it remains constant during continuous discharge, the battery previously had $48 \\cdot \\frac{1}{12} \\mathrm{Ah}=4 \\mathrm{Ah}$ charged.", "answer": "4\\mathrm{Ah}", "source": "olympiads", "problem_type": "Algebra", "uuid": "5e0354af-e3c8-55b2-8a4f-071457e4e43f"}
+{"problem": "Problem 6.8. In the country of Dragonia, there live red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Moreover, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:\n\n- 1st head: “To my left is a green dragon.”\n- 2nd head: “To my right is a blue dragon.”\n- 3rd head: “There is no red dragon next to me.”\n\nWhat is the maximum number of red dragons that could have been at the table?", "solution": "Answer: 176.\n\nSolution. Consider an arbitrary red dragon. To the right of this dragon, at least one head must tell the truth. Note that the 1st and 3rd heads cannot tell the truth (since there is a red dragon to the left), so the 2nd head must tell the truth, and to the right of this dragon, there must be a blue dragon. Now consider the dragon to the left of the chosen red dragon. By similar reasoning, we understand that only the 1st head can tell the truth, and to the left of this dragon, there must be a green one.\n\nFrom the previous reasoning, we get that for any red dragon, one position to the right there is definitely a blue dragon, and one position to the left - a green one. For each red dragon, we combine these three dragons into one group. Note that any dragon can be in at most one such group. Then we get that there are no more than $530 / 3$ < 177, i.e., no more than 176 such groups. Therefore, there are no more than 176 red dragons.\n\nNow let's provide an example. Arrange 176 red dragons in a circle, and in 175 gaps between them, place a green and a blue dragon: ...RGRB..., and in one gap, place two blue and two green dragons: ...RBGBGR.... It is easy to check that all conditions are met: each dragon has at least one head that will tell the truth, and there are exactly 530 dragons in total.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_8af0c885427e3e323cf9g-17.jpg?height=492&width=525&top_left_y=852&top_left_x=464)\n\n## 7th grade", "answer": "176", "source": "olympiads", "problem_type": "Logic and Puzzles", "uuid": "6c252a36-7d8e-5376-95d9-ada49d61667a"}
+{"problem": "|\n\nThere are 85 balloons in the room - red and blue. It is known that: 1) at least one of the balloons is red; 2) in every arbitrarily chosen pair of balloons, at least one is blue. How many red balloons are in the room?\n\n#", "solution": "Think about whether there can be two red balls in the room.\n\n## Solution\n\nSince among any two balls, one is blue, there cannot be two red balls in the room. Therefore, there are 84 blue balloons and 1 red balloon in the room.\n\n## Answer\n\n1 ball.", "answer": "1", "source": "olympiads", "problem_type": "Logic and Puzzles", "uuid": "4671698a-54f7-52fb-988c-460378917299"}
+{"problem": "Example 4 Find the integer solution of the equation $\\left[\\frac{x}{1!}\\right]+\\left[\\frac{x}{2!}\\right]+\\left[\\frac{x}{3!}\\right]+\\cdots+$ $\\left[\\frac{x}{10!}\\right]=1001$.", "solution": "Solution: $\\because$ the solution $x$ of the equation is a natural number, and $xx+\\frac{x}{2}+\\frac{x}{6}+\\frac{x}{24}+\\frac{x}{120}-4 \\\\\n=\\frac{103}{60} x-4, \\therefore 583<x<585, x=584,\n\\end{array}\n$$\n\nVerification shows that $x=584$ is the unique solution of the equation.", "answer": "584", "source": "olympiads", "problem_type": "Number Theory", "uuid": "b243ea4d-43cd-560b-919d-ac59cbae4cb9"}
+{"problem": "Example 1 In $\\triangle ABC$, it is known that $x \\sin A + y \\sin B + z \\sin C = 0$. Find the value of $(y + z \\cos A)(z + x \\cos B)(x + y \\cos C) + (y \\cos A + z)(z \\cos B + x)(x \\cos C + y)$.", "solution": "In $\\triangle A B C$, $\\sin C=\\sin (A+B)=\\sin A \\cos B+\\cos A \\sin B$. Substituting the known conditions, we get $x \\sin A+y \\sin B+z(\\sin A \\cos B+\\cos A \\sin B)=0$, which simplifies to $\\sin A(x+z \\cos B)=-\\sin B(y+z \\cos A)$.\nSimilarly, $\\sin B(y+x \\cos C)=-\\sin C(z+x \\cos B)$, $\\sin C(z+y \\cos A)=-\\sin A(x+y \\cos C)$,\nMultiplying the three equations, we have:\n$(x+z \\cos B)(y+x \\cos C)(z+y \\cos A)=-(y+z \\cos A)(z+x \\cos B)(x+y \\cos C)$, which simplifies to $(y+z \\cos A)(z+x \\cos B)(x+y \\cos C)+(y \\cos A+z)(z \\cos B+x)(x \\cos C+y)=0$.", "answer": "0", "source": "olympiads", "problem_type": "Algebra", "uuid": "80c5345a-b5f8-5bef-8c9f-f624dc51008a"}
+{"problem": "2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$.", "solution": "2. Solution. Since the desired integer solution $x$ is divisible by 10 and 12, it is divisible by 60, hence it can be written in the form $x=60 t, t \\in Z$. Substitute $x=60 t$ into the original equation:\n\n$3600 t^{2}+60 b t-9600=0$. Express $b: b=\\frac{160}{t}-60 t$. For $b$ to be an integer, $t$ must be a divisor of the number 160. $160=2^{5} \\cdot 5$. The number $160=2^{5} \\cdot 5$ has 12 positive divisors: $1,2,2^{2}, 2^{3}, 2^{4}, 2^{5}, 5,2 \\cdot 5,2^{2} \\cdot 5,2^{3} \\cdot 5,2^{4} \\cdot 5,2^{5} \\cdot 5$. Taking the sign into account, we get 24 divisors. Each of them corresponds to a solution $x=60 t$ and an integer $b=\\frac{160}{t}-60 t$. Therefore, the number of integers $b$ that satisfy the condition of the problem is 24. Since $b$ decreases as $t$ increases, the maximum $b$ corresponds to the smallest $t=-160$, so $b_{\\max }=-1+60 \\cdot 160=9599$.", "answer": "9599", "source": "olympiads", "problem_type": "Algebra", "uuid": "622e4608-b0d7-5071-90bd-fa39367cb1ac"}
+{"problem": "Task B-2.1. (20 points) For complex numbers $z, w$ such that $|z|=|w|=|z-w|$, calculate $\\left(\\frac{z}{w}\\right)^{99}$.", "solution": "First solution. Let $u=\\frac{z}{w}=x+y i$. Then $|u|=1,|u-1|=1$.\n\nThen $x^{2}+y^{2}=1,(x-1)^{2}+y^{2}=1$.\n\nBy subtracting these equations, we get $x=\\frac{1}{2}, y= \\pm \\frac{\\sqrt{3}}{2}$, so $u=\\frac{1}{2} \\pm \\frac{\\sqrt{3}}{2} i$.\n\nNow,\n\n$$\n\\left(\\frac{z}{w}\\right)^{99}=u^{99}=\\left[\\left(\\frac{1}{2} \\pm \\frac{\\sqrt{3}}{2} i\\right)^{3}\\right]^{33}=\\left(\\frac{1}{8} \\pm \\frac{3 \\sqrt{3}}{8} i-\\frac{9}{8} \\mp \\frac{3 \\sqrt{3}}{8} i\\right)^{33}=(-1)^{33}=-1\n$$", "answer": "-1", "source": "olympiads", "problem_type": "Algebra", "uuid": "bcfb9529-7521-5cae-8078-82e957941ea5"}
+{"problem": "4.9 On a $10 \\times 10$ grid paper, there are 11 horizontal grid lines and 11 vertical grid lines. A line segment connecting two adjacent nodes on the same straight line is called a \"link segment\". How many link segments need to be erased at least, so that at most 3 link segments remain at each node?", "solution": "[Solution] Removing the dotted links in the right figure satisfies the requirements of the problem, and there are 41 dotted links in the figure, so the minimum value required does not exceed 41.\n\nOn the other hand, there are 81 internal nodes on the grid paper, each with 4 links. Therefore, at least one link must be erased from each node to meet the requirement. Since each link connects two nodes, at least 41 links must be erased.\nIn summary, the minimum number of links that need to be erased is 41.", "answer": "41", "source": "olympiads", "problem_type": "Combinatorics", "uuid": "8253de0b-ede9-5630-a1d1-7d0ae1ae5cd2"}
+{"problem": "9. Point $Y$ lies on line segment $X Z$ such that $X Y=5$ and $Y Z=3$. Point $G$ lies on line $X Z$ such that there exists a triangle $A B C$ with centroid $G$ such that $X$ lies on line $B C, Y$ lies on line $A C$, and $Z$ lies on line $A B$. Compute the largest possible value of $X G$.", "solution": "Answer:\n$\\frac{20}{3}$\n\nSolution: The key claim is that we must have $\\frac{1}{X G}+\\frac{1}{Y G}+\\frac{1}{Z G}=0$ (in directed lengths).\nWe present three proofs of this fact.\nProof 1: By a suitable affine transformation, we can assume without loss of generality that $A B C$ is equilateral. Now perform an inversion about $G$ with radius $G A=G B=G C$. Then the images of $X, Y, Z$ (call them $\\left.X^{\\prime}, Y^{\\prime}, Z^{\\prime}\\right)$ lie on $(G B C),(G A C),(G A B)$, so they are the feet of the perpendiculars from $A_{1}, B_{1}, C_{1}$ to line $X Y Z$, where $A_{1}, B_{1}, C_{1}$ are the respective antipodes of $G$ on $(G B C),(G A C),(G A B)$. But now $A_{1} B_{1} C_{1}$ is an equilateral triangle with medial triangle $A B C$, so its centroid is $G$. Now the centroid of (degenerate) triangle $X^{\\prime} Y^{\\prime} Z^{\\prime}$ is the foot of the perpendicular of the centroid of $A_{1} B_{1} C_{1}$ onto the line, so it is $G$. Thus $X^{\\prime} G+Y^{\\prime} G+Z^{\\prime} G=0$, which yields the desired claim.\n\nProof 2: Let $W$ be the point on line $X Y Z$ such that $W G=2 G X$ (in directed lengths). Now note that $(Y, Z ; G, W)$ is a harmonic bundle, since projecting it through $A$ onto $B C$ gives $\\left(B, C ; M_{B C}, \\infty_{B C}\\right)$. By harmonic bundle properties, this yields that $\\frac{1}{Y G}+\\frac{1}{Z G}=\\frac{2}{W G}$ (in directed lengths), which gives the desired.\n\nProof 3: Let $P \\neq G$ be an arbitrary point on the line $X Y Z$. Now, in directed lengths and signed areas, $\\frac{G P}{G X}=\\frac{[G B P]}{[G B X]}=\\frac{[G C P]}{[G C X]}$, so $\\frac{G P}{G X}=\\frac{[G B P]-[G C P]}{[G B X]-[G C X]}=\\frac{[G B P]-[G C P]}{[G B C]}=\\frac{3([G B P]-[G C P])}{[A B C]}$. Writing analogous equations for $\\frac{G P}{G Y}$ and $\\frac{G P}{G Z}$ and summing yields $\\frac{G P}{G X}+\\frac{G P}{G Y}+\\frac{G P}{G Z}=0$, giving the desired.\nWith this lemma, we may now set $X G=g$ and know that\n$$\n\\frac{1}{g}+\\frac{1}{g-5}+\\frac{1}{g-8}=0\n$$\n\nSolving the quadratic gives the solutions $g=2$ and $g=\\frac{20}{3}$; the latter hence gives the maximum (it is not difficult to construct an example for which $X G$ is indeed 20/3).", "answer": "\\frac{20}{3}", "source": "olympiads", "problem_type": "Geometry", "uuid": "66849bc3-7be9-5bd8-8927-21c8941a4567"}
+{"problem": "5. Let $a$ and $n$ be natural numbers, and it is known that $a^{n}-2014$ is a number with $2014$ digits. Find the smallest natural number $k$ such that $a$ cannot be a $k$-digit number.", "solution": "Solution. Let $a$ be a $k$-digit number, then $10^{k-1} \\leq a2013$, i.e., $k>45$.\n\nWe will check for $k$ starting from 46, whether the condition of the absence of an integer in the specified interval is met. We will find that\n\n(for $k=46$) $2013 / 462013$ );\n\n(for $k=47$) $2013 / 472013$ );\n\n(for $k=48$) $2013 / 482013$ );\n\nbut (for $k=49$) $2013 / 49 > 41$ (since $\\quad 41 \\cdot 49 = 45^2 - 1^2 = 2025 - 16 < 2013$).\n\nThus, the interval $[2013 / 49, 2013 / 48)$ does not contain any integers, and for $k=49$, there are no suitable $n$.\n\n## Answer: $\\mathbf{k}=49$.\n\n## Criteria.\n\nThe problem is complex, so don't be stingy with points for reasonable thoughts. Any answer in the vicinity of $\\sqrt{2014}$, which is preceded by something reasonable, is worth points. A point is deducted for the absence of what is in small print.", "answer": "49", "source": "olympiads", "problem_type": "Number Theory", "uuid": "77cba4e7-5a90-5533-9f4b-220b85918184"}
+{"problem": "Let there be a figure with $9$ disks and $11$ edges, as shown below.\n\nWe will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$, and $1$ in disk $I$. Find the minimum sum of all real numbers written in $11$ edges.", "solution": "1. **Model the problem as an electrical circuit:**\n   - We are given a graph with 9 disks (nodes) and 11 edges.\n   - We need to assign real numbers to each disk such that the sum of the squares of the differences between the numbers on the disks connected by edges is minimized.\n   - We are required to set the number on disk \\(A\\) to \\(0\\) and on disk \\(I\\) to \\(1\\).\n\n2. **Interpret the problem using electrical circuit theory:**\n   - Consider each disk as a node in an electrical circuit.\n   - Assign a voltage to each node, with disk \\(A\\) at \\(0V\\) and disk \\(I\\) at \\(1V\\).\n   - Each edge represents a resistor of \\(1 \\Omega\\).\n\n3. **Apply the concept of minimizing power in the circuit:**\n   - The goal is to minimize the sum of the squares of the differences, which is equivalent to minimizing the total power dissipated in the circuit.\n   - The power dissipated in a resistor \\(R\\) with voltage difference \\(V\\) is given by \\(P = \\frac{V^2}{R}\\).\n   - For a resistor of \\(1 \\Omega\\), the power dissipated is \\(P = V^2\\).\n\n4. **Calculate the equivalent resistance of the circuit:**\n   - The total resistance between nodes \\(A\\) and \\(I\\) is given as \\(\\frac{33}{20} \\Omega\\).\n   - This is derived from the specific configuration of the graph and the resistances.\n\n5. **Determine the total power in the circuit:**\n   - The total voltage across the circuit is \\(1V\\) (from \\(A\\) to \\(I\\)).\n   - The total power dissipated in the circuit is given by:\n     \\[\n     P = \\frac{V^2}{R_{\\text{eq}}} = \\frac{1^2}{\\frac{33}{20}} = \\frac{20}{33}\n     \\]\n\n6. **Conclusion:**\n   - The minimum sum of the squares of the differences between the numbers on the disks connected by edges is \\(\\frac{20}{33}\\).\n\nThe final answer is \\(\\boxed{\\frac{20}{33}}\\).", "answer": "\\frac{20}{33}", "source": "aops_forum", "problem_type": "Calculus", "uuid": "69191341-fb5c-54c6-bc21-3d80836f8875"}
+{"problem": "There is a regular $17$-gon $\\mathcal{P}$ and its circumcircle $\\mathcal{Y}$ on the plane. \nThe vertices of $\\mathcal{P}$ are coloured in such a way that $A,B \\in \\mathcal{P}$ are of diff\u000berent colour, if the shorter arc connecting $A$ and $B$ on $\\mathcal{Y}$ has $2^k+1$ vertices, for some $k \\in \\mathbb{N},$ including $A$ and $B.$ \nWhat is the least number of colours which suffi\u000eces?", "solution": "1. **Identify the relevant values of \\(2^k + 1\\):**\n   Since the minor arc \\(AB\\) has at most 9 vertices on it, the relevant values of \\(2^k + 1\\) are \\(3, 5,\\) and \\(9\\). This is because:\n   \\[\n   2^1 + 1 = 3, \\quad 2^2 + 1 = 5, \\quad 2^3 + 1 = 9\n   \\]\n\n2. **Enumerate the vertices:**\n   Enumerate the vertices of the 17-gon as \\(V_0, V_1, \\ldots, V_{16}\\).\n\n3. **Determine the conditions for different colors:**\n   We have that \\(V_i\\) and \\(V_j\\) are of different colors if \\(i - j \\equiv \\pm 2, \\pm 4, \\pm 8 \\pmod{17}\\). This is derived from the relevant values of \\(2^k + 1\\).\n\n4. **Establish a lower bound for the number of colors:**\n   Since \\(V_0, V_2, V_4\\) must have pairwise distinct colors, we have \\(n \\geq 3\\).\n\n5. **Check if 4 colors suffice:**\n   If there are 4 colors \\(A, B, C,\\) and \\(D\\), we can try to color the vertices such that the condition is satisfied. One possible coloring is:\n   \\[\n   \\{ V_0, V_1, \\cdots, V_{16} \\} = \\{ A, D, B, B, C, C, D, D, B, B, C, A, D, D, B, B, C \\}\n   \\]\n   This coloring ensures that no two vertices \\(V_i\\) and \\(V_j\\) with \\(i - j \\equiv \\pm 2, \\pm 4, \\pm 8 \\pmod{17}\\) have the same color. Therefore, \\(n \\leq 4\\).\n\n6. **Prove that 3 colors are insufficient:**\n   Assume there exists a solution with \\(n = 3\\). Without loss of generality, let \\(V_0 = A\\) and \\(V_2 = B\\). Then:\n   \\[\n   V_4 = C, \\quad V_6 = A, \\quad V_8 = B, \\quad V_{10} = C, \\quad V_{12} = A, \\quad V_{14} = B, \\quad V_{16} = C\n   \\]\n   Continuing this pattern, we find that \\(V_{13} = A\\), which contradicts the assumption that \\(n = 3\\) is a solution because \\(V_0\\) and \\(V_{13}\\) would be the same color, violating the condition for different colors.\n\nTherefore, the least number of colors required is \\(4\\).\n\nThe final answer is \\(\\boxed{4}\\).", "answer": "4", "source": "aops_forum", "problem_type": "Combinatorics", "uuid": "ec0462f1-31cb-5753-b048-823b60e8ac48"}
+{"problem": "Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius?", "solution": "1. **Identify the roots of the polynomial:**\n   The polynomial given is \\(x^3 - 27x^2 + 222x - 540\\). The roots of this polynomial are the side lengths \\(a\\), \\(b\\), and \\(c\\) of the triangle \\(ABC\\).\n\n2. **Sum and product of the roots:**\n   By Vieta's formulas, for the polynomial \\(x^3 - 27x^2 + 222x - 540\\):\n   - The sum of the roots \\(a + b + c = 27\\).\n   - The product of the roots \\(abc = 540\\).\n\n3. **Inradius and circumradius formulas:**\n   - The inradius \\(r\\) of a triangle is given by:\n     \\[\n     r = \\frac{\\text{Area}}{s}\n     \\]\n     where \\(s\\) is the semi-perimeter of the triangle, \\(s = \\frac{a + b + c}{2}\\).\n   - The circumradius \\(R\\) of a triangle is given by:\n     \\[\n     R = \\frac{abc}{4 \\cdot \\text{Area}}\n     \\]\n\n4. **Product of inradius and circumradius:**\n   - The product of the inradius and circumradius is:\n     \\[\n     r \\cdot R = \\left(\\frac{\\text{Area}}{s}\\right) \\cdot \\left(\\frac{abc}{4 \\cdot \\text{Area}}\\right)\n     \\]\n     Simplifying this expression:\n     \\[\n     r \\cdot R = \\frac{\\text{Area} \\cdot abc}{s \\cdot 4 \\cdot \\text{Area}} = \\frac{abc}{4s}\n     \\]\n\n5. **Substitute the known values:**\n   - The semi-perimeter \\(s\\) is:\n     \\[\n     s = \\frac{a + b + c}{2} = \\frac{27}{2}\n     \\]\n   - The product of the roots \\(abc = 540\\).\n\n6. **Calculate the product \\(r \\cdot R\\):**\n   \\[\n   r \\cdot R = \\frac{abc}{4s} = \\frac{540}{4 \\cdot \\frac{27}{2}} = \\frac{540}{54} = 10\n   \\]\n\nThe final answer is \\(\\boxed{10}\\).", "answer": "10", "source": "aops_forum", "problem_type": "Geometry", "uuid": "54b47736-865d-5c1b-b404-586268ae6139"}
+{"problem": "Variant 9.5.2. Sides $A D$ and $D C$ of the inscribed quadrilateral $A B C D$ are equal. A point $X$ is marked on side $B C$ such that $A B = B X$. It is known that $\\angle B = 32^{\\circ}$ and $\\angle X D C = 52^{\\circ}$.\n(a) (1 point) How many degrees does the angle $A X C$ measure?\n\n(b) (3 points) How many degrees does the angle $A C B$ measure?\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_1f4981519479532effc1g-13.jpg?height=311&width=709&top_left_y=220&top_left_x=372)", "solution": "Answer: (a) $106^{\\circ}$. (b) $48^{\\circ}$.", "answer": "106", "source": "olympiads", "problem_type": "Geometry", "uuid": "cfc8164c-37e4-54f2-970f-e1f34174ff46"}
+{"problem": "10.024. The lengths of the parallel sides of the trapezoid are 25 and 4 cm, and the lengths of the non-parallel sides are 20 and 13 cm. Find the height of the trapezoid.", "solution": "## Solution.\n\nGiven $B C=4 \\text{ cm}, A D=25 \\text{ cm}, A B=20 \\text{ cm}, C D=13 \\text{ cm}$ (Fig. 10.24). Draw $B E \\perp A D$ and $C F \\perp A D$. Let $B E=C F=h, A E=x$, $F D=y$. Then from $\\triangle A B E$ and $\\triangle C F D$ we find $h^{2}=20^{2}-x^{2}=13^{2}-y^{2}$. Considering that $y=25-4-x=21-x$, we have $20^{2}-x^{2}=13^{2}-(21-x)^{2}$ or $42 x=672$, from which $x=16$ (cm). Therefore, $h=\\sqrt{20^{2}-16^{2}}=12$ (cm).\n\nAnswer: 12 cm.", "answer": "12", "source": "olympiads", "problem_type": "Geometry", "uuid": "a065ae49-8a7c-5a6d-8c48-e652a2042578"}
+{"problem": "7.2. Find the GCD of all numbers obtained by all possible permutations of the digits of the number 202120222023\n\nSolution. By the divisibility rule, all these numbers are divisible by 9 (the sum of the digits is 18). A sufficient condition to prove that there are no other numbers is that the difference between any two such numbers is also divisible by the GCD. For example, 222222100032 - 222222100023 = 9 is divisible by 9 and cannot be greater than 9.", "solution": "Answer: 9.\n\n## Criteria\n\n7 points - complete solution;\n\n3 points - GCD found, but not proven that there can be no other divisors.", "answer": "9", "source": "olympiads", "problem_type": "Number Theory", "uuid": "f17a8128-6a24-5cf7-999c-92fafc6772e7"}
+{"problem": "Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \\in \\{1,2,3,4,5\\}$ and $i \\neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$?", "solution": "1. **Define the problem and the sums:**\n   We are given five distinct real numbers \\(a_1, a_2, a_3, a_4, a_5\\). We need to consider all possible sums of the form \\(a_i + a_j\\) where \\(i \\neq j\\). There are \\(\\binom{5}{2} = 10\\) such sums since each pair \\((i, j)\\) is considered once.\n\n2. **Calculate the number of distinct sums:**\n   We need to find the smallest possible number of distinct sums, denoted as \\(m\\).\n\n3. **Example with specific values:**\n   Let's try the values \\(a_1 = 0, a_2 = 1, a_3 = 2, a_4 = 3, a_5 = 4\\). The sums are:\n   \\[\n   \\begin{aligned}\n   &0+1 = 1, \\\\\n   &0+2 = 2, \\\\\n   &0+3 = 3, \\\\\n   &0+4 = 4, \\\\\n   &1+2 = 3, \\\\\n   &1+3 = 4, \\\\\n   &1+4 = 5, \\\\\n   &2+3 = 5, \\\\\n   &2+4 = 6, \\\\\n   &3+4 = 7.\n   \\end{aligned}\n   \\]\n   The distinct sums are \\{1, 2, 3, 4, 5, 6, 7\\}, so \\(m = 7\\).\n\n4. **Check if \\(m = 6\\) is possible:**\n   Suppose \\(a_1 = 0\\) (without loss of generality, since adding a constant to each \\(a_i\\) does not change the number of distinct sums). Let the remaining numbers be \\(a, b, c, d\\) in increasing order. The sums are:\n   \\[\n   \\begin{aligned}\n   &0 + a = a, \\\\\n   &0 + b = b, \\\\\n   &0 + c = c, \\\\\n   &0 + d = d, \\\\\n   &a + b, \\\\\n   &a + c, \\\\\n   &a + d, \\\\\n   &b + c, \\\\\n   &b + d, \\\\\n   &c + d.\n   \\end{aligned}\n   \\]\n   For \\(m = 6\\), we need some sums to be equal. Assume \\(a + b = d\\). Then the sums are:\n   \\[\n   \\{a, b, c, d, a + c, a + d, b + c, b + d, c + d\\}.\n   \\]\n   Since \\(a + b = d\\), we have:\n   \\[\n   \\{a, b, c, d, a + c, d + a, b + c, b + d, c + d\\}.\n   \\]\n   We need to check if we can have only 6 distinct sums. If \\(a + c = b + d\\), then \\(a + c = b + (a + b) = a + 2b\\), so \\(c = 2b\\). This leads to contradictions as we need all numbers to be distinct.\n\n5. **Conclusion:**\n   After trying different configurations and ensuring all numbers are distinct, we find that the smallest possible value of \\(m\\) is indeed 7.\n\nThe final answer is \\(\\boxed{7}\\).", "answer": "7", "source": "aops_forum", "problem_type": "Combinatorics", "uuid": "93026f85-b5b7-5474-8263-958bfe22fd1c"}
+{"problem": "3. Given a convex quadrilateral $ABCD$ with area $P$. We extend side $AB$ beyond $B$ to $A_1$ such that $\\overline{AB}=\\overline{BA_1}$, then $BC$ beyond $C$ to $B_1$ such that $\\overline{BC}=\\overline{CB_1}$, then $CD$ beyond $D$ to $C_1$ so that $\\overline{CD}=\\overline{DC_1}$, and $DA$ beyond $A$ to $D_1$ such that $\\overline{DA}=\\overline{AD_1}$. What is the area of quadrilateral $A_1B_1C_1D_1$?", "solution": "Solution. We connect $A$ with $C_{1}$, $B$ with $D_{1}$, $C$ with $A_{1}$, and $D$ with $B_{1}$. Then,\n\n$$\n\\begin{aligned}\n& P_{\\triangle A B C}=P_{I}=P_{I I} \\\\\n& P_{\\triangle B C D}=P_{I I I}=P_{I V} \\\\\n& P_{\\triangle C D A}=P_{V}=P_{V I} \\\\\n& P_{\\triangle D A B}=P_{V I I}=P_{I I I}\n\\end{aligned}\n$$\n\nwhere\n\n$$\n\\begin{array}{ll}\nP_{I}=P_{\\Delta A_{1} C}, & P_{I I}=P_{\\Delta C A_{1} B_{1}} \\\\\nP_{I I I}=P_{\\Delta D C B_{1}}, & P_{I V}=P_{\\Delta C_{1} D B_{1}} \\\\\nP_{V}=P_{\\Delta C_{1} D A}, & P_{V I}=P_{\\Delta C_{1} D_{1} A} \\\\\nP_{I I I}=P_{\\Delta A D_{1} B}, & P_{V I I I}=P_{\\Delta B D_{1} A_{1}}\n\\end{array}\n$$\n\n![](https://cdn.mathpix.com/cropped/2024_06_05_ea01241e897ed9d0948dg-02.jpg?height=507&width=641&top_left_y=708&top_left_x=838)\n\nWe have:\n\n$$\n\\begin{aligned}\nP_{\\Delta A B C}+P_{\\Delta B C D}+P_{\\Delta C D A}+P_{\\Delta D A B} & =P_{I}+P_{I I I}+P_{V}+P_{I I}= \\\\\n& =P_{I I}+P_{I V}+P_{V I}+P_{V I I I}=2 P \\\\\nP_{1}=P+\\left(P_{I}+P_{I I}+P_{I I I}+P_{I V}+P_{V}+P_{I I}\\right. & \\left.+P_{V I I}+P_{I I I}\\right)=P+(2 P+2 P)=5 P\n\\end{aligned}\n$$", "answer": "5P", "source": "olympiads", "problem_type": "Geometry", "uuid": "070e4893-92fd-50bb-8c2b-f3c54801336d"}
+{"problem": "6.7. $f(x)=4 x^{3}-\\frac{5}{x^{7}}-8 \\sqrt[6]{x^{5}}+10$.\n\n6.7. $f(x)=4 x^{3}-\\frac{5}{x^{7}}-8 \\sqrt[6]{x^{5}}+10$.", "solution": "Solution.\n\n$$\n\\begin{aligned}\n& f^{\\prime}(x)=\\left(4 x^{3}-5 x^{-7}-8 x^{\\frac{5}{6}}+10\\right)^{\\prime}=\\left[\\left((x)^{n}\\right)^{\\prime}=n x^{n-1},(\\text { const })^{\\prime}=0\\right]= \\\\\n& =4\\left(x^{3}\\right)^{\\prime}-5\\left(x^{-7}\\right)^{\\prime}-8\\left(x^{\\frac{5}{6}}\\right)^{\\prime}+(10)^{\\prime}=4 \\cdot 3 x^{3-1}-5 \\cdot(-7) x^{-7-1}- \\\\\n& -8 \\cdot \\frac{5}{6} x^{\\frac{5}{6}-1}+0=12 x^{2}+35 x^{-8}-\\frac{20}{3} x^{-\\frac{1}{6}}=12 x^{2}+\\frac{35}{x^{8}}-\\frac{20}{3 \\sqrt[6]{x}} . \\\\\n& \\quad \\text { The derivative is defined for } x \\in(0 ;+\\infty) .\n\\end{aligned}\n$$", "answer": "12x^{2}+\\frac{35}{x^{8}}-\\frac{20}{3\\sqrt[6]{x}}", "source": "olympiads", "problem_type": "Algebra", "uuid": "4e10be47-9530-5395-8f0e-b29053c59d61"}
+{"problem": "1. Let $a, b, c, d$ be integers, and $a<2b, b<3c, c<4d$. Given $d<100$, then the maximum possible value of $a$ is ( ).\n(A) 2367\n(B) 2375\n(C) 2391\n(D) 2399", "solution": "$-.1 .(\\mathrm{A})$.\nThen, if $d=99$, we have $c<4 \\times 99=395$. Taking $c=395$, we get $b<3 \\times 395=1185$. Taking $b=1184$, we get $a<2368$. Therefore, the greatest integer for $a$ is 2367.", "answer": "2367", "source": "cn_contest", "problem_type": "Inequalities", "uuid": "6811d79b-bdc4-5428-8f42-9a1e6c7adca4"}
+{"problem": "7.242. $\\left(16 \\cdot 5^{2 x-1}-2 \\cdot 5^{x-1}-0.048\\right) \\lg \\left(x^{3}+2 x+1\\right)=0$.", "solution": "Solution.\n\nDomain: $x^{3}+2 x+1>0$.\n\nFrom the condition $16 \\cdot 5^{2 x-1}-2^{x-1}-0.048=0$ or $\\lg \\left(x^{3}+2 x+1\\right)=0$. Rewrite the first equation as\n\n$\\frac{16}{5} \\cdot 5^{2 x}-\\frac{2}{5} \\cdot 5^{x}-0.048=0 \\Leftrightarrow 16 \\cdot 5^{2 x}-2 \\cdot 5^{x}-0.24=0$.\n\nSolving this equation as a quadratic in terms of $5^{x}$, we get $5^{x}=-\\frac{3}{40}$ (no solutions), or $5^{x}=5^{-1} \\Leftrightarrow x_{1}=-1$ (does not satisfy the domain). From the second equation, we have\n\n$x^{3}+2 x+1=1 \\Leftrightarrow x^{3}+2 x=0 \\Leftrightarrow x\\left(x^{2}+2\\right)=0, x_{3}=0, x^{2}+2 \\neq 0$.\n\nAnswer: 0.", "answer": "0", "source": "olympiads", "problem_type": "Algebra", "uuid": "2bfb42a5-11c9-5aee-b6df-edb9b8efee0e"}
+{"problem": "5. Let the complex numbers $z_{1}, z_{2}$ satisfy $\\left|z_{1}\\right|=\\left|z_{1}+z_{2}\\right|=3,\\left|z_{1}-z_{2}\\right|=3 \\sqrt{3}$, then $\\log _{3}\\left|\\left(z_{1} \\overline{z_{2}}\\right)^{2000}+\\left(\\overline{z_{1}} z_{2}\\right)^{2000}\\right|=$ $\\qquad$ .", "solution": "4000\n5. 【Analysis and Solution】Solution 1 From the given conditions, we have\n$$\n\\begin{array}{l}\n9=\\left|z_{1}+z_{2}\\right|^{2}=\\left|z_{1}\\right|^{2}+\\left|z_{2}\\right|^{2}+z_{1} \\bar{z}_{2}+\\bar{z}_{1} z_{2}, \\\\\n27=\\left|z_{1}-z_{2}\\right|^{2}=\\left|z_{1}\\right|^{2}+\\left|z_{2}\\right|^{2}-\\left(z_{1} \\bar{z}_{2}+\\bar{z}_{1} z_{2}\\right) .\n\\end{array}\n$$\n\nSince $\\left|z_{1}\\right|=3$, it follows that $\\left|z_{2}\\right|=3, z_{1} \\bar{z}_{2}+\\bar{z}_{1} z_{2}=-9$, and $\\left|z_{1} \\bar{z}_{2}\\right|+| \\bar{z}_{1} z_{2} \\mid=9$.\nLet $z_{1} \\bar{z}_{2}=9(\\cos \\theta+i \\sin \\theta)$,\n$$\n\\bar{z}_{1} z_{2}=9(\\cos \\theta-i \\sin \\theta) \\text {. }\n$$\n$-9=z_{1} \\bar{z}_{2}+z_{1} \\bar{z}_{2}=18 \\cos \\theta$, which gives\n$$\n\\cos \\theta=-\\frac{1}{2} \\text {. }\n$$\n\nThus, $z_{1} \\bar{z}_{2}=9 \\omega$ or $z_{1} \\bar{z}_{2}=9 \\omega^{2}$,\nwhere $\\omega=-\\frac{1}{2}+\\frac{\\sqrt{3}}{2} i$.\nWhen $z_{1} \\bar{z}_{2}=9 \\omega$, we get\n$$\n\\left(z_{1} \\bar{z}_{2}\\right)^{2000}+\\left(\\bar{z}_{1} z_{2}\\right)^{2000}=-9^{2000} \\text {, }\n$$\n\nTherefore, $\\log _{3}\\left|\\left(z_{1} \\bar{z}_{2}\\right)^{2000}+\\left(\\bar{z}_{1} z_{2}\\right)^{2000}\\right|=4000$.\nWhen $z_{1} \\bar{z}_{2}=9 \\omega^{2}$, we get the same result, so the answer is 4000.\nSolution 2 Consider the geometric meaning of complex numbers. Let the points corresponding to the complex numbers $z_{1}, z_{2}$ in the complex plane be $A, B$, and let $C$ be the midpoint of $A B$. From the conditions,\n$$\n\\begin{array}{l}\n|O A|=\\left|z_{1}\\right|=3, \\\\\n|O C|=\\frac{1}{2}\\left|z_{1}+z_{2}\\right|=\\frac{3}{2},\n\\end{array}\n$$\n$|A C|=\\frac{1}{2}\\left|z_{1}-z_{2}\\right|=\\frac{3}{2} \\sqrt{3}$, thus it is easy to see that $\\triangle O A B$ is an equilateral triangle. Without loss of generality, assume $z_{2}=z_{1} e^{i \\frac{\\pi}{3}}$, substituting this in gives $\\log _{3}\\left|\\left(z_{1} \\bar{z}_{2}\\right)^{2000}+\\left(\\bar{z}_{1} z_{2}\\right)^{2000}\\right|=4000$.", "answer": "4000", "source": "olympiads", "problem_type": "Algebra", "uuid": "33d55892-4916-5da3-a46d-d4c242e74b6b"}
+{"problem": "II. (25 points) If $2 x+y \\geqslant 1$. Try to find the minimum value of the function $u=y^{2}-2 y$ $+x^{2}+4 x$.", "solution": "$\\therefore$ From $u=y^{2}-2 y+x^{2}+4 x$ completing the square, we get\n$$\n(x+2)^{2}+(y-1)^{2}=u+5 \\text {. }\n$$\n\nSince $(x+2)^{2}+(y-1)^{2}$ can be regarded as the square of the distance from point $P(x, y)$ to the fixed point $(-2,1)$. The constraint $2 x+y \\geqslant 1$ indicates that point $P$ is in the region $G$ which is the right upper area of the line $l: 2 x+y=1$. Therefore, the problem is transformed into finding the shortest distance from point $A(-2,1)$ to the region $G$.\nFrom the right figure, it can be seen that the distance from point $A$ to\n\nline $I$ is the minimum value of the distance from point $A$ to any point in region $G$.\n$$\n\\begin{aligned}\nd & =\\frac{|2 \\cdot(-2)+1-1|}{\\sqrt{2^{2}+1^{2}}} \\\\\n& =\\frac{4}{\\sqrt{5}}, \\\\\n& \\therefore d^{2}=\\frac{16}{5} \\text {, i.e., } u+5=\\frac{16}{5}\n\\end{aligned}\n$$\n$\\therefore u=-\\frac{9}{5}$. That is, the minimum value of $u$ is $-\\frac{9}{5}$.", "answer": "-\\frac{9}{5}", "source": "cn_contest", "problem_type": "Algebra", "uuid": "06e46eda-c480-5626-8e91-ffebd1a3fd68"}
+{"problem": "3.40 A red pencil costs 27 kopecks, a blue one - 23 kopecks. No more than 9 rubles 40 kopecks can be spent on purchasing pencils. It is necessary to purchase the maximum possible total number of red and blue pencils. At the same time, the number of red pencils should be as few as possible, but the number of blue pencils should not differ from the number of red pencils by more than 10. How many red and how many blue pencils should be purchased under the given conditions", "solution": "3.40 Let $x$ red and $y$ blue pencils be bought. According to the conditions, $27 x+23 y \\leqslant 940$ and $y-x \\leqslant 10$. Let's construct the lines\n\n![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-108.jpg?height=597&width=625&top_left_y=113&top_left_x=287)\n\n\\[\n\\begin{aligned}\n& 27 x+23 y=940 \\\\\n& y-x=10\n\\end{aligned}\n\\]\n\nFrom Fig. 3.8, it is clear that these lines intersect at point $A$, the coordinates of which satisfy equations (1) and (2), and at this point, the maximum possible sum $x+y$ is achieved. Solving the system (1), (2) and considering that the numbers $x$ and $y$ are natural, we get $x=14, y=24$.\n\nAnswer: 14 red and 24 blue pencils.", "answer": "14", "source": "olympiads", "problem_type": "Algebra", "uuid": "962f8132-4ca0-5e0f-b6b6-b040f23d5aa7"}
+{"problem": "Given $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\n$$\\frac{x}{1+y+z x}+\\frac{y}{1+z+x y}+\\frac{z}{1+x+y z}=\\frac{3}{x+y+z} .$$", "solution": "3. It is not difficult to prove: $\\frac{x}{1+y+z x} \\leqslant \\frac{1}{x+y+z}$, $\\frac{y}{1+z+x y} \\leqslant \\frac{1}{x+y+z}$, $\\frac{z}{1+x+y z} \\leqslant \\frac{1}{x+y+z}$. Therefore, if the equality holds, it is easy to get $x=y=z=1$.\n\nNote: This problem can also be solved as follows: First prove $\\frac{x}{1+y+z x} \\leqslant \\frac{x}{x+y+z}$, etc., so we can get $x+y+z \\geqslant 3$, hence $x=y=z=1$.", "answer": "x=y=z=1", "source": "inequalities", "problem_type": "Algebra", "uuid": "c01ee438-4f54-5585-bbf8-4d10362f295a"}
+{"problem": "Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\\theta$. The maximum possible value for $\\tan \\theta$ can be expressed in the form $\\dfrac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.\n\n[i]Proposed by James Lin[/i]", "solution": "1. **Given Information and Setup:**\n   - We have integers \\(a_1, a_2, a_3, a_4\\) with distinct absolute values.\n   - Points in the coordinate plane are \\(A_1 = (a_1, a_1^2)\\), \\(A_2 = (a_2, a_2^2)\\), \\(A_3 = (a_3, a_3^2)\\), and \\(A_4 = (a_4, a_4^2)\\).\n   - Lines \\(A_1A_2\\) and \\(A_3A_4\\) intersect on the \\(y\\)-axis at an acute angle \\(\\theta\\).\n   - We need to find the maximum possible value for \\(\\tan \\theta\\) expressed as \\(\\frac{m}{n}\\) for relatively prime positive integers \\(m\\) and \\(n\\), and then find \\(100m + n\\).\n\n2. **Slope Calculation:**\n   - The slope of line \\(A_1A_2\\) is:\n     \\[\n     m_1 = \\frac{a_2^2 - a_1^2}{a_2 - a_1} = a_2 + a_1\n     \\]\n   - The slope of line \\(A_3A_4\\) is:\n     \\[\n     m_2 = \\frac{a_4^2 - a_3^2}{a_4 - a_3} = a_4 + a_3\n     \\]\n\n3. **Intersection on the \\(y\\)-axis:**\n   - Since the lines intersect on the \\(y\\)-axis, their \\(x\\)-coordinates at the intersection point are zero. Thus, the \\(y\\)-intercepts of the lines are:\n     \\[\n     c_1 = a_1^2 \\quad \\text{and} \\quad c_2 = a_3^2\n     \\]\n\n4. **Angle Between Lines:**\n   - The tangent of the angle \\(\\theta\\) between the two lines is given by:\n     \\[\n     \\tan \\theta = \\left| \\frac{m_1 - m_2}{1 + m_1 m_2} \\right| = \\left| \\frac{(a_1 + a_2) - (a_3 + a_4)}{1 + (a_1 + a_2)(a_3 + a_4)} \\right|\n     \\]\n\n5. **Maximizing \\(\\tan \\theta\\):**\n   - Let \\(x = a_1 + a_2\\) and \\(y = a_3 + a_4\\). We need to maximize:\n     \\[\n     f(x, y) = \\left| \\frac{x - y}{1 + xy} \\right|\n     \\]\n\n6. **Case Analysis:**\n   - **Case 1: \\(x, y > 0\\)**\n     - Here, \\(x, y \\geq 1\\) since they are integers. In this case, \\(|f(x, y)| < 1\\), which cannot exceed our current maximum.\n   - **Case 2: \\(x > 0, y < 0\\)**\n     - The numerator is positive, and the denominator is negative, so \\(|f(x, y)|\\) cannot be our maximum.\n   - **Case 3: \\(x < 0, y > 0\\)**\n     - Let \\(x = -u\\) and \\(y = v\\). We want to maximize:\n       \\[\n       \\frac{u + v}{uv - 1}\n       \\]\n     - We need to check when:\n       \\[\n       \\frac{u + v}{uv - 1} > \\frac{3}{2} \\implies 3(uv - 1) < 2(u + v) \\implies 3uv - 3 < 2u + 2v \\implies 3uv - 2u - 2v < 3\n       \\]\n     - If \\(u, v > 1\\), then it's impossible because \\(3uv - 2u - 2v \\geq 3\\) is false. Thus, we can let \\(v = 1\\) and need \\(u < 5\\).\n\n7. **Checking Specific Values:**\n   - For \\(u = 4\\) and \\(v = 1\\):\n     - We have \\(a_1 = -6, a_2 = 2, a_3 = 4, a_4 = -3\\).\n     - This gives:\n       \\[\n       \\tan \\theta = \\left| \\frac{(-6 + 2) - (4 - 3)}{1 + (-6 + 2)(4 - 3)} \\right| = \\left| \\frac{-4 - 1}{1 - 4} \\right| = \\left| \\frac{-5}{-3} \\right| = \\frac{5}{3}\n       \\]\n\nThe final answer is \\(\\boxed{503}\\).", "answer": "503", "source": "aops_forum", "problem_type": "Geometry", "uuid": "079495cf-64e8-565d-ab82-523255814ae6"}
+{"problem": "7. (10 points) Two wheels rotate, meshed with each other, around fixed axes passing through the centers of wheels $A$ and $B$. The radii of the wheels differ by a factor of three. The smaller wheel makes 30 revolutions per minute. Determine how many seconds the larger wheel spends on one revolution?\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_31c3376e8d47665bbf90g-04.jpg?height=403&width=480&top_left_y=958&top_left_x=868)", "solution": "Answer: $6 s$\n\nSolution. The speeds of points lying on the edge of the wheels are the same.\n\n(2 points)\n\nThe distances traveled by these points differ by three times.\n\nThe small wheel spends on one revolution: $t_{\\text {small }}=\\frac{1 \\text { min }}{30 \\text { revolutions }}=\\frac{60 s}{30 \\text { revolutions }}=2 s$. (3 points)\n\nTherefore, the large wheel spends on one revolution:\n\n$t_{\\text {large }}=3 t_{\\text {small }}=3 \\cdot 2=6 s$.", "answer": "6", "source": "olympiads", "problem_type": "Algebra", "uuid": "1c1dd0c5-08ce-59f7-b12f-4275c9d41efb"}
+{"problem": "Two is $10 \\%$ of $x$ and $20 \\%$ of $y$. What is $x - y$?\n$(\\mathrm {A}) \\ 1 \\qquad (\\mathrm {B}) \\ 2 \\qquad (\\mathrm {C})\\ 5 \\qquad (\\mathrm {D}) \\ 10 \\qquad (\\mathrm {E})\\ 20$", "solution": "$2 = \\frac {1}{10}x \\Longrightarrow x = 20,\\quad 2 = \\frac{1}{5}y \\Longrightarrow y = 10,\\quad x-y = 20 - 10=10  \\mathrm{(D)}$.", "answer": "10", "source": "amc_aime", "problem_type": "Algebra", "uuid": "5fafd2ee-f7d4-5e50-8a7e-7fe29cc17f17"}
+{"problem": "A father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to the third child, and so on. It turns out that each child receives the same amount of property. The father has $\\qquad$ children.", "solution": "(Method 1)\nLet the father's property be $x$ yuan;\nSince the eldest and the second son receive the same amount of property, we have\n$$\n1000+(x-1000) \\times \\frac{1}{10}=2000+\\left[(x-1000) \\times\\left(1-\\frac{1}{10}\\right)-2000\\right] \\times \\frac{1}{10} ;\n$$\n\nSolving this, we get $x=81000$;\nThe eldest son receives $1000+(81000-1000) \\times \\frac{1}{10}=9000$ yuan;\nThat is, each child receives 9000 yuan;\nThis father has a total of $81000 \\div 9000=9$ children.\n(Method 2)\nSince each child receives the same amount of property;\nThe eldest and the second son receive the same amount of property;\nThe eldest son receives 1000 yuan plus $\\frac{1}{10}$ of the remaining property (let's denote this remaining as $A$); the second son receives 2000 yuan plus $\\frac{1}{10}$ of the remaining property (let's denote this remaining as $B$); thus, $\\frac{1}{10}$ of $A$ is 1000 yuan more than $\\frac{1}{10}$ of $B$;\n$A$ is $1000 \\div \\frac{1}{10}=10000$ yuan more than $B$;\nFrom $A$ to $B$, it decreases by $\\frac{1}{10}$ of $A$ and 2000 yuan;\nTherefore, $\\frac{1}{10}$ of $A$ is $10000-2000=8000$ yuan;\n$A$ is $8000 \\div \\frac{1}{10}=80000$ yuan;\nThis father's property is $1000+80000=81000$ yuan;\nThe eldest son receives $1000+(81000-1000) \\times \\frac{1}{10}=9000$ yuan;\nThat is, each child receives 9000 yuan;\nThis father has a total of $81000 \\div 9000=9$ children.\n(Method 3)\nLet this father have $n$ children;\nThen the second-to-last child receives $1000(n-1)$ yuan plus $\\frac{1}{10}$ of the remainder;\nThe last child receives $1000n$ yuan;\nThese $1000n$ yuan are $\\frac{9}{10}$ of what remains after the second-to-last child receives $1000(n-1)$; each child receives the same amount of property;\nFrom the equality of the property received by the second-to-last and the last child, we have\n$$\n1000(n-1)+1000n \\times \\frac{1}{9}=1000n \\text {; }\n$$\n\nSolving this, we get $n=9$;\nThis father has a total of 9 children.", "answer": "9", "source": "olympiads", "problem_type": "Algebra", "uuid": "16dbe1c7-62c1-5abf-88b4-a4d83a5c8276"}
+{"problem": "7.1. Solve the equation $\\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \\cdot 2 \\cdot \\ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a solution is a triplet $(n, k, l)$; if solutions differ in at least one component, they are considered different.", "solution": "Answer: 10 (all triples of solutions $(n, k, l):(3,1,2),(3,2,1),(4,3,3)$).\n\nSolution. Note that if $k4$, then $n!>4 \\cdot(n-1)!\\geqslant 2 \\cdot(k!+l!)$, so there are no such solutions. If $n=2$, we get $1=k!+l!$ - no solutions; if $n=3$, the equation $3=k!+l!$ has two solutions: $k=1, l=2$ and $k=2, l=1$; if $n=4$, the equation $12=k!+l!$ gives one more solution $k=l=3$.", "answer": "10", "source": "olympiads", "problem_type": "Number Theory", "uuid": "09902996-6e6f-5507-94c7-f8b8045a702b"}
+{"problem": "Example 13 Given that $a$, $b$, and $c$ are all positive integers, and the parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A$ and $B$. If the distances from $A$ and $B$ to the origin are both less than 1, find the minimum value of $a+b+c$.\n(1996, National Junior High School Mathematics League)", "solution": "Solution: Let $A\\left(x_{1}, 0\\right)$ and $B\\left(x_{2}, 0\\right)$, and $x_{1} < 0 < x_{2}$,  \nthen $x_{1} < 0, \\\\ \n\\therefore b > 2 \\sqrt{a c} . \\\\ \n\\text{Also, } \\because |O A| = |x_{1}| > 1$. Therefore, the parabola opens upwards, and when $x = -1$, $y > 0$, so $a(-1)^{2} + b(-1) + c > 0$, which means $b < a + c + 1$. Since $b > 2 \\sqrt{a c}$, we have $2 \\sqrt{a c} + 1 < a + c$,  \ni.e., $(\\sqrt{a} - \\sqrt{c})^{2} > 1$.  \nFrom equation (2), we get $\\sqrt{a} - \\sqrt{c} > 1$,  \n\ni.e., $\\sqrt{a} > \\sqrt{c} + 1$.  \nThus, $a > (\\sqrt{c} + 1)^{2} \\geq (\\sqrt{1} + 1)^{2} = 4$.  \nSolving, we get $a \\geq 5$.  \n$$\n\\begin{array}{l}\n\\text{Also, } \\because b > 2 \\sqrt{a c} \\geq 2 \\sqrt{5 \\times 1} > 4, \\\\\n\\therefore b \\geq 5 .\n\\end{array}\n$$\n\nWhen $a = 5, b = 5, c = 1$, the parabola $y = 5 x^{2} + 5 x + 1$ satisfies the given conditions.  \nTherefore, the minimum value of $a + b + c$ is 11.", "answer": "11", "source": "cn_contest", "problem_type": "Algebra", "uuid": "145da507-b5fa-5f63-b6dd-5097b644c349"}
+{"problem": "Example 2 Let $x \\in \\mathbf{R}$, try to find the minimum value of the function $f(x)=\\left(x^{2}+4 x+5\\right)\\left(x^{2}+4 x+2\\right)+2 x^{2}+8 x+1$.", "solution": "Let $u=x^{2}+4 x$, then\n$$\nf(x)=g(u)=(u+5)(u+2)+2 u+1=u^{2}+9 u+11 \\text {. }\n$$\n\nBy the properties of quadratic functions, when $u=-\\frac{9}{2}$, $g(u)$ reaches its minimum value, and when $u>-\\frac{9}{2}$, $g(u)$ is a strictly increasing function of $u$. Since $u=x^{2}+4 x=(x+2)^{2}-4$, we know that $u \\geqslant-4$, and when $x=-2$, $u=-4$. Therefore, when $x=-2$, $f(x)$ reaches its minimum value, which is -9.", "answer": "-9", "source": "olympiads", "problem_type": "Algebra", "uuid": "7e8b3d00-0ae4-5fe2-9545-c37a5bf4b6e3"}
+{"problem": "8. (15 points) A thin ray of light falls on a thin converging lens at a distance of $x=10$ cm from its optical center. The angle between the incident ray and the plane of the lens $\\alpha=45^{\\circ}$, and the angle between the refracted ray and the plane of the lens $\\beta=30^{\\circ}$. Determine its focal length.", "solution": "Answer: $\\approx 13.7$ cm.\n\nSolution. Parallel rays intersect at the focus, so $F$ is the focus of the given lens.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_7a4bc87492c06fc34cd6g-06.jpg?height=480&width=302&top_left_y=1502&top_left_x=957)\n\nIn triangle $O A F$: angle $F A O=30^{\\circ}$, angle $O F A=15^{\\circ}$, angle $A O F=135^{\\circ}$.\n\nTherefore, $\\frac{O F}{\\sin 30^{\\circ}}=\\frac{A O}{\\sin 15^{\\circ}}$.\n\nWe obtain that the focal length:\n\n$F C=O F \\sin 45^{\\circ}=A O \\frac{\\sin 30^{\\circ}}{\\sin 15^{\\circ}} \\sin 45^{\\circ} \\approx 13.7 \\mathrm{~cm}$.\n\n## Multidisciplinary Engineering Olympiad \"Star\" in Natural Sciences\n\n10th grade\n\nFinal Round\n\n$2022-2023$\n\nVersion 2\n\nProblems, answers, and evaluation criteria", "answer": "13.7", "source": "olympiads", "problem_type": "Geometry", "uuid": "78b0db16-409f-5e5d-9861-d426afbf5dc6"}
+{"problem": "The equation of line $\\ell_1$ is $24x-7y = 319$ and the equation of line $\\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\\ell_1$ and $\\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.\n\n[i]Proposed by Christopher Cheng[/i]\n\n[hide=Solution][i]Solution. [/i] $\\boxed{6}$\nNote that $(20,23)$ is the intersection of the lines $\\ell_1$ and $\\ell_2$. Thus, we only care about lattice points on the the two lines that are an integer distance away from $(20,23)$. Notice that $7$ and $24$ are part of the Pythagorean triple $(7,24,25)$ and $5$ and $12$ are part of the Pythagorean triple $(5,12,13)$. Thus, points on $\\ell_1$ only satisfy the conditions when $n$ is divisible by $25$ and points on $\\ell_2$ only satisfy the conditions when $n$ is divisible by $13$. Therefore, $a$ is just the number of positive integers less than $2023$ that are divisible by both $25$ and $13$. The LCM of $25$ and $13$ is $325$, so the answer is $\\boxed{6}$.[/hide]", "solution": "1. **Intersection of Lines**:\n   We start by finding the intersection of the lines $\\ell_1$ and $\\ell_2$. The equations are:\n   \\[\n   24x - 7y = 319\n   \\]\n   \\[\n   12x - 5y = 125\n   \\]\n   To find the intersection, we solve this system of linear equations. We can use the method of elimination or substitution. Here, we will use elimination.\n\n2. **Elimination Method**:\n   Multiply the second equation by 2 to align the coefficients of $x$:\n   \\[\n   24x - 10y = 250\n   \\]\n   Now subtract the second equation from the first:\n   \\[\n   (24x - 7y) - (24x - 10y) = 319 - 250\n   \\]\n   Simplifying, we get:\n   \\[\n   3y = 69 \\implies y = 23\n   \\]\n   Substitute $y = 23$ back into the first equation:\n   \\[\n   24x - 7(23) = 319 \\implies 24x - 161 = 319 \\implies 24x = 480 \\implies x = 20\n   \\]\n   Therefore, the intersection point is $(20, 23)$.\n\n3. **Distance from $(20, 23)$**:\n   We need to find lattice points on $\\ell_1$ and $\\ell_2$ that are an integer distance $n$ from $(20, 23)$. The distance formula is:\n   \\[\n   d = \\sqrt{(x - 20)^2 + (y - 23)^2}\n   \\]\n   For $d$ to be an integer, $(x - 20)^2 + (y - 23)^2$ must be a perfect square, say $k^2$. Thus:\n   \\[\n   (x - 20)^2 + (y - 23)^2 = n^2\n   \\]\n\n4. **Lattice Points on $\\ell_1$**:\n   The line $\\ell_1$ can be rewritten in slope-intercept form:\n   \\[\n   y = \\frac{24}{7}x - \\frac{319}{7}\n   \\]\n   Notice that $24$ and $7$ are part of the Pythagorean triple $(7, 24, 25)$. Therefore, the distance $n$ must be a multiple of $25$ for there to be lattice points on $\\ell_1$.\n\n5. **Lattice Points on $\\ell_2$**:\n   The line $\\ell_2$ can be rewritten in slope-intercept form:\n   \\[\n   y = \\frac{12}{5}x - \\frac{125}{5}\n   \\]\n   Notice that $12$ and $5$ are part of the Pythagorean triple $(5, 12, 13)$. Therefore, the distance $n$ must be a multiple of $13$ for there to be lattice points on $\\ell_2$.\n\n6. **Common Multiples**:\n   We need $n$ to be a multiple of both $25$ and $13$. The least common multiple (LCM) of $25$ and $13$ is:\n   \\[\n   \\text{LCM}(25, 13) = 325\n   \\]\n   We need to count the multiples of $325$ that are less than $2023$:\n   \\[\n   \\left\\lfloor \\frac{2023}{325} \\right\\rfloor = 6\n   \\]\n\nConclusion:\nThe number of positive integer values $n$ less than $2023$ such that there exists a lattice point on both $\\ell_1$ and $\\ell_2$ at a distance $n$ from $(20, 23)$ is $\\boxed{6}$.", "answer": "6", "source": "aops_forum", "problem_type": "Number Theory", "uuid": "b325fc9f-7ff8-5556-976b-f72348f99405"}
+{"problem": "## 128. Math Puzzle $1 / 76$\n\nA circus gave 200 performances in the last season, all of which were sold out. The number of seats in the circus tent is three times the fourth part of the number of performances given.\n\na) How many program leaflets were printed if one fourth of the visitors bought a leaflet?\n\nb) How many Marks were additionally collected from the entrance fees for the animal show if it was visited by half of the visitors and the entrance fee was 0.30 M?", "solution": "From $(200: 4) \\cdot 3=150$ it follows that the circus has over 150 seats.\n\na) $(200 \\cdot 150): 4=7500$; 7500 program booklets were sold during the season.\n\nb) $(200 \\cdot 150): 2=15000$ and $30 \\cdot 15000=450000$; from the animal show, €450,000 was earned.", "answer": "450000", "source": "olympiads", "problem_type": "Logic and Puzzles", "uuid": "350ffb62-b012-5c3c-adcf-6591fca675be"}
+{"problem": "Determine the value of $2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)$.", "solution": "Solution 1\n\nMultiplying through, we obtain\n\n$$\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)=16+\\frac{16}{2}+\\frac{16}{4}+\\frac{16}{8}+\\frac{16}{16}=16+8+4+2+1=31\n$$\n\n## Solution 2\n\nUsing a common denominator inside the parentheses, we obtain\n\n$$\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)=16\\left(\\frac{16}{16}+\\frac{8}{16}+\\frac{4}{16}+\\frac{2}{16}+\\frac{1}{16}\\right)=16\\left(\\frac{31}{16}\\right)=31\n$$\n\nANSWER: 31", "answer": "31", "source": "olympiads", "problem_type": "Algebra", "uuid": "39ed5515-9c2e-5efa-bc64-afc56279b4fe"}
+{"problem": "5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\\sqrt{1+a^{2}}+\\sqrt{1+2 b}$ the integer part is", "solution": "5. 2\n\nTranslate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.", "answer": "2", "source": "olympiads", "problem_type": "Algebra", "uuid": "30f87501-618e-5c88-94ef-7c3ded50115e"}
+{"problem": "1.57. The vertices of a rectangle inscribed in a circle divide it into four arcs. Find the distance from the midpoint of one of the larger arcs to the vertices of the rectangle, if its sides are 24 and 7 cm.", "solution": "1.57. Since $A C$ is the diameter of the circle (Fig. 1.44), then $R=0.5 \\sqrt{24^{2}+7^{2}}=12.5 \\quad$ (cm). In $\\triangle B O F$ we have $O F=$ $=\\sqrt{O B^{2}-B F^{2}}=\\sqrt{12.5^{2}-12^{2}}=3.5$ (cm); hence, $M F=12.5-3.5=9$ (cm), $M K=12.5+3.5=16$ (cm). From $\\triangle M B F$ and $\\triangle M A K$ we find the required distances: $\\quad M B=\\sqrt{M F^{2}+B F^{2}}=\\sqrt{9^{2}+12^{2}}=15(\\mathrm{~cm})$, $M A=\\sqrt{M K^{2}+K A^{2}}=\\sqrt{16^{2}+12^{2}}=20(\\mathrm{~cm})$.\n\nAnswer: 15 and 20 cm.", "answer": "15", "source": "olympiads", "problem_type": "Geometry", "uuid": "ff8623f2-0efa-534f-bd28-e6c8b7c6dae3"}
+{"problem": "15. As shown in the figure, the area of square $\\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, and the overlapping area of the two squares is 1 square centimeter. Therefore, the area of the shaded part is $\\qquad$ square centimeters.", "solution": "【Answer】 72\n【Analysis】Key point: Skillful area calculation\nThe area of the square is 196, so the side length is 14. The overlapping area is 1, so the side length is 1; the area of the larger square is 4 times that of the smaller square, so the side length of the larger square is twice that of the smaller square, and the sum of the side lengths of the larger and smaller squares is $14+1=15$.\nTherefore, the side length of the smaller square is $15 \\div 3=5$, and the side length of the larger square is $5 \\times 2=10$.\nThe area of the smaller rectangle is $(5-1) \\times(10-1)=36$, so the area of the two smaller rectangles is $36 \\times 2=72\\left(\\mathrm{~cm}^{2}\\right)$.", "answer": "72", "source": "olympiads", "problem_type": "Geometry", "uuid": "a62b9e76-5993-59bd-81e5-342b61459c26"}
+{"problem": "321. $\\sqrt{2 x+5}+\\sqrt{x-1}=8$.\n\nTranslate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.\n\n321. $\\sqrt{2 x+5}+\\sqrt{x-1}=8$.", "solution": "Solution. Isolate one of the radicals and square both sides of the equation:\n\n$$\n(\\sqrt{2 x+5})^{2}=(8-\\sqrt{x-1})^{2} ; 2 x+5=64-16 \\sqrt{x-1}+(x-1)\n$$\n\nMove $16 \\sqrt{x-1}$ to the left side, and all other terms to the right side:\n\n$$\n16 \\sqrt{x-1}=64+x-1-2 x-5 ; 16 \\sqrt{x-1}=58-x\n$$\n\nSquare both sides of the equation again:\n\n$(16 \\sqrt{x-1})^{2}=(58-x)^{2} ; 256 x-256=3364-116 x+x^{2} ; x^{2}-372 x+3620=0$.\n\nSolve the resulting quadratic equation. Since $a=1, b=-372$, $c=3620, \\quad D=b^{2}-4 a c=(-372)^{2}-4 \\cdot 1 \\cdot 3620=138384-14480=123904$; $\\sqrt{D}=352$, then\n\n$$\nx_{1}=\\frac{-b-\\sqrt{D}}{2 a}=\\frac{372-352}{2}=10 ; \\quad x_{2}=\\frac{-b+\\sqrt{D}}{2 a}=\\frac{372+352}{2}=362\n$$\n\nCheck both values $x_{1}=10 ; x_{2}=362$.\n\nIf $x=10$, then $\\sqrt{2 \\cdot 10+5}+\\sqrt{10-1}=8 ; \\sqrt{25}+\\sqrt{9}=8 ; 5+3=8$; $8=8$\n\nIf $x=362$, then $\\sqrt{2 \\cdot 362+5}+\\sqrt{362-1}=8 ; \\sqrt{729}+\\sqrt{361}=8 ; 27+$ $+19 \\neq 8$; therefore, $x=362$ is an extraneous root.\n\nThe root of the given equation is $x=10$.", "answer": "10", "source": "olympiads", "problem_type": "Algebra", "uuid": "512b5c06-d4f8-5506-afbf-d88196ff995d"}
+{"problem": "# Problem 3.\n\nA polynomial $P(x)$ with integer coefficients has the properties\n\n$$\nP(1)=2019, \\quad P(2019)=1, \\quad P(k)=k,\n$$\n\nwhere the number $k$ is an integer. Find this number $k$.\n\n#", "solution": "# Solution.\n\nSince the polynomial $P(x)$ has integer coefficients, $P(a)-P(b)$ is divisible by $a-b$ for any integers $a$ and $b$.\n\nWe get that\n\n$$\n\\begin{gathered}\nP(k)-P(1)=(k-2019) \\text { is divisible by }(k-1), \\\\\nP(k)-P(2019)=(k-1) \\text { is divisible by }(k-2019) .\n\\end{gathered}\n$$\n\nThis can only be true if $|k-1|=|k-2019|$.\n\nThe solution to the obtained equation is $k=1010$.\n\nAnswer: $\\quad k=1010$.", "answer": "1010", "source": "olympiads", "problem_type": "Algebra", "uuid": "e3651185-feff-5ded-b50a-063f8af01839"}
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_rollout_manifest.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_rollout_manifest.json
new file mode 100644
index 0000000000000000000000000000000000000000..05bc1c58ed227f18f05ce547e4682916ad6d3b42
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_rollout_manifest.json
@@ -0,0 +1,17 @@
+{
+  "groups": 96,
+  "splits": {
+    "math_train": 72,
+    "math_dev": 24
+  },
+  "correct": 6,
+  "incorrect": 186,
+  "loops": 221,
+  "parse_fail": 0,
+  "benchmark_family_rows": 0,
+  "reward": "math_verify exact symbolic equivalence; reference +1, correct rollout +1, wrong -0.5, loop extra -0.5",
+  "budgie_correct": 6,
+  "budgie_incorrect": 186,
+  "budgie_loops": 150,
+  "reference_loops_ignored": 71
+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_rollouts_scored.jsonl b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_rollouts_scored.jsonl
new file mode 100644
index 0000000000000000000000000000000000000000..29f743cc65e11a0cb6a5f5cae555ded8e7eccc3d
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/math_rollouts_scored.jsonl
@@ -0,0 +1,96 @@
+{"id": "math_000", "row": {"problem": "Lines $l_1^{}$ and $l_2^{}$ both pass through the origin and make first-quadrant angles of $\\frac{\\pi}{70}$ and $\\frac{\\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l^{}_{}$, the transformation $R(l)^{}_{}$ produces another line as follows: $l^{}_{}$ is reflected in $l_1^{}$, and the resulting line is reflected in $l_2^{}$. Let $R^{(1)}(l)=R(l)^{}_{}$ and $R^{(n)}(l)^{}_{}=R\\left(R^{(n-1)}(l)\\right)$. Given that $l^{}_{}$ is the line $y=\\frac{19}{92}x^{}_{}$, find the smallest positive integer $m^{}_{}$ for which $R^{(m)}(l)=l^{}_{}$.", "solution": "Let $l$ be a line that makes an angle of $\\theta$ with the positive $x$-axis. Let $l'$ be the reflection of $l$ in $l_1$, and let $l''$ be the reflection of $l'$ in $l_2$. \nThe angle between $l$ and $l_1$ is $\\theta - \\frac{\\pi}{70}$, so the angle between $l_1$ and $l'$ must also be $\\theta - \\frac{\\pi}{70}$. Thus,  $l'$ makes an angle of $\\frac{\\pi}{70}-\\left(\\theta-\\frac{\\pi}{70}\\right) = \\frac{\\pi}{35}-\\theta$ with the positive $x$-axis.  \nSimilarly, since the angle between $l'$ and $l_2$ is $\\left(\\frac{\\pi}{35}-\\theta\\right)-\\frac{\\pi}{54}$, the angle between $l''$ and the positive $x$-axis is $\\frac{\\pi}{54}-\\left(\\left(\\frac{\\pi}{35}-\\theta\\right)-\\frac{\\pi}{54}\\right) = \\frac{\\pi}{27}-\\frac{\\pi}{35}+\\theta = \\frac{8\\pi}{945} + \\theta$. \nThus, $R(l)$ makes an $\\frac{8\\pi}{945} + \\theta$ angle with the positive $x$-axis. So $R^{(n)}(l)$ makes an $\\frac{8n\\pi}{945} + \\theta$ angle with the positive $x$-axis. \nTherefore, $R^{(m)}(l)=l$ iff $\\frac{8m\\pi}{945}$ is an integral multiple of $\\pi$. Thus, $8m \\equiv 0\\pmod{945}$. Since $\\gcd(8,945)=1$, $m \\equiv 0 \\pmod{945}$, so the smallest positive integer $m$ is $\\boxed{945}$.", "answer": "945", "source": "amc_aime", "problem_type": "Geometry", "uuid": "8791f647-68c1-5dec-b9f3-b235014b5116"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nLines $l_1^{}$ and $l_2^{}$ both pass through the origin and make first-quadrant angles of $\\frac{\\pi}{70}$ and $\\frac{\\pi}{54}$ radians, respectively, with the positive x-axis. For any line $l^{}_{}$, the transformation $R(l)^{}_{}$ produces another line as follows: $l^{}_{}$ is reflected in $l_1^{}$, and the resulting line is reflected in $l_2^{}$. Let $R^{(1)}(l)=R(l)^{}_{}$ and $R^{(n)}(l)^{}_{}=R\\left(R^{(n-1)}(l)\\right)$. Given that $l^{}_{}$ is the line $y=\\frac{19}{92}x^{}_{}$, find the smallest positive integer $m^{}_{}$ for which $R^{(m)}(l)=l^{}_{}$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "First, we need to find the possible values of $R(l)$. We can do this by using the given information about the angles $l_1^{}$ and $l_2^{}$.\n\nLet's start by finding $R(l_1)$. For a line $l_1$ to make an angle of $\\frac{\\pi}{70}$ with the positive x-axis, $R(l_1)$ must be parallel to the x-axis. This means that $R(l_1)$ is a horizontal line.\n\nNow, let's find $R(l_2)$. For a line $l_2$ to make an angle of $\\frac{\\pi}{54}$ with the positive x-axis, $R(l_2)$ must be a horizontal line. This means that $R(l_2)$ is also a horizontal line.\n\nUsing the reflection rule for parallel lines, we can find the inverse of $R(l_1)$ and $R(l_2)$.\n\nFor line $l_1$ to make an angle of $\\frac{\\pi}{70}$ with the x-axis, it must be parallel to the x-axis. This means that $R(l_1)$ is a horizontal line.\n\nFor line $l_2$ to make an angle of $\\frac{\\pi}{54}$ with the x-axis, it must be parallel to the x-axis. This means that $R(l_2)$ is a horizontal line.\n\nNow, let's find $R^{(n)}(l)$. For $n = 1$, $R^{(1)}(l) = R(l)$. To find $R^{(n)}(l)$, we need to find a point", "correct": false, "reward": -1, "repetition": {"words": 229, "unique_ratio": 0.2445414847161572, "max_4gram_count": 4, "max_sentence_count": 2, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "Since the graph of $y=\\tan 70x$ passes through the origin, we know that\n\\[ \\tan 70x = 0 \\Rightarrow 70x = k\\pi \\Rightarrow x = \\frac{k\\pi}{70} \\]\n\nSubstituting $x = \\frac{k\\pi}{70}$ into the equation $y = \\tan 70x$, we have\n\\[ y = \\tan\\left(\\frac{k\\pi}{70}\\right) = \\frac{19}{92} \\left(\\frac{k\\pi}{70}\\right)^2 \\]\n\nSimplifying, we get\n\\[ y = \\frac{19}{92}\\left(\\frac{k^2\\pi^2}{400k\\pi}\\right) = \\frac{19}{92}\\left(\\frac{k^2}{400}\\right) = \\frac{19}{32} \\left(\\frac{k}{100}\\right) = \\frac{19}{32}\\left(\\frac{k}{100}\\right) \\]\n\nThe first line passes through the origin and has a slope of $-\\frac{19}{32}$. Since the first line is the first quadrant line, we can take its slope as $-1$ and its equation as $y = -x + b$. The line $y = \\frac{19}{92}x^2$ passes through the first quadrant line, so we can take $b = -1$.\n\nThe line $y = \\frac{19}{32}x^2$ passes through the second quadrant line and has a slope of $\\frac{19}{", "correct": false, "reward": -1, "repetition": {"words": 179, "unique_ratio": 0.2849162011173184, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Let $l$ be a line that makes an angle of $\\theta$ with the positive $x$-axis. Let $l'$ be the reflection of $l$ in $l_1$, and let $l''$ be the reflection of $l'$ in $l_2$. \nThe angle between $l$ and $l_1$ is $\\theta - \\frac{\\pi}{70}$, so the angle between $l_1$ and $l'$ must also be $\\theta - \\frac{\\pi}{70}$. Thus,  $l'$ makes an angle of $\\frac{\\pi}{70}-\\left(\\theta-\\frac{\\pi}{70}\\right) = \\frac{\\pi}{35}-\\theta$ with the positive $x$-axis.  \nSimilarly, since the angle between $l'$ and $l_2$ is $\\left(\\frac{\\pi}{35}-\\theta\\right)-\\frac{\\pi}{54}$, the angle between $l''$ and the positive $x$-axis is $\\frac{\\pi}{54}-\\left(\\left(\\frac{\\pi}{35}-\\theta\\right)-\\frac{\\pi}{54}\\right) = \\frac{\\pi}{27}-\\frac{\\pi}{35}+\\theta = \\frac{8\\pi}{945} + \\theta$. \nThus, $R(l)$ makes an $\\frac{8\\pi}{945} + \\theta$ angle with the positive $x$-axis. So $R^{(n)}(l)$ makes an $\\frac{8n\\pi}{945} + \\theta$ angle with the positive $x$-axis. \nTherefore, $R^{(m)}(l)=l$ iff $\\frac{8m\\pi}{945}$ is an integral multiple of $\\pi$. Thus, $8m \\equiv 0\\pmod{945}$. Since $\\gcd(8,945)=1$, $m \\equiv 0 \\pmod{945}$, so the smallest positive integer $m$ is $\\boxed{945}$.", "correct": true, "reward": 1.0, "repetition": {"words": 217, "unique_ratio": 0.2626728110599078, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_001", "row": {"problem": "9. Given that $M N$ is a moving chord of the circumcircle of equilateral $\\triangle A B C$ with side length $2 \\sqrt{6}$, $M N=4$, and $P$ is a moving point on the sides of $\\triangle A B C$. Then the maximum value of $\\overrightarrow{M P} \\cdot \\overrightarrow{P N}$ is", "solution": "9.4.\n\nLet the circumcenter of $\\triangle ABC$ be $O$.\nIt is easy to find that the radius of $\\odot O$ is $r=2 \\sqrt{2}$.\nAlso, $MN=4$, so $\\triangle OMN$ is an isosceles right triangle, and\n$$\n\\begin{aligned}\n& \\overrightarrow{O M} \\cdot \\overrightarrow{O N}=0,|\\overrightarrow{O M}+\\overrightarrow{O N}|=4 . \\\\\n& \\text { Let }|\\overrightarrow{O P}|=x . \\\\\n& \\text { Then } \\sqrt{2} \\leqslant x \\leqslant 2 \\sqrt{2} \\text {, and } \\\\\n& \\overrightarrow{M P} \\cdot \\overrightarrow{P N}=(\\overrightarrow{O P}-\\overrightarrow{O M}) \\cdot(\\overrightarrow{O N}-\\overrightarrow{O P}) \\\\\n& =-\\overrightarrow{O P}{ }^{2}+(\\overrightarrow{O M}+\\overrightarrow{O N}) \\cdot \\overrightarrow{O P}-\\overrightarrow{O M} \\cdot \\overrightarrow{O N} \\\\\n= & -x^{2}+(\\overrightarrow{O M}+\\overrightarrow{O N}) \\cdot \\overrightarrow{O P} \\\\\n\\leqslant & -x^{2}+|\\overrightarrow{O M}+\\overrightarrow{O N}||\\overrightarrow{O P}| \\\\\n= & -x^{2}+4 x=4-(x-2)^{2} \\leqslant 4,\n\\end{aligned}\n$$\n\nThe equality holds if and only if $x=2$ and $\\overrightarrow{O P}$ is in the same direction as $\\overrightarrow{O M}+\\overrightarrow{O N}$.\nTherefore, the maximum value of $\\overrightarrow{M P} \\cdot \\overrightarrow{P N}$ is 4.", "answer": "4", "source": "cn_contest", "problem_type": "Geometry", "uuid": "fea23b7f-e22e-521d-be6d-4c469b94d965"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n9. Given that $M N$ is a moving chord of the circumcircle of equilateral $\\triangle A B C$ with side length $2 \\sqrt{6}$, $M N=4$, and $P$ is a moving point on the sides of $\\triangle A B C$. Then the maximum value of $\\overrightarrow{M P} \\cdot \\overrightarrow{P N}$ is", "candidates": [{"slot": 0, "origin": "budgie", "answer": "This problem is an application of complex numbers in geometry. Given a fixed complex number $z$, when a point $P$ on the sides of an equilateral triangle $\\triangle ABC$ is moving, the problem can be solved by finding the maximum value of the dot product between $\\overrightarrow{M P} \\cdot \\overrightarrow{P N}$.\n\nHere's the step-by-step reasoning:\n\nLet $z_1 = z_0 = 1$. Then $z_1$ has a magnitude of $1$, and $|\\overrightarrow{M P}| = 2 \\sqrt{3}$.\n\nLet $z_2 = z_0 = e^{i\\theta}$. Then $z_2$ has a magnitude of $2 \\sqrt{3} (\\cos \\theta - 1)$, and $|\\overrightarrow{M P}| = 2 \\sqrt{3} + 2 \\sqrt{3} = 4$.\n\nNow, let $z_3 = z_0 = e^{i2\\phi}$. Then $z_3$ has a magnitude of $2 \\sqrt{3} + 2 \\sqrt{3} = 4$, and $|\\overrightarrow{M P}| = 4$.\n\nTherefore, the maximum value of $\\overrightarrow{M P} \\cdot \\overrightarrow{P N}$ is $\\boxed{16}$.\n\nThis problem is best solved using complex numbers.", "correct": false, "reward": -1, "repetition": {"words": 163, "unique_ratio": 0.3987730061349693, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 1, "origin": "budgie", "answer": "The maximum value is achieved when $\\overrightarrow{P M} \\cdot \\overrightarrow{P N}$ is maximized.\n\nThis maximum value is $\\boxed{2\\sqrt{2}}$.\n\nFinal answer:\n#### 2√2", "correct": false, "reward": -0.5, "repetition": {"words": 27, "unique_ratio": 0.6666666666666666, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 2, "origin": "verified_reference", "answer": "9.4.\n\nLet the circumcenter of $\\triangle ABC$ be $O$.\nIt is easy to find that the radius of $\\odot O$ is $r=2 \\sqrt{2}$.\nAlso, $MN=4$, so $\\triangle OMN$ is an isosceles right triangle, and\n$$\n\\begin{aligned}\n& \\overrightarrow{O M} \\cdot \\overrightarrow{O N}=0,|\\overrightarrow{O M}+\\overrightarrow{O N}|=4 . \\\\\n& \\text { Let }|\\overrightarrow{O P}|=x . \\\\\n& \\text { Then } \\sqrt{2} \\leqslant x \\leqslant 2 \\sqrt{2} \\text {, and } \\\\\n& \\overrightarrow{M P} \\cdot \\overrightarrow{P N}=(\\overrightarrow{O P}-\\overrightarrow{O M}) \\cdot(\\overrightarrow{O N}-\\overrightarrow{O P}) \\\\\n& =-\\overrightarrow{O P}{ }^{2}+(\\overrightarrow{O M}+\\overrightarrow{O N}) \\cdot \\overrightarrow{O P}-\\overrightarrow{O M} \\cdot \\overrightarrow{O N} \\\\\n= & -x^{2}+(\\overrightarrow{O M}+\\overrightarrow{O N}) \\cdot \\overrightarrow{O P} \\\\\n\\leqslant & -x^{2}+|\\overrightarrow{O M}+\\overrightarrow{O N}||\\overrightarrow{O P}| \\\\\n= & -x^{2}+4 x=4-(x-2)^{2} \\leqslant 4,\n\\end{aligned}\n$$\n\nThe equality holds if and only if $x=2$ and $\\overrightarrow{O P}$ is in the same direction as $\\overrightarrow{O M}+\\overrightarrow{O N}$.\nTherefore, the maximum value of $\\overrightarrow{M P} \\cdot \\overrightarrow{P N}$ is 4.", "correct": true, "reward": 1.0, "repetition": {"words": 189, "unique_ratio": 0.2804232804232804, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_dev"}
+{"id": "math_002", "row": {"problem": "25. Anna, Bridgit and Carol run in a $100 \\mathrm{~m}$ race. When Anna finishes, Bridgit is $16 \\mathrm{~m}$ behind her and when Bridgit finishes, Carol is $25 \\mathrm{~m}$ behind her. The girls run at constant speeds throughout the race. How far behind was Carol when Anna finished?\nA $37 \\mathrm{~m}$\nB $41 \\mathrm{~m}$\nC $50 \\mathrm{~m}$\nD $55 \\mathrm{~m}$\nE $60 \\mathrm{~m}$", "solution": "25. A Carol finishes 25 metres behind Bridgit, so she travels 75 metres while Bridgit runs 100 metres. Therefore she runs 3 metres for every 4 metres Bridgit runs. When Anna finishes, Bridgit has run 84 metres, so that at that time Carol has run $\\frac{3}{4} \\times 84$ metres $=63$ metres. Hence Carol finishes $(100-63)$ metres $=37$ metres behind Anna.", "answer": "37\\mathrm{~}", "source": "olympiads", "problem_type": "Algebra", "uuid": "f7bf0e57-7e91-5810-bab6-d1e4ad0d4461"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n25. Anna, Bridgit and Carol run in a $100 \\mathrm{~m}$ race. When Anna finishes, Bridgit is $16 \\mathrm{~m}$ behind her and when Bridgit finishes, Carol is $25 \\mathrm{~m}$ behind her. The girls run at constant speeds throughout the race. How far behind was Carol when Anna finished?\nA $37 \\mathrm{~m}$\nB $41 \\mathrm{~m}$\nC $50 \\mathrm{~m}$\nD $55 \\mathrm{~m}$\nE $60 \\mathrm{~m}$", "candidates": [{"slot": 0, "origin": "budgie", "answer": "1. **Understanding the problem:** We have a race with 100 meters, and the runners are running at constant speeds. We need to determine how far behind Carol was when Anna finished.\n\n2. **Relative Speed and Time:** Since the runners are running at constant speeds, we can assume that the time taken by each runner to complete the race is constant. This means we can use relative speed and time to find out how much distance each runner covers in the time it takes for them to complete the race.\n\n3. **First runner's Time:** When Anna finishes the race, the first runner has already run for 100 meters. Since they are running at constant speeds, the first runner will cover 100 meters in the same amount of time.\n\n4. **Second runner's Time:** When Anna finishes the race, the second runner has only 90 meters left to run. Since they are running at constant speeds, the second runner will cover 90 meters in the same amount of time.\n\n5. **Distance Calculation for Anna's dye run:** When Anna finishes the race, the second runner has already run for 90 meters. Since they are running at constant speeds, the second runner covers a distance of 90 meters in the same amount of time. Therefore, the distance between Carol and Anna is 100 meters minus 90 meters, which is 10 meters.\n\nIn summary, when Anna finishes, Carol is 10 meters behind.\n\nFinal answer:\n#### 10", "correct": false, "reward": -1, "repetition": {"words": 239, "unique_ratio": 0.3598326359832636, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "1. **Identify the speeds of Anna and Bridgit:**\n   - Since Anna finishes the race in $37 \\mathrm{~m}$, her speed is $37 \\mathrm{~m/ \\text{min}}$.\n   - Bridgit finishes the race in $41 \\mathrm{~m}$, so her speed is $41 \\mathrm{~m/ \\text{min}}$.\n\n2. **Determine the time taken by each girl to finish:**\n   - Time taken by Anna: $37 \\mathrm{~m/ \\text{min}}$.\n   - Time taken by Bridgit: $41 \\mathrm{~m/ \\text{min}}$.\n\n3. **Calculate the distance each girl covers after the start of the race:**\n   - Distance covered by Anna: $37 \\mathrm{~m/ \\text{min}} \\times 30 \\mathrm{min} = 1110 \\mathrm{ \\text{ m}}$.\n   - Distance covered by Bridgit: $41 \\mathrm{~m/ \\text{min}} \\times 30 \\mathrm{min} = 1230 \\mathrm{ \\text{ m}}$.\n\n4. **Determine the time taken by Carol to finish:**\n   - Sits for $100 \\mathrm{ \\text{ m}}$ at a constant speed.\n   - Carol's speed: $100 \\mathrm{ \\text{ m}} \\times \\frac{1}{37} \\mathrm{ \\text{ min}} = \\frac{100}{37} \\mathrm{ \\text{ min}} = \\frac{100}{37} \\times \\frac{1}{3} \\mathrm", "correct": false, "reward": -1, "repetition": {"words": 163, "unique_ratio": 0.31901840490797545, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "25. A Carol finishes 25 metres behind Bridgit, so she travels 75 metres while Bridgit runs 100 metres. Therefore she runs 3 metres for every 4 metres Bridgit runs. When Anna finishes, Bridgit has run 84 metres, so that at that time Carol has run $\\frac{3}{4} \\times 84$ metres $=63$ metres. Hence Carol finishes $(100-63)$ metres $=37$ metres behind Anna.", "correct": true, "reward": 1.0, "repetition": {"words": 63, "unique_ratio": 0.5079365079365079, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_003", "row": {"problem": "4. The sequence $\\left\\{x_{n}\\right\\}$ is defined as follows:\n$$\nx_{1}=\\frac{2}{3}, x_{n+1}=\\frac{x_{n}}{2(2 n+1) x_{n}+1}\\left(n \\in \\mathbf{Z}_{+}\\right) \\text {. }\n$$\n\nThen $x_{1}+x_{2}+\\cdots+x_{2014}=$ $\\qquad$", "solution": "4. $\\frac{4028}{4029}$.\n\nBy definition, $\\frac{1}{x_{n+1}}=2(2 n+1)+\\frac{1}{x_{n}}$.\nLet $y_{n}=\\frac{1}{x_{n}}$.\nThus, $y_{1}=\\frac{3}{2}$, and $y_{n+1}-y_{n}=2(2 n+1)$.\nTherefore, $y_{n}=\\left(y_{n}-y_{n-1}\\right)+\\left(y_{n-1}-y_{n-2}\\right)+\\cdots+$ $\\left(y_{2}-y_{1}\\right)+y_{1}$\n$=2[(2 n-1)+(2 n-3)+\\cdots+3]+\\frac{3}{2}$\n$=2\\left(n^{2}-1\\right)+\\frac{3}{2}=\\frac{1}{2}(2 n+1)(2 n-1)$,\n$\\sum_{i=1}^{2014} x_{i}=\\sum_{i=1}^{2014} \\frac{1}{y_{i}}=\\sum_{i=1}^{2014} \\frac{2}{(2 i+1)(2 i-1)}$\n$=\\sum_{i=1}^{2014}\\left(\\frac{1}{2 i-1}-\\frac{1}{2 i+1}\\right)$\n$=1-\\frac{1}{4029}=\\frac{4028}{4029}$.", "answer": "\\frac{4028}{4029}", "source": "olympiads", "problem_type": "Algebra", "uuid": "8c80f4bc-5be2-599b-b225-3c48179b1052"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n4. The sequence $\\left\\{x_{n}\\right\\}$ is defined as follows:\n$$\nx_{1}=\\frac{2}{3}, x_{n+1}=\\frac{x_{n}}{2(2 n+1) x_{n}+1}\\left(n \\in \\mathbf{Z}_{+}\\right) \\text {. }\n$$\n\nThen $x_{1}+x_{2}+\\cdots+x_{2014}=$ $\\qquad$", "candidates": [{"slot": 0, "origin": "budgie", "answer": "1. Given the sequence $\\left\\{x_{n}\\right\\}$ is defined by\n   \\[\n   x_{1} = \\frac{2}{3}, \\quad x_{n+1} = \\frac{x_{n}}{2(2n + 1)x_{n} + 1}, \\quad n \\in \\mathbb{Z}_{+}\n   \\]\n\n2. To find the sum of the first $2014$ terms, we can start by calculating the first few terms to identify any patterns.\n\n3. Calculate the first few terms:\n   - First term: $x_{1} = \\frac{2}{3}$\n   - Second term: $x_{2} = \\frac{\\frac{2}{3}}{2(2 \\cdot 1 + 1) \\left(\\frac{2}{3}\\right) + 1} = \\frac{1}{3}$\n   - Third term: $x_{3} = \\frac{\\frac{1}{3}}{2(2 \\cdot 2 + 1) \\left(\\frac{1}{3}\\right) + 1} = \\frac{1}{9}$\n   - Fourth term: $x_{4} = \\frac{\\frac{1}{9}}{2(2 \\cdot 4 + 1) \\left(\\frac{1}{3}\\right) + 1} = \\frac{1}{7}$\n\n4. Notice that the denominator of $x_{n+1}$ is $2(2n + 1)x_{n} + 1$.\n\n5. Notice that the sum $x_{1} + x_{2} + \\cdots + x_{2014}$ can be rewritten as:\n   \\[\n   x_{1} +", "correct": false, "reward": -1, "repetition": {"words": 170, "unique_ratio": 0.3, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "1. The sequence $\\{x_n\\}$ is defined by:\n$$\nx_1 = \\frac{2}{3}, \\quad x_{n+1} = \\frac{x_n}{2(2n+1)x_n+1}\n$$\n2. We need to find $x_1 + x_2 + \\cdots + x_{2014}$.\n3. We start by computing the first few terms of the sequence to identify any patterns:\n   - $x_2 = \\frac{\\frac{2}{3}}{2(2\\cdot2+1) \\cdot \\frac{2}{3}+1} = \\frac{1}{24}$\n   - $x_3 = \\frac{\\frac{1}{24}}{2(2\\cdot3+1) \\cdot \\frac{1}{24}+1} = \\frac{1}{216}$\n   - $x_4 = \\frac{\\frac{1}{216}}{\\frac{2}{3} \\cdot \\frac{26}{3}+1} = \\frac{24}{216} = \\frac{1}{9}$\n4. Notice that $x_2 = x_3 = x_4 = \\frac{1}{9}$.\n5. Now, we need to find the sum of the squares of these values:\n   - $x_1^2 = \\left(\\frac{2}{3}\\right)^2 = \\frac{4}{9}$\n   - $x_2^2 = \\left(\\frac{1}{9}\\right)^2 = \\frac{1}{81}$\n   - $x_3^2 = \\left(\\frac{1}{9}\\right)^", "correct": false, "reward": -1, "repetition": {"words": 168, "unique_ratio": 0.27976190476190477, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "4. $\\frac{4028}{4029}$.\n\nBy definition, $\\frac{1}{x_{n+1}}=2(2 n+1)+\\frac{1}{x_{n}}$.\nLet $y_{n}=\\frac{1}{x_{n}}$.\nThus, $y_{1}=\\frac{3}{2}$, and $y_{n+1}-y_{n}=2(2 n+1)$.\nTherefore, $y_{n}=\\left(y_{n}-y_{n-1}\\right)+\\left(y_{n-1}-y_{n-2}\\right)+\\cdots+$ $\\left(y_{2}-y_{1}\\right)+y_{1}$\n$=2[(2 n-1)+(2 n-3)+\\cdots+3]+\\frac{3}{2}$\n$=2\\left(n^{2}-1\\right)+\\frac{3}{2}=\\frac{1}{2}(2 n+1)(2 n-1)$,\n$\\sum_{i=1}^{2014} x_{i}=\\sum_{i=1}^{2014} \\frac{1}{y_{i}}=\\sum_{i=1}^{2014} \\frac{2}{(2 i+1)(2 i-1)}$\n$=\\sum_{i=1}^{2014}\\left(\\frac{1}{2 i-1}-\\frac{1}{2 i+1}\\right)$\n$=1-\\frac{1}{4029}=\\frac{4028}{4029}$.", "correct": true, "reward": 1.0, "repetition": {"words": 148, "unique_ratio": 0.14864864864864866, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_004", "row": {"problem": "Example 6 The rules of a \"level-up game\" stipulate: On the $n$-th level, a die must be rolled $n$ times. If the sum of the points obtained from these $n$ rolls is greater than $2^{n}$, the level is considered passed. Questions:\n(1) What is the maximum number of levels a person can pass in this game?\n(2) What is the probability that he can pass the first three levels consecutively?\n(Note: A die is a uniform cube with points numbered $1,2,3,4,5,6$ on its faces. The number of points on the face that lands up after rolling the die is the result of the roll.)", "solution": "Solution: Since the dice is a uniform cube, the possibility of each number appearing after a roll is equal.\n(1) Because the maximum number that can appear on a single dice is 6, and $6 \\times 1 > 2^4, 6 \\times 5 < 2$.\nTherefore, when $n \\geqslant 5$, the sum of the numbers that appear after rolling the dice $n$ times cannot be greater than 2, which means this is an impossible event, and the probability of passing is 0.\nSo, the maximum number of consecutive levels that can be passed is 4.\n(2) Let event $A_{n}$ be \"failing the $n$-th level\", then the complementary event $\\overline{A_{n}}$ is \"passing the $n$-th level\". In the $n$-th level game, the total number of basic events is $6^n$.\nLevel 1: The number of basic events contained in event $A_{1}$ is 2 (i.e., the cases where the number is 1 or 2),\nSo the probability of passing this level is: $P\\left(\\overline{A_{1}}\\right)=1-P\\left(A_{1}\\right)=1-\\frac{2}{6}=\\frac{2}{3}$\nLevel 2: The number of basic events contained in event $A_{2}$ is the sum of the number of positive integer solutions to the equation $x+y=a$ when $a$ takes the values $2,3,4$. That is, $C_{1}^{3}+C_{3}^{1}+C_{3}^{1}=1+2+3=6$ (cases).\nSo the probability of passing this level is $P\\left(\\bar{A}_{2}\\right)=1-P\\left(A_{2}\\right)=1-\\frac{6}{6^{2}}=\\frac{5}{6}$\nLevel 3: The number of basic events contained in event $A_{3}$ is the sum of the number of positive integer solutions to the equation $x+y+z=a$ when $a$ takes the values $3,4,5,6,7,8$. That is, $C_{2}^{3}+C_{3}^{3}+C_{4}^{2}+C_{5}^{3}+C_{6}^{3}+C_{5}^{2}=1+3+6+10+15+21=56$ (cases).\nSo the probability of passing this level is: $P\\left(\\bar{A}_{3}\\right)=1-P\\left(A_{3}\\right)=1-\\frac{56}{6^{3}}=\\frac{20}{27}$\nTherefore, the probability of passing the first 3 levels consecutively is: $P\\left(\\bar{A}_{1}\\right) \\cdot P\\left(\\bar{A}_{2}\\right) \\cdot P\\left(\\bar{A}_{3}\\right)=\\frac{2}{3} \\times \\frac{5}{6} \\times \\frac{20}{27}=\\frac{100}{243}$", "answer": "\\frac{100}{243}", "source": "olympiads", "problem_type": "Combinatorics", "uuid": "96c9bf5f-cffe-505b-80c0-e679e10d2cf9"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nExample 6 The rules of a \"level-up game\" stipulate: On the $n$-th level, a die must be rolled $n$ times. If the sum of the points obtained from these $n$ rolls is greater than $2^{n}$, the level is considered passed. Questions:\n(1) What is the maximum number of levels a person can pass in this game?\n(2) What is the probability that he can pass the first three levels consecutively?\n(Note: A die is a uniform cube with points numbered $1,2,3,4,5,6$ on its faces. The number of points on the face that lands up after rolling the die is the result of the roll.)", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The best answer is 12.\n\nFinal answer:\n#### 12", "correct": false, "reward": -0.5, "repetition": {"words": 8, "unique_ratio": 0.75, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "The minimum possible outcome is rolling a 1 on one of the dice, resulting in 1 point. The maximum possible outcome is rolling an 8 on one of the dice, resulting in 8 points.\n\nThe probability of rolling a 1 on one die is $\\frac{1}{6}$, the probability of rolling an 8 on one die is $\\frac{1}{6}$. The probability of rolling an 8 on one die and then rolling a 1 on the next roll is $\\frac{1}{6} \\times \\frac{1}{6} = \\frac{1}{36}$.\n\nTherefore, the maximum number of levels is 6, and the probability that a person can pass the first three levels consecutively is $\\frac{1}{36} \\times \\frac{1}{6} = \\frac{1}{216}$.\n\nFinal answer:\n#### \\frac{1}{216}", "correct": false, "reward": -1, "repetition": {"words": 126, "unique_ratio": 0.3333333333333333, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "Solution: Since the dice is a uniform cube, the possibility of each number appearing after a roll is equal.\n(1) Because the maximum number that can appear on a single dice is 6, and $6 \\times 1 > 2^4, 6 \\times 5 < 2$.\nTherefore, when $n \\geqslant 5$, the sum of the numbers that appear after rolling the dice $n$ times cannot be greater than 2, which means this is an impossible event, and the probability of passing is 0.\nSo, the maximum number of consecutive levels that can be passed is 4.\n(2) Let event $A_{n}$ be \"failing the $n$-th level\", then the complementary event $\\overline{A_{n}}$ is \"passing the $n$-th level\". In the $n$-th level game, the total number of basic events is $6^n$.\nLevel 1: The number of basic events contained in event $A_{1}$ is 2 (i.e., the cases where the number is 1 or 2),\nSo the probability of passing this level is: $P\\left(\\overline{A_{1}}\\right)=1-P\\left(A_{1}\\right)=1-\\frac{2}{6}=\\frac{2}{3}$\nLevel 2: The number of basic events contained in event $A_{2}$ is the sum of the number of positive integer solutions to the equation $x+y=a$ when $a$ takes the values $2,3,4$. That is, $C_{1}^{3}+C_{3}^{1}+C_{3}^{1}=1+2+3=6$ (cases).\nSo the probability of passing this level is $P\\left(\\bar{A}_{2}\\right)=1-P\\left(A_{2}\\right)=1-\\frac{6}{6^{2}}=\\frac{5}{6}$\nLevel 3: The number of basic events contained in event $A_{3}$ is the sum of the number of positive integer solutions to the equation $x+y+z=a$ when $a$ takes the values $3,4,5,6,7,8$. That is, $C_{2}^{3}+C_{3}^{3}+C_{4}^{2}+C_{5}^{3}+C_{6}^{3}+C_{5}^{2}=1+3+6+10+15+21=56$ (cases).\nSo the probability of passing this level is: $P\\left(\\bar{A}_{3}\\right)=1-P\\left(A_{3}\\right)=1-\\frac{56}{6^{3}}=\\frac{20}{27}$\nTherefore, the probability of passing the first 3 levels consecutively is: $P\\left(\\bar{A}_{1}\\right) \\cdot P\\left(\\bar{A}_{2}\\right) \\cdot P\\left(\\bar{A}_{3}\\right)=\\frac{2}{3} \\times \\frac{5}{6} \\times \\frac{20}{27}=\\frac{100}{243}$", "correct": true, "reward": 1.0, "repetition": {"words": 405, "unique_ratio": 0.2518518518518518, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_005", "row": {"problem": "6. (1999 American High School Mathematics Exam) A regular tetrahedron has two spheres, one inscribed and one circumscribed. Additionally, there are four smaller spheres between each face of the tetrahedron and the circumscribed sphere, each tangent to the center of the face. Point $P$ is a point inside the circumscribed sphere. The probability that $P$ falls within one of the five smaller spheres is closest to ( ).\nA. 0\nB. 0.1\nC. 0.2\nD. 0.3\nE. 0.4", "solution": "6. C. Reason: Let the edge length of the regular tetrahedron be \\( a \\). It is known that the radius of its inscribed sphere is \\( R_{1} = \\frac{\\sqrt{6}}{12} a \\), and the radius of its circumscribed sphere is \\( R_{2} = \\frac{\\sqrt{6}}{4} a \\).\n\nIf the radius of the sphere that is tangent to the circumscribed sphere and the faces of the tetrahedron is \\( r \\), then\n\\[\nr = \\frac{R_{2} - R_{1}}{2} = \\frac{\\sqrt{6}}{12} a \\text{.}\n\\]\n\nThus, the required probability is\n\\[\n\\frac{\\frac{4}{3} \\pi R_{1}^{3} + 4 \\cdot \\frac{4}{3} \\pi r^{3}}{\\frac{4}{3} \\pi R_{2}^{3}} = \\frac{5 \\left( \\frac{1}{12} \\right)^{3}}{\\left( \\frac{1}{4} \\right)^{3}} = \\frac{5}{27} \\approx 0.19.\n\\]", "answer": "0.19", "source": "olympiads", "problem_type": "Geometry", "uuid": "6539a719-572d-506d-9260-e0bd1489885e"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n6. (1999 American High School Mathematics Exam) A regular tetrahedron has two spheres, one inscribed and one circumscribed. Additionally, there are four smaller spheres between each face of the tetrahedron and the circumscribed sphere, each tangent to the center of the face. Point $P$ is a point inside the circumscribed sphere. The probability that $P$ falls within one of the five smaller spheres is closest to ( ).\nA. 0\nB. 0.1\nC. 0.2\nD. 0.3\nE. 0.4", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find the probability that a randomly placed point falls within one of the five smaller spheres, we can approach the problem by considering the scenario in which the point $P$ is placed in the center of the largest sphere. Here's how we can break it down:\n\n1. The largest sphere has a radius of 1. This is because if we place a point in this sphere, the maximum distance to any other point on the sphere is 1.\n\n2. The remaining smaller spheres are arranged in a regular tetrahedron with two inscribed spheres. Each face of the tetrahedron is a sphere, and there are four smaller spheres between each pair of faces. Since the tetrahedron is regular, each face has a smaller sphere (tangent to the center of the face) and a center sphere (radius 1).\n\n3. Since there are five smaller spheres, and each face of the tetrahedron is a sphere, one of these smaller spheres is inscribed in the circumscribed sphere (radius 1), and the other four are circumscribed (radius 1).\n\n4. Each of these smaller spheres is tangent to the center of the face of the tetrahedron. Therefore, a randomly placed point on the circumscribed sphere (radius 1) will fall within one of these smaller spheres.\n\nNow, let's consider the total number of possible outcomes. Each of the five smaller spheres has a radius of 1, and there are four of them. Therefore, the total number of possible outcomes for the tetrahedron is $5^4$.\n\nHowever, we need to be careful not to overcount the cases where a point falls within a smaller sphere. For example, if a point is placed in the center of a smaller sphere but not on the edge of the tetrahedron, then it falls within the smaller sphere.\n\nTo find the probability, we need to divide the number", "correct": false, "reward": -1, "repetition": {"words": 307, "unique_ratio": 0.3127035830618892, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 1, "origin": "budgie", "answer": "There are $24$ possible outcomes for the tetrahedron, and only one of them has the point $P$ inside. The probability that $P$ is one of the five smaller spheres is $\\frac{1}{5}$.\n\nSo, the answer is 0.1.\n\nFinal answer: B", "correct": false, "reward": -0.5, "repetition": {"words": 42, "unique_ratio": 0.7380952380952381, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 2, "origin": "verified_reference", "answer": "6. C. Reason: Let the edge length of the regular tetrahedron be \\( a \\). It is known that the radius of its inscribed sphere is \\( R_{1} = \\frac{\\sqrt{6}}{12} a \\), and the radius of its circumscribed sphere is \\( R_{2} = \\frac{\\sqrt{6}}{4} a \\).\n\nIf the radius of the sphere that is tangent to the circumscribed sphere and the faces of the tetrahedron is \\( r \\), then\n\\[\nr = \\frac{R_{2} - R_{1}}{2} = \\frac{\\sqrt{6}}{12} a \\text{.}\n\\]\n\nThus, the required probability is\n\\[\n\\frac{\\frac{4}{3} \\pi R_{1}^{3} + 4 \\cdot \\frac{4}{3} \\pi r^{3}}{\\frac{4}{3} \\pi R_{2}^{3}} = \\frac{5 \\left( \\frac{1}{12} \\right)^{3}}{\\left( \\frac{1}{4} \\right)^{3}} = \\frac{5}{27} \\approx 0.19.\n\\]", "correct": true, "reward": 1.0, "repetition": {"words": 129, "unique_ratio": 0.37209302325581395, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_006", "row": {"problem": "8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).\n\nTranslate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.", "solution": "8. Solution: $\\because 7 \\equiv-1(\\bmod 4), \\therefore 7^{7^{7}} \\equiv$ $-1(\\bmod 4)(98$ sevens$)$, let $7^{\\gamma^{\\prime 7}}=4 k+3(98$ sevens$), k \\in \\mathbf{Z}^{+}$, then $7^{7^{\\prime 7}}=7^{4 k+3}$ (99 sevens), $\\because 7^{4} \\equiv$ $1(\\bmod 100), \\therefore 7^{4 k} \\equiv 1(\\bmod 100), 7^{7^{\\prime 2}} \\equiv 7^{4 k+3}$ $\\equiv 7^{3} \\equiv 43(\\bmod 100)$. Let $7^{\\gamma^{7}}=7^{100 m+43}$ (100 sevens), $m \\in \\mathbf{Z}^{+}$.\n$$\n\\because 7^{4} \\equiv 2401(\\bmod 10000), \\therefore 7^{8} \\equiv 4801\n$$\n$(\\bmod 10000), 7^{16} \\equiv 9601(\\bmod 10000)$,\n$$\n7^{32} \\equiv 9201(\\bmod 10000), \\quad 7^{64} \\equiv 8401\n$$\n$(\\bmod 10000), 7^{100} \\equiv 7^{4} \\times 7^{32} \\times 7^{64} \\equiv 2401 \\times 9201$ $\\times 8401 \\equiv 1(\\bmod 10000), 7^{100 m} \\equiv 1(\\bmod 10000)$, $7^{100 m+43} \\equiv 7^{43} \\equiv 7^{3} \\times 7^{8} \\times 7^{32} \\equiv 2343$ $(\\bmod 10000), \\therefore 7^{\\gamma^{7}}(100$ sevens$) \\equiv 7^{100 m+43} \\equiv$ $2343(\\bmod 10000), 7^{7^{77}}(100$ sevens$)$ has the last four digits as 2343.", "answer": "2343", "source": "olympiads", "problem_type": "Number Theory", "uuid": "5c348b9a-119c-5e81-8cef-b2905b202dbd"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n8. Find the last four digits of $7^{7^{-7}}$ (100 sevens).\n\nTranslate the above text into English, please keep the original text's line breaks and format, and output the translation result directly.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The problem asks for the last four digits of $7^{7^{-7}}$.\n\nFirst, let's find the value of $7^{7^{-7}}$.\n\nWe can rewrite $7^{7^{-7}}$ as $(7^7)^{7^{-7}}$. Using the power of a power property, we have:\n\n$$7^{7^{-7}} = (7^7)^{7^{-7}} = 7^7 = 823543$$\n\nNow, we need to find the last four digits of $823543$. This can be done by looking at the last four digits of the number.\n\nThe last four digits of $823543$ are 354.\n\nTherefore, the last four digits of $7^{7^{-7}}$ are 354.\n\nFinal answer:\n#### 354", "correct": false, "reward": -1, "repetition": {"words": 98, "unique_ratio": 0.3877551020408163, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "The last four digits of $7^7^{-7} = 7^{7^{-7}}$ are $7^{7^{-7}} = 7^{7^{-7+7}} = 7^{7^{-6}} = 7^{7^{-6+4}} = 7^{7^{-6}} = 7^{7^{-6+6}} = 7^{7^{-6}} = 0777777", "correct": false, "reward": -1, "repetition": {"words": 37, "unique_ratio": 0.2702702702702703, "max_4gram_count": 9, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "8. Solution: $\\because 7 \\equiv-1(\\bmod 4), \\therefore 7^{7^{7}} \\equiv$ $-1(\\bmod 4)(98$ sevens$)$, let $7^{\\gamma^{\\prime 7}}=4 k+3(98$ sevens$), k \\in \\mathbf{Z}^{+}$, then $7^{7^{\\prime 7}}=7^{4 k+3}$ (99 sevens), $\\because 7^{4} \\equiv$ $1(\\bmod 100), \\therefore 7^{4 k} \\equiv 1(\\bmod 100), 7^{7^{\\prime 2}} \\equiv 7^{4 k+3}$ $\\equiv 7^{3} \\equiv 43(\\bmod 100)$. Let $7^{\\gamma^{7}}=7^{100 m+43}$ (100 sevens), $m \\in \\mathbf{Z}^{+}$.\n$$\n\\because 7^{4} \\equiv 2401(\\bmod 10000), \\therefore 7^{8} \\equiv 4801\n$$\n$(\\bmod 10000), 7^{16} \\equiv 9601(\\bmod 10000)$,\n$$\n7^{32} \\equiv 9201(\\bmod 10000), \\quad 7^{64} \\equiv 8401\n$$\n$(\\bmod 10000), 7^{100} \\equiv 7^{4} \\times 7^{32} \\times 7^{64} \\equiv 2401 \\times 9201$ $\\times 8401 \\equiv 1(\\bmod 10000), 7^{100 m} \\equiv 1(\\bmod 10000)$, $7^{100 m+43} \\equiv 7^{43} \\equiv 7^{3} \\times 7^{8} \\times 7^{32} \\equiv 2343$ $(\\bmod 10000), \\therefore 7^{\\gamma^{7}}(100$ sevens$) \\equiv 7^{100 m+43} \\equiv$ $2343(\\bmod 10000), 7^{7^{77}}(100$ sevens$)$ has the last four digits as 2343.", "correct": true, "reward": 1.0, "repetition": {"words": 196, "unique_ratio": 0.22448979591836735, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_007", "row": {"problem": "# Problem 5. (3 points)\n\nIn trapezoid $A B C D$, the lateral side $B C$ is equal to the diagonal $B D$. On the smaller arc $A B$ of the circumscribed circle of triangle $A B C$, a point $E$ is chosen such that $B C=B E$. Find the angle $\\angle A E D$.", "solution": "Answer: $90^{\\circ}$.\n\n## Solution:\n\nFrom the isosceles triangle $B C D$ and the parallelism of $A B$ and $C D$, we get $\\angle B C D = \\angle B D C = \\angle D B A = \\alpha$.\n\nLet the line $C D$ intersect the circumcircle of triangle $A B C$ at point $F$. Then $B C F A$ is an inscribed, i.e., isosceles, trapezoid, from which the arcs $B C, B E$, and $A F$ are equal. Hence, $\\angle B C E = \\angle A C F = \\beta$.\n\n$\\angle E B A = \\angle E C A = \\gamma$, since these angles subtend the same arc.\n\n$\\angle B C A = \\angle B C E + \\angle E C A = \\beta + \\gamma$.\n\n$\\angle E B D = \\angle E B A + \\angle D B A = \\gamma + \\alpha$. Therefore, in the isosceles triangle $E B D$, the equality $\\angle B D E = \\angle B E D = \\frac{180^{\\circ} - \\alpha - \\gamma}{2}$ holds.\n\nMoreover, $\\alpha = \\angle B C D = \\angle B C F = \\angle B C A + \\angle A C F = 2 \\beta + \\gamma$.\n\n$\\angle A E D = \\angle B E A - \\angle B E D = (180^{\\circ} - \\angle B C A) - \\frac{180^{\\circ} - \\alpha - \\gamma}{2} = 180^{\\circ} - \\beta - \\gamma - 90^{\\circ} + \\frac{2 \\beta + 2 \\gamma}{2} = 90^{\\circ}$", "answer": "90", "source": "olympiads", "problem_type": "Geometry", "uuid": "02f1d413-9526-5c96-a94d-ea128382c247"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n# Problem 5. (3 points)\n\nIn trapezoid $A B C D$, the lateral side $B C$ is equal to the diagonal $B D$. On the smaller arc $A B$ of the circumscribed circle of triangle $A B C$, a point $E$ is chosen such that $B C=B E$. Find the angle $\\angle A E D$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "First, we need to understand the properties of the circumscribed circle of a triangle and the specific geometric constructions involved.\n\nIn a triangle, the circumscribed circle (circumcircle) is the circle that passes through all three vertices of the triangle. This circle is the circumcircle of the triangle.\n\nGiven that $B C = B E$, and considering the circumcircle properties in a triangle (which is a special case of a cyclic quadrilateral), we can infer several things:\n\n1. Since $B C = B E$, it follows that $B E = C E$. This means that $B, E, C$ are collinear, and angle $B E C$ is a straight line.\n\n2. The circumscribed circle's center (circumcenter) must be equidistant from the vertices of the triangle, meaning $E$ must be the circumcenter of triangle $A B C$.\n\n3. In a triangle, the circumcenter is effectively the center of the circumcircle, without the need for the circumcenter to be the circumcenter itself.\n\nThe key insight here is recognizing that because $B E = C E$, it directly follows from the properties of circumcircle in a triangle that $B E$ is the circumradius of triangle $B C D$. This is because the circumradius is the distance from the circumcenter to any vertex, and since $E$ is the circumcenter, it's the unique point that is equidistant from the other vertices of the triangle. Therefore, $\\angle B E C$ is a straight line, making it a right angle.\n\nThus, the angle $\\angle A E D$ is a right angle because it is the complement of $\\angle B E C$ (or its vertical angle, depending on the direction of the line segment $B E$), making it a right angle.", "correct": false, "reward": -1, "repetition": {"words": 277, "unique_ratio": 0.33212996389891697, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To solve this problem, we will use the properties of the circumscribed circle and inscribed angles.\n\n1. Since $B C = B E$, the angle $\\angle B C E = 2 \\angle B E C$.\n2. The arc $A B$ of the circumscribed circle of triangle $A B C$ is equal to the arc $A C$. Therefore, the angle $\\angle A E C$ is twice the angle $\\angle B E C$.\n3. Since $\\angle B E C = \\frac{1}{2} \\angle B C E$ and $\\angle B C E = 2 \\angle B E C$, we have $\\angle A E C = 2 \\angle B E C = 2 \\cdot 2 \\angle B E C = \\frac{1}{2} \\angle B E C$.\n4. Substituting this value into (2), we find that $\\angle A E C = 2 \\angle B E C = 2 \\cdot 2 \\angle B E C = 4 \\angle B E C$.\n5. Since $\\angle B C E = \\frac{1}{2} \\angle B E C$, we can find $\\angle A E C = 2 \\angle B E C$.\n6. Also, since the sum of angles in a triangle is $180^\\circ$, we have $\\angle A D C + \\angle B E D = 180^\\circ - \\angle B E C$.\n7. Therefore, $\\angle A E D + 2 \\angle B E E = 180^\\circ - \\angle B E C$.\n8. Since $\\angle B E C = 4 \\angle B E E$, we have $\\angle A E D + 2 \\angle B E E = 180^\\circ -", "correct": false, "reward": -1, "repetition": {"words": 242, "unique_ratio": 0.2066115702479339, "max_4gram_count": 15, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Answer: $90^{\\circ}$.\n\n## Solution:\n\nFrom the isosceles triangle $B C D$ and the parallelism of $A B$ and $C D$, we get $\\angle B C D = \\angle B D C = \\angle D B A = \\alpha$.\n\nLet the line $C D$ intersect the circumcircle of triangle $A B C$ at point $F$. Then $B C F A$ is an inscribed, i.e., isosceles, trapezoid, from which the arcs $B C, B E$, and $A F$ are equal. Hence, $\\angle B C E = \\angle A C F = \\beta$.\n\n$\\angle E B A = \\angle E C A = \\gamma$, since these angles subtend the same arc.\n\n$\\angle B C A = \\angle B C E + \\angle E C A = \\beta + \\gamma$.\n\n$\\angle E B D = \\angle E B A + \\angle D B A = \\gamma + \\alpha$. Therefore, in the isosceles triangle $E B D$, the equality $\\angle B D E = \\angle B E D = \\frac{180^{\\circ} - \\alpha - \\gamma}{2}$ holds.\n\nMoreover, $\\alpha = \\angle B C D = \\angle B C F = \\angle B C A + \\angle A C F = 2 \\beta + \\gamma$.\n\n$\\angle A E D = \\angle B E A - \\angle B E D = (180^{\\circ} - \\angle B C A) - \\frac{180^{\\circ} - \\alpha - \\gamma}{2} = 180^{\\circ} - \\beta - \\gamma - 90^{\\circ} + \\frac{2 \\beta + 2 \\gamma}{2} = 90^{\\circ}$", "correct": true, "reward": 1.0, "repetition": {"words": 214, "unique_ratio": 0.2523364485981308, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_008", "row": {"problem": "Problem 6. (Option 1).\n\nIn the right triangle $ABC: \\angle ACB=90^{\\circ}, AC=6, BC=4$. On the line $BC$, a point $D (CD>BD)$ is marked such that $\\angle ADC=45^{\\circ}$. On the line $AD$, a point $E$ is marked such that the perimeter of triangle $CBE$ is the smallest possible. Then, on the line $DC$, a point $F$ is marked such that the perimeter of triangle $AFE$ is the smallest possible. Find $CF$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_3fdcc7ef9dfdd3dab8ecg-07.jpg?height=846&width=557&top_left_y=228&top_left_x=241)\n\n## Construction and Proof:\n\nFor convenience, construct a square ACDK (AD is its diagonal, since $\\angle ADC=45^{\\circ}$. Mark point $B_1$ on side $DK (B_1D=BD \\Rightarrow B_1B \\perp AD)$.\n\nDraw line $B_1C$, which intersects $AD$ at point $E$.\n\nThe perimeter of triangle $CBE$ is the smallest because among all possible points $E_1$ on line $AD$, the sum of the lengths of segments $B_1E + EC$ is the smallest ($B_1E + EC < B_1E_1 + E_1C$ - triangle inequality) and $B_1E = EB$. Similarly, mark point $A_1$ on side $AC$ ($A_1C = AC$). Draw line $A_1E$, which intersects $CD$ at point $F$.\n\nThe perimeter of triangle $AFE$ is the smallest because among all possible points $F_1$ on line $AD$, the sum of the lengths of segments $A_1F + EF$ is the smallest ($A_1F + EF < A_1F_1 + F_1E$ - triangle inequality) and $A_1F = FA$.", "solution": "# Solution:\n\n$C D=A C=6 \\Rightarrow B D=B_{1} D=6-4=2$. Since triangles $\\mathrm{ACE}$ and $\\mathrm{B}_{1} \\mathrm{ED}$\n\nare similar ( $\\angle A E C=\\angle B_{1} E D$ - vertical angles and $\\angle E A C=\\angle B_{1} D E=45^{\\circ}$ ), then\n\n$E C: B_{1} E=A C: D B_{1}=A E: E D=6: 2=3: 1$\n\nLet $E H \\perp D C \\Rightarrow$ right triangles ACD and EHD are similar ( $\\angle A C D-$ common), then $C D: H D=A C: E H=A D: E D=4: 1 \\Rightarrow E H=H D=1.5 \\Rightarrow C H=4.5$.\n\nRight triangles FHE and $\\mathrm{FCA}_{1}$ are similar $\\left(\\angle E F H=\\angle A_{1} F C\\right.$ vertical angles) $)$\n\nthen $C F: H F=A_{1} F: E F=A_{1} C: E H=6: 1.5=4: 1 ; C H=4.5 \\Rightarrow C F=0.8 \\cdot 4.5=3.6$\n\nAnswer: 3.6 .", "answer": "3.6", "source": "olympiads", "problem_type": "Geometry", "uuid": "090bc9bd-3980-510f-bf7d-7a0c75345340"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nProblem 6. (Option 1).\n\nIn the right triangle $ABC: \\angle ACB=90^{\\circ}, AC=6, BC=4$. On the line $BC$, a point $D (CD>BD)$ is marked such that $\\angle ADC=45^{\\circ}$. On the line $AD$, a point $E$ is marked such that the perimeter of triangle $CBE$ is the smallest possible. Then, on the line $DC$, a point $F$ is marked such that the perimeter of triangle $AFE$ is the smallest possible. Find $CF$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_3fdcc7ef9dfdd3dab8ecg-07.jpg?height=846&width=557&top_left_y=228&top_left_x=241)\n\n## Construction and Proof:\n\nFor convenience, construct a square ACDK (AD is its diagonal, since $\\angle ADC=45^{\\circ}$. Mark point $B_1$ on side $DK (B_1D=BD \\Rightarrow B_1B \\perp AD)$.\n\nDraw line $B_1C$, which intersects $AD$ at point $E$.\n\nThe perimeter of triangle $CBE$ is the smallest because among all possible points $E_1$ on line $AD$, the sum of the lengths of segments $B_1E + EC$ is the smallest ($B_1E + EC < B_1E_1 + E_1C$ - triangle inequality) and $B_1E = EB$. Similarly, mark point $A_1$ on side $AC$ ($A_1C = AC$). Draw line $A_1E$, which intersects $CD$ at point $F$.\n\nThe perimeter of triangle $AFE$ is the smallest because among all possible points $F_1$ on line $AD$, the sum of the lengths of segments $A_1F + EF$ is the smallest ($A_1F + EF < A_1F_1 + F_1E$ - triangle inequality) and $A_1F = FA$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "1000", "correct": false, "reward": -0.5, "repetition": {"words": 1, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "## Step 1:  To solve this problem, we can start by considering the properties of right triangles and the given conditions.\n## Step 2:  First, we can use the fact that $\\angle ADC = 45^{\\circ}$ to derive an equation involving the lengths of the sides of the triangle.\n## Step 3:  Specifically, we can use the similarity of triangles $\\triangle ABC$ and $\\triangle ADC$ to derive an equation involving the lengths of the sides of the triangle.\n## Step 4:  After simplifying the equations, we can find the value of $\\cos \\angle ADC$ and then determine the value of $\\cos \\angle ADC + \\cos \\angle ADC$.\n## Step 5:  By solving these equations, we can find the value of the expression and subsequently determine the value of the expression.\n## Step 6:  By following these steps, we can construct a triangle with sides $8, 4, 6$ and verify that this triangle satisfies the given conditions.\n## Step 7:  By verifying the conditions, we can determine the value of the expression.\n## Step 8:  Finally, using the value of the expression, we can calculate the value of the expression and subsequently the value of the third side of the triangle.\n\nThe final answer is: $\\boxed{50}$", "correct": false, "reward": -1, "repetition": {"words": 196, "unique_ratio": 0.37755102040816324, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "# Solution:\n\n$C D=A C=6 \\Rightarrow B D=B_{1} D=6-4=2$. Since triangles $\\mathrm{ACE}$ and $\\mathrm{B}_{1} \\mathrm{ED}$\n\nare similar ( $\\angle A E C=\\angle B_{1} E D$ - vertical angles and $\\angle E A C=\\angle B_{1} D E=45^{\\circ}$ ), then\n\n$E C: B_{1} E=A C: D B_{1}=A E: E D=6: 2=3: 1$\n\nLet $E H \\perp D C \\Rightarrow$ right triangles ACD and EHD are similar ( $\\angle A C D-$ common), then $C D: H D=A C: E H=A D: E D=4: 1 \\Rightarrow E H=H D=1.5 \\Rightarrow C H=4.5$.\n\nRight triangles FHE and $\\mathrm{FCA}_{1}$ are similar $\\left(\\angle E F H=\\angle A_{1} F C\\right.$ vertical angles) $)$\n\nthen $C F: H F=A_{1} F: E F=A_{1} C: E H=6: 1.5=4: 1 ; C H=4.5 \\Rightarrow C F=0.8 \\cdot 4.5=3.6$\n\nAnswer: 3.6 .", "correct": true, "reward": 1.0, "repetition": {"words": 174, "unique_ratio": 0.2413793103448276, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_009", "row": {"problem": "Problem 10.7. The graph of the function $f(x)=\\frac{1}{12} x^{2}+a x+b$ intersects the $O x$ axis at points $A$ and $C$, and the $O y$ axis at point $B$, as shown in the figure. It turned out that for the point $T$ with coordinates $(3 ; 3)$, the condition $T A=T B=T C$ is satisfied. Find $b$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-39.jpg?height=321&width=517&top_left_y=195&top_left_x=468)", "solution": "Answer: -6.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-39.jpg?height=359&width=614&top_left_y=600&top_left_x=420)\n\nFig. 11: to the solution of problem 10.7\n\nSolution. Let point $A$ have coordinates $\\left(x_{1} ; 0\\right)$, and point $C$ have coordinates $\\left(x_{2} ; 0\\right)$. From the condition, it is clear that $x_{1} < 0$ and $x_{2} > 0$. Since $x_{1}$ and $x_{2}$ are the roots of the quadratic trinomial $f(x)$, by Vieta's theorem, we have $x_{1} \\cdot x_{2} = 12b$, from which we get $b = \\frac{x_{1} \\cdot x_{2}}{12} < 0$.\n\nLet $H$ be the point with coordinates $(3 ; 0)$ (Fig. 11). Clearly, in the isosceles triangle $A T C$, the segment $T H$ is the height, and therefore it is also the median. Thus, $3 - x_{1} = A H = H C = x_{2} - 3$, from which we get $x_{1} = 6 - x_{2}$.\n\nLet $M$ be the point with coordinates $(0, 3)$. Since $T H = T M = 3$ and $T A = T B$, the right triangles $A T H$ and $B T M$ are equal by the leg and hypotenuse. Therefore, $3 - x_{1} = H A = M B = 3 - b$, that is, $x_{1} = b = \\frac{x_{1} \\cdot x_{2}}{12}$ (by Vieta's theorem), from which we find $x_{2} = 12$. Finally, $x_{1} = 6 - x_{2} = 6 - 12 = -6$ and $b = x_{1} = -6$.\n\nAnother solution. As in the previous solution, let the abscissas of points $A$ and $C$ be $x_{1}$ and $x_{2}$, respectively; we will also use the fact that point $B$ has coordinates $(0 ; b)$. Immediately, we understand that $O A = |x_{1}| = -x_{1}$, $O C = |x_{2}| = x_{2}$, and $O B = |b| = -b$.\n\nLet's find the second intersection of the circle with the y-axis, let this be point $D$ with coordinates $(0 ; d)$ (Fig. 12). The chords $A C$ and $B D$ of the circle intersect at the origin $O$; from the properties of the circle, we know that $O A \\cdot O C = O B \\cdot O D$. We get $-x_{1} \\cdot x_{2} = -b \\cdot d$, from which, replacing $x_{1} \\cdot x_{2}$ with $12b$ by Vieta's theorem, we get $d = 12$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-40.jpg?height=521&width=571&top_left_y=83&top_left_x=437)\n\nFig. 12: to the solution of problem 10.7\n\nIt remains to note that triangle $BTD$ is isosceles, and the midpoint of its base, point $M$, has coordinates $(0 ; 3)$. Reflecting point $D(0 ; 12)$ relative to it, we get $B(0 ; -6)$.", "answer": "-6", "source": "olympiads", "problem_type": "Algebra", "uuid": "4f14788f-fd45-586e-a5e9-c4653ba17885"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nProblem 10.7. The graph of the function $f(x)=\\frac{1}{12} x^{2}+a x+b$ intersects the $O x$ axis at points $A$ and $C$, and the $O y$ axis at point $B$, as shown in the figure. It turned out that for the point $T$ with coordinates $(3 ; 3)$, the condition $T A=T B=T C$ is satisfied. Find $b$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-39.jpg?height=321&width=517&top_left_y=195&top_left_x=468)", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Let's start by considering the geometry of the graph from the provided image. The function is a parabola opening upwards, with its vertex at the origin. The equation is $\\frac{1}{12} x^2 + ax + b = 0$, which we can rewrite as $y = -\\frac{1}{12} x^2 - ax + b$. This is a downward-opening parabola, and its axis of symmetry is the x-axis.\n\nThe point T with coordinates $(3, 3)$ lies on the axis of symmetry. Since the parabola is symmetric about this line, the point T is also on the axis of symmetry. Therefore, the axis of symmetry of the parabola is the $y$-axis. This implies that the equation of the axis of symmetry is $y = b$.\n\nNow, we need to find the intersection points of the graph with the $O y$ axis. The point $A$ is the intersection of the parabola with the $y$ axis, which is the line $x = 0$. Therefore, we can substitute $x = 0$ into the equation $y = b$ to find the value of $b$.\n\n$$b = \\boxed{3}$$", "correct": false, "reward": -1, "repetition": {"words": 176, "unique_ratio": 0.3806818181818182, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "Since the graph of the function $f(x)=\\frac{1}{12} x^{2}+a x+b$ intersects the $O x$ axis at points $A$ and $C$, and the $O y$ axis at point $B$, we have the following system of equations:\n$$\n\\begin{cases}\nf(A)=\\frac{1}{12} a+b=0 \\\\\nf(C)=\\frac{1}{12} a+b=0 \\\\\nf(B)=\\frac{1}{12} a+b=0\n\\end{cases}\n$$\n\nSimplifying the equations, we obtain:\n$$\n\\begin{cases}\na=\\frac{1}{3} \\\\\nb=-\\frac{12}{3}\n\\end{cases}\n$$\n\nThus, $a=\\boxed{\\frac{1}{3}}$.\n\nFor the point $T$ with coordinates $(3 ; 3)$, the condition $T A=T B=T C$ holds, which is satisfied.", "correct": false, "reward": -1, "repetition": {"words": 112, "unique_ratio": 0.4017857142857143, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Answer: -6.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-39.jpg?height=359&width=614&top_left_y=600&top_left_x=420)\n\nFig. 11: to the solution of problem 10.7\n\nSolution. Let point $A$ have coordinates $\\left(x_{1} ; 0\\right)$, and point $C$ have coordinates $\\left(x_{2} ; 0\\right)$. From the condition, it is clear that $x_{1} < 0$ and $x_{2} > 0$. Since $x_{1}$ and $x_{2}$ are the roots of the quadratic trinomial $f(x)$, by Vieta's theorem, we have $x_{1} \\cdot x_{2} = 12b$, from which we get $b = \\frac{x_{1} \\cdot x_{2}}{12} < 0$.\n\nLet $H$ be the point with coordinates $(3 ; 0)$ (Fig. 11). Clearly, in the isosceles triangle $A T C$, the segment $T H$ is the height, and therefore it is also the median. Thus, $3 - x_{1} = A H = H C = x_{2} - 3$, from which we get $x_{1} = 6 - x_{2}$.\n\nLet $M$ be the point with coordinates $(0, 3)$. Since $T H = T M = 3$ and $T A = T B$, the right triangles $A T H$ and $B T M$ are equal by the leg and hypotenuse. Therefore, $3 - x_{1} = H A = M B = 3 - b$, that is, $x_{1} = b = \\frac{x_{1} \\cdot x_{2}}{12}$ (by Vieta's theorem), from which we find $x_{2} = 12$. Finally, $x_{1} = 6 - x_{2} = 6 - 12 = -6$ and $b = x_{1} = -6$.\n\nAnother solution. As in the previous solution, let the abscissas of points $A$ and $C$ be $x_{1}$ and $x_{2}$, respectively; we will also use the fact that point $B$ has coordinates $(0 ; b)$. Immediately, we understand that $O A = |x_{1}| = -x_{1}$, $O C = |x_{2}| = x_{2}$, and $O B = |b| = -b$.\n\nLet's find the second intersection of the circle with the y-axis, let this be point $D$ with coordinates $(0 ; d)$ (Fig. 12). The chords $A C$ and $B D$ of the circle intersect at the origin $O$; from the properties of the circle, we know that $O A \\cdot O C = O B \\cdot O D$. We get $-x_{1} \\cdot x_{2} = -b \\cdot d$, from which, replacing $x_{1} \\cdot x_{2}$ with $12b$ by Vieta's theorem, we get $d = 12$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_b46fbd582cc3a82460aeg-40.jpg?height=521&width=571&top_left_y=83&top_left_x=437)\n\nFig. 12: to the solution of problem 10.7\n\nIt remains to note that triangle $BTD$ is isosceles, and the midpoint of its base, point $M$, has coordinates $(0 ; 3)$. Reflecting point $D(0 ; 12)$ relative to it, we get $B(0 ; -6)$.", "correct": true, "reward": 1.0, "repetition": {"words": 450, "unique_ratio": 0.2822222222222222, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_010", "row": {"problem": "## 3. Math Puzzle $8 / 65$\n\nA crane is being transported to a construction site. To measure the length of the crane, Peter walks from the tip of the crane to the end and takes 15 steps (step length 80 $\\mathrm{cm}$).\n\nTo walk back from the end to the tip of the crane while the train is moving at a constant speed, he needs 75 steps at the same walking speed.\n\nHow long is the crane?", "solution": "With $v=$ speed, $s=$ distance, $t_{1}=$ time for 15 steps, $t_{2}=$ time for 75 steps, and $l=$ crane length, we have $l-v t_{1}=12 \\mathrm{~m}, l+v t_{2}=60 \\mathrm{~m}$.\n\nIt holds that $\\frac{t_{1}}{t_{2}}=\\frac{1}{5} \\rightarrow t_{1}=\\frac{1}{5} t_{2}$.\n\nThus, $l-v \\cdot \\frac{1}{5} t_{2}=12 \\mathrm{~m}$ and $l+v t_{2}=60 \\mathrm{~m}$. Multiplying the first equation by 5 and adding both equations yields $6 l=120 \\mathrm{~m}, l=20 \\mathrm{~m}$. The length of the crane is $20 \\mathrm{~m}$.", "answer": "20\\mathrm{~}", "source": "olympiads", "problem_type": "Logic and Puzzles", "uuid": "d350c85b-5988-5321-93ae-87b2dbeadca2"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n## 3. Math Puzzle $8 / 65$\n\nA crane is being transported to a construction site. To measure the length of the crane, Peter walks from the tip of the crane to the end and takes 15 steps (step length 80 $\\mathrm{cm}$).\n\nTo walk back from the end to the tip of the crane while the train is moving at a constant speed, he needs 75 steps at the same walking speed.\n\nHow long is the crane?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Let $x$ be the length of theosal-O-L in centimeters.\n\nWe can set up a proportion based on the given information:\n\\[ \\frac{10}{x} = \\frac{75}{x + 75} \\]\n\nCross-multiply and solve for $x$:\n\\[ 10(x + 75) = 75x \\]\n\n\\[ 10x + 750 = 75x \\]\n\nSubtract $75x$ from both sides:\n\\[ -65x + 750 = 0 \\]\n\n\\[ 65x = 750 \\]\n\n\\[ x = 750 : 65 = \\boxed{10} \\]\n\nFinal answer:\n#### 10", "correct": false, "reward": -0.5, "repetition": {"words": 60, "unique_ratio": 0.7, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 1, "origin": "budgie", "answer": "Peter walks from the tip of thebinding sign to the end of thedeliberation sign, which is 150 steps long.\n\nSince his speed is constant, he will take 150 steps to the end of the Deliberation sign.\n\nThis is also the length of the Garcia-Petersen sign.\n\nTherefore, the length of theWA sign is 150 steps.\n\nThe answer is: $\\boxed{150}$", "correct": false, "reward": -0.5, "repetition": {"words": 60, "unique_ratio": 0.55, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 2, "origin": "verified_reference", "answer": "With $v=$ speed, $s=$ distance, $t_{1}=$ time for 15 steps, $t_{2}=$ time for 75 steps, and $l=$ crane length, we have $l-v t_{1}=12 \\mathrm{~m}, l+v t_{2}=60 \\mathrm{~m}$.\n\nIt holds that $\\frac{t_{1}}{t_{2}}=\\frac{1}{5} \\rightarrow t_{1}=\\frac{1}{5} t_{2}$.\n\nThus, $l-v \\cdot \\frac{1}{5} t_{2}=12 \\mathrm{~m}$ and $l+v t_{2}=60 \\mathrm{~m}$. Multiplying the first equation by 5 and adding both equations yields $6 l=120 \\mathrm{~m}, l=20 \\mathrm{~m}$. The length of the crane is $20 \\mathrm{~m}$.", "correct": true, "reward": 1.0, "repetition": {"words": 105, "unique_ratio": 0.42857142857142855, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142115623759233}], "split": "math_train"}
+{"id": "math_011", "row": {"problem": "## Zadatak B-1.2.\n\nNa slici su prikazani pravilni peterokut. $A B C D E$ i kvadrat $A B F G$. Odredite mjeru kuta $F A D$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_30_d455efeee432fadf0574g-01.jpg?height=234&width=257&top_left_y=1842&top_left_x=797)", "solution": "## Rješenje.\n\nZbroj mjera unutrašnjih kutova pravilnog peterokuta iznosi $(5-2) 180^{\\circ}=540^{\\circ}$ pa je mjera unutrašnjeg kuta pravilnog peterokuta jednaka $\\frac{540^{\\circ}}{5}=108^{\\circ}$.\n\nBudući da je trokut $A E D$ jednakokračan, $\\varangle D A E=\\frac{180^{\\circ}-108^{\\circ}}{2}=36^{\\circ}$\n\nKut $\\varangle B A F=45^{\\circ}$ jer je to kut između dijagonale i stranice kvadrata.\n\nTada je $\\varangle F A D=\\varangle B A E-\\varangle B A F-\\varangle D A E=108^{\\circ}-45^{\\circ}-36^{\\circ}=27^{\\circ}$.", "answer": "27", "source": "olympiads", "problem_type": "Geometry", "uuid": "95bcc8e7-a809-533d-8cbd-45e190531b31"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n## Zadatak B-1.2.\n\nNa slici su prikazani pravilni peterokut. $A B C D E$ i kvadrat $A B F G$. Odredite mjeru kuta $F A D$.\n\n![](https://cdn.mathpix.com/cropped/2024_05_30_d455efeee432fadf0574g-01.jpg?height=234&width=257&top_left_y=1842&top_left_x=797)", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The distance from $C$ to $F$ is:\n\\[ CF = \\sqrt{(s - 0)^2 + (s - 0)^2} = \\sqrt{s^2 + s^2} = s\\sqrt{2} \\]\n\nThe only possible value of $s$ is 1, but this would mean that $A = B$ or $A = C$ or $A = D$ or $B = D$ or $F = A$ and $E = C$. However, this would mean that $A = F$ and $E = D$, which is a contradiction, so $s = 1$ is the only possible solution.\n\nThe distance from $F$ to $G$ is:\n\\[ FG = \\sqrt{(s - 0)^2 + (s - 0)^2} = s\\sqrt{2} \\]\n\nThe possible values of $s$ are 1 and 2, and the distances from $C$ to $F$ and $FG$ are the same. In this case, the area $AEFG$ is:\n\\[ AEFG = \\frac{1}{2} \\cdot 1 \\cdot s\\sqrt{2} = \\frac{s\\sqrt{2}}{2} \\]\n\nThe area of the figure $AEFG$ is $2AEFG = 2 \\cdot \\frac{s\\sqrt{2}}{2} = s\\sqrt{2}$.\n\nThe final answer is:\n#### \\frac{\\sqrt{2}s}{2}", "correct": false, "reward": -1, "repetition": {"words": 163, "unique_ratio": 0.3006134969325153, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "The problem mentions a rectangle in which $A B F G$ is a square. We need to determine the all possible positions of the square.\n\nTo start solving this problem, let's consider the different steps in the process.\n\nFirst, we need to identify the vertices of the square. The side length of the square is equal to the side length of the rectangle.\n\nLet's denote the side length of the rectangle as $x$. Then, the side length of the square is $x$.\n\nNow, we have a square with a side length of $x$, so its top vertex is $x$ units above the bottom face of the rectangle.\n\nNext, we can consider the other vertices of the square. The bottom face of the rectangle is a square with a side length of $x$. The top vertices of this square are $x$ units above the bottom face of the rectangle.\n\nNow, we have a square with a side length of $x$, so its bottom face is a square with a side length of $x$. The top vertices of this square are $x$ units above the bottom face of the rectangle.\n\nNext, we can consider the other vertices of the square. The left face of the rectangle is a square with a side length of $x$. The bottom vertices of this square are $x$ units above the left face of the rectangle.\n\nFinally, we can consider the other vertices of the square. The right face of the rectangle is a square with a side length of $x$. The top vertices of this square are $x$ units above the right face of the rectangle.\n\nNow, we have a square with a side length of $x$, so its top face is a square with a side length of $x$. The left face of the rectangle is a square with a side length of $x", "correct": false, "reward": -1, "repetition": {"words": 309, "unique_ratio": 0.18122977346278318, "max_4gram_count": 9, "max_sentence_count": 2, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "## Rješenje.\n\nZbroj mjera unutrašnjih kutova pravilnog peterokuta iznosi $(5-2) 180^{\\circ}=540^{\\circ}$ pa je mjera unutrašnjeg kuta pravilnog peterokuta jednaka $\\frac{540^{\\circ}}{5}=108^{\\circ}$.\n\nBudući da je trokut $A E D$ jednakokračan, $\\varangle D A E=\\frac{180^{\\circ}-108^{\\circ}}{2}=36^{\\circ}$\n\nKut $\\varangle B A F=45^{\\circ}$ jer je to kut između dijagonale i stranice kvadrata.\n\nTada je $\\varangle F A D=\\varangle B A E-\\varangle B A F-\\varangle D A E=108^{\\circ}-45^{\\circ}-36^{\\circ}=27^{\\circ}$.", "correct": true, "reward": 1.0, "repetition": {"words": 96, "unique_ratio": 0.4791666666666667, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_012", "row": {"problem": "The area of the region in the $xy$-plane satisfying the inequality \\[\\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, \\, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1\\] is $k\\pi$, for some integer $k$. Find $k$.\n\n[i]Proposed by Michael Tang[/i]", "solution": "To find the area of the region in the $xy$-plane satisfying the inequality \n\\[\n\\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, \\, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1,\n\\]\nwe need to analyze the given inequality step by step.\n\n1. **Understanding the Inequality:**\n   The inequality involves two expressions:\n   \\[\n   \\max\\left(\\frac{x^2 + y^2}{4n^2}, 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1}\\right) \\le 1.\n   \\]\n   This means that for some integer $n$ in the range $1 \\le n \\le 10$, the maximum of the two expressions must be less than or equal to 1.\n\n2. **Breaking Down the Inequality:**\n   For the maximum of the two expressions to be less than or equal to 1, both expressions must individually be less than or equal to 1:\n   \\[\n   \\frac{x^2 + y^2}{4n^2} \\le 1 \\quad \\text{and} \\quad 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\le 1.\n   \\]\n\n3. **Simplifying Each Expression:**\n   - For the first expression:\n     \\[\n     \\frac{x^2 + y^2}{4n^2} \\le 1 \\implies x^2 + y^2 \\le 4n^2.\n     \\]\n   - For the second expression:\n     \\[\n     2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\le 1 \\implies \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\ge 1 \\implies x^2 + y^2 \\ge 4n^2 - 4n + 1.\n     \\]\n\n4. **Combining the Conditions:**\n   Therefore, for some $n$ in the range $1 \\le n \\le 10$, the point $(x, y)$ must satisfy:\n   \\[\n   4n^2 - 4n + 1 \\le x^2 + y^2 \\le 4n^2.\n   \\]\n   This describes a ring (annulus) in the $xy$-plane with inner radius $\\sqrt{4n^2 - 4n + 1}$ and outer radius $2n$.\n\n5. **Calculating the Area of Each Annulus:**\n   The area of an annulus with inner radius $r_1$ and outer radius $r_2$ is given by:\n   \\[\n   \\pi (r_2^2 - r_1^2).\n   \\]\n   For our specific case:\n   \\[\n   r_2 = 2n \\quad \\text{and} \\quad r_1 = \\sqrt{4n^2 - 4n + 1}.\n   \\]\n   Therefore, the area of the annulus is:\n   \\[\n   \\pi \\left((2n)^2 - (4n^2 - 4n + 1)\\right) = \\pi \\left(4n^2 - (4n^2 - 4n + 1)\\right) = \\pi (4n - 1).\n   \\]\n\n6. **Summing the Areas for $n$ from 1 to 10:**\n   We need to sum the areas of the annuli for $n = 1, 2, \\ldots, 10$:\n   \\[\n   \\sum_{n=1}^{10} \\pi (4n - 1) = \\pi \\sum_{n=1}^{10} (4n - 1).\n   \\]\n   The sum inside the parentheses is:\n   \\[\n   \\sum_{n=1}^{10} (4n - 1) = 4 \\sum_{n=1}^{10} n - \\sum_{n=1}^{10} 1 = 4 \\left(\\frac{10 \\cdot 11}{2}\\right) - 10 = 4 \\cdot 55 - 10 = 220 - 10 = 210.\n   \\]\n\n7. **Final Area:**\n   Therefore, the total area is:\n   \\[\n   210\\pi.\n   \\]\n\nThe final answer is $\\boxed{210}$.", "answer": "210", "source": "aops_forum", "problem_type": "Inequalities", "uuid": "36667977-8bb5-552e-8fcc-d328f2a797c0"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nThe area of the region in the $xy$-plane satisfying the inequality \\[\\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, \\, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1\\] is $k\\pi$, for some integer $k$. Find $k$.\n\n[i]Proposed by Michael Tang[/i]", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Since $x, y \\le 1$, we can simplify the inequality to:\n\\[ \\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1 \\]\n\nWe can simplify the inequality to:\n\\[ \\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1 \\]\n\nSince $n$ is a positive integer, we can simplify the inequality to:\n\\[ \\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1 \\]\n\nWe can simplify the inequality to:\n\\[ \\min_{1 \\le n \\le 10} \\frac{x^2+y^2}{4n^2} \\le 1 + \\frac{2 - \\frac{x^2+y^2}{4n^2-4n+1}}{4n^2} \\]\n\nSimplifying the inequality to:\n\\[ 1 \\le \\frac{x^2+y^2}{4n^2} + \\frac{2 - \\frac{x^2+y^2}{4n^2-4n", "correct": false, "reward": -1, "repetition": {"words": 170, "unique_ratio": 0.14705882352941177, "max_4gram_count": 10, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "1. **Identify the Function:**\n   The function to analyze is $f(x) = \\min_{1 \\leqslant n \\leqslant 10} \\max\\left(\\frac{x^2 + y^2}{4n^2}, \\, 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1}\\right)$.\n\n2. **Simplify the Function:**\n   We can simplify $f(x)$ by considering the expressions $\\frac{x^2 + y^2}{4n^2}$ and $\\frac{x^2 + y^2}{4n^2 - 4n + 1}$. Let's denote these as $p(x, y)$ for $x, y \\in \\mathbb{R}$.\n\n3. **Analyze the Sums:**\n   The function $f(x)$ can be rewritten as:\n   \\[\n   f(x) = \\min_{1 \\leqslant n \\leqslant 10} \\max\\left(\\frac{x^2 + y^2}{4n^2}, \\, 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1}\\right) = \\min_{n=1}^\\infty \\max\\left(\\frac{1}{4n^2}, \\, 2 - \\frac{1}{4n^2 - 4n + 1}\\right) = \\max\\left(\\frac{1}{4(1)}, \\, \\frac{2}{4(1) - 1}\\right) = \\max\\left(\\frac{1}{4}, \\frac{1}{3}\\right)\n   \\", "correct": false, "reward": -1, "repetition": {"words": 159, "unique_ratio": 0.2641509433962264, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "To find the area of the region in the $xy$-plane satisfying the inequality \n\\[\n\\min_{1 \\le n \\le 10} \\max\\left(\\frac{x^2+y^2}{4n^2}, \\, 2 - \\frac{x^2+y^2}{4n^2-4n+1}\\right) \\le 1,\n\\]\nwe need to analyze the given inequality step by step.\n\n1. **Understanding the Inequality:**\n   The inequality involves two expressions:\n   \\[\n   \\max\\left(\\frac{x^2 + y^2}{4n^2}, 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1}\\right) \\le 1.\n   \\]\n   This means that for some integer $n$ in the range $1 \\le n \\le 10$, the maximum of the two expressions must be less than or equal to 1.\n\n2. **Breaking Down the Inequality:**\n   For the maximum of the two expressions to be less than or equal to 1, both expressions must individually be less than or equal to 1:\n   \\[\n   \\frac{x^2 + y^2}{4n^2} \\le 1 \\quad \\text{and} \\quad 2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\le 1.\n   \\]\n\n3. **Simplifying Each Expression:**\n   - For the first expression:\n     \\[\n     \\frac{x^2 + y^2}{4n^2} \\le 1 \\implies x^2 + y^2 \\le 4n^2.\n     \\]\n   - For the second expression:\n     \\[\n     2 - \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\le 1 \\implies \\frac{x^2 + y^2}{4n^2 - 4n + 1} \\ge 1 \\implies x^2 + y^2 \\ge 4n^2 - 4n + 1.\n     \\]\n\n4. **Combining the Conditions:**\n   Therefore, for some $n$ in the range $1 \\le n \\le 10$, the point $(x, y)$ must satisfy:\n   \\[\n   4n^2 - 4n + 1 \\le x^2 + y^2 \\le 4n^2.\n   \\]\n   This describes a ring (annulus) in the $xy$-plane with inner radius $\\sqrt{4n^2 - 4n + 1}$ and outer radius $2n$.\n\n5. **Calculating the Area of Each Annulus:**\n   The area of an annulus with inner radius $r_1$ and outer radius $r_2$ is given by:\n   \\[\n   \\pi (r_2^2 - r_1^2).\n   \\]\n   For our specific case:\n   \\[\n   r_2 = 2n \\quad \\text{and} \\quad r_1 = \\sqrt{4n^2 - 4n + 1}.\n   \\]\n   Therefore, the area of the annulus is:\n   \\[\n   \\pi \\left((2n)^2 - (4n^2 - 4n + 1)\\right) = \\pi \\left(4n^2 - (4n^2 - 4n + 1)\\right) = \\pi (4n - 1).\n   \\]\n\n6. **Summing the Areas for $n$ from 1 to 10:**\n   We need to sum the areas of the annuli for $n = 1, 2, \\ldots, 10$:\n   \\[\n   \\sum_{n=1}^{10} \\pi (4n - 1) = \\pi \\sum_{n=1}^{10} (4n - 1).\n   \\]\n   The sum inside the parentheses is:\n   \\[\n   \\sum_{n=1}^{10} (4n - 1) = 4 \\sum_{n=1}^{10} n - \\sum_{n=1}^{10} 1 = 4 \\left(\\frac{10 \\cdot 11}{2}\\right) - 10 = 4 \\cdot 55 - 10 = 220 - 10 = 210.\n   \\]\n\n7. **Final Area:**\n   Therefore, the total area is:\n   \\[\n   210\\pi.\n   \\]\n\nThe final answer is $\\boxed{210}$.", "correct": true, "reward": 1.0, "repetition": {"words": 458, "unique_ratio": 0.2336244541484716, "max_4gram_count": 12, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_013", "row": {"problem": "Four. (30 points) Given the sequence $\\left\\{a_{n}\\right\\}$ satisfies\n$$\na_{1}=\\frac{1}{2}, a_{n}=2 a_{n} a_{n+1}+3 a_{n+1}\\left(n \\in \\mathbf{N}_{+}\\right) \\text {. }\n$$\n(1) Find the general term formula of the sequence $\\left\\{a_{n}\\right\\}$;\n(2) If the sequence $\\left\\{b_{n}\\right\\}$ satisfies $b_{n}=1+\\frac{1}{a_{n}}\\left(n \\in \\mathbf{N}_{+}\\right)$, and for any positive integer $n(n \\geqslant 2)$, the inequality\n$$\n\\sum_{k=1}^{n} \\frac{1}{n+\\log _{3} b_{k}}>\\frac{m}{24}\n$$\n\nalways holds, find the maximum value of the integer $m$.", "solution": "(1) From\n$$\n\\begin{array}{l}\na_{n}=2 a_{n} a_{n+1}+3 a_{n+1} \\\\\n\\Rightarrow a_{n+1}=\\frac{a_{n}}{2 a_{n}+3} \\Rightarrow a_{n}>0 . \\\\\n\\text { Hence } \\frac{1}{a_{n+1}}-\\frac{3}{a_{n}}=2 \\Rightarrow \\frac{1}{a_{n+1}}+1=3\\left(\\frac{1}{a_{n}}+1\\right) .\n\\end{array}\n$$\n\nTherefore, $\\left\\{\\frac{1}{a_{n}}+1\\right\\}$ is a geometric sequence with the first term 3 and common ratio 3.\nThus, $\\frac{1}{a_{n}}+1=3^{n} \\Rightarrow a_{n}=\\frac{1}{3^{n}-1}\\left(n \\in \\mathbf{N}_{+}\\right)$.\n(2) From (1), we get $b_{n}=1+\\frac{1}{a_{n}}=3^{n}$.\nTherefore, $\\log _{3} b_{k}=k(k=1,2, \\cdots)$.\nHence, $\\sum_{k=1}^{n} \\frac{1}{n+\\log _{3} b_{k}}>\\frac{m}{24}$\n$\\Leftrightarrow \\frac{1}{n+1}+\\frac{1}{n+2}+\\cdots+\\frac{1}{n+n}>\\frac{m}{24}$.\nLet $f(n)=\\frac{1}{n+1}+\\frac{1}{n+2}+\\cdots+\\frac{1}{n+n}$. Then\n$$\n\\begin{array}{l}\nf(n+1)-f(n) \\\\\n=\\frac{1}{2 n+1}+\\frac{1}{2 n+2}-\\frac{1}{n+1} \\\\\n=\\frac{1}{2 n+1}-\\frac{1}{2 n+2}>0 .\n\\end{array}\n$$\n\nTherefore, $f(n)$ is monotonically increasing.\nHence, $f(n)_{\\min }=f(2)=\\frac{1}{3}+\\frac{1}{4}=\\frac{7}{12}$.\nThus, $\\frac{7}{12}>\\frac{m}{24} \\Rightarrow m<14$.\nTherefore, the maximum integer value of $m$ is 13.", "answer": "13", "source": "cn_contest", "problem_type": "Algebra", "uuid": "3929973c-a32c-5fed-884e-cdaf87e29bbc"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nFour. (30 points) Given the sequence $\\left\\{a_{n}\\right\\}$ satisfies\n$$\na_{1}=\\frac{1}{2}, a_{n}=2 a_{n} a_{n+1}+3 a_{n+1}\\left(n \\in \\mathbf{N}_{+}\\right) \\text {. }\n$$\n(1) Find the general term formula of the sequence $\\left\\{a_{n}\\right\\}$;\n(2) If the sequence $\\left\\{b_{n}\\right\\}$ satisfies $b_{n}=1+\\frac{1}{a_{n}}\\left(n \\in \\mathbf{N}_{+}\\right)$, and for any positive integer $n(n \\geqslant 2)$, the inequality\n$$\n\\sum_{k=1}^{n} \\frac{1}{n+\\log _{3} b_{k}}>\\frac{m}{24}\n$$\n\nalways holds, find the maximum value of the integer $m$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "(1) For $n=1$, we have $a_{1}=2 a_{2}+3 a_{3}$, which simplifies to $2a_{2}=a_{3}$, therefore $a_{2}=a_{3}=\\frac{1}{2}$.\n\nFor $n\\geqslant 2$, the recursive formula $a_{n}=2 a_{n} a_{n+1}+3 a_{n+1}$ can be simplified to $a_{n+1}=a_{n}$.\n\nHence, $a_{n+1}=a_{n}$.\n\nTherefore, for all $n\\in \\mathbb{N}_{+}$, $a_{n}=\\frac{1}{n}$.\n\n(2) For $n=1$, we have $a_{1}=\\frac{1}{2}$, and for all $n\\in \\mathbb{N}_{+}$, $b_{n}=1+\\frac{1}{a_{n}}(n \\in \\mathbb{N}_{+})$.\n\nWe can rewrite $b_{n}$ as follows:\n\n$$b_{n}=1+\\frac{1}{a_{n}}\\cdot n \\in \\mathbb{N}_{+}$$\n\nThe given inequality becomes:\n\n$$1+\\frac{1}{a_{1}}\\cdot\\left(1+\\frac{1}{a_{2}}\\cdot\\left(1+\\frac{1}{a_{3}}\\cdot\\left(1+\\frac{1}{a_{4}}\\cdot\\left(1+\\frac{1}{a_{5}}\\cdot\\left(1+\\frac{1}{a_{6}}\\cdot\\", "correct": false, "reward": -1, "repetition": {"words": 168, "unique_ratio": 0.2261904761904762, "max_4gram_count": 8, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "(1) When $n=1$, $\\frac{1}{a_{1}}$, $a_{2}$, $a_{3}$, $\\ldots$ ;\nWhen $n=2$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $a_{3}$, $\\frac{1}{a_{3}}$, $\\ldots$;\nWhen $n=3$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $a_{4}$, $\\frac{1}{a_{4}}$, $\\ldots$;\nWhen $n=4$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $\\frac{1}{a_{3}}$, $\\frac{1}{a_{4}}$;\nWhen $n=5$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $\\frac{1}{a_{3}}$, $\\frac{1}{a_{4}}$;\n...\nWhen $n=49$, $\\frac{1}{a_{1}}$, $\\frac{1}{a_{2}}$, $\\frac{1}{a_{3}}$, $\\frac{1}{a_{4}}$...\n$\\end{aligned}$\n\n(2) Since for any positive integer $n(n \\geqslant 2)$, the inequality\n$\\sum_{k=1}^{n} \\frac{1}{n+\\log _{3} b_{k}}>\\frac{m}{24}$\n\nalways holds,\n\n$\\therefore$ The sequence $\\left\\{b_{n}\\right\\}$ is monotonically increasing,\n\n$\\therefore b_{1}<", "correct": false, "reward": -1, "repetition": {"words": 149, "unique_ratio": 0.24161073825503357, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "(1) From\n$$\n\\begin{array}{l}\na_{n}=2 a_{n} a_{n+1}+3 a_{n+1} \\\\\n\\Rightarrow a_{n+1}=\\frac{a_{n}}{2 a_{n}+3} \\Rightarrow a_{n}>0 . \\\\\n\\text { Hence } \\frac{1}{a_{n+1}}-\\frac{3}{a_{n}}=2 \\Rightarrow \\frac{1}{a_{n+1}}+1=3\\left(\\frac{1}{a_{n}}+1\\right) .\n\\end{array}\n$$\n\nTherefore, $\\left\\{\\frac{1}{a_{n}}+1\\right\\}$ is a geometric sequence with the first term 3 and common ratio 3.\nThus, $\\frac{1}{a_{n}}+1=3^{n} \\Rightarrow a_{n}=\\frac{1}{3^{n}-1}\\left(n \\in \\mathbf{N}_{+}\\right)$.\n(2) From (1), we get $b_{n}=1+\\frac{1}{a_{n}}=3^{n}$.\nTherefore, $\\log _{3} b_{k}=k(k=1,2, \\cdots)$.\nHence, $\\sum_{k=1}^{n} \\frac{1}{n+\\log _{3} b_{k}}>\\frac{m}{24}$\n$\\Leftrightarrow \\frac{1}{n+1}+\\frac{1}{n+2}+\\cdots+\\frac{1}{n+n}>\\frac{m}{24}$.\nLet $f(n)=\\frac{1}{n+1}+\\frac{1}{n+2}+\\cdots+\\frac{1}{n+n}$. Then\n$$\n\\begin{array}{l}\nf(n+1)-f(n) \\\\\n=\\frac{1}{2 n+1}+\\frac{1}{2 n+2}-\\frac{1}{n+1} \\\\\n=\\frac{1}{2 n+1}-\\frac{1}{2 n+2}>0 .\n\\end{array}\n$$\n\nTherefore, $f(n)$ is monotonically increasing.\nHence, $f(n)_{\\min }=f(2)=\\frac{1}{3}+\\frac{1}{4}=\\frac{7}{12}$.\nThus, $\\frac{7}{12}>\\frac{m}{24} \\Rightarrow m<14$.\nTherefore, the maximum integer value of $m$ is 13.", "correct": true, "reward": 1.0, "repetition": {"words": 252, "unique_ratio": 0.2222222222222222, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_014", "row": {"problem": "Example 7 How many positive integer factors does 20! have?", "solution": "Analyze writing 20! in its standard factorization form $n=\\beta_{1}^{a_{1}} \\beta_{2}^{a_{2}} \\cdots \\beta_{k}^{q_{k}}$, and then using $r(n)=\\left(\\alpha_{1}+1\\right)\\left(\\alpha_{2}+1\\right) \\cdots$ $\\left(\\alpha_{k}+1\\right)$ to calculate the result.\nSolution Since the prime numbers less than 20 are $2,3,5,7,11,13,17,19$, in the standard factorization of 20!, the highest power of 2 $=\\left[\\frac{20}{2}\\right]+\\left[\\frac{20}{4}\\right]+\\left[\\frac{20}{8}\\right]+\\left[\\frac{20}{16}\\right]=18$,\nthe highest power of 3 $=\\left[\\frac{20}{3}\\right]+\\left[\\frac{20}{9}\\right]=8$,\nthe highest power of 5 $=\\left[\\frac{20}{5}\\right]=4$,\nthe highest powers of $7,11,13,17,19$ are $2,1,1,1,1$ respectively.\nThus, the standard factorization of 20! is\n$$20!=2^{18} \\cdot 3^{8} \\cdot 5^{4} \\cdot 7^{2} \\cdot 11 \\cdot 13 \\cdot 17 \\cdot 19$$\n\nTherefore, the number of positive divisors $r(20!)$ of 20! is:\n$$\\begin{aligned}\nr(20!) & =(18+1)(8+1)(4+1)(2+1)(1+1)(1+1)(1+1)(1+1) \\\\\n& =19 \\cdot 9 \\cdot 5 \\cdot 3 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2 \\\\\n& =41040\n\\end{aligned}$$\n\nThus, the number of positive divisors of $20!$ is 41040.", "answer": "41040", "source": "number_theory", "problem_type": "Number Theory", "uuid": "38902e28-3cc8-5f40-a032-89bdea866710"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nExample 7 How many positive integer factors does 20! have?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find the number of positive integer factors of 20!, we need to find the prime factorization of 20! and then use the formula for the number of factors.\n\nThe prime factorization of 20! is the product of all prime numbers less than or equal to 20.\n\nThe prime numbers less than or equal to 20 are 2, 3, 5, 7, 11, 13, 17, 19.\n\nThe factor 20! has 11 factors (including 1 and 20).\nThe number of factors of 20! is 11.\n\nFinal answer:\n#### 11", "correct": false, "reward": -0.5, "repetition": {"words": 86, "unique_ratio": 0.47674418604651164, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "To find the number of positive integer factors of 20!, we can use the formula for the number of factors of a number. This formula is based on the prime factorization of the number.\n\nThe formula is:\n(a + 1)(b + 1)...(c + 1) = 20!\n\nWe need to find the prime factorization of 20! and then plug in the values for (a + 1), (b + 1), and (c + 1) to calculate the number of factors.\n\nNow, let's find the prime factorization of 20!:\n\nFor 20!, the prime factorization is:\n20! = 20 × 19 × 18 × ... × 1\n\nThis is not a prime number, but we can see that 20! is a product of numbers from 2 to 20.\n\nWe can write 20! as:\n20! = 2 × 19 × 18 × 17 × 16 × 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1\n\nNow, let's divide this number by the prime factors of 20! to find the number of factors:\n\n(2 × 19 × 18 × 17 × 16 × 15 × 14 × 13 × 12 × 11 × 10 × 9 × 8 × 7 × 6 × 5 × 4 × 3 × 2 × 1) / (", "correct": false, "reward": -1, "repetition": {"words": 176, "unique_ratio": 0.3522727272727273, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "Analyze writing 20! in its standard factorization form $n=\\beta_{1}^{a_{1}} \\beta_{2}^{a_{2}} \\cdots \\beta_{k}^{q_{k}}$, and then using $r(n)=\\left(\\alpha_{1}+1\\right)\\left(\\alpha_{2}+1\\right) \\cdots$ $\\left(\\alpha_{k}+1\\right)$ to calculate the result.\nSolution Since the prime numbers less than 20 are $2,3,5,7,11,13,17,19$, in the standard factorization of 20!, the highest power of 2 $=\\left[\\frac{20}{2}\\right]+\\left[\\frac{20}{4}\\right]+\\left[\\frac{20}{8}\\right]+\\left[\\frac{20}{16}\\right]=18$,\nthe highest power of 3 $=\\left[\\frac{20}{3}\\right]+\\left[\\frac{20}{9}\\right]=8$,\nthe highest power of 5 $=\\left[\\frac{20}{5}\\right]=4$,\nthe highest powers of $7,11,13,17,19$ are $2,1,1,1,1$ respectively.\nThus, the standard factorization of 20! is\n$$20!=2^{18} \\cdot 3^{8} \\cdot 5^{4} \\cdot 7^{2} \\cdot 11 \\cdot 13 \\cdot 17 \\cdot 19$$\n\nTherefore, the number of positive divisors $r(20!)$ of 20! is:\n$$\\begin{aligned}\nr(20!) & =(18+1)(8+1)(4+1)(2+1)(1+1)(1+1)(1+1)(1+1) \\\\\n& =19 \\cdot 9 \\cdot 5 \\cdot 3 \\cdot 2 \\cdot 2 \\cdot 2 \\cdot 2 \\\\\n& =41040\n\\end{aligned}$$\n\nThus, the number of positive divisors of $20!$ is 41040.", "correct": true, "reward": 1.0, "repetition": {"words": 225, "unique_ratio": 0.28, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_dev"}
+{"id": "math_015", "row": {"problem": "Francisco has 3 daughters: Alina, Valentina, and Civela. An interesting fact is that all three daughters were born on March 18. Today, March 18, 2014, is their birthday. Noting another curious fact, Francisco says:\n\n- Alina, your age is now double the age of Valentina.\n\na) Show that this could never have happened before and that, after the next birthday of Valentina, it will never happen again.\n\nThen, Alina, who was very clever, exclaimed:\n\n- Dad, exactly 10 years ago, the age of one of us three was double the age of another, and in 10 years, the same fact will happen again!\n\nb) It is known that the oldest of the daughters is over 30 years old. How old is Civela?", "solution": "Solution\n\na) It is clear that Alina is older than Valentina. Let's call the difference in age between Alina and Valentina $d$. Since they were born on the same day of the year, $d$ does not vary (note that this would not be true if they were born on different days, as in that case, the difference would change on the birthday of each one).\n\nIf Valentina's age today is $\\alpha$, then Alina's age must be $2 \\alpha$, so the age difference between the two is given by:\n\n$$\nd=2 \\alpha-\\alpha\n$$\n\nWe conclude that $d=\\alpha$, so Alina is $2 d$ years old and Valentina is $d$ years old.\n\nSuch a relationship between the ages of Alina and Valentina could not have happened before, as Valentina was younger than $d$ years old. It also will not happen again, as Valentina will be older than $d$ years.\n\nb) Let's call the age of the youngest daughter $A$, the age of the middle daughter $B$, and the age of the oldest daughter $C$. Let $x=B-A$ be the age difference between the two youngest, and $y=C-B$ be the age difference between the two oldest daughters. The age difference between the youngest and the oldest is therefore $C-A=x+y$. Since they were born on the same day of the year, these differences do not vary.\n\nIt is known that in each of the years 2004, 2014, and 2024, the age of one of the daughters was, is, or will be double the age of one of the others. For this, there are only 3 possibilities:\n\n(i) $B$ is double $A$.\n\n(ii) $C$ is double $B$.\n\n(iii) $C$ is double $A$.\n\nBy the reasoning used in part a), we know that each situation can only occur in one of the three years. Therefore, the three situations: (i), (ii), and (iii) must happen in the three years: 2004, 2014, and 2024, but we do not yet know in what order. Also, from part a), we know that:\n\n- When (i) happens, $B$ must be equal to $2 x$ and $A$ must be equal to $x$. Moreover, $C$ is equal to $B+y$, that is, $C$ must be equal to $2 x+y$.\n- When (ii) happens, $B=y, C=2 y$, and therefore, $A=y-x$.\n- When (iii) happens, $C=2(x+y), A=x+y$, and therefore, $B=2 x+y$.\n\nObserve that the ages of $A$ in cases (i), (ii), and (iii) are\n\n$$\nx, \\quad y-x \\quad \\text { and } \\quad x+y\n$$\n\nrespectively. Since $x<x+y$ and $y-x<x+y$, we conclude that $x+y$ is the largest of the three ages and, therefore, (iii) must happen in the year 2024. Only remains to see in which of the two years 2004 or 2014 each of the events in (i) and (ii) happens. Let's assume that (i) and (ii) happen in the years 2014 and 2004, respectively. Then, the ages of $A$ in the years 2004, 2014, and 2024 would be $y-x, x$, and $x+y$, respectively. Thus, we would have the equations\n\n$$\ny-x+10=x \\quad \\text { and } \\quad x+10=x+y\n$$\n\nSolving, we would have $x=10$ and $y=10$. But in that case, $C$ would be today equal to $2 x+y=30$. But the oldest daughter is more than 30 years old, as stated in the problem. We discard, then, this possibility.\n\nWe know then that (i) and (ii) happen in the years 2004 and 2014, respectively. Then, the ages of $A$ in the years 2004, 2014, and 2024 are $x, y-x$, and $x+y$, respectively. Therefore, we have that\n\n$$\nx+10=y-x \\quad \\text { and } \\quad y-x+10=x+y .\n$$\n\nSolving, we get $x=5$ and $y=20$. Finally, the ages of Francisco's daughters today are equal to\n\n$$\ny-x=15, \\quad y=20 \\quad \\text { and } \\quad 2 y=40\n$$\n\nSince Alina's age must be double Valentina's age, the ages of Alina and Valentina must be 40 and 20, respectively. Therefore, Civela is 15 years old today.", "answer": "15", "source": "olympiads", "problem_type": "Logic and Puzzles", "uuid": "9e1d477e-bfd3-5d98-a9ef-b7c0570221fb"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nFrancisco has 3 daughters: Alina, Valentina, and Civela. An interesting fact is that all three daughters were born on March 18. Today, March 18, 2014, is their birthday. Noting another curious fact, Francisco says:\n\n- Alina, your age is now double the age of Valentina.\n\na) Show that this could never have happened before and that, after the next birthday of Valentina, it will never happen again.\n\nThen, Alina, who was very clever, exclaimed:\n\n- Dad, exactly 10 years ago, the age of one of us three was double the age of another, and in 10 years, the same fact will happen again!\n\nb) It is known that the oldest of the daughters is over 30 years old. How old is Civela?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "a) To show that this could never have happened before and that, after the next birthday of\"{ parent_id }\", it will never happen again, we need to find a counterexample.\n\nConsider the following scenario:\n\n- The oldest daughter is 49 years old.\n- The youngest daughter is 24 years old.\n- The oldest daughter is 48 years old.\n- The youngest daughter is 24 years old.\n\nAt the next birthday of their father, as mentioned before, each daughter's age at that time is halved, and in 10 years, this will occur two times.\n\n- The oldest daughter will be 14 years old, and the youngest daughter will be 8 years old.\n- The oldest daughter will be 12 years old, and the youngest daughter will be 6 years old.\n- The youngest daughter will be 6 years old, and the oldest daughter will be 4 years old.\n\nIn this case, the oldest daughter is 6 years old.\n\nb) To determine if it is possible for the oldest daughter to be over 30 years old, we can see that she will be 30 years old only if she is 30 - 24 = 6 years old.\n\nSince the oldest daughter is 6 years old, we know that the youngest daughter is 24 - 6 = 18 years old.\n\nTherefore, Máiding will be 24 - 6 = 18 years old, and his son will be 18 + 6 = 24 years old.\n\nIn conclusion, there is a misunderstanding in the problem, as the statement \"All three daughters were born on March 18\" is not true. The age of the youngest daughter", "correct": false, "reward": -1, "repetition": {"words": 255, "unique_ratio": 0.36470588235294116, "max_4gram_count": 4, "max_sentence_count": 2, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "a) 10 years ago, Alina was 36 years old.\n2014 - 10 = 2004.\n2004 / 3 = 668.\nAlina's age is now double the age of Valentina, who was 36 years old 10 years ago.\nA 10-year-old daughter, the product of socks and the age of each sock is 10! The socks are 6 feet, 2 feet each.\nHence, Alina is 6 times 2 feet old.\nAlina is now 36 * 2 = 72 years old.\nIn 10 years, Alina will be 72 + 10 = 82 years old.\nThus, in 10 years, Traditional will be 82 x 3 = 216 years old.\nThe oldest of the daughters will be 216 - 10 = 206 years old in 10 years.\nThis fact can never have happened before and it can never happen again.\nb) \nThe oldest of the daughters is over 30 years old...\n#### 30", "correct": false, "reward": -1, "repetition": {"words": 137, "unique_ratio": 0.4306569343065693, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution\n\na) It is clear that Alina is older than Valentina. Let's call the difference in age between Alina and Valentina $d$. Since they were born on the same day of the year, $d$ does not vary (note that this would not be true if they were born on different days, as in that case, the difference would change on the birthday of each one).\n\nIf Valentina's age today is $\\alpha$, then Alina's age must be $2 \\alpha$, so the age difference between the two is given by:\n\n$$\nd=2 \\alpha-\\alpha\n$$\n\nWe conclude that $d=\\alpha$, so Alina is $2 d$ years old and Valentina is $d$ years old.\n\nSuch a relationship between the ages of Alina and Valentina could not have happened before, as Valentina was younger than $d$ years old. It also will not happen again, as Valentina will be older than $d$ years.\n\nb) Let's call the age of the youngest daughter $A$, the age of the middle daughter $B$, and the age of the oldest daughter $C$. Let $x=B-A$ be the age difference between the two youngest, and $y=C-B$ be the age difference between the two oldest daughters. The age difference between the youngest and the oldest is therefore $C-A=x+y$. Since they were born on the same day of the year, these differences do not vary.\n\nIt is known that in each of the years 2004, 2014, and 2024, the age of one of the daughters was, is, or will be double the age of one of the others. For this, there are only 3 possibilities:\n\n(i) $B$ is double $A$.\n\n(ii) $C$ is double $B$.\n\n(iii) $C$ is double $A$.\n\nBy the reasoning used in part a), we know that each situation can only occur in one of the three years. Therefore, the three situations: (i), (ii), and (iii) must happen in the three years: 2004, 2014, and 2024, but we do not yet know in what order. Also, from part a), we know that:\n\n- When (i) happens, $B$ must be equal to $2 x$ and $A$ must be equal to $x$. Moreover, $C$ is equal to $B+y$, that is, $C$ must be equal to $2 x+y$.\n- When (ii) happens, $B=y, C=2 y$, and therefore, $A=y-x$.\n- When (iii) happens, $C=2(x+y), A=x+y$, and therefore, $B=2 x+y$.\n\nObserve that the ages of $A$ in cases (i), (ii), and (iii) are\n\n$$\nx, \\quad y-x \\quad \\text { and } \\quad x+y\n$$\n\nrespectively. Since $x<x+y$ and $y-x<x+y$, we conclude that $x+y$ is the largest of the three ages and, therefore, (iii) must happen in the year 2024. Only remains to see in which of the two years 2004 or 2014 each of the events in (i) and (ii) happens. Let's assume that (i) and (ii) happen in the years 2014 and 2004, respectively. Then, the ages of $A$ in the years 2004, 2014, and 2024 would be $y-x, x$, and $x+y$, respectively. Thus, we would have the equations\n\n$$\ny-x+10=x \\quad \\text { and } \\quad x+10=x+y\n$$\n\nSolving, we would have $x=10$ and $y=10$. But in that case, $C$ would be today equal to $2 x+y=30$. But the oldest daughter is more than 30 years old, as stated in the problem. We discard, then, this possibility.\n\nWe know then that (i) and (ii) happen in the years 2004 and 2014, respectively. Then, the ages of $A$ in the years 2004, 2014, and 2024 are $x, y-x$, and $x+y$, respectively. Therefore, we have that\n\n$$\nx+10=y-x \\quad \\text { and } \\quad y-x+10=x+y .\n$$\n\nSolving, we get $x=5$ and $y=20$. Finally, the ages of Francisco's daughters today are equal to\n\n$$\ny-x=15, \\quad y=20 \\quad \\text { and } \\quad 2 y=40\n$$\n\nSince Alina's age must be double Valentina's age, the ages of Alina and Valentina must be 40 and 20, respectively. Therefore, Civela is 15 years old today.", "correct": true, "reward": 1.0, "repetition": {"words": 683, "unique_ratio": 0.22108345534407028, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_016", "row": {"problem": "4. Let the sequence $\\left\\{a_{n}\\right\\}$ satisfy $a_{1}=0, a_{2}=1$, and for all $n \\geqslant 3, a_{n}$ is the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \\cdots, a_{n}$ that forms an arithmetic sequence. Find $a_{2014}$.", "solution": "4. First, prove a lemma using mathematical induction.\n\nLemma A non-negative integer appears in the sequence if and only if its ternary expansion contains only 0 and 1.\nProof It is obvious that the proposition holds for 0.\nAssume the proposition holds for all non-negative integers less than \\( N \\), and consider \\( N \\).\n\nIf the ternary expansion of \\( N \\) contains the digit 2, replace all occurrences of 2 with 0 to get the number \\( N_{0} \\); replace all occurrences of 2 with 1 to get the number \\( N_{1} \\). Thus, the ternary expansions of \\( N_{0} \\) and \\( N_{1} \\) do not contain the digit 2.\nBy the induction hypothesis, the numbers \\( N_{0} \\) and \\( N_{1} \\) are in the sequence.\nSince \\( N_{0} \\), \\( N_{1} \\), and \\( N \\) form an arithmetic sequence, \\( N \\) does not appear in the sequence. If the ternary expansion of \\( N \\) does not contain the digit 2, then we need to prove that \\( N \\) must appear in the sequence. If not, then there exist terms \\( N_{0} \\) and \\( N_{1} \\) in the sequence such that \\( N_{0} \\), \\( N_{1} \\), and \\( N \\) form an arithmetic sequence. Let the common difference be \\( d \\), and \\( 3^{k} \\| d \\). Then the ternary expansions of these three numbers have the same lowest \\( k-1 \\) digits, and the \\( k \\)-th digit is different for each pair, so one of the numbers must have a 2 in the \\( k \\)-th digit, which contradicts the assumption.\nReturning to the original problem.\nWe only need to find the 2014th non-negative integer whose ternary expansion does not contain the digit 2.\n\nNotice that under this restriction, the ternary carry-over method is equivalent to binary, so we just need to write 2013 (note \\( a_{1}=0 \\)) in binary \\((11111011101)_{2}\\) and then convert it to ternary \\( a_{2014}=88327 \\).", "answer": "88327", "source": "cn_contest", "problem_type": "Combinatorics", "uuid": "27ef0a0a-88ec-50b0-a32a-0ae07e230bed"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n4. Let the sequence $\\left\\{a_{n}\\right\\}$ satisfy $a_{1}=0, a_{2}=1$, and for all $n \\geqslant 3, a_{n}$ is the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \\cdots, a_{n}$ that forms an arithmetic sequence. Find $a_{2014}$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "From the problem, we know that $a_{1}=0$ and $a_{2}=1$.\n\nFor $n \\geqslant 3$, we have the following condition:\n$$a_{n} = a_{1} + (n-2)d$$\n\nwhere $d$ is the common difference.\n\nGiven that $a_{n}$ must be the smallest positive integer greater than $a_{n-1}$ such that there is no subsequence of $a_{1}, a_{2}, \\cdots, a_{n}$ that forms an arithmetic sequence, we know that $d < a_{n-1}$.\n\nTherefore, we have the following inequalities:\n\\begin{align*}\na_{n} &= a_{1} + (n-2)d \\\\\na_{n-1} &= a_{1} + (n-3)d \\\\\na_{n-2} &= a_{1} + (n-4)d\n\\end{align*}\n\nAdding these two inequalities together, we get:\n$$a_{n} + a_{n-2} &= (a_{1} + (n-2)d) + (a_{1} + (n-4)d) \\\\\n&= 2a_{1} + (n-3)d$$\n\nSince $a_{1} = 0$, we have:\n$$2a_{1} + (n-3)d = 0$$\n\nDividing both sides by 2, we get:\n$$a_{1} + \\frac{n-3}{2}d = 0$$\n\nSince $a_{1} = 0$, we have:\n$$0 + \\", "correct": false, "reward": -1, "repetition": {"words": 170, "unique_ratio": 0.34705882352941175, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "Assume, by the method of differences, that $a_{n-1} = b_{n-1} - b_{n-2}$, where $b_{n-1}$ is the largest index such that $a_{n-1} > b_{n-2}$.\n\nThen, $a_{n} = b_{n-1} + b_{n-2}$, and the subsequence $a_{n-1}, a_{n-2}, \\cdots$ is the arithmetic sequence $b_{n-1} - b_{n-2}, b_{n-2} - b_{n-1}, \\cdots$.\n\nTherefore, $a_{n-1} > a_{n-2} > b_{n-3} > b_{n-4} \\cdots$, which contradicts $a_{n} = b_{n-1} + b_{n-2}$.\n\nHence, $a_{n-1} = b_{1} + b_{2} + \\cdots + b_{n-n}$.\n\nSince $a_{1} = 0$, the sequence $\\{a_{n}\\}$ is a geometric sequence with a common ratio of $b_{1} + b_{2} + \\cdots + b_{n-n}$.\n\nWe know that $a_{n-1} = b_{1} + b_{2} + \\cdots + b_{n-n-1}$.\n\nBy the property of geometric sequences, we have:\n\n$$a_{n} = b_{1} + (b_{1} + b_{2} + \\cdots + b_{n-n}) = b_{1} + (b_{1} + (b_{1} +", "correct": false, "reward": -1, "repetition": {"words": 170, "unique_ratio": 0.23529411764705882, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "4. First, prove a lemma using mathematical induction.\n\nLemma A non-negative integer appears in the sequence if and only if its ternary expansion contains only 0 and 1.\nProof It is obvious that the proposition holds for 0.\nAssume the proposition holds for all non-negative integers less than \\( N \\), and consider \\( N \\).\n\nIf the ternary expansion of \\( N \\) contains the digit 2, replace all occurrences of 2 with 0 to get the number \\( N_{0} \\); replace all occurrences of 2 with 1 to get the number \\( N_{1} \\). Thus, the ternary expansions of \\( N_{0} \\) and \\( N_{1} \\) do not contain the digit 2.\nBy the induction hypothesis, the numbers \\( N_{0} \\) and \\( N_{1} \\) are in the sequence.\nSince \\( N_{0} \\), \\( N_{1} \\), and \\( N \\) form an arithmetic sequence, \\( N \\) does not appear in the sequence. If the ternary expansion of \\( N \\) does not contain the digit 2, then we need to prove that \\( N \\) must appear in the sequence. If not, then there exist terms \\( N_{0} \\) and \\( N_{1} \\) in the sequence such that \\( N_{0} \\), \\( N_{1} \\), and \\( N \\) form an arithmetic sequence. Let the common difference be \\( d \\), and \\( 3^{k} \\| d \\). Then the ternary expansions of these three numbers have the same lowest \\( k-1 \\) digits, and the \\( k \\)-th digit is different for each pair, so one of the numbers must have a 2 in the \\( k \\)-th digit, which contradicts the assumption.\nReturning to the original problem.\nWe only need to find the 2014th non-negative integer whose ternary expansion does not contain the digit 2.\n\nNotice that under this restriction, the ternary carry-over method is equivalent to binary, so we just need to write 2013 (note \\( a_{1}=0 \\)) in binary \\((11111011101)_{2}\\) and then convert it to ternary \\( a_{2014}=88327 \\).", "correct": true, "reward": 1.0, "repetition": {"words": 304, "unique_ratio": 0.3815789473684211, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_017", "row": {"problem": "3. It is known that for some positive coprime numbers $m$ and $n$, the numbers $m+1947 n$ and $n+1947 m$ have a common prime divisor $d>9$. Find the smallest possible value of the number $d$ under these conditions.", "solution": "Answer: $d_{\\min }=139$.\n\nFor example,\n\n$$\n\\begin{aligned}\n& m=138, n=1 \\rightarrow 1947 m+n=1947 \\cdot 138+1=139 \\cdot 1933 \\\\\n& m+1947 n=138+1947=139 \\cdot 15\n\\end{aligned}\n$$", "answer": "139", "source": "olympiads", "problem_type": "Number Theory", "uuid": "381718c3-4fbf-5607-ac47-c87e50bb9dcf"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n3. It is known that for some positive coprime numbers $m$ and $n$, the numbers $m+1947 n$ and $n+1947 m$ have a common prime divisor $d>9$. Find the smallest possible value of the number $d$ under these conditions.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To determine the smallest positive integer $d$ for which the two numbers $m+1947n$ and $n+1947m$ share a common prime divisor $d > 9$, we need to examine the properties of the greatest common divisor (GCD) of the two numbers.\n\nLet's express the two numbers in terms of their GCD $d$. We can write:\n\\[ m = d \\times a \\]\n\\[ n = d \\times b \\]\nwhere $a$ and $b$ are integers and $a$ and $b$ are relatively prime (i.e., their GCD is 1).\n\nNow, we can express $m+1947n$ and $n+1947m$ in terms of $a$ and $b$:\n\\[ m+1947n = d(a+b) \\]\n\\[ n+1947m = d(a+b) \\]\n\nSince $a$ and $b$ are relatively prime, we can write $a = 1$ and $b = 1$, as they are relatively prime.\n\nSubstituting these values into the equations, we get:\n\\[ m+1947n = d(1+1) = 2d \\]\n\\[ n+1947m = d(1+1) = d \\]\n\nNow, we can see that $d$ must be equal to either $2$ or $d = 2$.\n\nIf $d = 2$, then $m+1947n$ and $n+1947m$ share a common divisor $2$ with $d=2$, which is $2", "correct": false, "reward": -1, "repetition": {"words": 181, "unique_ratio": 0.3756906077348066, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "1. Let's denote $k$ as the common prime divisor of $m$ and $n$. We can express $m+1947 n$ and $n+1947 m$ in terms of $k$.\n\n\\[ m+1947 n = k(m+n) \\]\n\n\\[ n+1947 m = k(n+m) \\]\n\nSince $m$ and $n$ are coprime, $k$ is a positive integer. Also, since $m$ and $n$ are positive integers, $k$ must also be a positive integer.\n\nWe have the following system of equations:\n\n\\[ m+n = kn \\]\n\n\\[ m+n = kn+km \\]\n\nAdding these two equations, we get:\n\n\\[ 2mn = 2kn + 2k \\cdot kn \\]\n\n\\[ (m+n)^2 = 2kn(k+1) \\]\n\nSince $m+n$ and $n+m$ are both coprime, $k(k+1)$ is a prime number. Therefore, $2kn$ must be a prime number as well.\n\nSince $k$ is a positive integer, $2n$ must also be a prime number. This implies that $n$ is a prime number.\n\nNow, since $m$ and $n$ are coprime, $m$ and $n$ must each be a prime number. Let $m = p$ and $n = p$, where $p$ is a positive integer.\n\nSubstituting $m = p$ and $n = p$ into the first equation, we get:\n\n\\[ p + p", "correct": false, "reward": -1, "repetition": {"words": 184, "unique_ratio": 0.33695652173913043, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Answer: $d_{\\min }=139$.\n\nFor example,\n\n$$\n\\begin{aligned}\n& m=138, n=1 \\rightarrow 1947 m+n=1947 \\cdot 138+1=139 \\cdot 1933 \\\\\n& m+1947 n=138+1947=139 \\cdot 15\n\\end{aligned}\n$$", "correct": true, "reward": 1.0, "repetition": {"words": 33, "unique_ratio": 0.5454545454545454, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_018", "row": {"problem": "6. (10 points) As shown in the figure, the area of square $A B C D$ is 840 square centimeters, $A E=E B, B F=2 F C, D F$ intersects $E C$ at $G$. Then the area of quadrilateral $A E G D$ is square centimeters.", "solution": "510\n【Analysis】This figure easily leads one to think about finding the areas of triangles $BCE$, $CDF$, and $CGF$. The most challenging part is finding the area of $\\triangle CGF$. According to the given conditions, we should draw $GM \\perp BC$ with the intersection point at $M$. This forms two sets of similar triangles corresponding to $\\triangle BCE$ and $\\triangle CDF$. By using the property that the ratio of the areas of similar triangles is equal to the square of the similarity ratio, we can find the area of $\\triangle CGF$. Finally, based on the figure, we can calculate the area of the required shape.\n\n【Solution】Draw $GM \\perp BC$ with the intersection point at $M$,\n$$\n\\begin{array}{l}\n\\therefore \\triangle FMG \\sim \\triangle FCD \\Rightarrow FM: FC = MG: CD, \\\\\n\\because BF = 2FC \\Rightarrow BC = 3FC, \\\\\n\\therefore MG = 3FM, \\\\\n\\because \\triangle CGM: \\triangle CEB \\Rightarrow CM: CB = GM: BE, \\\\\nBC = 2BE, \\\\\n\\therefore GM = \\frac{1}{2} CM = 3FM \\Rightarrow CM = 6FM, \\\\\n\\therefore FM: FC = 1: 7, \\\\\nCM: CB = 2: 7, \\\\\nS \\triangle BCE = \\frac{1}{4} \\times \\square ABCD = 210, \\\\\nS \\triangle CGM = 4 \\div 49 \\times 210 = 840 \\div 49 = 120 \\div 7, \\\\\nS \\triangle CDF = S \\square ABCD \\div 6 = 140, \\\\\nS \\triangle MGF = 140 \\times 1 \\div 49 = 140 \\div 49 = 20 \\div 7, \\\\\nS \\triangle CGM + \\triangle MGF = 120 \\div 7 + 20 \\div 7 = 20, \\\\\n840 - 210 - 140 + 20 = 510 \\text{ (square centimeters). }\n\\end{array}\n$$\n\nTherefore, the area of quadrilateral $AEGD$ is 510 square centimeters.", "answer": "510", "source": "olympiads", "problem_type": "Geometry", "uuid": "2a224068-3c17-5d14-ad3a-426aec3cae2b"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n6. (10 points) As shown in the figure, the area of square $A B C D$ is 840 square centimeters, $A E=E B, B F=2 F C, D F$ intersects $E C$ at $G$. Then the area of quadrilateral $A E G D$ is square centimeters.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To solve this problem, we need to find the length of $AE$ and $ED$ and use them to calculate the area of quadrilateral $A E G D$. We can do this by using the properties of similar triangles and the given values.\n\nLet's denote $AE = x$, $EB = y$, $BC = z$, $DF = w$, $DF$ intersects $AC$ at $G$. By the properties of similar triangles, we have:\n\n$$\\frac{AE}{EB} = \\frac{DF}{BC}$$\n\nSubstitute the given values for $AE$, $EB$, $BC$, and $DF$:\n\n$$\\frac{x}{y} = \\frac{w}{z}$$\n\nNow, we can simplify this equation:\n\n$$xz = yw$$\n\nWe also know that $x = \\frac{2}{3}y$, $y = \\frac{1}{2}z$, and $w = \\frac{3}{4}z$.\n\nSubstitute these values into the equation:\n\n$$\\frac{2}{3}y\\cdot\\frac{3}{4}z = y\\cdot\\frac{1}{2}z\\cdot\\frac{3}{4}z$$\n\nSimplify this equation:\n\n$$\\frac{2}{3}\\cdot\\frac{3}{4}z\\cdot\\frac{3}{4}z = y\\cdot\\frac{y}{2}z$$\n\n$z\\cdot z = y^2$\n\nSince the area of square $ABCDE$ is $840$ square centimeters, we can write the area of quadrilateral $A E G D$ as:\n\n$840 = x^2", "correct": false, "reward": -1, "repetition": {"words": 193, "unique_ratio": 0.3626943005181347, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "Let $x=AE$. Then $BE=x-10$. Since $BE=200$, we have $x=200+10=210$. Therefore, $DE=210-x=210-210=0$. The area of quadrilateral $A E G D$ is the sum of the areas of square $A B C D$ and triangles $A B E$ and $A E G D$ and quarter-circle $F C E$. We have $[A B C D] + [A B E] + [A AE G] + [A DE G] + [F C E] = 840 + [A E G D] + 1900 + 3400 + [210*(10-210)/2].$ Therefore, the area of the region is $x^2=10000-3318000=14200$. So, the area of quadrilateral $A E G D$ is $\\boxed{14200}$.", "correct": false, "reward": -1, "repetition": {"words": 112, "unique_ratio": 0.38392857142857145, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "510\n【Analysis】This figure easily leads one to think about finding the areas of triangles $BCE$, $CDF$, and $CGF$. The most challenging part is finding the area of $\\triangle CGF$. According to the given conditions, we should draw $GM \\perp BC$ with the intersection point at $M$. This forms two sets of similar triangles corresponding to $\\triangle BCE$ and $\\triangle CDF$. By using the property that the ratio of the areas of similar triangles is equal to the square of the similarity ratio, we can find the area of $\\triangle CGF$. Finally, based on the figure, we can calculate the area of the required shape.\n\n【Solution】Draw $GM \\perp BC$ with the intersection point at $M$,\n$$\n\\begin{array}{l}\n\\therefore \\triangle FMG \\sim \\triangle FCD \\Rightarrow FM: FC = MG: CD, \\\\\n\\because BF = 2FC \\Rightarrow BC = 3FC, \\\\\n\\therefore MG = 3FM, \\\\\n\\because \\triangle CGM: \\triangle CEB \\Rightarrow CM: CB = GM: BE, \\\\\nBC = 2BE, \\\\\n\\therefore GM = \\frac{1}{2} CM = 3FM \\Rightarrow CM = 6FM, \\\\\n\\therefore FM: FC = 1: 7, \\\\\nCM: CB = 2: 7, \\\\\nS \\triangle BCE = \\frac{1}{4} \\times \\square ABCD = 210, \\\\\nS \\triangle CGM = 4 \\div 49 \\times 210 = 840 \\div 49 = 120 \\div 7, \\\\\nS \\triangle CDF = S \\square ABCD \\div 6 = 140, \\\\\nS \\triangle MGF = 140 \\times 1 \\div 49 = 140 \\div 49 = 20 \\div 7, \\\\\nS \\triangle CGM + \\triangle MGF = 120 \\div 7 + 20 \\div 7 = 20, \\\\\n840 - 210 - 140 + 20 = 510 \\text{ (square centimeters). }\n\\end{array}\n$$\n\nTherefore, the area of quadrilateral $AEGD$ is 510 square centimeters.", "correct": true, "reward": 1.0, "repetition": {"words": 249, "unique_ratio": 0.42971887550200805, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_019", "row": {"problem": "1. In tetrahedron $ABCD$, let $AB=1$, $CD=\\sqrt{3}$, the distance between lines $AB$ and $CD$ is 2, and the angle between them is $\\frac{\\pi}{3}$. Then the volume of tetrahedron $ABCD$ is ( ).\n(2003 National High School League Question)\nA. $\\frac{\\sqrt{3}}{2}$\nB. $\\frac{1}{2}$\nC. $\\frac{1}{3}$\nD. $\\frac{\\sqrt{3}}{3}$", "solution": "1. Choose B. Reason: The area of the parallelogram formed by $\\overrightarrow{A B}$ and $\\overrightarrow{C D}$ is $|\\overrightarrow{A B}| \\cdot|\\overrightarrow{C D}| \\cdot \\sin 60^{\\circ}=\\frac{3}{2}$. If a line $C E \\mathbb{E}$ is drawn through point $C$ parallel to $A B$, then $S_{\\triangle C D E}=\\frac{1}{2} \\cdot \\frac{3}{2}=\\frac{3}{4}$, and the volume of the prism $A B F-E C D$ with $\\triangle C D E$ as the base and $B C$ as the side edge is 3 times that of $V_{A B C D}$, and $V_{A B F-E C D}=S_{\\triangle C D E} \\cdot 2=\\frac{3}{2}$, hence $V_{A B C D}=\\frac{1}{2}$.", "answer": "\\frac{1}{2}", "source": "olympiads", "problem_type": "Geometry", "uuid": "33b39e35-cec5-5faf-a661-1defa78a43a2"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n1. In tetrahedron $ABCD$, let $AB=1$, $CD=\\sqrt{3}$, the distance between lines $AB$ and $CD$ is 2, and the angle between them is $\\frac{\\pi}{3}$. Then the volume of tetrahedron $ABCD$ is ( ).\n(2003 National High School League Question)\nA. $\\frac{\\sqrt{3}}{2}$\nB. $\\frac{1}{2}$\nC. $\\frac{1}{3}$\nD. $\\frac{\\sqrt{3}}{3}$", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The volume of a tetrahedron is given by the formula $V = \\frac{1}{3} \\times \\text{area of base} \\times \\text{height}$.\n\nSince $ABCD$ is a triangle, the height is the distance from the vertex $D$ to the line $CD$. We can find the area of triangle $ABC$ using the formula for the area of a triangle: $\\frac{1}{2} \\times \\text{base} \\times \\text{height}$.\n\nSince the distance from $D$ to $CD$ is 2, the base of the triangle is 2. Therefore, the area of triangle $ABC$ is:\n$$\\frac{1}{2} \\times 2 \\times 2 = 2$$\n\nNow, we need to find the volume of the tetrahedron. We can do this by multiplying the area of triangle $ABC$ by the height from $D$ to line $CD$.\n\n$$V = \\frac{1}{3} \\times 2 \\times 2 = \\frac{4}{3}$$\n\nTherefore, the volume of tetrahedron $ABCD$ is $P = \\frac{4}{3}$.\n\nThe answer is:\n\\[ \\boxed{\\frac{4}{3}} \\]\nFinal answer:\n#### \\frac{4}{3}", "correct": false, "reward": -1, "repetition": {"words": 160, "unique_ratio": 0.2875, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "1. Let $P$ be the foot of the perpendicular dropped from point $A$ to the plane of tetrahedron $ABCD$. Since the height of the tetrahedron from $A$ to plane $ABCD$ is the perpendicular distance between $AB$ and the plane of $ABCD$, we have $AP = AB = 1$.\n2. Now, let $|AP| = 1$, $|BP| = 2$, and $|CD| = \\sqrt{3}$. Then, by the Pythagorean theorem, we have $|BP| = \\sqrt{|BP|^2 + |BP|^2} = \\sqrt{4} = 2$.\n3. The volume of a tetrahedron is given by the formula $V = \\frac{1}{3} \\times \\text{area of base} \\times \\text{height}$. Here, the area of the base is the area of triangle $ABP$, and the height is the distance from $P$ to the plane $ABCD$.\n4. Therefore, the volume of tetrahedron $ABCD$ is $V = \\frac{1}{3} \\times 1 \\times 1 \\times \\frac{\\sqrt{3}}{2} = \\boxed{\\frac{\\sqrt{3}}{6}}$.\nFinal answer:\n#### \\frac{\\sqrt{3}}{6}", "correct": false, "reward": -1, "repetition": {"words": 152, "unique_ratio": 0.35526315789473684, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "1. Choose B. Reason: The area of the parallelogram formed by $\\overrightarrow{A B}$ and $\\overrightarrow{C D}$ is $|\\overrightarrow{A B}| \\cdot|\\overrightarrow{C D}| \\cdot \\sin 60^{\\circ}=\\frac{3}{2}$. If a line $C E \\mathbb{E}$ is drawn through point $C$ parallel to $A B$, then $S_{\\triangle C D E}=\\frac{1}{2} \\cdot \\frac{3}{2}=\\frac{3}{4}$, and the volume of the prism $A B F-E C D$ with $\\triangle C D E$ as the base and $B C$ as the side edge is 3 times that of $V_{A B C D}$, and $V_{A B F-E C D}=S_{\\triangle C D E} \\cdot 2=\\frac{3}{2}$, hence $V_{A B C D}=\\frac{1}{2}$.", "correct": true, "reward": 1.0, "repetition": {"words": 129, "unique_ratio": 0.37209302325581395, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_020", "row": {"problem": "3. (6 points) A construction company was building a tunnel. When $\\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \\%$ and reduced the working hours to $80 \\%$ of the original. As a result, it took a total of 185 days to complete the tunnel. If they had not used the new equipment and continued at the original speed, it would have taken $\\qquad$ days to complete the tunnel.", "solution": "3. (6 points) A construction company builds a tunnel. When $\\frac{1}{3}$ of the tunnel is completed at the original speed, new equipment is used, increasing the construction speed by $20 \\%$ and reducing the daily working hours to $80 \\%$ of the original. As a result, the tunnel is completed in 185 days. If the new equipment was not used and the original speed was maintained, it would take $\\qquad$ 180 days.\n\n【Solution】Solution: $\\left(1-\\frac{1}{3}\\right) \\div[(1+20 \\%) \\times 80 \\%]$\n$$\n\\begin{array}{l}\n=\\frac{2}{3} \\div[120 \\% \\times 80 \\%], \\\\\n=\\frac{2}{3} \\div \\frac{24}{25} \\\\\n=\\frac{25}{36} ; \\\\\n185 \\div\\left(\\frac{1}{3}+\\frac{25}{36}\\right) \\\\\n=185 \\div \\frac{37}{36}, \\\\\n=180 \\text { (days). }\n\\end{array}\n$$\n\nAnswer: If the original speed was maintained, it would take 180 days.\nTherefore, the answer is: 180.", "answer": "180", "source": "olympiads", "problem_type": "Algebra", "uuid": "1d57f4b3-c516-5522-91cc-a60dd91f6f66"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n3. (6 points) A construction company was building a tunnel. When $\\frac{1}{3}$ of the tunnel was completed at the original speed, they started using new equipment, which increased the construction speed by $20 \\%$ and reduced the working hours to $80 \\%$ of the original. As a result, it took a total of 185 days to complete the tunnel. If they had not used the new equipment and continued at the original speed, it would have taken $\\qquad$ days to complete the tunnel.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Let $x$ be the original speed of the tunnel. When $\\frac{1}{3}$ of the tunnel was completed at the original speed, they would have used $5x$ days to complete the tunnel. When they used the new equipment and increased the speed of the tunnel by $20\\%$ and reduced the working hours to $80\\%$ of the original speed, they would have used $1.2x$ days to complete the tunnel.\n\nIt is given that the total time taken is 185 days, so we have the equation $5x - 1.2x = 185$. Solving this equation, we get $8.8x = 185$, which gives $x = 22.25$.\n\nTherefore, if they had not used the new equipment and continued at the original speed, it would have taken a total of 109 days to complete the tunnel.\n\nFinal answer:\n#### 109", "correct": false, "reward": -1, "repetition": {"words": 133, "unique_ratio": 0.47368421052631576, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 1, "origin": "budgie", "answer": "The original speed was $150\\%$ of the tunnel in $150^{\\text{days}}$;\nWhen $\\frac{1}{3}$ of the tunnel was completed at the original speed, it took a total of $150^{\\text{days}} + 80^{\\text{days}} = 230^{\\text{days}}$;\nIf they had not used the new equipment, the construction speed would have been $\\frac{1}{2}$ of the original speed in $150^{\\text{days} + 20^{\\text{days}}} = 180^{\\text{days}}$;\nIn total, it would have taken $150^{\\text{days}} + 230^{\\text{days}} = 380^{\\text{days}}$\n\nFinal answer:\n#### 380", "correct": false, "reward": -0.5, "repetition": {"words": 88, "unique_ratio": 0.45454545454545453, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 2, "origin": "verified_reference", "answer": "3. (6 points) A construction company builds a tunnel. When $\\frac{1}{3}$ of the tunnel is completed at the original speed, new equipment is used, increasing the construction speed by $20 \\%$ and reducing the daily working hours to $80 \\%$ of the original. As a result, the tunnel is completed in 185 days. If the new equipment was not used and the original speed was maintained, it would take $\\qquad$ 180 days.\n\n【Solution】Solution: $\\left(1-\\frac{1}{3}\\right) \\div[(1+20 \\%) \\times 80 \\%]$\n$$\n\\begin{array}{l}\n=\\frac{2}{3} \\div[120 \\% \\times 80 \\%], \\\\\n=\\frac{2}{3} \\div \\frac{24}{25} \\\\\n=\\frac{25}{36} ; \\\\\n185 \\div\\left(\\frac{1}{3}+\\frac{25}{36}\\right) \\\\\n=185 \\div \\frac{37}{36}, \\\\\n=180 \\text { (days). }\n\\end{array}\n$$\n\nAnswer: If the original speed was maintained, it would take 180 days.\nTherefore, the answer is: 180.", "correct": true, "reward": 1.0, "repetition": {"words": 142, "unique_ratio": 0.44366197183098594, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_dev"}
+{"id": "math_021", "row": {"problem": "Exercise 8. The price (in euros) of a diamond corresponds to its mass (in grams) squared and then multiplied by 100. The price (in euros) of a crystal corresponds to three times its mass (in grams). Martin and Théodore unearth a treasure composed of precious stones which are either diamonds or crystals, and the total value is $5000000 €$. They cut each precious stone in half, and each takes one half of each stone. The total value of Martin's stones is $2000000 €$. In euros, what was the total initial value of the diamonds contained in the treasure?\n\nOnly a numerical answer is expected here.", "solution": "Solution to Exercise 8: Let $r$ be the number of stones that are diamonds, and let $d_{1}, \\ldots, d_{r}$ be the respective masses of the diamonds unearthed. We also denote $c$ as the mass of crystal unearthed. We aim to determine the value of $100\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)$.\n\nSince the total value of the precious stones unearthed is 5000000, we have\n\n$$\n100\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)+3 c=5000000\n$$\n\nSince Martin receives half of the total mass of diamonds and half of the total mass of crystal, and the value of Martin's share is 2000000, we also have\n\n$$\n100\\left[\\left(\\frac{d_{1}}{2}\\right)^{2}+\\ldots+\\left(\\frac{d_{r}}{2}\\right)^{2}\\right]+3 \\cdot \\frac{c}{2}=2000000\n$$\n\nBy subtracting twice (2) from (1), we get\n\n$$\n100\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)-2 \\times 25\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)+3 c-2 \\times 3 \\cdot \\frac{c}{2}=5000000-2 \\times 2000000\n$$\n\nwhich simplifies to $50\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)=1000000$. Therefore, $100\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)=2000000$, and the initial total value of the diamonds in the treasure is 2000000.\n\nGrader Comments: The exercise was well done by the majority. The mistakes made are mainly due to poor modeling of the problem statement. Several students simply wrote a numerically incorrect result and thus received no points for this first exercise, which was the simplest in the high school subject. Even if only a numerical answer is expected, it is recommended to write down the reasoning to earn some points.", "answer": "2000000", "source": "olympiads", "problem_type": "Algebra", "uuid": "d1ae3c87-5993-573c-891f-1453c3724013"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nExercise 8. The price (in euros) of a diamond corresponds to its mass (in grams) squared and then multiplied by 100. The price (in euros) of a crystal corresponds to three times its mass (in grams). Martin and Théodore unearth a treasure composed of precious stones which are either diamonds or crystals, and the total value is $5000000 €$. They cut each precious stone in half, and each takes one half of each stone. The total value of Martin's stones is $2000000 €$. In euros, what was the total initial value of the diamonds contained in the treasure?\n\nOnly a numerical answer is expected here.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Let's denote the masses of the diamonds in the treasure as $x$ and the masses of the crystals as $y$. We know that $x + y = 5000000", "correct": false, "reward": -0.5, "repetition": {"words": 26, "unique_ratio": 0.6538461538461539, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": -1.4142115623759233}, {"slot": 1, "origin": "budgie", "answer": "The price of a diamond corresponds to its mass (in grams) squared and then multiplied by 100.\n\nThe price of a crystal corresponds to three times its mass (in grams).\n\nThe total value of Martin's stones is 2000000", "correct": true, "reward": 1.0, "repetition": {"words": 38, "unique_ratio": 0.6842105263157895, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 0.7071057811879616}, {"slot": 2, "origin": "verified_reference", "answer": "Solution to Exercise 8: Let $r$ be the number of stones that are diamonds, and let $d_{1}, \\ldots, d_{r}$ be the respective masses of the diamonds unearthed. We also denote $c$ as the mass of crystal unearthed. We aim to determine the value of $100\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)$.\n\nSince the total value of the precious stones unearthed is 5000000, we have\n\n$$\n100\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)+3 c=5000000\n$$\n\nSince Martin receives half of the total mass of diamonds and half of the total mass of crystal, and the value of Martin's share is 2000000, we also have\n\n$$\n100\\left[\\left(\\frac{d_{1}}{2}\\right)^{2}+\\ldots+\\left(\\frac{d_{r}}{2}\\right)^{2}\\right]+3 \\cdot \\frac{c}{2}=2000000\n$$\n\nBy subtracting twice (2) from (1), we get\n\n$$\n100\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)-2 \\times 25\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)+3 c-2 \\times 3 \\cdot \\frac{c}{2}=5000000-2 \\times 2000000\n$$\n\nwhich simplifies to $50\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)=1000000$. Therefore, $100\\left(d_{1}^{2}+\\ldots+d_{r}^{2}\\right)=2000000$, and the initial total value of the diamonds in the treasure is 2000000.\n\nGrader Comments: The exercise was well done by the majority. The mistakes made are mainly due to poor modeling of the problem statement. Several students simply wrote a numerically incorrect result and thus received no points for this first exercise, which was the simplest in the high school subject. Even if only a numerical answer is expected, it is recommended to write down the reasoning to earn some points.", "correct": true, "reward": 1.0, "repetition": {"words": 286, "unique_ratio": 0.3881118881118881, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": 0.7071057811879616}], "split": "math_dev"}
+{"id": "math_022", "row": {"problem": "5. 5.1. Find the largest natural number in which all digits are different and any two adjacent digits differ by 6 or 7.", "solution": "Answer: 60718293.\n\n## Solution.\n\nWe will map each digit from 0 to 9 to a vertex in a graph and connect the vertices with an edge if the corresponding digits differ by 6 or 7.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_8a1d5b347fbc679ac0eeg-5.jpg?height=448&width=590&top_left_y=524&top_left_x=819)\n\nWe see that the vertices corresponding to the digits 4 and 5 are not connected to anything, so the maximum number that can be obtained will be 8 digits long. The vertices corresponding to the digits 3 and 6 can only be the first or last in the number. Since we need to find the maximum number, it is uniquely determined by moving along the edges of the graph and will be equal to 60718293.", "answer": "60718293", "source": "olympiads", "problem_type": "Number Theory", "uuid": "7edacb04-aea8-5a67-b4b4-4a1930dcb4df"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n5. 5.1. Find the largest natural number in which all digits are different and any two adjacent digits differ by 6 or 7.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Okay, let me think about this. I need to find the largest natural number where all digits are different and adjacent digits differ by 6 or 7. Hmm. So, the idea is that this number has to have as many digits as possible, with all digits unique and the adjacency condition being 6 or 7. \n\nAlright, so first, let's understand the problem. We need a number that uses digits, with all different and adjacent digits differing by 6 or 7. Let's denote the digits as a, b, c, d, e, where a ≠ b, a ≠ c, a ≠ d, b ≠ c, b ≠ d, c ≠ d, and all digits are different. The question is asking for the maximum possible number under these constraints.\n\nSo, the problem is about constructing such a number with as many digits as possible, with no two adjacent digits differing by 6 or 7. To maximize the number, we want to use as many 6s as possible and 7s as much as possible. However, we have a restriction on the digits: they can't repeat digits. So, we need to find a way to arrange the digits while keeping the adjacency rule and all digits different.\n\nLet me think about how to approach this. One way to maximize the number is to use as many 6s as possible, but with some remainder. For example, if we have two 6s next to each other, we can have 6s with one digit in between, but then the digits would have to be different. However, the next digit should have to be different from both 6s. But if we have two 6s next to each other, the digits would have to be the same, which is not allowed.\n\nAlternatively, maybe we", "correct": false, "reward": -1, "repetition": {"words": 288, "unique_ratio": 0.3819444444444444, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To find the largest natural number where all digits are different and any two adjacent digits differ by either 6 or 7, we can approach this by considering the constraints imposed by the digits and the divisibility rules.\n\nA natural number has the form $abcdefg$, where each digit is a single digit from 0 to 9. We need to satisfy the following conditions:\n1. All digits are different.\n2. Any two adjacent digits differ by either 6 or 7.\n3. The number is as large as possible.\n\nConsidering the constraints, we can start by finding the largest possible number with the given constraints. Since the digits must all be different, the largest possible number would be a six-digit number with the first digit being 9.\n\nLet's consider the constraints again:\n\n1. The first digit cannot be 0, so it has 9 choices (1 through 9).\n2. The second digit cannot be the same as the first digit, so it has 8 choices (since one digit is already used).\n3. The third digit cannot be the same as the second digit, so it has 7 choices (since one digit is already used).\n4. The fourth digit cannot be the same as the third digit, so it has 6 choices (since two digits are already used).\n5. The fifth digit cannot be the same as the fourth digit, so it has 5 choices (since one digit is already used).\n6. The sixth digit cannot be the same as the fifth digit, so it has 4 choices (since two digits are already used).\n\nWe can initially iterate through all possible combinations of the digits and then check if they satisfy the conditions.\n\nHere is a starting point:\n\nFor the first digit, we have 9 choices.\nFor the", "correct": false, "reward": -1, "repetition": {"words": 295, "unique_ratio": 0.3220338983050847, "max_4gram_count": 5, "max_sentence_count": 2, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Answer: 60718293.\n\n## Solution.\n\nWe will map each digit from 0 to 9 to a vertex in a graph and connect the vertices with an edge if the corresponding digits differ by 6 or 7.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_8a1d5b347fbc679ac0eeg-5.jpg?height=448&width=590&top_left_y=524&top_left_x=819)\n\nWe see that the vertices corresponding to the digits 4 and 5 are not connected to anything, so the maximum number that can be obtained will be 8 digits long. The vertices corresponding to the digits 3 and 6 can only be the first or last in the number. Since we need to find the maximum number, it is uniquely determined by moving along the edges of the graph and will be equal to 60718293.", "correct": true, "reward": 1.0, "repetition": {"words": 132, "unique_ratio": 0.6287878787878788, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_023", "row": {"problem": "Example 12 Given $a, b, c, d \\in \\mathbf{R}^{+}$, try to find\n$f(a, b, c, d)=\\frac{a}{b+c+d}+\\frac{b}{c+d+a}+\\frac{c}{a+b+d}+\\frac{d}{a+b+c}$ the minimum value.", "solution": "Solution 1 Since directly eliminating the denominator is complicated, consider changing the problem.\nLet $b+c+d=A, c+d+a=B, d+a+b=C, a+b+c=D$.\nAdding these four equations, we get $a+b+c+d=\\frac{1}{3}(A+B+C+D)$.\nThus, we can find that $a=\\frac{1}{3}(B+C+D-2 A), b=\\frac{1}{3}(C+D+A-2 B)$,\n$$\nc=\\frac{1}{3}(D+A+B-2 C), d=\\frac{1}{3}(A+B+C-2 D).\n$$\n\nTherefore, the original expression $=\\frac{B+C+D-2 A}{3 A}+\\frac{C+D+A-2 B}{3 B}+\\frac{D+A+B-2 C}{3 C}+$\n$$\n\\frac{A+B+C-2 D}{3 D},\n$$\n\nwhich is $3 \\cdot f(a, b, c, d)+12$\n$$\n\\begin{array}{l}\n=\\frac{A+B+C+D}{A}+\\frac{A+B+C+D}{B}+\\frac{A+B+C+D}{C}+\\frac{A+B+C+D}{D} \\\\\n=(A+B+C+D) \\cdot\\left(\\frac{1}{A}+\\frac{1}{B}+\\frac{1}{C}+\\frac{1}{D}\\right) \\geqslant 16,\n\\end{array}\n$$\n\ni.e., $3 \\cdot f(a, b, c, d)+12 \\geqslant 16$, hence $f(a, b, c, d) \\geqslant \\frac{4}{3}$, where equality holds if and only if $A=$ $B=C=D$, i.e., $a=b=c=d$.\nTherefore, the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.\nSolution 2 Noting the weighted sum inequality, we have\n$$\n\\begin{aligned}\nf(a, b, c, d) & =\\frac{a^{2}}{a(b+c+d)}+\\frac{b^{2}}{b(a+c+d)}+\\frac{c^{2}}{c(a+b+d)}+\\frac{d^{2}}{d(a+b+c)} \\\\\n& \\geqslant \\frac{(a+b+c+d)^{2}}{2(a b+a c+a d+b c+b d+c d)} \\\\\n& =\\frac{(a+b+c+d)^{2}}{(a+b+c+d)^{2}-\\left(a^{2}+b^{2}+c^{2}+d^{2}\\right)}\n\\end{aligned}\n$$\n\nand $a^{2}+b^{2}+c^{2}+d^{2}=\\frac{a^{2}}{1}+\\frac{b^{2}}{1}+\\frac{c^{2}}{1}+\\frac{d^{2}}{1} \\geqslant \\frac{(a+b+c+d)^{2}}{4}$, thus\n$$\nf(a, b, c, d) \\geqslant \\frac{(a+b+c+d)^{2}}{(a+b+c+d)^{2}-\\frac{1}{4}(a+b+c+d)^{2}}=\\frac{4}{3} .\n$$\n\nwhere equality holds if and only if $a=b=c=d$, hence the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.\nSolution 3 Noting the Cauchy inequality or the weighted sum inequality, we have\n$$\n\\begin{aligned}\nf(a, b, c, d) & =\\frac{a+b+c+d}{b+c+d}+\\frac{a+b+c+d}{a+c+d}+\\frac{a+b+c+d}{a+b+d}+ \\\\\n& \\frac{a+b+c+d}{a+b+c}-4 \\\\\n& \\geqslant(a+b+c+d)\\left(\\frac{1}{b+c+d}+\\frac{1}{a+c+d}+\\frac{1}{a+b+d}+\\right. \\\\\n& \\left.\\frac{1}{a+b+c}\\right)-4 \\\\\n& \\geqslant(a+b+c+d) \\cdot \\frac{4^{2}}{3(a+b+c+d)}-4=\\frac{4}{3} .\n\\end{aligned}\n$$\n\nwhere equality holds if and only if $a=b=c=d$, hence the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.", "answer": "\\frac{4}{3}", "source": "olympiads", "problem_type": "Inequalities", "uuid": "10813f94-5443-58ca-9c8c-20d9f8a70b52"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nExample 12 Given $a, b, c, d \\in \\mathbf{R}^{+}$, try to find\n$f(a, b, c, d)=\\frac{a}{b+c+d}+\\frac{b}{c+d+a}+\\frac{c}{a+b+d}+\\frac{d}{a+b+c}$ the minimum value.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To determine the minimum value of $f(a, b, c, d)$, let's first consider each fraction in the expression $\\frac{a}{b+c+d} + \\frac{b}{c+d+a} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c}$.\n\nNotice that each fraction can be rewritten as:\n\n$$f(a, b, c, d) = \\frac{a}{b+c+d} + \\frac{b}{c+d+a} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c}$$\n\nSince the denominators are the same, we can combine the numerators:\n\n$$f(a, b, c, d) = \\frac{a}{b+c+d} + \\frac{b}{c+d+a} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c}$$\n\nNow, let's simplify the expression by finding a common denominator. Since the denominators are the same, the common denominator is the product of the denominators.\n\n$$f(a, b, c, d) = \\frac{a}{b+c+d} + \\frac{b}{c+d+a} + \\frac{c}{a+b+d} + \\frac{d}{a+b+c} = \\frac{ad+bc+ab}{(b+c+d)(a+b+d)}$$\n\nNow, let's simplify the numerator. Notice that $(b+c+d)(a+b+d) = ab + bc + cd + ad + bd + a", "correct": false, "reward": -1, "repetition": {"words": 187, "unique_ratio": 0.24598930481283424, "max_4gram_count": 8, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To find the minimum value of the given expression, we can use the method of Lagrange multipliers or recognize it as a known optimization problem.\n\nLet $f(x, y, z) = \\frac{x}{y+z+y} + \\frac{y}{z+x+z} + \\frac{z}{x+y+z}$, and $g(x, y, z) = \\frac{x}{y+z} + \\frac{y}{x+z} + \\frac{z}{x+y} + \\frac{w}{x+w}$, where $x, y, z, w > 0$ with $w > 0$.\n\nWe want to minimize $f(x, y, z, x) + g(x, y, z, y)$.\n\nSince $x, y, z, w > 0$, we can set $w = x + y + z$.\n\nNow, we have:\n\n$$f(x, y, z) = \\frac{x}{y+z+y} + \\frac{y}{z+x+z} + \\frac{z}{x+y+z} + \\frac{x+y+z}{x+y+z}$$\n\n$$g(x, y, z) = \\frac{x}{y+z} + \\frac{y}{x+z} + \\frac{z}{x+y} + \\frac{x+y+z}{x+y+z}$$\n\nTo minimize $f(x, y, z, x) + g(x, y, z, y)$, we want to minimize $f(x, y, z, x)$ and $g(x, y,", "correct": false, "reward": -1, "repetition": {"words": 179, "unique_ratio": 0.22346368715083798, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution 1 Since directly eliminating the denominator is complicated, consider changing the problem.\nLet $b+c+d=A, c+d+a=B, d+a+b=C, a+b+c=D$.\nAdding these four equations, we get $a+b+c+d=\\frac{1}{3}(A+B+C+D)$.\nThus, we can find that $a=\\frac{1}{3}(B+C+D-2 A), b=\\frac{1}{3}(C+D+A-2 B)$,\n$$\nc=\\frac{1}{3}(D+A+B-2 C), d=\\frac{1}{3}(A+B+C-2 D).\n$$\n\nTherefore, the original expression $=\\frac{B+C+D-2 A}{3 A}+\\frac{C+D+A-2 B}{3 B}+\\frac{D+A+B-2 C}{3 C}+$\n$$\n\\frac{A+B+C-2 D}{3 D},\n$$\n\nwhich is $3 \\cdot f(a, b, c, d)+12$\n$$\n\\begin{array}{l}\n=\\frac{A+B+C+D}{A}+\\frac{A+B+C+D}{B}+\\frac{A+B+C+D}{C}+\\frac{A+B+C+D}{D} \\\\\n=(A+B+C+D) \\cdot\\left(\\frac{1}{A}+\\frac{1}{B}+\\frac{1}{C}+\\frac{1}{D}\\right) \\geqslant 16,\n\\end{array}\n$$\n\ni.e., $3 \\cdot f(a, b, c, d)+12 \\geqslant 16$, hence $f(a, b, c, d) \\geqslant \\frac{4}{3}$, where equality holds if and only if $A=$ $B=C=D$, i.e., $a=b=c=d$.\nTherefore, the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.\nSolution 2 Noting the weighted sum inequality, we have\n$$\n\\begin{aligned}\nf(a, b, c, d) & =\\frac{a^{2}}{a(b+c+d)}+\\frac{b^{2}}{b(a+c+d)}+\\frac{c^{2}}{c(a+b+d)}+\\frac{d^{2}}{d(a+b+c)} \\\\\n& \\geqslant \\frac{(a+b+c+d)^{2}}{2(a b+a c+a d+b c+b d+c d)} \\\\\n& =\\frac{(a+b+c+d)^{2}}{(a+b+c+d)^{2}-\\left(a^{2}+b^{2}+c^{2}+d^{2}\\right)}\n\\end{aligned}\n$$\n\nand $a^{2}+b^{2}+c^{2}+d^{2}=\\frac{a^{2}}{1}+\\frac{b^{2}}{1}+\\frac{c^{2}}{1}+\\frac{d^{2}}{1} \\geqslant \\frac{(a+b+c+d)^{2}}{4}$, thus\n$$\nf(a, b, c, d) \\geqslant \\frac{(a+b+c+d)^{2}}{(a+b+c+d)^{2}-\\frac{1}{4}(a+b+c+d)^{2}}=\\frac{4}{3} .\n$$\n\nwhere equality holds if and only if $a=b=c=d$, hence the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.\nSolution 3 Noting the Cauchy inequality or the weighted sum inequality, we have\n$$\n\\begin{aligned}\nf(a, b, c, d) & =\\frac{a+b+c+d}{b+c+d}+\\frac{a+b+c+d}{a+c+d}+\\frac{a+b+c+d}{a+b+d}+ \\\\\n& \\frac{a+b+c+d}{a+b+c}-4 \\\\\n& \\geqslant(a+b+c+d)\\left(\\frac{1}{b+c+d}+\\frac{1}{a+c+d}+\\frac{1}{a+b+d}+\\right. \\\\\n& \\left.\\frac{1}{a+b+c}\\right)-4 \\\\\n& \\geqslant(a+b+c+d) \\cdot \\frac{4^{2}}{3(a+b+c+d)}-4=\\frac{4}{3} .\n\\end{aligned}\n$$\n\nwhere equality holds if and only if $a=b=c=d$, hence the minimum value of $f(a, b, c, d)$ is $\\frac{4}{3}$.", "correct": true, "reward": 1.0, "repetition": {"words": 539, "unique_ratio": 0.12244897959183673, "max_4gram_count": 36, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_024", "row": {"problem": "1. (6 points) Today is January 31, 2015, welcome to the 2015 \"Spring Cup\" Final. The calculation result of the expression $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$ is", "solution": "【Solution】Solve: $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$\n$$\n\\begin{array}{l}\n=\\frac{\\left(\\frac{2015}{1}+\\frac{2015}{0.31}\\right) \\times 0.31}{(1+0.31) \\times 0.31} \\\\\n=\\frac{(1+0.31) \\times 2015}{(1+0.31) \\times 0.31} \\\\\n=6500 ;\n\\end{array}\n$$\n\nTherefore, the answer is: 6500.", "answer": "6500", "source": "olympiads", "problem_type": "Algebra", "uuid": "46eb6c3b-bc5d-5fc0-8913-2c4a06359b71"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n1. (6 points) Today is January 31, 2015, welcome to the 2015 \"Spring Cup\" Final. The calculation result of the expression $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$ is", "candidates": [{"slot": 0, "origin": "budgie", "answer": "1. The expression is $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$.\n2. Simplify the numerator:\n$$\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31} = \\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{\\frac{31+0.31}{1}} = \\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{\\frac{31+0.31}{1}} = 2015 + 2015$$\n\nFinal answer:\n#### 2015", "correct": false, "reward": -1, "repetition": {"words": 61, "unique_ratio": 0.21311475409836064, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "2. (1 point) The problem involves calculating a complex expression.\n\nEliminating the denominator by multiplying both the numerator and denominator by the conjugate of the denominator.\n\n$$\n\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+\\frac{2015}{0.31}} = \\frac{2015}{1} + 3030\n$$\n\nNext, we calculate the denominator of the fraction by multiplying both the numerator and the denominator by the conjugate of the denominator, which is 1.\n\n$$\n1+\\frac{2015}{0.31} = 1 + 2015 = 2016\n$$\n\nNow that we have the full fraction, we can simplify it by dividing the numerator by 2015 and multiplying both the numerator and denominator by 1 (since 2015 is twice 1).\n\n$$\n\\frac{2015}{2015} + 3030 = \\frac{1}{2015} + 3030 = 3030 + 2015 = 5045\n$$\n\nThe result of the new problem is 5045.", "correct": false, "reward": -1, "repetition": {"words": 126, "unique_ratio": 0.3492063492063492, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "【Solution】Solve: $\\frac{\\frac{2015}{1}+\\frac{2015}{0.31}}{1+0.31}$\n$$\n\\begin{array}{l}\n=\\frac{\\left(\\frac{2015}{1}+\\frac{2015}{0.31}\\right) \\times 0.31}{(1+0.31) \\times 0.31} \\\\\n=\\frac{(1+0.31) \\times 2015}{(1+0.31) \\times 0.31} \\\\\n=6500 ;\n\\end{array}\n$$\n\nTherefore, the answer is: 6500.", "correct": true, "reward": 1.0, "repetition": {"words": 55, "unique_ratio": 0.34545454545454546, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_025", "row": {"problem": "Problem 4. A segment is divided by three points into four unequal parts. The distance between the midpoints of the inner parts is $6 \\mathrm{~cm}$, and the distance between the midpoints of the outer parts is $16 \\mathrm{~cm}$. What is the length of the given segment?", "solution": "Solution. Let the endpoints of the given segment be $A$ and $B$, and let the points $C, D, E$ divide it into four unequal parts. Let $M$ be the midpoint of\n\n![](https://cdn.mathpix.com/cropped/2024_06_05_dc4d34c0a9dd2a3ba572g-04.jpg?height=113&width=639&top_left_y=1155&top_left_x=570)\n\n$AC$, $N$ the midpoint of $CD$, $P$ the midpoint of $DE$, and $Q$ the midpoint of $EB$. Then, from the condition $\\overline{NP}=6 \\text{ cm}$ and $\\overline{MQ}=16 \\text{ cm}$ (see the diagram). Now, from $\\overline{NP}=6 \\text{ cm}$, we have that $\\overline{CD}+\\overline{DE}=2 \\cdot 6=12 \\text{ cm}$. From this and from $\\overline{MQ}=16 \\text{ cm}$, we get that $\\overline{MC}+\\overline{EQ}=16-12=4 \\text{ cm}$, and since $M$ is the midpoint of $AC$ and $Q$ is the midpoint of $EB$, it follows that $\\overline{AM}+\\overline{QB}=4 \\text{ cm}$. Finally, the length of the given segment is $\\overline{AB}=(\\overline{AM}+\\overline{QB})+\\overline{MQ}=4+16=20 \\text{ cm}$.", "answer": "20", "source": "olympiads", "problem_type": "Geometry", "uuid": "5c51b71b-267b-590e-be6a-eaf5657d85d3"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nProblem 4. A segment is divided by three points into four unequal parts. The distance between the midpoints of the inner parts is $6 \\mathrm{~cm}$, and the distance between the midpoints of the outer parts is $16 \\mathrm{~cm}$. What is the length of the given segment?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "1. Let $\\mathbf{a}, \\mathbf{b},$ and $\\mathbf{c}$ be the three points in space that divide the segment into four parts.\n2. The distance between the midpoints of the inner parts is $6 \\, \\mathrm{cm}$ and the distance between the midpoints of the outer parts is $16 \\, \\mathrm{cm}$.\n3. The distance between the midpoints of the outer parts can be expressed as the sum of the distances between each pair of points:\n    \\[\n    \\text{Distance} = \\text{distance} + \\left(\\frac{2}{\\pi}\\right) \\left(\\frac{2}{\\pi}\\right) + \\left(\\frac{2}{\\pi}\\right)^2\n    \\]\n4. Substitute the given distances:\n    \\[\n    \\text{Distance} = 6 \\, \\mathrm{cm} + \\frac{2}{\\pi} \\times 6 \\, \\mathrm{cm} + \\left(\\frac{2}{\\pi}\\right)^2 = 10 \\, \\mathrm{cm}\n    \\]\n5. Therefore, we have:\n    \\[\n    \\text{Distance} = 10 \\, \\mathrm{cm}\n    \\]\n6. The length of the given segment is the sum of the distances between each pair of points:\n    \\[\n    \\text{Length} = 16 \\, \\mathrm{cm}\n    \\]\n\n### Conclusion\n\\[\n\\boxed{16}\n\\]", "correct": false, "reward": -1, "repetition": {"words": 155, "unique_ratio": 0.36129032258064514, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "1. Let's denote a point $P$ as the center of the circle with a radius of $2 \\mathrm{~cm}$.\n2. From the problem, we know that the distances from $P$ to the midpoints of the inner part, the midpoint of the outer part, and the midpoint of the inner and outer part are all equal.\n3. For the midpoints of the inner parts, let the midpoint of the inner part be $M_{\\text{inner}}$ and the midpoint of the outer part be $M_{\\text{outer}}$.\n4. Given that the total length of segment $MN_{\\text{inner}}$ is $6 \\mathrm{~cm}$, and the total length of segment $MN_{\\text{outer}}$ is also $6 \\mathrm{~cm}$, we can establish the following equations:\n   \\[\n   |M_{\\text{inner}}| + |M_{\\text{outer}}| = 3\n   \\]\n5. Additionally, for the midpoints of the outer parts, the distance is given by:\n   \\[\n   |M_{\\text{outer}}| = \\frac{1}{2} |MN_{\\text{outer}}| = \\frac{1}{2} \\cdot 16 \\mathrm{~cm} = 8 \\mathrm{~cm}\n   \\]\n6. Using the first equation, we have:\n   \\[\n   |M_{\\text{inner}}| + |M_{\\text{outer}}| = 3\n   \\]\n7. Substitute the value of $|M_{\\text{inner}}| + |M_{\\text{outer}}|$ into the equation:\n   \\[\n   3 + 8 =", "correct": false, "reward": -1, "repetition": {"words": 190, "unique_ratio": 0.35789473684210527, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution. Let the endpoints of the given segment be $A$ and $B$, and let the points $C, D, E$ divide it into four unequal parts. Let $M$ be the midpoint of\n\n![](https://cdn.mathpix.com/cropped/2024_06_05_dc4d34c0a9dd2a3ba572g-04.jpg?height=113&width=639&top_left_y=1155&top_left_x=570)\n\n$AC$, $N$ the midpoint of $CD$, $P$ the midpoint of $DE$, and $Q$ the midpoint of $EB$. Then, from the condition $\\overline{NP}=6 \\text{ cm}$ and $\\overline{MQ}=16 \\text{ cm}$ (see the diagram). Now, from $\\overline{NP}=6 \\text{ cm}$, we have that $\\overline{CD}+\\overline{DE}=2 \\cdot 6=12 \\text{ cm}$. From this and from $\\overline{MQ}=16 \\text{ cm}$, we get that $\\overline{MC}+\\overline{EQ}=16-12=4 \\text{ cm}$, and since $M$ is the midpoint of $AC$ and $Q$ is the midpoint of $EB$, it follows that $\\overline{AM}+\\overline{QB}=4 \\text{ cm}$. Finally, the length of the given segment is $\\overline{AB}=(\\overline{AM}+\\overline{QB})+\\overline{MQ}=4+16=20 \\text{ cm}$.", "correct": true, "reward": 1.0, "repetition": {"words": 176, "unique_ratio": 0.4772727272727273, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_026", "row": {"problem": "6. The set of all real number pairs $(x, y)$ that satisfy $\\left\\{\\begin{array}{l}4^{-x}+27^{-y}=\\frac{5}{6} \\\\ \\log _{27} y-\\log _{4} x \\geqslant \\frac{1}{6} \\text { is } \\\\ 27^{y}-4^{x} \\leqslant 1\\end{array}\\right.$ $\\qquad$ . $\\qquad$", "solution": "6. $\\left\\{\\left(\\frac{1}{2}, \\frac{1}{3}\\right)\\right\\}$ Detailed Explanation: Let $a=4^{x}, b=27^{y}$, then $\\frac{1}{a}+\\frac{1}{b}=\\frac{5}{6}, a>\\frac{6}{5}, b>\\frac{6}{5}$. From $\\frac{1}{a}+\\frac{1}{b}=\\frac{5}{6}$, we get $b=\\frac{6 a}{5 a-6}$, then $\\frac{6 a}{5 a-6}-a \\leqslant 1$, solving this yields $a \\geqslant 2$ or $a>\\frac{6}{5}$, so $a \\geqslant 2$. Similarly, we get $b \\leqslant 3$. Therefore, $x \\geqslant \\frac{1}{2}, y \\leqslant \\frac{1}{3} . \\log _{27} y-\\log _{4} x \\leqslant \\log _{27} \\frac{1}{3}-\\log _{4} \\frac{1}{2}=\\frac{1}{6}$. So $x=\\frac{1}{2}, y=\\frac{1}{3}$.", "answer": "(\\frac{1}{2},\\frac{1}{3})", "source": "olympiads", "problem_type": "Algebra", "uuid": "01b0b161-74b6-5dd2-958d-c4fa3a8e18a3"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n6. The set of all real number pairs $(x, y)$ that satisfy $\\left\\{\\begin{array}{l}4^{-x}+27^{-y}=\\frac{5}{6} \\\\ \\log _{27} y-\\log _{4} x \\geqslant \\frac{1}{6} \\text { is } \\\\ 27^{y}-4^{x} \\leqslant 1\\end{array}\\right.$ $\\qquad$ . $\\qquad$", "candidates": [{"slot": 0, "origin": "budgie", "answer": "1. To solve the given system of equations, we can start by rewriting the second equation in terms of $y$:\n\n$$\n\\log_{27} y - \\log_{4} x = \\frac{27}{6}\n$$\n\n2. Using the change of base formula, we can rewrite the second equation in terms of $x$:\n\n$$\n\\frac{1}{y} - \\frac{\\log_{27} x}{\\log_{4} 27} = \\frac{27}{6}\n$$\n\n3. Notice that $\\log_{27} x = \\log_{4} x$ because $27 = 4^2$. Substituting this into the second equation, we get:\n\n$$\n\\frac{1}{y} - \\frac{\\log_{4} x}{\\log_{4} 27} = \\frac{27}{6}\n$$\n\n4. Simplify the equation further by cancelling out $y$ and $\\log_{4} x$:\n\n$$\n\\frac{1}{\\log_{4} x} = \\frac{27}{6} - \\frac{\\log_{4} x}{\\log_{4} 27}\n$$\n\n5. Let $\\log_{4} 27 = a$, where $a > 0$ since $27 > 1$. Substitute this into the equation:\n\n$$\n\\frac{1}{\\log_{4} x} = \\frac{27}{6} - \\frac{a}{\\log_{4} x}\n$$\n\n6. Multiply both sides of the equation by $\\log_{4} x$ to clear the fractions:\n\n$$\n1 = \\frac{27}{6} \\log_{", "correct": false, "reward": -1, "repetition": {"words": 171, "unique_ratio": 0.3216374269005848, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To solve this problem, we first need to find the range of values for $x$ and $y$ such that the given equation holds true.\n\nGiven:\n\\[ \\frac{1}{4^x} + \\frac{1}{27^y} = \\frac{5}{6} \\]\n\nTo eliminate the fractions, we can multiply both sides by $6^x 4^y$. We know that $6^x 4^y = (2^2 \\cdot 3)^x 2^y = 2^x \\cdot 3^x \\cdot 2^y = 2^x \\cdot 3^x \\cdot 2^y$.\n\nSo, we have:\n\\[ 6^x 4^y \\cdot \\frac{1}{4^x} + 6^x \\cdot 4^y \\cdot \\frac{1}{27^y} = 6^x \\cdot 4^y \\cdot \\frac{5}{6} \\]\n\nNow, we can simplify the expressions on the left side of the equation:\n\\[ 6^x 4^y + 6^x \\cdot 4^y = 2^x \\cdot 3^x \\cdot 2^y \\]\n\nWe can further simplify the expression on the right side of the equation:\n\\[ 6^x \\cdot 4^y \\cdot \\frac{5}{6} = 2^x \\cdot 3^x \\cdot 2^y \\]\n\nNow, we can rewrite the equation as:\n\\[ 2^x \\cdot 3^x \\cdot 2^y + 6^x \\cdot 4^y \\cdot \\frac", "correct": false, "reward": -1, "repetition": {"words": 193, "unique_ratio": 0.2694300518134715, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "6. $\\left\\{\\left(\\frac{1}{2}, \\frac{1}{3}\\right)\\right\\}$ Detailed Explanation: Let $a=4^{x}, b=27^{y}$, then $\\frac{1}{a}+\\frac{1}{b}=\\frac{5}{6}, a>\\frac{6}{5}, b>\\frac{6}{5}$. From $\\frac{1}{a}+\\frac{1}{b}=\\frac{5}{6}$, we get $b=\\frac{6 a}{5 a-6}$, then $\\frac{6 a}{5 a-6}-a \\leqslant 1$, solving this yields $a \\geqslant 2$ or $a>\\frac{6}{5}$, so $a \\geqslant 2$. Similarly, we get $b \\leqslant 3$. Therefore, $x \\geqslant \\frac{1}{2}, y \\leqslant \\frac{1}{3} . \\log _{27} y-\\log _{4} x \\leqslant \\log _{27} \\frac{1}{3}-\\log _{4} \\frac{1}{2}=\\frac{1}{6}$. So $x=\\frac{1}{2}, y=\\frac{1}{3}$.", "correct": true, "reward": 1.0, "repetition": {"words": 128, "unique_ratio": 0.2421875, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_027", "row": {"problem": "4. (6 points) A number when divided by 3 leaves a remainder of 2, when divided by 4 leaves a remainder of 3, and when divided by 5 leaves a remainder of 4. This number is $\\qquad$", "solution": "【Analysis】Interpreting \"divided by 3 leaves a remainder of 2, divided by 4 leaves a remainder of 3, divided by 5 leaves a remainder of 4\" as being 1 less than when divided by 3, 1 less than when divided by 4, and 1 less than when divided by 5, means this number is at least 1 less than the least common multiple (LCM) of 3, 4, and 5. Since 3, 4, and 5 are pairwise coprime, the LCM of these three numbers is their product. Find the LCM of 3, 4, and 5, then subtract 1.\n\n【Solution】Solution: $3 \\times 4 \\times 5-1$,\n$$\n\\begin{array}{l}\n=60-1, \\\\\n=59 ;\n\\end{array}\n$$\n\nAnswer: The number is 59.\nTherefore, the answer is: 59.\n【Comment】This problem mainly examines the method of finding the least common multiple (LCM) when three numbers are pairwise coprime: If three numbers are pairwise coprime, their LCM is the product of these three numbers.", "answer": "59", "source": "olympiads", "problem_type": "Number Theory", "uuid": "7cfc118d-0022-572d-bacc-abb0a3b763a4"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n4. (6 points) A number when divided by 3 leaves a remainder of 2, when divided by 4 leaves a remainder of 3, and when divided by 5 leaves a remainder of 4. This number is $\\qquad$", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Let the number be $n$. We can represent this situation using modular arithmetic.\n\n\\[ n \\equiv 2 \\mod 3 \\]\n\\[ n \\equiv 3 \\mod 4 \\]\n\\[ n \\equiv 4 \\mod 5 \\]\n\nFirst, we need to find a number that satisfies all these conditions. We can do this by looking for a number that is 1 more than a multiple of 3.\n\n\\[ n = 3k + 1 \\]\n\nwhere $k$ is a positive integer.\n\nNow, we need to find a number that also satisfies the other two conditions:\n\n\\[ n \\equiv 2 \\mod 4 \\]\n\\[ n \\equiv 3 \\mod 4 \\]\n\nSubstitute $n = 3k + 1$ into the second congruence:\n\n\\[ 3k + 1 \\equiv 3 \\mod 4 \\]\n\nSubtract 1 from both sides:\n\n\\[ 3k \\equiv 2 \\mod 4 \\]\n\nSubtract 2 from both sides:\n\n\\[ 3k - 2 \\equiv 0 \\mod 4 \\]\n\nMultiply both sides by the multiplicative inverse of 3 modulo 4 (which is also 3 since 3 is a factor of 4):\n\n\\[ k \\equiv 2 \\times 3 \\mod 4 \\]\n\\[ k \\equiv 6 \\mod 4 \\]\n\\[ k \\equiv 2 \\mod 4 \\]\n\nSo, we have found that $k \\equiv 2 \\mod 4$. This means that $k", "correct": false, "reward": -1, "repetition": {"words": 179, "unique_ratio": 0.39106145251396646, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To solve this problem, we can use the Chinese Remainder Theorem (CRT). The CRT states that if we have a system of congruences\n\n$$\n\\begin{cases}\nx \\equiv 0 \\pmod{m} \\\\\nx \\equiv 2 \\pmod{m} \\\\\nx \\equiv 3 \\pmod{m} \\\\\nx \\equiv 4 \\pmod{m}\n\\end{cases}\n$$\n\nwhere $m$ is an integer, then there exists a unique solution modulo $m$.\n\nFirst, we need to find a solution modulo $m$ for each congruence. We can do this by setting the right-hand sides of each congruence equal to 0.\n\nFor the first congruence, we have $x \\equiv 0 \\pmod{3}$. This means that $x$ leaves a remainder of 0 when divided by 3.\n\nFor the second congruence, we have $2x \\equiv 2 \\pmod{4}$. Dividing both sides by 2 gives us $x \\equiv 1 \\pmod{2}$.\n\nFor the third congruence, we have $3x \\equiv 3 \\pmod{5}$. Dividing both sides by 3 gives us $x \\equiv 1 \\pmod{5}$.\n\nFor the fourth congruence, we have $4x \\equiv 4 \\pmod{6}$. Dividing both sides by 4 gives us $x \\equiv 1 \\pmod{3}$.\n\nNow, we need to find a solution modulo $m$ that satisfies all the congruences. We can do this by finding an integer $x$ that satisfies", "correct": false, "reward": -1, "repetition": {"words": 206, "unique_ratio": 0.3640776699029126, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "【Analysis】Interpreting \"divided by 3 leaves a remainder of 2, divided by 4 leaves a remainder of 3, divided by 5 leaves a remainder of 4\" as being 1 less than when divided by 3, 1 less than when divided by 4, and 1 less than when divided by 5, means this number is at least 1 less than the least common multiple (LCM) of 3, 4, and 5. Since 3, 4, and 5 are pairwise coprime, the LCM of these three numbers is their product. Find the LCM of 3, 4, and 5, then subtract 1.\n\n【Solution】Solution: $3 \\times 4 \\times 5-1$,\n$$\n\\begin{array}{l}\n=60-1, \\\\\n=59 ;\n\\end{array}\n$$\n\nAnswer: The number is 59.\nTherefore, the answer is: 59.\n【Comment】This problem mainly examines the method of finding the least common multiple (LCM) when three numbers are pairwise coprime: If three numbers are pairwise coprime, their LCM is the product of these three numbers.", "correct": true, "reward": 1.0, "repetition": {"words": 158, "unique_ratio": 0.3670886075949367, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_028", "row": {"problem": "A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all possible values of $n$.\n\nA rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all possible values of $n$.", "solution": "Answer: $n=324$.\n\nLet $a b=n$ and $c d=n+76$, where $a, b$ and $c, d$ are the numbers of squares in each direction for the partitioning of the rectangle into $n$ and $n+76$ squares, respectively. Then $\\frac{a}{c}=\\frac{b}{d}$, or $a d=b c$. Denote $u=\\operatorname{gcd}(a, c)$ and $v=\\operatorname{gcd}(b, d)$, then there exist positive integers $x$ and $y$ such that $\\operatorname{gcd}(x, y)=1, a=u x, c=u y$ and $b=v x, d=v y$. Hence we have\n\n$$\nc d-a b=u v\\left(y^{2}-x^{2}\\right)=u v(y-x)(y+x)=76=2^{2} \\cdot 19 .\n$$\n\nSince $y-x$ and $y+x$ are positive integers of the same parity and $\\operatorname{gcd}(x, y)=1$, we have $y-x=1$ and $y+x=19$ as the only possibility, yielding $y=10, x=9$ and $u v=4$. Finally we have $n=a b=x^{2} u v=324$.", "answer": "324", "source": "olympiads_ref", "problem_type": "Number Theory", "uuid": "NaN"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nA rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all possible values of $n$.\n\nA rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares. Find all possible values of $n$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "A rectangle can be divided into $n$ equal squares. The same rectangle can also be divided into $n+76$ equal squares.\n\nThe key is that the area of the rectangle is equal to the area of a square.\n\nThe area of the rectangle is $A = n^2$ (since it's a rectangle).\n\nThe area of the square is $A = (n + 76)^2$ (since it's a square).\n\nSince the area of the rectangle is equal to the area of a square, we have $A = A$.\n\nThis means $n^2 = (n + 76)^2$.\n\nExpanding the equation, we get $n^2 = n^2 + 152n + 5916$.\n\nSubtracting $n^2$ from both sides, we get $0 = 152n + 5915$.\n\nSubtracting $5915$ from both sides, we get $-5910 = 152n$.\n\nDividing both sides by $152$, we get $n = -5910/152$.\n\nSimplifying the right side, we get $n = -58.5785794...$.\n\nSince $n$ must be a positive integer, the only possible value of $n$ is $n = -5910/152 \\approx -41.4083$.\n\nHowever, since $n$ must be a non-negative integer, we consider the next value of $n$, which is $n = -5910/152 = -58$.\n\nThe only possible value of $n$", "correct": false, "reward": -1, "repetition": {"words": 188, "unique_ratio": 0.32978723404255317, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "If a rectangle can be divided into $n$ equal squares, then the area of the rectangle must be a perfect square. Since the area is a perfect square, this means that the length and width of the rectangle must be powers of 2.\n\nLet $a$ and $b$ be the length and width of the rectangle, respectively. Then we have:\n\n\\[ a^2 = 2^n \\quad \\text{and} \\quad b^2 = 2^n + 76 \\]\n\nWe can rewrite these equations as:\n\n\\[ a = 2^x \\quad \\text{and} \\quad b = 2^y \\]\n\nwhere $x$ and $y$ are non-negative integers.\n\nSubstituting these expressions for $a$ and $b$, we get:\n\n\\[ (2^x)^2 = 2^n \\quad \\text{and} \\quad (2^y)^2 = 2^n + 76 \\]\n\nSimplifying these equations, we get:\n\n\\[ 2^{2x} = 2^n \\quad \\text{and} \\quad 2^{2y} = 2^n + 76 \\]\n\nDividing both sides of the first equation by $2^x$, we get:\n\n\\[ 2^{2x - x} = 2^n \\Rightarrow 2^x = 2^n \\Rightarrow x = n \\]\n\nSimilarly, dividing both sides of the second equation by $2^y$, we get:\n\n\\[ 2^{2y - y} = 2^n + 76 \\Rightarrow 2^y = 2^n +", "correct": false, "reward": -1, "repetition": {"words": 187, "unique_ratio": 0.3315508021390374, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Answer: $n=324$.\n\nLet $a b=n$ and $c d=n+76$, where $a, b$ and $c, d$ are the numbers of squares in each direction for the partitioning of the rectangle into $n$ and $n+76$ squares, respectively. Then $\\frac{a}{c}=\\frac{b}{d}$, or $a d=b c$. Denote $u=\\operatorname{gcd}(a, c)$ and $v=\\operatorname{gcd}(b, d)$, then there exist positive integers $x$ and $y$ such that $\\operatorname{gcd}(x, y)=1, a=u x, c=u y$ and $b=v x, d=v y$. Hence we have\n\n$$\nc d-a b=u v\\left(y^{2}-x^{2}\\right)=u v(y-x)(y+x)=76=2^{2} \\cdot 19 .\n$$\n\nSince $y-x$ and $y+x$ are positive integers of the same parity and $\\operatorname{gcd}(x, y)=1$, we have $y-x=1$ and $y+x=19$ as the only possibility, yielding $y=10, x=9$ and $u v=4$. Finally we have $n=a b=x^{2} u v=324$.", "correct": true, "reward": 1.0, "repetition": {"words": 169, "unique_ratio": 0.35502958579881655, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_029", "row": {"problem": "$[$ [Arithmetic. Mental calculation, etc.] $]$\n\nAuthors: Gaityerri G.A., Grierenko D.:\n\n2002 is a palindrome year, which means it reads the same backward as forward. The previous palindrome year was 11 years earlier (1991). What is the maximum number of non-palindrome years that can occur consecutively (between 1000 and 9999 years)?", "solution": "Let the current year be a palindrome of the form $\\overline{a b b a}$. When will the next such year occur? Consider two cases:\n\na) $b=9$ (year of the form $\\overline{a 99 a}, a<9)$. Then, 11 years later, another palindromic year will occur: $\\overline{(a+1) 00(a+1)}$. For example, the years 3993 and 4004.\n\nb) $b<9$. In this case, the next palindromic year will occur 110 years later: $\\overline{a(b+1)(b+1) a}$. For example, the years 9339 and 9449. Therefore, the maximum number of consecutive non-palindromic years is 109.\n\n## Answer\n\n109 years.", "answer": "109", "source": "olympiads", "problem_type": "Number Theory", "uuid": "effbcdc3-5088-5c32-8530-ff0fddfa69ea"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n$[$ [Arithmetic. Mental calculation, etc.] $]$\n\nAuthors: Gaityerri G.A., Grierenko D.:\n\n2002 is a palindrome year, which means it reads the same backward as forward. The previous palindrome year was 11 years earlier (1991). What is the maximum number of non-palindrome years that can occur consecutively (between 1000 and 9999 years)?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The maximum number of non-palindrome years is 999 years + 1 year = 1000 years.\n\nFinal answer:\n#### 1000", "correct": false, "reward": -0.5, "repetition": {"words": 17, "unique_ratio": 0.8235294117647058, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 1, "origin": "budgie", "answer": "The maximum number of non-palindrome years between 1000 and 9999 that can occur consecutively is 9999 - 1000 + 1 = 9000 years.", "correct": false, "reward": -0.5, "repetition": {"words": 21, "unique_ratio": 0.8571428571428571, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 2, "origin": "verified_reference", "answer": "Let the current year be a palindrome of the form $\\overline{a b b a}$. When will the next such year occur? Consider two cases:\n\na) $b=9$ (year of the form $\\overline{a 99 a}, a<9)$. Then, 11 years later, another palindromic year will occur: $\\overline{(a+1) 00(a+1)}$. For example, the years 3993 and 4004.\n\nb) $b<9$. In this case, the next palindromic year will occur 110 years later: $\\overline{a(b+1)(b+1) a}$. For example, the years 9339 and 9449. Therefore, the maximum number of consecutive non-palindromic years is 109.\n\n## Answer\n\n109 years.", "correct": true, "reward": 1.0, "repetition": {"words": 103, "unique_ratio": 0.46601941747572817, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142115623759233}], "split": "math_train"}
+{"id": "math_030", "row": {"problem": "Of the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie? \n\n$ \\text{(A)}\\ 10\\qquad\\text{(B)}\\ 20\\qquad\\text{(C)}\\ 30\\qquad\\text{(D)}\\ 50\\qquad\\text{(E)}\\ 72 $", "solution": "1. First, we need to determine the number of students who prefer each type of pie. We know:\n   - 12 students prefer chocolate pie.\n   - 8 students prefer apple pie.\n   - 6 students prefer blueberry pie.\n\n2. Calculate the total number of students who prefer chocolate, apple, or blueberry pie:\n   \\[\n   12 + 8 + 6 = 26\n   \\]\n\n3. Subtract this number from the total number of students in the class to find the number of students who prefer either cherry or lemon pie:\n   \\[\n   36 - 26 = 10\n   \\]\n\n4. Since half of the remaining students prefer cherry pie and the other half prefer lemon pie, we divide the remaining students by 2:\n   \\[\n   \\frac{10}{2} = 5\n   \\]\n   Therefore, 5 students prefer cherry pie.\n\n5. To find the number of degrees Richelle should use for cherry pie in her pie graph, we need to calculate the proportion of students who prefer cherry pie and then convert this proportion to degrees. The proportion of students who prefer cherry pie is:\n   \\[\n   \\frac{5}{36}\n   \\]\n\n6. Multiply this proportion by the total number of degrees in a circle (360 degrees):\n   \\[\n   \\frac{5}{36} \\times 360 = 50\n   \\]\n\n7. Therefore, Richelle should use 50 degrees for cherry pie in her pie graph.\n\nThe final answer is $\\boxed{50}$.", "answer": "50", "source": "aops_forum", "problem_type": "Combinatorics", "uuid": "4f89da07-3f71-5edc-85a6-7460e959c360"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nOf the 36 students in Richelle's class, 12 prefer chocolate pie, 8 prefer apple, and 6 prefer blueberry. Half of the remaining students prefer cherry pie and half prefer lemon. For Richelle's pie graph showing this data, how many degrees should she use for cherry pie? \n\n$ \\text{(A)}\\ 10\\qquad\\text{(B)}\\ 20\\qquad\\text{(C)}\\ 30\\qquad\\text{(D)}\\ 50\\qquad\\text{(E)}\\ 72 $", "candidates": [{"slot": 0, "origin": "budgie", "answer": "First, we need to find the total number of students in the class. We can do this by adding up the number of students who prefer chocolate, apple, blueberry, and cherry pie.\n\n\\[ \\text{Total} = \\text{chocolate} + \\text{apple} + \\text{blueberry} + \\text{cherry} = 12 + 8 + 6 + 3 = 27 \\]\n\nNext, we need to find the total number of students who prefer cherry pie. We can do this by subtracting the total number of students who prefer apple and blueberry pie from the total number of students.\n\n\\[ \\text{Total cherry} = \\text{Total} - \\text{Total apple} - \\text{Total blueberry} = 27 - 5 - 6 = 16 \\]\n\nFinally, we need to find the fraction of the class that prefer cherry pie. We can do this by dividing the total number of students who prefer cherry pie by the total number of students in the class.\n\n\\[ \\text{Fraction} = \\frac{\\text{Total cherry}}{\\text{Total students}} = \\frac{16}{27} \\]\n\nNow, we can find the fraction of the class that prefer cherry pie by multiplying this fraction by the total number of students in the class.\n\n\\[ \\text{Fraction} = \\frac{16}{27} \\times 27 = 6 \\]\n\nSo, the fraction of the class that prefer cherry pie is 6.\n\nThe answer is:\n\\[ \\boxed{6} \\]\nFinal answer:\n#### 6", "correct": false, "reward": -1, "repetition": {"words": 203, "unique_ratio": 0.24630541871921183, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "First, we need to find out how many students prefer cherry pie. We can do this by subtracting the number of students who prefer other pie types from the total number of students.\n\n$ \\text{Number of students who prefer cherry pie} = \\text{Total number of students} - \\text{Number of students who prefer other pie types} $$ $= 36 - 12 $ $= 24 $\n\nNow that we have the number of students who prefer cherry pie, we can find the percentage of students who prefer cherry pie. We can do this by dividing the number of students who prefer cherry pie by the total number of students and then multiplying by 100.\n\n$ \\text{Percentage of students who prefer cherry pie} = \\frac{\\text{Number of students who prefer cherry pie}}{\\text{Total number of students}} $$ $= \\frac{24}{36} $ $= \\frac{2}{3} $\n\nSo, the answer is 2/3 or 60%.\n\nFinal answer:\n#### 60", "correct": false, "reward": -1, "repetition": {"words": 145, "unique_ratio": 0.32413793103448274, "max_4gram_count": 8, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "1. First, we need to determine the number of students who prefer each type of pie. We know:\n   - 12 students prefer chocolate pie.\n   - 8 students prefer apple pie.\n   - 6 students prefer blueberry pie.\n\n2. Calculate the total number of students who prefer chocolate, apple, or blueberry pie:\n   \\[\n   12 + 8 + 6 = 26\n   \\]\n\n3. Subtract this number from the total number of students in the class to find the number of students who prefer either cherry or lemon pie:\n   \\[\n   36 - 26 = 10\n   \\]\n\n4. Since half of the remaining students prefer cherry pie and the other half prefer lemon pie, we divide the remaining students by 2:\n   \\[\n   \\frac{10}{2} = 5\n   \\]\n   Therefore, 5 students prefer cherry pie.\n\n5. To find the number of degrees Richelle should use for cherry pie in her pie graph, we need to calculate the proportion of students who prefer cherry pie and then convert this proportion to degrees. The proportion of students who prefer cherry pie is:\n   \\[\n   \\frac{5}{36}\n   \\]\n\n6. Multiply this proportion by the total number of degrees in a circle (360 degrees):\n   \\[\n   \\frac{5}{36} \\times 360 = 50\n   \\]\n\n7. Therefore, Richelle should use 50 degrees for cherry pie in her pie graph.\n\nThe final answer is $\\boxed{50}$.", "correct": true, "reward": 1.0, "repetition": {"words": 203, "unique_ratio": 0.3497536945812808, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_031", "row": {"problem": "# Problem 6. (3 points)\n\nThree runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.\n\nAt some point, the first runner meets the second. After 15 minutes, the second runner meets the third for the first time. Another 25 minutes later, the third runner meets the first for the first time.\n\nHow many minutes does it take for one runner to complete the entire track?", "solution": "# Answer: 80\n\n## Solution:\n\nLet the first runner meet the second, then after $a$ minutes the second runner meets the third for the first time, and after another $b$ minutes the third runner meets the first for the first time.\n\nLet the first and second runners meet at point $A$, the second and third at point $B$, and the first and third at point $C$. Additionally, let the first runner be at point $D$ at the moment the second and third runners meet.\n\nThen the points are arranged on the circle in the following order: $D, A, B, C$, where the first runner ran from $D$ to $A$ in $a$ minutes, the second runner ran from $A$ to $B$ in $a$ minutes, the third runner ran from $B$ to $C$ in $b$ minutes, and the first runner ran from $D$ to $C$ in $b$ minutes. These four segments together form the entire track, so it is covered in $2a + 2b$ minutes.", "answer": "80", "source": "olympiads", "problem_type": "Logic and Puzzles", "uuid": "62791263-04f8-508f-9886-2dd0d0a262be"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n# Problem 6. (3 points)\n\nThree runners are moving along a circular track at constant equal speeds. When two runners meet, they instantly turn around and start running in opposite directions.\n\nAt some point, the first runner meets the second. After 15 minutes, the second runner meets the third for the first time. Another 25 minutes later, the third runner meets the first for the first time.\n\nHow many minutes does it take for one runner to complete the entire track?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The problem is based on the two-way circular track.\n\nThe key here is understanding the behavior of the runners and the time it takes for them to complete a lap.\n\nThe runners are moving at different constant speeds, and each runner has to cover the same distance around the track.\n\nAt the first meeting, the third runner will have to cover the distance that the second runner covered in the first 15 minutes.\n\nAfter 25 minutes, the second runner will have to cover the same distance that the first runner covered in the first 15 minutes.\n\nTherefore, the second runner will have to cover a distance that the first runner covered in 25 minutes, and the third runner will have to cover a distance that the first runner covered in 15 minutes.\n\nThe third runner will have to cover the same distance that the first runner covered in 25 minutes, and the second runner will have to cover a distance that the first runner covered in 15 minutes.\n\nSo, the distance that the third runner needs to cover is 25 minutes.\n\nLet's denote the distance that the first runner covers as $s$. At the time of the first meeting, the second runner will have to cover $s-15$ meters.\n\nThe first runner will have to cover $s+15$ meters in the same amount of time.\n\nWe know that the third runner will have to cover 25 meters in the same amount of time.\n\nTherefore, we have:\n\n$$s+15 = 25$$\n\nSolving for $s$ gives us:\n\n$$s = 25 - 15 = 10$$\n\nSo, the first runner will have to cover 10 meters in the time it takes the second runner to", "correct": false, "reward": -1, "repetition": {"words": 278, "unique_ratio": 0.24100719424460432, "max_4gram_count": 10, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 1, "origin": "budgie", "answer": "The answer is 25 minutes.\n\nFinal answer:\n#### 25", "correct": false, "reward": -0.5, "repetition": {"words": 8, "unique_ratio": 0.75, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 2, "origin": "verified_reference", "answer": "# Answer: 80\n\n## Solution:\n\nLet the first runner meet the second, then after $a$ minutes the second runner meets the third for the first time, and after another $b$ minutes the third runner meets the first for the first time.\n\nLet the first and second runners meet at point $A$, the second and third at point $B$, and the first and third at point $C$. Additionally, let the first runner be at point $D$ at the moment the second and third runners meet.\n\nThen the points are arranged on the circle in the following order: $D, A, B, C$, where the first runner ran from $D$ to $A$ in $a$ minutes, the second runner ran from $A$ to $B$ in $a$ minutes, the third runner ran from $B$ to $C$ in $b$ minutes, and the first runner ran from $D$ to $C$ in $b$ minutes. These four segments together form the entire track, so it is covered in $2a + 2b$ minutes.", "correct": true, "reward": 1.0, "repetition": {"words": 160, "unique_ratio": 0.33125, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_032", "row": {"problem": "2. A helicopter and an airplane flew towards each other at the same time. By the time they met, the helicopter had flown $100 \\mathrm{~km}$ less than the airplane, and the helicopter reached the airplane's takeoff location 3 hours after the meeting. The airplane reached the helicopter's takeoff location 1 hour and 20 minutes after the meeting. Determine the speeds of the helicopter and the airplane, and the distance between the takeoff locations.", "solution": "Solution. Let the helicopter fly $x \\mathrm{~km}$ to the meeting. Then the airplane flew $(x+100) \\mathrm{~km}$ to the meeting. The speed of the helicopter is $\\frac{x+100}{3} \\mathrm{~km} / \\mathrm{h}$, and the speed of the airplane is $\\frac{x}{1 \\frac{1}{3}} \\mathrm{~km} / \\mathrm{h}$. Thus, the time the helicopter flew from its takeoff point to the meeting is $x: \\frac{x+100}{3}=\\frac{3 x}{x+100}$ hours, and the time the airplane flew from its takeoff point to the meeting is $\\frac{1 \\frac{1}{3}(x+100)}{x}$ hours. They flew the same amount of time to the meeting, so\n\n$$\n\\frac{1 \\frac{1}{3}(x+100)}{x}=\\frac{3 x}{x+100}, \\text { i.e. }\\left(\\frac{x+100}{x}\\right)^{2}=\\frac{9}{4}\n$$\n\nFrom this, since $\\frac{9}{4}=\\left(\\frac{3}{2}\\right)^{2}=\\left(-\\frac{3}{2}\\right)^{2}$, we get\n\n$$\n\\frac{x+100}{x}=\\frac{3}{2} \\text { or } \\frac{x+100}{x}=-\\frac{3}{2} \\text {. }\n$$\n\nHowever, $\\frac{x+100}{x}>0$, so $\\frac{x+100}{x}=\\frac{3}{2}$, from which we get $x=200 \\mathrm{~km}$.\n\nThus, the helicopter flew $200 \\mathrm{~km}$, the airplane flew $300 \\mathrm{~km}$, the speed of the helicopter is $100 \\mathrm{~km} / \\mathrm{h}$, and the speed of the airplane is $150 \\mathrm{~km} / \\mathrm{h}$. The distance between the takeoff points is $500 \\mathrm{~km}$.", "answer": "500\\mathrm{~}", "source": "olympiads", "problem_type": "Algebra", "uuid": "3ed05b96-c2f2-5248-aa03-084302cdd1b5"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n2. A helicopter and an airplane flew towards each other at the same time. By the time they met, the helicopter had flown $100 \\mathrm{~km}$ less than the airplane, and the helicopter reached the airplane's takeoff location 3 hours after the meeting. The airplane reached the helicopter's takeoff location 1 hour and 20 minutes after the meeting. Determine the speeds of the helicopter and the airplane, and the distance between the takeoff locations.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "1. Let $v$ be the speed of the airplane and $s$ be the speed of the would-be youth.\n2. We know that the distance between their travel locations is the same, so the distance that the airplane must have flown is the difference between the two speeds: $V_{\\text{ Artillery}} = s - v$.\n\n\\[\n\\text{Distance covered by airplane} = V_{\\text{airplane}} - V_{\\text{ appealing}} = (s - v) - \\frac{s}{2}\n\\]\n\n\\[\n\\text{Distance covered by would-be youth} = s - v = s - v + \\frac{s}{2}\n\\]\n\nAccording to the problem, when the countries meet, the airplane has flown $100$ km less than the would-be youth. This gives us the equation:\n\n\\[\ns - v = s - v + \\frac{s}{2} - 100\n\\]\n\nSimplify and solve for $v$:\n\n\\[\n\\frac{s}{2} - 100 = 0\n\\]\n\n\\[\ns = 200\n\\]\n\n\\[\ns - v = 200 - v\n\\]\n\n\\[\n\\frac{s}{2} - 100 = 0\n\\]\n\n\\[\ns = 200\n\\]\n\nSo, the speeds of the airplane and the would-be youth are 200 km/h and 100 km/h, respectively.\n\n\\[\ns = 200 \\text{ km/h}\n\\]\n\n#### 2. Next", "correct": false, "reward": -1, "repetition": {"words": 166, "unique_ratio": 0.3614457831325301, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "The key information is that the glued-up vegetable plane flew by 100 km less than the other to reach the other's position, and the other plane took 3 hours to reach the other's position after meeting. The other plane also took 1 hour and 20 minutes to reach the other's position after meeting.\n\nFirst, we need to find the distances the two planes traveled to reach each other. The distance is equal to the speed multiplied by the time.\n\nThe first plane flew by 100 km less than the second plane, so it flew 100 km less than the second plane, which means it flew 100 km + 100 km = 200 km.\n\nThe second plane flew by 3 hours after the first plane, which means it flew 3 hours + 3 hours = 6 hours.\n\nThe total distance the two planes traveled is 200 km + 200 km = 400 km.\n\nTo find the speed of the second plane, we divide the distance by the time taken.\n\nThe speed of the second plane is 400 km / 6 hours = 66.67 km/h.\n\nThe distance between theALH (the Lfriend and theaughter) and the LUSD (the Attempt) is 400 km - 200 km = 200 km.\n\nThe speed of the L movieer is 200 km / 4 hours = 50 km/h.\n\nThe answer is:\n#### 50", "correct": false, "reward": -1, "repetition": {"words": 216, "unique_ratio": 0.3287037037037037, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution. Let the helicopter fly $x \\mathrm{~km}$ to the meeting. Then the airplane flew $(x+100) \\mathrm{~km}$ to the meeting. The speed of the helicopter is $\\frac{x+100}{3} \\mathrm{~km} / \\mathrm{h}$, and the speed of the airplane is $\\frac{x}{1 \\frac{1}{3}} \\mathrm{~km} / \\mathrm{h}$. Thus, the time the helicopter flew from its takeoff point to the meeting is $x: \\frac{x+100}{3}=\\frac{3 x}{x+100}$ hours, and the time the airplane flew from its takeoff point to the meeting is $\\frac{1 \\frac{1}{3}(x+100)}{x}$ hours. They flew the same amount of time to the meeting, so\n\n$$\n\\frac{1 \\frac{1}{3}(x+100)}{x}=\\frac{3 x}{x+100}, \\text { i.e. }\\left(\\frac{x+100}{x}\\right)^{2}=\\frac{9}{4}\n$$\n\nFrom this, since $\\frac{9}{4}=\\left(\\frac{3}{2}\\right)^{2}=\\left(-\\frac{3}{2}\\right)^{2}$, we get\n\n$$\n\\frac{x+100}{x}=\\frac{3}{2} \\text { or } \\frac{x+100}{x}=-\\frac{3}{2} \\text {. }\n$$\n\nHowever, $\\frac{x+100}{x}>0$, so $\\frac{x+100}{x}=\\frac{3}{2}$, from which we get $x=200 \\mathrm{~km}$.\n\nThus, the helicopter flew $200 \\mathrm{~km}$, the airplane flew $300 \\mathrm{~km}$, the speed of the helicopter is $100 \\mathrm{~km} / \\mathrm{h}$, and the speed of the airplane is $150 \\mathrm{~km} / \\mathrm{h}$. The distance between the takeoff points is $500 \\mathrm{~km}$.", "correct": true, "reward": 1.0, "repetition": {"words": 243, "unique_ratio": 0.23045267489711935, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_033", "row": {"problem": "14. Let $k$ be a positive integer, such that for any positive numbers $a, b, c$ satisfying the inequality $k(a b+b c+c a)>5\\left(a^{2}+b^{2}\\right.$ $+c^{2}$), there must exist a triangle with side lengths $a, b, c$. Find the minimum value of $k$.", "solution": "14. $k(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right) \\geqslant 5(a b+a c+b c), k \\leqslant 5$ when, the inequality has no solution. When $k=6$, the inequality becomes $6(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right)$. Without loss of generality, assume $c=\\max (a, b, c)$, transform the inequality into $5 c^{2}-6(a+b) c+5\\left(a^{2}+b^{2}\\right)-6 a b0$, $5\\left(a^{2}+b^{2}\\right)-6 a b>0$, so the quadratic equation corresponding to the inequality has two positive roots $c_{1}, c_{2}$, the solution to the quadratic inequality in $c$ is $c \\in\\left(c_{1}, c_{2}\\right)$. If $c_{2} \\leqslant a+b$, then $c6$ when, $k(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right)$ has solutions, but they cannot form a triangle, so the only $k$ that satisfies the condition is $k=6$.", "answer": "6", "source": "olympiads", "problem_type": "Inequalities", "uuid": "562e031a-910d-5c5d-b73e-56e9bc99849c"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n14. Let $k$ be a positive integer, such that for any positive numbers $a, b, c$ satisfying the inequality $k(a b+b c+c a)>5\\left(a^{2}+b^{2}\\right.$ $+c^{2}$), there must exist a triangle with side lengths $a, b, c$. Find the minimum value of $k$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find the minimum value of $k$ that satisfies the given condition, we need to analyze the inequality and identify the conditions under which a triangle with side lengths $a, b, c$ exists.\n\nGiven:\n\\[ k(a b + b c + c a) > 5(a^{2} + b^{2} + c^{2}) \\]\n\nWe can rewrite the inequality as:\n\\[ k(a b + b c + c a) > 5 a^{2} + 5 b^{2} + 5 c^{2} \\]\n\nTo simplify the analysis, divide both sides by $a^{2} + b^{2} + c^{2}$ (which is non-zero), yielding:\n\\[ \\frac{k(a b + b c + c a)}{a^{2} + b^{2} + c^{2}} > \\frac{5 a^{2} + 5 b^{2} + 5 c^{2}}{4} \\]\n\nWe want the left-hand side (LHS) to be greater than $k$ for some positive $a, b, c$. To achieve this, we must have:\n\\[ \\frac{5 a^{2} + 5 b^{2} + 5 c^{2}}{4} < k \\]\n\nMultiplying through by 4 gives us:\n\\[ 5 a^{2} + 5 b^{2} + 5 c^{2} < 4k \\]\n\nTo ensure the inequality holds for any $a, b, c$, we need to find the smallest $k$ that satisfies this condition.\n\nOne approach is to find the smallest $k$ that makes the left-hand side positive and the inequality", "correct": false, "reward": -1, "repetition": {"words": 205, "unique_ratio": 0.3463414634146341, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "The inequality can be rewritten as $k > \\frac{5}{a^{2}+b^{2}} + \\frac{5}{c^{2}}$.\n\nNow, we can see that if the triangle side lengths satisfy this inequality, the triangle must have one of its sides of length less than the sum of the other two.\n\nLet $s = a + b$ and $t = c$.\n\nThen, we have $k > \\frac{{s}^{2}}{4} + \\frac{{t}^{2}}{4}$.\n\nSince $s > t$, we have $k > \\frac{{s}^{2}}{4}$.\n\nThis means that $\\frac{{s}^{2}}{4} + \\frac{{t}^{2}}{4} \\geq k$ must hold true.\n\nEquality holds when $s = k$ and $t = \\frac{{s}^{2}}{4}$.\n\nSince $s > t$, we can choose $k = s^{2}$.\n\nTherefore, the minimum value of $k$ is $s^{2}$.\n\nThe answer is:\n\\[ \\boxed{s^{2}} \\]\n\nFinal answer:\n#### s^2", "correct": false, "reward": -1, "repetition": {"words": 132, "unique_ratio": 0.4318181818181818, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "14. $k(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right) \\geqslant 5(a b+a c+b c), k \\leqslant 5$ when, the inequality has no solution. When $k=6$, the inequality becomes $6(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right)$. Without loss of generality, assume $c=\\max (a, b, c)$, transform the inequality into $5 c^{2}-6(a+b) c+5\\left(a^{2}+b^{2}\\right)-6 a b0$, $5\\left(a^{2}+b^{2}\\right)-6 a b>0$, so the quadratic equation corresponding to the inequality has two positive roots $c_{1}, c_{2}$, the solution to the quadratic inequality in $c$ is $c \\in\\left(c_{1}, c_{2}\\right)$. If $c_{2} \\leqslant a+b$, then $c6$ when, $k(a b+a c+b c)>5\\left(a^{2}+b^{2}+c^{2}\\right)$ has solutions, but they cannot form a triangle, so the only $k$ that satisfies the condition is $k=6$.", "correct": true, "reward": 1.0, "repetition": {"words": 175, "unique_ratio": 0.3028571428571429, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_034", "row": {"problem": "6. As shown in Figure 2, let $P$ be a point inside the equilateral $\\triangle ABC$ with side length 12. Draw perpendiculars from $P$ to the sides $BC$, $CA$, and $AB$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $PD: PE: PF = 1: 2: 3$. Then, the area of quadrilateral $BDPF$ is", "solution": "$$\n6.11 \\sqrt{3} \\text {. }\n$$\n\nFor the equilateral $\\triangle ABC$ with a height of $12 \\times \\frac{\\sqrt{3}}{2}=6 \\sqrt{3}$, then\n$$\n\\begin{array}{l}\nP D+P E+P F=6 \\sqrt{3} . \\\\\n\\text { Also, } P D: P E: P F=1: 2: 3, \\text { so, } \\\\\nP D=\\sqrt{3}, P E=2 \\sqrt{3}, P F=3 \\sqrt{3} .\n\\end{array}\n$$\n\nAs shown in Figure 7, draw\n$P S / / B C$, intersecting $A B$ at $S$.\nSince $\\angle F S P=\\angle B$ $=60^{\\circ}$, therefore,\n$$\n\\begin{array}{l}\nF S=P F \\cot 60^{\\circ}=3, \\\\\nP S=6, \\\\\nB D=P S+P D \\cot 60^{\\circ}=7 .\n\\end{array}\n$$\n\nThus, $S_{\\text {quadrilateral BDPF }}=S_{\\triangle P S}+S_{\\text {trapezoid BDPS }}$\n$$\n=\\frac{1}{2} \\times 3 \\times 3 \\sqrt{3}+\\frac{6+7}{2} \\times \\sqrt{3}=11 \\sqrt{3} .\n$$", "answer": "11 \\sqrt{3}", "source": "cn_contest", "problem_type": "Geometry", "uuid": "2a4c0374-d9ea-538e-8e0f-0811359d6cf6"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n6. As shown in Figure 2, let $P$ be a point inside the equilateral $\\triangle ABC$ with side length 12. Draw perpendiculars from $P$ to the sides $BC$, $CA$, and $AB$, with the feet of the perpendiculars being $D$, $E$, and $F$ respectively. Given that $PD: PE: PF = 1: 2: 3$. Then, the area of quadrilateral $BDPF$ is", "candidates": [{"slot": 0, "origin": "budgie", "answer": "1. Given that $PD: PE: PF = 1: 2: 3$, denote $PD = x$, $PE = 2x$, and $PF = 3x$. Since $PD + PE + PF = 12$, we have:\n   \\[ x + 2x + 3x = 12 \\Rightarrow 6x = 12 \\Rightarrow x = 2 \\]\n   \\[ PD = 2, PE = 4, PF = 9 \\]\n\n2. Since $\\triangle PEF$ is a right triangle with $PD$ as its hypotenuse, we can deduce that $PE = PF = 4$, making it an equilateral triangle.\n\n3. In an equilateral triangle, all sides are equal, and in a right triangle, the hypotenuse is $\\sqrt{3}$ times longer than each leg. Therefore, the length of each leg is:\n   \\[ PE = PF = 4\\sqrt{3} \\]\n\n4. The area of a triangle is given by:\n   \\[ \\text{Area} = \\frac{1}{2} \\times \\text{base} \\times \\text{height} \\]\n   where the base and height are the legs of the triangle.\n\n5. Let's denote the area of $\\triangle PEF$ as $[PEF]$. Using the Pythagorean theorem in $\\triangle PEF$:\n   \\[ [PEF] = \\left(\\frac{3}{4}\\right)^2 \\]\n\n6. From the given ratio $PD: PE: PF = 1:2:3$, we have:\n   \\[ PD = x = 2 \\times 1, PE = 2x = 4, PF = 3x = 9 \\]", "correct": false, "reward": -1, "repetition": {"words": 180, "unique_ratio": 0.37777777777777777, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "1. From the problem, it is clear that the ratio of the areas of triangles $\\triangle PBD$ and $\\triangle PFE$ is equal to the ratio of their bases, $\\frac{1}{2}$.\n2. The area of a triangle is given by the formula $S_{\\triangle} = \\frac{1}{2}ab$, where $a$ and $b$ are the lengths of the legs of the triangle.\n3. Therefore, we have $S_{\\triangle PBD} = \\frac{1}{4}S_{\\triangle ABC}$.\n4. Since $PD: PE: PF = 1: 2: 3$, we have $\\frac{PD}{PE} = \\frac{1}{2}$ and $\\frac{PE}{PF} = \\frac{2}{3}$.\n5. Therefore, $\\frac{PD}{PE} = \\frac{1}{3}$.\n6. Since the area of $\\triangle ABC$ is $S_{\\triangle ABC} = \\frac{1}{2}bc$, we have $\\frac{1}{2}bc = \\frac{1}{3}bc$, which implies $bc = 2$.\n7. Let the altitudes of $\\triangle ABC$ from $A$ and $B$ to $BC$ and $CA$ be $h_{a}$ and $h_{b}$, respectively. Then we have $h_{a} + h_{b} = h_{c}$.\n8. We also know that $\\frac{h_{a}}{h_{b}} = \\frac{2}{3}$, which means $h_{b} = \\frac{2}{3}h_{a}$.\n9. By the area formula, we have $\\frac", "correct": false, "reward": -1, "repetition": {"words": 195, "unique_ratio": 0.3230769230769231, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "$$\n6.11 \\sqrt{3} \\text {. }\n$$\n\nFor the equilateral $\\triangle ABC$ with a height of $12 \\times \\frac{\\sqrt{3}}{2}=6 \\sqrt{3}$, then\n$$\n\\begin{array}{l}\nP D+P E+P F=6 \\sqrt{3} . \\\\\n\\text { Also, } P D: P E: P F=1: 2: 3, \\text { so, } \\\\\nP D=\\sqrt{3}, P E=2 \\sqrt{3}, P F=3 \\sqrt{3} .\n\\end{array}\n$$\n\nAs shown in Figure 7, draw\n$P S / / B C$, intersecting $A B$ at $S$.\nSince $\\angle F S P=\\angle B$ $=60^{\\circ}$, therefore,\n$$\n\\begin{array}{l}\nF S=P F \\cot 60^{\\circ}=3, \\\\\nP S=6, \\\\\nB D=P S+P D \\cot 60^{\\circ}=7 .\n\\end{array}\n$$\n\nThus, $S_{\\text {quadrilateral BDPF }}=S_{\\triangle P S}+S_{\\text {trapezoid BDPS }}$\n$$\n=\\frac{1}{2} \\times 3 \\times 3 \\sqrt{3}+\\frac{6+7}{2} \\times \\sqrt{3}=11 \\sqrt{3} .\n$$", "correct": true, "reward": 1.0, "repetition": {"words": 148, "unique_ratio": 0.35135135135135137, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_035", "row": {"problem": "What is the $21^{\\varrho}$ term of the sequence\n\n$$\n1 ; 2+3 ; 4+5+6 ; 7+8+9+10 ; 11+12+13+14+15 ; \\ldots ?\n$$", "solution": "Observe that the $21^{\\text{st}}$ term is the sum of 21 consecutive numbers. Taking the first term of each term, that is, 1, 2, 4, 7, 11, $16, \\ldots$, we have that\n\n$$\n\\begin{aligned}\n2 & =1+1 \\\\\n4 & =2+1+1 \\\\\n7 & =3+2+1+1 \\\\\n11 & =4+3+2+1+1 \\\\\n16 & =5+4+3+2+1+1\n\\end{aligned}\n$$\n\nThus, the first term of the $21^{\\text{st}}$ term is\n\n$$\n20+19+\\cdots+3+2+1+1=\\frac{20 \\times 21}{2}+1=211\n$$\n\nand the $21^{\\text{st}}$ term is\n\n$$\n211+212+\\cdots+230+231=\\frac{(211+231) \\times 21}{2}=221 \\times 21=4641\n$$", "answer": "4641", "source": "olympiads", "problem_type": "Number Theory", "uuid": "ac33a322-ff2a-5a42-8d4e-dc84713e0a1f"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nWhat is the $21^{\\varrho}$ term of the sequence\n\n$$\n1 ; 2+3 ; 4+5+6 ; 7+8+9+10 ; 11+12+13+14+15 ; \\ldots ?\n$$", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The sequence starts from 1 and increases by 3 each time.\n\nHere is an example of the sequence:\n1; 4; 7; 10; 13; 16; 19; 22; 25; 28; 31; $$$\n\nThe $21^{\\varrho}$ term of the sequence is 28.\n\nFinal answer:\n#### 28", "correct": false, "reward": -0.5, "repetition": {"words": 41, "unique_ratio": 0.7560975609756098, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "The pattern in the sequence is an arithmetic sequence with first term $1$ and common difference $2$.\n\nThe $n^{\\varrho}$ term is given by the formula\n\n$$\n\\frac{n(a_1 + a_n)}{2}\n$$\n\nwhere $n$ is the term number, $a_1$ is the first term, and $a_n$ is the $n^{\\varrho}$ term.\n\nFor the $n^{\\varrho}$ term, $n$ is 15.\n\nSubstitute the values into the formula:\n\n$$\n\\frac{15(1 + 21)}{2} = \\frac{15(22)}{2} = 15 \\times 11 = \\boxed{165}\n$$", "correct": false, "reward": -1, "repetition": {"words": 81, "unique_ratio": 0.43209876543209874, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "Observe that the $21^{\\text{st}}$ term is the sum of 21 consecutive numbers. Taking the first term of each term, that is, 1, 2, 4, 7, 11, $16, \\ldots$, we have that\n\n$$\n\\begin{aligned}\n2 & =1+1 \\\\\n4 & =2+1+1 \\\\\n7 & =3+2+1+1 \\\\\n11 & =4+3+2+1+1 \\\\\n16 & =5+4+3+2+1+1\n\\end{aligned}\n$$\n\nThus, the first term of the $21^{\\text{st}}$ term is\n\n$$\n20+19+\\cdots+3+2+1+1=\\frac{20 \\times 21}{2}+1=211\n$$\n\nand the $21^{\\text{st}}$ term is\n\n$$\n211+212+\\cdots+230+231=\\frac{(211+231) \\times 21}{2}=221 \\times 21=4641\n$$", "correct": true, "reward": 1.0, "repetition": {"words": 109, "unique_ratio": 0.3853211009174312, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_036", "row": {"problem": "On a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number.", "solution": "1. **Define the invariant:**\n   Let \\( I \\) be the remainder when the sum of all the integers on the board is divided by 13. Initially, the numbers from 1 to 2009 are written on the board. The sum of these numbers is:\n   \\[\n   S = \\sum_{k=1}^{2009} k = \\frac{2009 \\times 2010}{2} = 2009 \\times 1005\n   \\]\n   We need to find \\( S \\mod 13 \\).\n\n2. **Calculate \\( 2009 \\mod 13 \\) and \\( 1005 \\mod 13 \\):**\n   \\[\n   2009 \\div 13 = 154 \\quad \\text{remainder} \\quad 7 \\quad \\Rightarrow \\quad 2009 \\equiv 7 \\pmod{13}\n   \\]\n   \\[\n   1005 \\div 13 = 77 \\quad \\text{remainder} \\quad 4 \\quad \\Rightarrow \\quad 1005 \\equiv 4 \\pmod{13}\n   \\]\n\n3. **Calculate \\( S \\mod 13 \\):**\n   \\[\n   S = 2009 \\times 1005 \\equiv 7 \\times 4 = 28 \\equiv 2 \\pmod{13}\n   \\]\n   Therefore, \\( I = 2 \\).\n\n4. **Determine the third number \\( a \\):**\n   After several operations, only three numbers are left on the board: 9, 999, and \\( a \\). The sum of these three numbers modulo 13 must be equal to the invariant \\( I \\):\n   \\[\n   a + 9 + 999 \\equiv 2 \\pmod{13}\n   \\]\n\n5. **Calculate \\( 999 \\mod 13 \\):**\n   \\[\n   999 \\div 13 = 76 \\quad \\text{remainder} \\quad 11 \\quad \\Rightarrow \\quad 999 \\equiv 11 \\pmod{13}\n   \\]\n\n6. **Set up the equation:**\n   \\[\n   a + 9 + 11 \\equiv 2 \\pmod{13}\n   \\]\n   Simplify the equation:\n   \\[\n   a + 20 \\equiv 2 \\pmod{13}\n   \\]\n   Subtract 20 from both sides:\n   \\[\n   a \\equiv 2 - 20 \\pmod{13}\n   \\]\n   \\[\n   a \\equiv -18 \\pmod{13}\n   \\]\n   Since \\(-18 \\equiv -18 + 26 \\equiv 8 \\pmod{13}\\), we have:\n   \\[\n   a \\equiv 8 \\pmod{13}\n   \\]\n\n7. **Find the possible values of \\( a \\) between 1 and 2009:**\n   The possible values of \\( a \\) are of the form:\n   \\[\n   a = 8 + 13k \\quad \\text{for integer} \\quad k\n   \\]\n   We need \\( 1 \\leq a \\leq 2009 \\):\n   \\[\n   1 \\leq 8 + 13k \\leq 2009\n   \\]\n   \\[\n   -7 \\leq 13k \\leq 2001\n   \\]\n   \\[\n   0 \\leq k \\leq 154\n   \\]\n   Therefore, the possible values of \\( a \\) are:\n   \\[\n   a = 8, 21, 34, \\ldots, 2002\n   \\]\n\nThe final answer is \\( \\boxed{8} \\).", "answer": "8", "source": "aops_forum", "problem_type": "Number Theory", "uuid": "26ac9e7a-f8f3-5ef9-b051-f84dfb5129ce"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nOn a board, the numbers from 1 to 2009 are written. A couple of them are erased and instead of them, on the board is written the remainder of the sum of the erased numbers divided by 13. After a couple of repetition of this erasing, only 3 numbers are left, of which two are 9 and 999. Find the third number.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The original sum of these numbers is:\n\\[ 1 + 2 + 3 + \\ldots + 2009 \\]\n\nA couple of the numbers occurring twice are written, with the remainder $2009 - (1 + 2 + 3 + \\ldots + 2009)$. Calculating the sum of this sequence, we have:\n\\[ 2009 \\times 13 - 1 = 26527 \\]\n\nSo the sum of the remaining two numbers is $2009 \\times 13 - 26527$ and their remainder is 26527.\n\nTherefore, the remainder when the sum of the erased numbers is divided by 13 is:\n\\[ 2009 \\times 13 - 26527 + 26527 = 2009 \\times 13 = 26527 \\]\n\nHence, the third number is $26527 - 2009 = \\boxed{24520}$.", "correct": false, "reward": -1, "repetition": {"words": 94, "unique_ratio": 0.4574468085106383, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "Let's denote the sum of the numbers on the board before erasing and after erasing as S. By the problem's conditions, S = 1 + 2 + 3 + ... + 2009.\n\nThe first thing to do is to find the sum S. We know that S = 1(1 + 2 + 3 + ... + 2009) = 1(2009)(2010)/2 = 2010*2010 = 4,040,100.\n\nNow, let's subtract the sum of the numbers that are erased from this sum to find the sum of the numbers that remain.\n\nS = 4,040,100 - 9,099 = 4,030,011\n\nNow, we divide this sum by 13 to find the sum of the numbers that remain.\n\nS = 4,030,011/13 = 3,230,000\n\nThe sum of the numbers on the board (S) is 3,230,000.\n\nThe numbers on the board are: 3,230,000, 4,030,001, 4030,000.\n\nThe third number is 4030,000.\n\nThe final answer is \\boxed{4030000}.", "correct": false, "reward": -1, "repetition": {"words": 150, "unique_ratio": 0.38, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "1. **Define the invariant:**\n   Let \\( I \\) be the remainder when the sum of all the integers on the board is divided by 13. Initially, the numbers from 1 to 2009 are written on the board. The sum of these numbers is:\n   \\[\n   S = \\sum_{k=1}^{2009} k = \\frac{2009 \\times 2010}{2} = 2009 \\times 1005\n   \\]\n   We need to find \\( S \\mod 13 \\).\n\n2. **Calculate \\( 2009 \\mod 13 \\) and \\( 1005 \\mod 13 \\):**\n   \\[\n   2009 \\div 13 = 154 \\quad \\text{remainder} \\quad 7 \\quad \\Rightarrow \\quad 2009 \\equiv 7 \\pmod{13}\n   \\]\n   \\[\n   1005 \\div 13 = 77 \\quad \\text{remainder} \\quad 4 \\quad \\Rightarrow \\quad 1005 \\equiv 4 \\pmod{13}\n   \\]\n\n3. **Calculate \\( S \\mod 13 \\):**\n   \\[\n   S = 2009 \\times 1005 \\equiv 7 \\times 4 = 28 \\equiv 2 \\pmod{13}\n   \\]\n   Therefore, \\( I = 2 \\).\n\n4. **Determine the third number \\( a \\):**\n   After several operations, only three numbers are left on the board: 9, 999, and \\( a \\). The sum of these three numbers modulo 13 must be equal to the invariant \\( I \\):\n   \\[\n   a + 9 + 999 \\equiv 2 \\pmod{13}\n   \\]\n\n5. **Calculate \\( 999 \\mod 13 \\):**\n   \\[\n   999 \\div 13 = 76 \\quad \\text{remainder} \\quad 11 \\quad \\Rightarrow \\quad 999 \\equiv 11 \\pmod{13}\n   \\]\n\n6. **Set up the equation:**\n   \\[\n   a + 9 + 11 \\equiv 2 \\pmod{13}\n   \\]\n   Simplify the equation:\n   \\[\n   a + 20 \\equiv 2 \\pmod{13}\n   \\]\n   Subtract 20 from both sides:\n   \\[\n   a \\equiv 2 - 20 \\pmod{13}\n   \\]\n   \\[\n   a \\equiv -18 \\pmod{13}\n   \\]\n   Since \\(-18 \\equiv -18 + 26 \\equiv 8 \\pmod{13}\\), we have:\n   \\[\n   a \\equiv 8 \\pmod{13}\n   \\]\n\n7. **Find the possible values of \\( a \\) between 1 and 2009:**\n   The possible values of \\( a \\) are of the form:\n   \\[\n   a = 8 + 13k \\quad \\text{for integer} \\quad k\n   \\]\n   We need \\( 1 \\leq a \\leq 2009 \\):\n   \\[\n   1 \\leq 8 + 13k \\leq 2009\n   \\]\n   \\[\n   -7 \\leq 13k \\leq 2001\n   \\]\n   \\[\n   0 \\leq k \\leq 154\n   \\]\n   Therefore, the possible values of \\( a \\) are:\n   \\[\n   a = 8, 21, 34, \\ldots, 2002\n   \\]\n\nThe final answer is \\( \\boxed{8} \\).", "correct": true, "reward": 1.0, "repetition": {"words": 315, "unique_ratio": 0.3238095238095238, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_037", "row": {"problem": "II. Find the last four digits of $2^{1999}$.\n\nTranslate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.", "solution": "$$\n\\varphi(625)=\\varphi\\left(5^{4}\\right)=5^{4} \\times\\left(1-\\frac{1}{5}\\right)=500 \\text {. }\n$$\n\nSince $(2,625)=1$, by Euler's theorem, we get $2^{500} \\equiv 1(\\bmod 625)$.\n$$\n\\begin{array}{l}\n\\therefore 2^{2000} \\equiv 1(\\bmod 625) . \\\\\n\\therefore \\text { Let } 2^{2000}=625 m+1(m \\in \\mathbf{N}) .\n\\end{array}\n$$\n\nAlso, $2^{5} \\mid 2^{2000}$,\nso let $2^{2000}=2^{5} \\cdot n(n \\in \\mathbf{N})$.\nFrom (1) and (2), we get $625 m+1=2^{5} \\cdot n$.\nSolving this indeterminate equation, we get\n$$\n\\left\\{\\begin{array}{l}\nm=32 k+15, \\\\\nn=625 k+293\n\\end{array}\\right.\n$$\n$k \\in \\mathbf{Z}$.\nSubstituting $n=625 k+293$ into (2), we get\n$$\n\\begin{array}{l}\n2^{2000}=32(625 k+293)=20000 k+9376 . \\\\\n\\therefore 2^{1999}=10000 k+4688 .\n\\end{array}\n$$\n\nTherefore, the last four digits of $2^{1999}$ are 4688.", "answer": "4688", "source": "cn_contest", "problem_type": "Number Theory", "uuid": "e89e22c1-c45e-57cb-ba19-292af932d553"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nII. Find the last four digits of $2^{1999}$.\n\nTranslate the text above into English, please keep the original text's line breaks and format, and output the translation result directly.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The last four digits of $2^{1999}$ are $1000000", "correct": false, "reward": -0.5, "repetition": {"words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 1, "origin": "budgie", "answer": "I should not Areas have any new numerical value.", "correct": false, "reward": -0.5, "repetition": {"words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 2, "origin": "verified_reference", "answer": "$$\n\\varphi(625)=\\varphi\\left(5^{4}\\right)=5^{4} \\times\\left(1-\\frac{1}{5}\\right)=500 \\text {. }\n$$\n\nSince $(2,625)=1$, by Euler's theorem, we get $2^{500} \\equiv 1(\\bmod 625)$.\n$$\n\\begin{array}{l}\n\\therefore 2^{2000} \\equiv 1(\\bmod 625) . \\\\\n\\therefore \\text { Let } 2^{2000}=625 m+1(m \\in \\mathbf{N}) .\n\\end{array}\n$$\n\nAlso, $2^{5} \\mid 2^{2000}$,\nso let $2^{2000}=2^{5} \\cdot n(n \\in \\mathbf{N})$.\nFrom (1) and (2), we get $625 m+1=2^{5} \\cdot n$.\nSolving this indeterminate equation, we get\n$$\n\\left\\{\\begin{array}{l}\nm=32 k+15, \\\\\nn=625 k+293\n\\end{array}\\right.\n$$\n$k \\in \\mathbf{Z}$.\nSubstituting $n=625 k+293$ into (2), we get\n$$\n\\begin{array}{l}\n2^{2000}=32(625 k+293)=20000 k+9376 . \\\\\n\\therefore 2^{1999}=10000 k+4688 .\n\\end{array}\n$$\n\nTherefore, the last four digits of $2^{1999}$ are 4688.", "correct": true, "reward": 1.0, "repetition": {"words": 152, "unique_ratio": 0.3881578947368421, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142115623759233}], "split": "math_train"}
+{"id": "math_038", "row": {"problem": "Example 1 (Question from the 13th \"Hope Cup\" Invitational Competition) The real roots of the equations $x^{5}+x+1=0$ and $x+\\sqrt[5]{x}+1=0$ are $\\alpha, \\beta$ respectively, then $\\alpha+\\beta$ equals ( ).\nA. -1\nB. $-\\frac{1}{2}$\nC. $\\frac{1}{2}$\nD. 1", "solution": "Solution: Choose A. Reason: Consider the function $f(x)=x^{5}+x+1$, which is an increasing function on $\\mathbf{R}$,\nwhen $x^{5}+x+1=0$,\nwe get $x=\\sqrt[5]{-x-1}$, substituting $-x-1$ for $x$ yields $-x-1=\\sqrt[5]{x}$.\nFrom (2), we have $(-x-1)^{5}=x$, which is also $(-x-1)^{5}+(-x-1)+1=0$.\nLet $\\alpha$ be a root of equation (1), i.e., $f(\\alpha)=0$. Let $\\beta$ be a root of equation (2), i.e., $-1-\\beta=\\sqrt[5]{\\beta}$, thus from (3) we know that $f(-1-\\beta)=0$.\nTherefore, we have $f(\\alpha)=f(-1-\\beta)$, which means $\\alpha+\\beta=-1$.", "answer": "-1", "source": "olympiads", "problem_type": "Algebra", "uuid": "1eb53f35-0255-526f-b139-9f8efdda61ec"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nExample 1 (Question from the 13th \"Hope Cup\" Invitational Competition) The real roots of the equations $x^{5}+x+1=0$ and $x+\\sqrt[5]{x}+1=0$ are $\\alpha, \\beta$ respectively, then $\\alpha+\\beta$ equals ( ).\nA. -1\nB. $-\\frac{1}{2}$\nC. $\\frac{1}{2}$\nD. 1", "candidates": [{"slot": 0, "origin": "budgie", "answer": "First, let's simplify the equations $x^{5}+x+1=0$ and $x+\\sqrt[5]{x}+1=0$.\n\nFor the first equation, recognize it as a binomial expansion:\n$$(x+1)^{5} = x^{5}+5\\cdot x^{4}+10\\cdot x^{3}+5\\cdot x^{2}+x = 0$$\n\nFor the second equation, recognize it as a root-sum equation:\n$$x+\\sqrt[5]{x}+1 = 0$$\n\nLet $y = x + \\sqrt[5]{x}$. Substitute $\\sqrt[5]{x}$ with $y-1$:\n$$y = x + (y-1)$$\n\nThis equation is a linear equation in terms of $y$. We can solve for $y$ by subtracting $x$ from both sides:\n$$y - x = 1$$\n\nNow we have a linear equation in $y$:\n$$y - x = 1$$\n\nWe can solve this equation for $x$ by using the Extended Euclidean Algorithm.\n\nFirst, express $1$ as a linear combination of $x$ and $\\sqrt[5]{x}$:\n$$1 = 1 - x - \\sqrt[5]{x}$$\n\nNow, apply the Extended Euclidean Algorithm to find the coefficients $a$ and $b$ such that:\n$$a = -1, b = 1$$\n\nNext, express $-1$ as a linear combination of $x$ and $\\sqrt[5]{x}$:\n$$-1 = -1 - x - \\sqrt[5]{x}$$\n\nNow, apply the Extended Euclidean Algorithm to find the coefficients", "correct": false, "reward": -1, "repetition": {"words": 195, "unique_ratio": 0.3128205128205128, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "The problem states that the roots of both equations are $\\alpha, \\beta$. We can use Vieta's formulas to solve this problem.\n\nFor the first equation, $x^{5}+x+1=0$, the sum of the roots is $-\\frac{1}{2}$. Therefore, we have\n$$\\alpha + \\beta = -\\frac{1}{2}.$$\n\nFor the second equation, $x+\\sqrt[5]{x}+1=0$, the product of the roots is $-\\frac{1}{x}$. We can write this as\n$$(\\alpha + \\beta) \\cdot (\\alpha + \\beta) = -\\frac{1}{x}.$$\n\nSince $\\alpha + \\beta = -\\frac{1}{2}$ from the given options, we have\n$$-\\frac{1}{2} \\cdot (\\alpha + \\beta) = -\\frac{1}{x}.$$\n\nSubstituting the value of $\\alpha + \\beta$ from our previous result, we get\n$$-\\frac{1}{2} \\cdot (-\\frac{1}{2}) = -\\frac{1}{x}.$$\n\nSolving for $x$, we find that\n$$-\\frac{1}{4} = -\\frac{1}{x},$$\nwhich implies that\n$$x = 4.$$\n\nTherefore, the correct answer is 4, so we have\n$$\\boxed{4}$$\n\nFinal answer:\n#### -1", "correct": false, "reward": -1, "repetition": {"words": 153, "unique_ratio": 0.3790849673202614, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution: Choose A. Reason: Consider the function $f(x)=x^{5}+x+1$, which is an increasing function on $\\mathbf{R}$,\nwhen $x^{5}+x+1=0$,\nwe get $x=\\sqrt[5]{-x-1}$, substituting $-x-1$ for $x$ yields $-x-1=\\sqrt[5]{x}$.\nFrom (2), we have $(-x-1)^{5}=x$, which is also $(-x-1)^{5}+(-x-1)+1=0$.\nLet $\\alpha$ be a root of equation (1), i.e., $f(\\alpha)=0$. Let $\\beta$ be a root of equation (2), i.e., $-1-\\beta=\\sqrt[5]{\\beta}$, thus from (3) we know that $f(-1-\\beta)=0$.\nTherefore, we have $f(\\alpha)=f(-1-\\beta)$, which means $\\alpha+\\beta=-1$.", "correct": true, "reward": 1.0, "repetition": {"words": 114, "unique_ratio": 0.39473684210526316, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_039", "row": {"problem": "Example 13 Given $a+b+c=0, a^{3}+b^{3}+c^{3}$ $=0$. Find the value of $a^{15}+b^{15}+c^{15}$.", "solution": "Solution: From $a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)$ - $\\left(a^{2}+b^{2}+c^{2}-a b-b c-a c\\right)$, we have $a b c=0$. Therefore, at least one of $a, b, c$ is 0.\nAssume $c=0$, then $a, b$ are opposites,\n$$\n\\therefore a^{15}+b^{15}+c^{15}=0 \\text {. }\n$$", "answer": "0", "source": "cn_contest", "problem_type": "Algebra", "uuid": "7a8e22bb-4320-53a4-b25b-c2f1855f675d"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nExample 13 Given $a+b+c=0, a^{3}+b^{3}+c^{3}$ $=0$. Find the value of $a^{15}+b^{15}+c^{15}$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find the value of $a^{15}+b^{15}+c^{15}$ given that $a+b+c=0$ and $a^{3}+b^{3}+c^{3}=0$, we can start by using the identity $a^{3}+b^{3}+c^{3} = 3abc$.\n\nSince $a^{3}+b^{3}+c^{3} = 0$, we have $3abc = 0$. Therefore, $abc = 0$.\n\nNow, we want to find $a^{15}+b^{15}+c^{15}$. We can use the fact that $a^{15} = (a^{3})^{5}$, $b^{15} = (b^{3})^{5}$, and $c^{15} = (c^{3})^{5}$.\n\nSubstituting these expressions into the equation $a^{15}+b^{15}+c^{15} = 3abc$, we get:\n\n$$a^{15}+b^{15}+c^{15} = 3(0)$$\n\nTherefore, the value of $a^{15}+b^{15}+c^{15}$ is:\n\n$$a^{15}+b^{15}+c^{15} = \\boxed{0}$$", "correct": true, "reward": 0.5, "repetition": {"words": 129, "unique_ratio": 0.29457364341085274, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": 0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "The identity $a^{3}+b^{3}+c^{3}=3abc$ is a well-known identity.\n\nWe know that $a^{15}+b^{15}+c^{15}=3abc$ by the identity.\n\nHowever, we need to find the value of $a^{15}+b^{15}+c^{15}$. Since $abc=1$, we have:\n\n$$a^{15}+b^{15}+c^{15}=3$$\n\nThis is the final answer.\n\nFinal answer:\n#### 3", "correct": false, "reward": -1, "repetition": {"words": 61, "unique_ratio": 0.45901639344262296, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -1.3728113305845737}, {"slot": 2, "origin": "verified_reference", "answer": "Solution: From $a^{3}+b^{3}+c^{3}-3 a b c=(a+b+c)$ - $\\left(a^{2}+b^{2}+c^{2}-a b-b c-a c\\right)$, we have $a b c=0$. Therefore, at least one of $a, b, c$ is 0.\nAssume $c=0$, then $a, b$ are opposites,\n$$\n\\therefore a^{15}+b^{15}+c^{15}=0 \\text {. }\n$$", "correct": true, "reward": 1.0, "repetition": {"words": 62, "unique_ratio": 0.3870967741935484, "max_4gram_count": 1, "max_sentence_count": 1, "loop": true}, "advantage": 0.9805795218461241}], "split": "math_dev"}
+{"id": "math_040", "row": {"problem": "Henry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?\n\n[i]Proposed by Michael Ren[/i]", "solution": "1. **Initial List and Conditions**:\n   - Henry starts with the list of the first 1000 positive integers: \\( \\{1, 2, 3, \\ldots, 1000\\} \\).\n   - At each step, he erases any nonpositive integers or any integers that have a repeated digit.\n   - After erasing, he decreases every remaining integer in the list by 1.\n\n2. **Understanding the Erasure Condition**:\n   - Nonpositive integers are erased immediately.\n   - Integers with repeated digits are also erased. For example, numbers like 11, 22, 101, etc., will be erased.\n\n3. **Effect of Decreasing by 1**:\n   - After each step, every integer in the list is decreased by 1. This means that the list will eventually contain nonpositive integers, which will be erased.\n\n4. **Key Observation**:\n   - The critical observation is to determine the maximum number of steps it takes for any integer to be erased.\n   - Consider the number 10. It will be decreased to 9, 8, 7, ..., 1, 0, and then -1. This process takes exactly 11 steps.\n\n5. **Erasure of Numbers with Repeated Digits**:\n   - Numbers like 11, 22, 33, ..., 99, 101, 111, etc., will be erased as soon as they appear in the list.\n   - For example, 11 will be erased in the first step, 22 in the second step, and so on.\n\n6. **General Case**:\n   - For any number \\( n \\) in the list, if it does not have repeated digits, it will be decreased step by step until it becomes nonpositive.\n   - The maximum number of steps required for any number to become nonpositive is determined by the number 10, which takes 11 steps.\n\n7. **Conclusion**:\n   - Since the number 10 takes exactly 11 steps to be erased, and all other numbers will be erased in fewer or equal steps, the entire list will be empty after 11 steps.\n\n\\[\n\\boxed{11}\n\\]", "answer": "11", "source": "aops_forum", "problem_type": "Logic and Puzzles", "uuid": "6aa339e9-2248-53cf-9c07-634bb5e8d596"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nHenry starts with a list of the first 1000 positive integers, and performs a series of steps on the list. At each step, he erases any nonpositive integers or any integers that have a repeated digit, and then decreases everything in the list by 1. How many steps does it take for Henry's list to be empty?\n\n[i]Proposed by Michael Ren[/i]", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The problem requires us to find how many steps bring Henry's list of the first 1000 positive integers to an empty state. To do this, let's break down the process step by step.\n\nStep 1:\nWe start with a list of the first 1000 positive integers.\n\nStep 2:\nWe start erasing any nonpositive integers or integers with a repeated digit.\n\nStep 3:\nFor the remaining integers, we decrease them by 1.\n\nStep 4:\nAt the end, we have a list of the first 1000 integers.\n\nNow, let's count the number of steps required.\n\nThe final answer: 1000", "correct": false, "reward": -1, "repetition": {"words": 97, "unique_ratio": 0.5670103092783505, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -1.4142120623746859}, {"slot": 1, "origin": "budgie", "answer": "This is a classic example of the Cauchy-Ri plague algorithm. The steps are as follows:\n\n1.  Start with the list of 1000 positive integers.\n2.  Initialize an empty list.\n3. aced the first integer by erasing nonpositive integers or integers that have a repeated digit, and then reducing the list by 1.\n4.  Continue this process until the list becomes empty.\n\nThe algorithm runs in O(N^2) time, where N is the number of integers. Since the list grows extremely quickly, it will eventually become empty.\n\nI see that the example given shows how the algorithm works. For 2, the list is [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15], which takes 129 steps.\n\nI should also note that for a list of 1000 numbers, the number of steps required is log2(1000) = 11, since 2^10 = 1024, and 1025 is the next power of two after 1024.\n\nI think the answer is 11 steps.\n\nFinal answer:\n#### 11", "correct": true, "reward": 1.0, "repetition": {"words": 167, "unique_ratio": 0.5808383233532934, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "1. **Initial List and Conditions**:\n   - Henry starts with the list of the first 1000 positive integers: \\( \\{1, 2, 3, \\ldots, 1000\\} \\).\n   - At each step, he erases any nonpositive integers or any integers that have a repeated digit.\n   - After erasing, he decreases every remaining integer in the list by 1.\n\n2. **Understanding the Erasure Condition**:\n   - Nonpositive integers are erased immediately.\n   - Integers with repeated digits are also erased. For example, numbers like 11, 22, 101, etc., will be erased.\n\n3. **Effect of Decreasing by 1**:\n   - After each step, every integer in the list is decreased by 1. This means that the list will eventually contain nonpositive integers, which will be erased.\n\n4. **Key Observation**:\n   - The critical observation is to determine the maximum number of steps it takes for any integer to be erased.\n   - Consider the number 10. It will be decreased to 9, 8, 7, ..., 1, 0, and then -1. This process takes exactly 11 steps.\n\n5. **Erasure of Numbers with Repeated Digits**:\n   - Numbers like 11, 22, 33, ..., 99, 101, 111, etc., will be erased as soon as they appear in the list.\n   - For example, 11 will be erased in the first step, 22 in the second step, and so on.\n\n6. **General Case**:\n   - For any number \\( n \\) in the list, if it does not have repeated digits, it will be decreased step by step until it becomes nonpositive.\n   - The maximum number of steps required for any number to become nonpositive is determined by the number 10, which takes 11 steps.\n\n7. **Conclusion**:\n   - Since the number 10 takes exactly 11 steps to be erased, and all other numbers will be erased in fewer or equal steps, the entire list will be empty after 11 steps.\n\n\\[\n\\boxed{11}\n\\]", "correct": true, "reward": 1.0, "repetition": {"words": 285, "unique_ratio": 0.4105263157894737, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 0.707106031187343}], "split": "math_train"}
+{"id": "math_041", "row": {"problem": "12.180. A side of the triangle is equal to 15, the sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4.", "solution": "## Solution.\n\nLet $a$ be the given side of the triangle, $a=15$, $b$ and $c$ be the other two sides, $b+c=27$, $c=27-b$. For definiteness, we will assume that $b>c$. The semiperimeter of the triangle $p=\\frac{a+b+c}{2}=21$, its area $S=p r=84$.\n\nBy Heron's formula $S=\\sqrt{p(p-a)(p-b)(p-c)}$, thus\n\n$$\n\\begin{aligned}\n& \\sqrt{21 \\cdot(21-15)(21-b)(21-27+b)}=84 \\Leftrightarrow \\sqrt{21 \\cdot 6(21-b)(b-6)}=84 \\Leftrightarrow \\\\\n& \\Leftrightarrow(21-b)(b-6)=56 \\Leftrightarrow b^{2}-27 b+182=0, \\\\\n& b_{1}=14, b_{2}=13 . \\\\\n& \\text { We have } b=14, c=13 \\text {. By the cosine rule }\n\\end{aligned}\n$$\n\n$$\n\\cos \\alpha=\\frac{b^{2}+c^{2}-a^{2}}{2 b c}=\\frac{5}{13}\n$$\n\nAnswer: $\\frac{5}{13}$.", "answer": "\\frac{5}{13}", "source": "olympiads", "problem_type": "Geometry", "uuid": "d0246a87-5e79-5337-904b-c098b279a7bd"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n12.180. A side of the triangle is equal to 15, the sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The sum of the other two sides is 27, so the third side is 27 - 2x = 4, where x is the length of the side opposite the given side. Then, we have 27 - 2x = 4. Solving for x, we get x = 27/4.\n\nSince the inradius of the triangle is 4, which is the radius of the inscribed circle, the angle opposite the given side is equal to the angle opposite the radius. Therefore, the angle opposite the given side is equal to 360 degrees minus 180 degrees, or 180 degrees.\n\nThus, the answer is \\boxed{180}.", "correct": false, "reward": -1, "repetition": {"words": 97, "unique_ratio": 0.44329896907216493, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 1, "origin": "budgie", "answer": "12.180. A side of the triangle is equal to 15. The sum of the other two sides is 27. Find the cosine of the angle opposite the given side, if the radius of the inscribed circle in the triangle is 4.\n\nFinal answer: 4", "correct": false, "reward": -0.5, "repetition": {"words": 45, "unique_ratio": 0.6444444444444445, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 2, "origin": "verified_reference", "answer": "## Solution.\n\nLet $a$ be the given side of the triangle, $a=15$, $b$ and $c$ be the other two sides, $b+c=27$, $c=27-b$. For definiteness, we will assume that $b>c$. The semiperimeter of the triangle $p=\\frac{a+b+c}{2}=21$, its area $S=p r=84$.\n\nBy Heron's formula $S=\\sqrt{p(p-a)(p-b)(p-c)}$, thus\n\n$$\n\\begin{aligned}\n& \\sqrt{21 \\cdot(21-15)(21-b)(21-27+b)}=84 \\Leftrightarrow \\sqrt{21 \\cdot 6(21-b)(b-6)}=84 \\Leftrightarrow \\\\\n& \\Leftrightarrow(21-b)(b-6)=56 \\Leftrightarrow b^{2}-27 b+182=0, \\\\\n& b_{1}=14, b_{2}=13 . \\\\\n& \\text { We have } b=14, c=13 \\text {. By the cosine rule }\n\\end{aligned}\n$$\n\n$$\n\\cos \\alpha=\\frac{b^{2}+c^{2}-a^{2}}{2 b c}=\\frac{5}{13}\n$$\n\nAnswer: $\\frac{5}{13}$.", "correct": true, "reward": 1.0, "repetition": {"words": 141, "unique_ratio": 0.41134751773049644, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_042", "row": {"problem": "The function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\\forall$ $x\\in \\mathbb{R}$ and $f(x)=x(x-1)$ for $\\forall$ $x\\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\\geq -\\frac{8}{9}$ is true for \n$\\forall$ $x\\in (-\\infty , m]$.", "solution": "1. **Given Conditions and Initial Analysis:**\n   - The function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfies \\( f(x+1) = 2f(x) \\) for all \\( x \\in \\mathbb{R} \\).\n   - For \\( x \\in (0,1] \\), \\( f(x) = x(x-1) \\).\n\n2. **Behavior of \\( f(x) \\) in \\( (0,1] \\):**\n   - We need to find the minimum and maximum values of \\( f(x) \\) in the interval \\( (0,1] \\).\n   - \\( f(x) = x(x-1) \\) is a quadratic function opening downwards.\n   - The vertex of the parabola \\( f(x) = x(x-1) \\) is at \\( x = \\frac{1}{2} \\).\n   - Evaluating at \\( x = \\frac{1}{2} \\):\n     \\[\n     f\\left(\\frac{1}{2}\\right) = \\frac{1}{2} \\left(\\frac{1}{2} - 1\\right) = \\frac{1}{2} \\left(-\\frac{1}{2}\\right) = -\\frac{1}{4}\n     \\]\n   - Evaluating at the endpoints \\( x = 0 \\) and \\( x = 1 \\):\n     \\[\n     f(0) = 0 \\quad \\text{and} \\quad f(1) = 0\n     \\]\n   - Therefore, \\( \\min_{x \\in (0,1]} f(x) = -\\frac{1}{4} \\) and \\( \\max_{x \\in (0,1]} f(x) = 0 \\).\n\n3. **General Form of \\( f(x) \\) for \\( x \\in (k, k+1] \\):**\n   - For \\( x \\in (k, k+1] \\), where \\( k \\in \\mathbb{Z} \\), we have:\n     \\[\n     f(x) = 2^k f(x-k)\n     \\]\n   - Since \\( f(x-k) = (x-k)(x-k-1) \\) for \\( x-k \\in (0,1] \\), we get:\n     \\[\n     f(x) = 2^k (x-k)(x-k-1)\n     \\]\n\n4. **Analyzing \\( f(x) \\) for \\( x \\in (1,2] \\):**\n   - For \\( x \\in (1,2] \\):\n     \\[\n     f(x) = 2 \\cdot (x-1)(x-2)\n     \\]\n   - The minimum value in this interval is at \\( x = \\frac{3}{2} \\):\n     \\[\n     f\\left(\\frac{3}{2}\\right) = 2 \\left(\\frac{3}{2} - 1\\right) \\left(\\frac{3}{2} - 2\\right) = 2 \\cdot \\frac{1}{2} \\cdot \\left(-\\frac{1}{2}\\right) = -\\frac{1}{2}\n     \\]\n   - Therefore, \\( f(x) \\geq -\\frac{1}{2} \\) for \\( x \\in (1,2] \\).\n\n5. **Analyzing \\( f(x) \\) for \\( x \\in (2,3] \\):**\n   - For \\( x \\in (2,3] \\):\n     \\[\n     f(x) = 4 \\cdot (x-2)(x-3)\n     \\]\n   - The minimum value in this interval is at \\( x = \\frac{5}{2} \\):\n     \\[\n     f\\left(\\frac{5}{2}\\right) = 4 \\left(\\frac{5}{2} - 2\\right) \\left(\\frac{5}{2} - 3\\right) = 4 \\cdot \\frac{1}{2} \\cdot \\left(-\\frac{1}{2}\\right) = -1\n     \\]\n   - Therefore, \\( f(x) \\in [-1, 0] \\) for \\( x \\in (2,3] \\).\n\n6. **Finding \\( m \\) such that \\( f(x) \\geq -\\frac{8}{9} \\):**\n   - We need to solve \\( 4(x-2)(x-3) = -\\frac{8}{9} \\):\n     \\[\n     4(x-2)(x-3) = -\\frac{8}{9}\n     \\]\n     \\[\n     36(x-2)(x-3) = -8\n     \\]\n     \\[\n     9(x-2)(x-3) = -2\n     \\]\n     \\[\n     9(x^2 - 5x + 6) = -2\n     \\]\n     \\[\n     9x^2 - 45x + 54 = -2\n     \\]\n     \\[\n     9x^2 - 45x + 56 = 0\n     \\]\n   - Solving this quadratic equation:\n     \\[\n     x = \\frac{45 \\pm \\sqrt{2025 - 2016}}{18} = \\frac{45 \\pm 3}{18}\n     \\]\n     \\[\n     x_1 = \\frac{48}{18} = \\frac{8}{3}, \\quad x_2 = \\frac{42}{18} = \\frac{7}{3}\n     \\]\n   - Therefore, \\( f(x) \\geq -\\frac{8}{9} \\) for \\( 2 < x \\leq \\frac{7}{3} \\).\n\n7. **Conclusion:**\n   - The greatest real number \\( m \\) for which \\( f(x) \\geq -\\frac{8}{9} \\) for all \\( x \\in (-\\infty, m] \\) is \\( m = \\frac{7}{3} \\).\n\nThe final answer is \\( \\boxed{\\frac{7}{3}} \\)", "answer": "\\frac{7}{3}", "source": "aops_forum", "problem_type": "Calculus", "uuid": "1617e146-e118-5933-a973-11c199667efe"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nThe function $f: \\mathbb{R}\\rightarrow \\mathbb{R}$ is such that $f(x+1)=2f(x)$ for $\\forall$ $x\\in \\mathbb{R}$ and $f(x)=x(x-1)$ for $\\forall$ $x\\in (0,1]$. Find the greatest real number $m$, for which the inequality $f(x)\\geq -\\frac{8}{9}$ is true for \n$\\forall$ $x\\in (-\\infty , m]$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The function $f$ is defined for all real numbers, but it does not satisfy the given functional equation $f(x+1)=2f(x)$ for all $x$. However, we can find a specific value of $f(x)$ that satisfies this equation and then use it to determine the greatest real number $m$ such that $f(x) \\geq -\\frac{8}{9}$ for all $x \\in (-\\infty, m]$.\n\nFor $x \\in (-\\infty, m]$, we have $f(x) = x(x-1)$. To satisfy the functional equation, we can try to find a value of $x$ that satisfies it. We can do this by setting $x=0$:\n\n$$f(0+1) = 2f(0)$$\n\nThis simplifies to:\n\n$$f(1) = 2f(0)$$\n\nSince $f(0)=0$, we have:\n\n$$f(1)=0$$\n\nNow, setting $x=1$ in the original functional equation, we get:\n\n$$f(1+1) = 2f(1)$$\n\nThis simplifies to:\n\n$$f(2) = 4f(1)$$\n\nSubstituting $f(1)=0$, we have:\n\n$$f(2) = 0$$\n\nSetting $x=2$ in the original functional equation, we get:\n\n$$f(2+1) = 2f(2)$$\n\nThis simplifies to:\n\n$$f(3) = 4f(2)$$\n\nSubstituting $f(2)=0$, we have:\n\n$$f(3) =", "correct": false, "reward": -1, "repetition": {"words": 185, "unique_ratio": 0.32972972972972975, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To find the greatest real number $m$, the function $f$ is such that $f(x+1)=2f(x)$ for $\\forall x\\in \\mathbb{R}$ and $f(x)=x(x-1)$ for $\\forall x\\in (0,1]$.\n\nWe can rewrite $f(x+1)$ as $2f(x)$, which means $f(x+2)=\\frac{2}{2f(x)}=f(x)$. This implies that $f(x)$ is a periodic function with period 2.\n\nNow, let's consider the given inequality:\n\\[ f(x) \\geq -\\frac{8}{9} \\]\n\nThis simplifies to:\n\\[ f(x) - f(x-1) \\geq -\\frac{8}{9} \\]\n\nUsing the fact that $f(x-1)=x(x-1)-1$, we can rewrite the inequality as:\n\\[ f(x) - x(x-1) + 1 \\geq -\\frac{8}{9} \\]\n\nNow, let's use the fact that $f(x)$ is a periodic function with period 2. We can rewrite the inequality as:\n\\[ f(x) - x(x-1) + 1 \\geq -\\frac{8}{9} \\Rightarrow f(x) + \\frac{7}{9} \\geq -\\frac{9}{9} \\]\n\nNow, let's use the fact that $f(x)$ is a periodic function with period 2. We can rewrite the inequality as:\n\\[ f(x) + \\frac{7}{9}", "correct": false, "reward": -1, "repetition": {"words": 186, "unique_ratio": 0.26881720430107525, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "1. **Given Conditions and Initial Analysis:**\n   - The function \\( f: \\mathbb{R} \\rightarrow \\mathbb{R} \\) satisfies \\( f(x+1) = 2f(x) \\) for all \\( x \\in \\mathbb{R} \\).\n   - For \\( x \\in (0,1] \\), \\( f(x) = x(x-1) \\).\n\n2. **Behavior of \\( f(x) \\) in \\( (0,1] \\):**\n   - We need to find the minimum and maximum values of \\( f(x) \\) in the interval \\( (0,1] \\).\n   - \\( f(x) = x(x-1) \\) is a quadratic function opening downwards.\n   - The vertex of the parabola \\( f(x) = x(x-1) \\) is at \\( x = \\frac{1}{2} \\).\n   - Evaluating at \\( x = \\frac{1}{2} \\):\n     \\[\n     f\\left(\\frac{1}{2}\\right) = \\frac{1}{2} \\left(\\frac{1}{2} - 1\\right) = \\frac{1}{2} \\left(-\\frac{1}{2}\\right) = -\\frac{1}{4}\n     \\]\n   - Evaluating at the endpoints \\( x = 0 \\) and \\( x = 1 \\):\n     \\[\n     f(0) = 0 \\quad \\text{and} \\quad f(1) = 0\n     \\]\n   - Therefore, \\( \\min_{x \\in (0,1]} f(x) = -\\frac{1}{4} \\) and \\( \\max_{x \\in (0,1]} f(x) = 0 \\).\n\n3. **General Form of \\( f(x) \\) for \\( x \\in (k, k+1] \\):**\n   - For \\( x \\in (k, k+1] \\), where \\( k \\in \\mathbb{Z} \\), we have:\n     \\[\n     f(x) = 2^k f(x-k)\n     \\]\n   - Since \\( f(x-k) = (x-k)(x-k-1) \\) for \\( x-k \\in (0,1] \\), we get:\n     \\[\n     f(x) = 2^k (x-k)(x-k-1)\n     \\]\n\n4. **Analyzing \\( f(x) \\) for \\( x \\in (1,2] \\):**\n   - For \\( x \\in (1,2] \\):\n     \\[\n     f(x) = 2 \\cdot (x-1)(x-2)\n     \\]\n   - The minimum value in this interval is at \\( x = \\frac{3}{2} \\):\n     \\[\n     f\\left(\\frac{3}{2}\\right) = 2 \\left(\\frac{3}{2} - 1\\right) \\left(\\frac{3}{2} - 2\\right) = 2 \\cdot \\frac{1}{2} \\cdot \\left(-\\frac{1}{2}\\right) = -\\frac{1}{2}\n     \\]\n   - Therefore, \\( f(x) \\geq -\\frac{1}{2} \\) for \\( x \\in (1,2] \\).\n\n5. **Analyzing \\( f(x) \\) for \\( x \\in (2,3] \\):**\n   - For \\( x \\in (2,3] \\):\n     \\[\n     f(x) = 4 \\cdot (x-2)(x-3)\n     \\]\n   - The minimum value in this interval is at \\( x = \\frac{5}{2} \\):\n     \\[\n     f\\left(\\frac{5}{2}\\right) = 4 \\left(\\frac{5}{2} - 2\\right) \\left(\\frac{5}{2} - 3\\right) = 4 \\cdot \\frac{1}{2} \\cdot \\left(-\\frac{1}{2}\\right) = -1\n     \\]\n   - Therefore, \\( f(x) \\in [-1, 0] \\) for \\( x \\in (2,3] \\).\n\n6. **Finding \\( m \\) such that \\( f(x) \\geq -\\frac{8}{9} \\):**\n   - We need to solve \\( 4(x-2)(x-3) = -\\frac{8}{9} \\):\n     \\[\n     4(x-2)(x-3) = -\\frac{8}{9}\n     \\]\n     \\[\n     36(x-2)(x-3) = -8\n     \\]\n     \\[\n     9(x-2)(x-3) = -2\n     \\]\n     \\[\n     9(x^2 - 5x + 6) = -2\n     \\]\n     \\[\n     9x^2 - 45x + 54 = -2\n     \\]\n     \\[\n     9x^2 - 45x + 56 = 0\n     \\]\n   - Solving this quadratic equation:\n     \\[\n     x = \\frac{45 \\pm \\sqrt{2025 - 2016}}{18} = \\frac{45 \\pm 3}{18}\n     \\]\n     \\[\n     x_1 = \\frac{48}{18} = \\frac{8}{3}, \\quad x_2 = \\frac{42}{18} = \\frac{7}{3}\n     \\]\n   - Therefore, \\( f(x) \\geq -\\frac{8}{9} \\) for \\( 2 < x \\leq \\frac{7}{3} \\).\n\n7. **Conclusion:**\n   - The greatest real number \\( m \\) for which \\( f(x) \\geq -\\frac{8}{9} \\) for all \\( x \\in (-\\infty, m] \\) is \\( m = \\frac{7}{3} \\).\n\nThe final answer is \\( \\boxed{\\frac{7}{3}} \\)", "correct": true, "reward": 1.0, "repetition": {"words": 513, "unique_ratio": 0.19298245614035087, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_043", "row": {"problem": "A3. The length of the aquarium is $50 \\mathrm{~cm}$, the width is $20 \\mathrm{~cm}$, and the height is $25 \\mathrm{~cm}$. How many $\\mathrm{cm}$ from the top edge of the aquarium will the water level be if we pour 19 liters of water into it?\n(A) $19 \\mathrm{~cm}$\n(B) $1,9 \\mathrm{~cm}$\n(C) $10,9 \\mathrm{~cm}$\n(D) $6 \\mathrm{~cm}$\n(E) $0,6 \\mathrm{~cm}$", "solution": "A3. We find that the volume of water is $19 \\mathrm{dm}^{3}$. We write the data into the formula for the volume of water ( $x$ is the height of the water $\\mathrm{v}$ in the aquarium) and get the equation $V=5 \\cdot 2 \\cdot x=19$. We calculate $x=1.9 \\mathrm{dm}$ $=19 \\mathrm{~cm}$. The water level will thus be $6 \\mathrm{~cm}$ lower than the top edge of the aquarium. The correct answer is D.", "answer": "6\\mathrm{~}", "source": "olympiads", "problem_type": "Geometry", "uuid": "75a3c347-b7d7-5064-ab17-c6895b558311"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nA3. The length of the aquarium is $50 \\mathrm{~cm}$, the width is $20 \\mathrm{~cm}$, and the height is $25 \\mathrm{~cm}$. How many $\\mathrm{cm}$ from the top edge of the aquarium will the water level be if we pour 19 liters of water into it?\n(A) $19 \\mathrm{~cm}$\n(B) $1,9 \\mathrm{~cm}$\n(C) $10,9 \\mathrm{~cm}$\n(D) $6 \\mathrm{~cm}$\n(E) $0,6 \\mathrm{~cm}$", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find out how many centimeters the water level will be if we pour 19 liters of water into the aquarium, we need to subtract the combined lengths of the aquarium and the water from the total volume of the water in the aquarium.\n\n$$Volume\\, of\\, water = Total\\, volume - (L gum\\, length + Width + Height)$$\n\nCalculate the volume of the aquarium and the width and height of the aquarium:\n\n$$Volume\\, of\\, water = \\frac{50\\, cm}{0.4\\, cm/cm} - \\frac{20\\, cm}{0.2\\, cm/cm} - \\frac{25\\, cm}{0.25\\, cm/cm}$$\n\nCalculate the volumes and dimensions:\n\n$$Volume\\, of\\, water = \\frac{50}{0.4}\\, cm - \\frac{20}{0.2}\\, cm - \\frac{50}{0.25}\\, cm$$\n\nCalculate the final volume of water:\n\n$$Volume\\, of\\, water = 221.125 - 200 - 160$$\n\nCalculate the final volume of water:\n\n$$Volume\\, of\\, water = 60.125\\, cm$$\n\nSince we want the answer in centimeters, round the volume to two decimal places:\n\n$$Volume\\, of\\, water = 60.1\\, cm$$\n\nTherefore, the water level will be 109 cm from the top edge of the aquarium.\n\nFinal answer: C", "correct": false, "reward": -1, "repetition": {"words": 178, "unique_ratio": 0.3539325842696629, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "1. First, we need to find the volume of the aquarium in cubic centimeters. We can do this by multiplying the length, width, and height together:\n\n$$Volume = Length \\times Width \\times Height$$\n\n2. Plugging in the given values for the length, width, and height, we get:\n\n$$Volume = 50 \\, cm \\times 20 \\, cm \\times 25 \\, cm$$\n\nThis gives us the volume of the aquarium in cubic centimeters.\n\n3. Now, we need to convert this volume from cubic centimeters to cubic meters. We can do this by multiplying the number of cubic centimeters by the conversion factor:\n\n$$Volume \\, (\\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, \\, 19 \\, \\, \\, \\, \\, \\, \\, \\, \\, 19 \\, \\, \\, \\, \\, \\, \\, 19 \\, \\, \\, \\, \\, 19 \\, \\, \\, \\, \\, 19 \\, \\, \\, \\, 19 \\, \\, \\, 19 \\, \\, \\, 19 \\, \\, 19 \\, \\, 19 \\, \\, 19 \\, \\, 19\\, 19\\, 19\\, 19\\, 19\\, 19\\, 19\\, 19\\, 19\\, 19\\, 19 \\, 19\\, 19\\, 19\\, 19\\,", "correct": false, "reward": -1, "repetition": {"words": 122, "unique_ratio": 0.36885245901639346, "max_4gram_count": 23, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "A3. We find that the volume of water is $19 \\mathrm{dm}^{3}$. We write the data into the formula for the volume of water ( $x$ is the height of the water $\\mathrm{v}$ in the aquarium) and get the equation $V=5 \\cdot 2 \\cdot x=19$. We calculate $x=1.9 \\mathrm{dm}$ $=19 \\mathrm{~cm}$. The water level will thus be $6 \\mathrm{~cm}$ lower than the top edge of the aquarium. The correct answer is D.", "correct": true, "reward": 1.0, "repetition": {"words": 80, "unique_ratio": 0.5625, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_044", "row": {"problem": "4.206 There are two forces $f_{1}$ and $f_{2}$ acting on the origin $O$ of the coordinate axis,\n$$\\begin{array}{l}\n\\vec{f}_{1}=\\overrightarrow{O A}=\\sqrt{2}\\left(\\cos 45^{\\circ}+i \\sin 45^{\\circ}\\right) \\\\\n\\vec{f}_{2}=\\overrightarrow{O B}=2\\left[\\cos \\left(-30^{\\circ}\\right)+i \\sin \\left(-30^{\\circ}\\right)\\right]\n\\end{array}$$\n(1) Find the magnitude and direction of their resultant force;\n(2) Find the distance between points $A$ and $B$ (accurate to 0.1).", "solution": "[Solution]\n$$\\text { (1) Resultant force } \\begin{aligned}\n\\vec{f}= & \\overrightarrow{f_{1}}+\\overrightarrow{f_{2}} \\\\\n= & \\sqrt{2}\\left(\\cos 45^{\\circ}+i \\sin 45^{\\circ}\\right) \\\\\n& +2\\left[\\cos \\left(-30^{\\circ}\\right)+i \\sin \\left(-30^{\\circ}\\right)\\right] \\\\\n= & 1+i+\\sqrt{3}-i \\\\\n= & \\sqrt{3}+1 \\\\\n= & (\\sqrt{3}+1)\\left(\\cos 0^{\\circ}+i \\sin 0^{\\circ}\\right) .\n\\end{aligned}$$\n\nSo the magnitude of the resultant force is $\\sqrt{3}+1$, and its direction is the same as the $x$-axis.\n(2) Since $\\overrightarrow{O A}=1+i$, the Cartesian coordinates of point $A$ are $(1,1)$, and $\\overrightarrow{O B}=\\sqrt{3}-i$, so the Cartesian coordinates of point $B$ are $(\\sqrt{3},-1)$.\n$$\\text { Therefore } \\begin{aligned}\n|A B| & =\\sqrt{(\\sqrt{3}-1)^{2}+(-1-1)^{2}} \\\\\n& =\\sqrt{8-2 \\sqrt{3}} \\approx 2.1\n\\end{aligned}$$", "answer": "2.1", "source": "inequalities", "problem_type": "Algebra", "uuid": "e93a8cbd-336b-5aa7-9e39-b81d5037510d"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n4.206 There are two forces $f_{1}$ and $f_{2}$ acting on the origin $O$ of the coordinate axis,\n$$\\begin{array}{l}\n\\vec{f}_{1}=\\overrightarrow{O A}=\\sqrt{2}\\left(\\cos 45^{\\circ}+i \\sin 45^{\\circ}\\right) \\\\\n\\vec{f}_{2}=\\overrightarrow{O B}=2\\left[\\cos \\left(-30^{\\circ}\\right)+i \\sin \\left(-30^{\\circ}\\right)\\right]\n\\end{array}$$\n(1) Find the magnitude and direction of their resultant force;\n(2) Find the distance between points $A$ and $B$ (accurate to 0.1).", "candidates": [{"slot": 0, "origin": "budgie", "answer": "(1) The magnitude of the resultant force is equal to the product of their magnitudes.\n$$|\\vec{f}_{1}|=\\left|\\overrightarrow{O A}\\right|=\\sqrt{r^2+2r=2r\\sqrt{2}}$$\n$$|\\vec{f}_{2}|=\\left|\\overrightarrow{O B}\\right|=\\sqrt{4r^2+4r^2=8r^2}$$\n(2) The sum of the magnitudes of the two forces is 0.\n$$|\\vec{f}_{1}|+\\left|\\vec{f}_{2}\\right|= \\sqrt{2}+8$$\n$$|\\vec{f}_{1}|+|\\overrightarrow{O A}\\r|+|\\vec{f}_{2}|= 2\\sqrt{2}+8$$\n$$|\\vec{f}_{1}|+|\\overrightarrow{O B}\\r|+|\\vec{f}_{2}|= 2\\sqrt{2}+8$$\n\nFinal answer:\n#### 2", "correct": false, "reward": -1, "repetition": {"words": 99, "unique_ratio": 0.3333333333333333, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "(1) The magnitude of the resultant force is given by $|f_{1}|=\\sqrt{2}\\left(\\cos 45^{\\circ}+i \\sin 45^{\\circ}\\right)$. Its magnitude is $\\sqrt{2}$. Its direction is from the origin to $O$ by a vector of magnitude $\\sqrt{2}$ units, which is a $45^{\\circ}$ angle from the positive $x$-axis.\n\n(2) Let $P$ be the point representing the difference between $A$ and $B$. Then, the distance $|A O B|$ is given by $\\sqrt{V_{A O B}^2+V_{B O A}} = \\sqrt{2}+1$.\n\n(1) If we set $A O B=0$, the distance $|A B|$ is equal to the distance $|A O|$.\n\n$$|A B| = \\sqrt{V_{A O B}^2+V_{B O A}} = \\sqrt{2+1} = \\boxed{2}$$\n\n(2) The problem requires finding the distance between points $A$ and $B$, which is equivalent to finding the distance between the origin and point $A$.\n\n$$Distance \\, AB = \\sqrt{A O^2 + O B^2} = \\boxed{\\frac{\\sqrt{29}}{2}}$$\n\nFinal answer:\n#### 29", "correct": false, "reward": -1, "repetition": {"words": 168, "unique_ratio": 0.3630952380952381, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "[Solution]\n$$\\text { (1) Resultant force } \\begin{aligned}\n\\vec{f}= & \\overrightarrow{f_{1}}+\\overrightarrow{f_{2}} \\\\\n= & \\sqrt{2}\\left(\\cos 45^{\\circ}+i \\sin 45^{\\circ}\\right) \\\\\n& +2\\left[\\cos \\left(-30^{\\circ}\\right)+i \\sin \\left(-30^{\\circ}\\right)\\right] \\\\\n= & 1+i+\\sqrt{3}-i \\\\\n= & \\sqrt{3}+1 \\\\\n= & (\\sqrt{3}+1)\\left(\\cos 0^{\\circ}+i \\sin 0^{\\circ}\\right) .\n\\end{aligned}$$\n\nSo the magnitude of the resultant force is $\\sqrt{3}+1$, and its direction is the same as the $x$-axis.\n(2) Since $\\overrightarrow{O A}=1+i$, the Cartesian coordinates of point $A$ are $(1,1)$, and $\\overrightarrow{O B}=\\sqrt{3}-i$, so the Cartesian coordinates of point $B$ are $(\\sqrt{3},-1)$.\n$$\\text { Therefore } \\begin{aligned}\n|A B| & =\\sqrt{(\\sqrt{3}-1)^{2}+(-1-1)^{2}} \\\\\n& =\\sqrt{8-2 \\sqrt{3}} \\approx 2.1\n\\end{aligned}$$", "correct": true, "reward": 1.0, "repetition": {"words": 141, "unique_ratio": 0.3262411347517731, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_045", "row": {"problem": "Prove that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily  distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?", "solution": "1. **Restate the problem in a more manageable form:**\n   We need to prove that the number \\(1\\) can be represented as a sum of a finite number \\(n\\) of real numbers, each less than \\(1\\), and each containing only the digits \\(0\\) and \\(7\\) in their decimal representation. We also need to find the least possible value of \\(n\\).\n\n2. **Reformulate the problem using a known rational number:**\n   Consider the rational number \\(\\frac{1}{7} = 0.\\overline{142857}\\). We need to express \\(\\frac{1}{7}\\) as a sum of \\(n\\) real numbers, each less than \\(\\frac{1}{7}\\) and containing only the digits \\(0\\) and \\(1\\) in their decimal representation.\n\n3. **Assume \\(n < 10\\) and find the least value of \\(n\\):**\n   We assume that all the real numbers are positive and that \\(n < 10\\). The least value of \\(n\\) is found to be \\(8\\), for which we have the identity:\n   \\[\n   \\frac{1}{7} = 0.\\overline{1} + 0.\\overline{011111} + 0.\\overline{010111} + 0.\\overline{010111} + 0.\\overline{000111} + 0.\\overline{000101} + 0.\\overline{000101} + 0.\\overline{000100}.\n   \\]\n\n4. **Consider the case with both positive and negative real numbers:**\n   Suppose now that \\(\\frac{1}{7}\\) is the sum of \\(i > 0\\) positive real numbers and \\(j > 0\\) negative real numbers, with \\(i + j \\leq 7\\). Denote by \\(u = 0.u_1u_2\\ldots\\) (respectively by \\(-v = -0.v_1v_2\\ldots\\)) the sum of the positive (respectively negative) real numbers, so that \\(\\frac{1}{7} + v = u\\). Note that \\(u\\) and \\(v\\) are necessarily strictly less than \\(1\\).\n\n5. **Analyze the constraints on \\(u\\) and \\(v\\):**\n   Since \\(i, j < 7\\), it follows that \\(v_6 \\geq 3\\), so that \\(j \\geq 3\\) and \\(i \\leq 4\\). Then, we must have \\(v_5 \\geq 4\\), so that \\(j \\geq 4\\) and \\(i \\leq 3\\). In this case, we find \\(u_3 \\geq 3\\) and thus \\(i = 3\\). This finally implies that \\(v_2 \\geq 6\\), which is impossible since \\(j = 4\\).\n\n6. **Conclude the least value of \\(n\\):**\n   The least value of \\(n\\) is therefore \\(8\\).\n\nThe final answer is \\( \\boxed{ 8 } \\).", "answer": "8", "source": "aops_forum", "problem_type": "Number Theory", "uuid": "1cd82e3d-2e7a-5eca-9571-62f5669824c4"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nProve that number $1$ can be represented as a sum of a finite number $n$ of real numbers, less than $1,$ not necessarily  distinct, which contain in their decimal representation only the digits $0$ and/or $7.$ Which is the least possible number $n$?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To prove that $1$ can be expressed as a sum of a finite number $n$ of real numbers (including 0 and 7), such that all the digits of the sum are $0$ and/or $7$, we can start by considering the representation of the number $1$ in base $10$.\n\nIn base $10$, the number $1$ can be represented as:\n\\[ 1 = 1_{10} \\]\n\nNow, let $n$ be the number of real numbers that contain the digits $0$ and/or $7$ in their decimal representation. We can express $1$ as a sum of these numbers in the following way:\n\\[ 1 = 1_{10} + x_{0} + x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} + x_{10} \\]\nwhere $x_{i}$ is the digit in the $i$-th decimal place.\n\nThe sum of the digits is:\n\\[ x_{0} + x_{1} + x_{2} + x_{3} + x_{4} + x_{5} + x_{6} + x_{7} + x_{8} + x_{9} + x_{10} = 1 + 1 + 1 + 1 + 1 + 1 + 1 = 8 \\]\n\nThe number $1$ can be represented as the sum of the digits $8$ times, not necessarily a single number. Therefore, $n = 8$.\n\nTo find the least possible number $n$, we need to find the smallest number", "correct": false, "reward": -1, "repetition": {"words": 206, "unique_ratio": 0.3300970873786408, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "1. **Apply the Pigeonhole Principle**: The principle states that if there are $n+k$ objects (in this case, $n$ real numbers and $k$ objects) and more than $n$ of them are contained in the same \" repeatable\" manner (in this case, having only two digits), then there must be at least two objects among the $n+k$ objects that share a digit.\n\n2. **Set up the problem**: We need to find the smallest $n$ such that there exists a sequence of $n$ real numbers with only two digits that includes at least two numbers.\n\n3. **Start by considering the case where $n$ is the smallest possible**: Median numbers are the smallest possible sequence of $n$ numbers. If $n$ is the smallest possible, the median would be a sequence of $n/2$ numbers, all of which would be the same or have a repeated decimal representation.\n\n4. **Consider the case where $n$ is not the smallest**: To minimize $n$, we need to distribute the remaining numbers among $n$ real numbers in a way that minimizes the total count of numbers. Since the median is the smallest, we want to minimize the number of distinct numbers.\n\n5. **Use the Pigeonhole Principle**: If we have fewer than $n$ distinct numbers, then by the Pigeonhole Principle, there must be at least two numbers that share the same digit. This means that there are repeated numbers in the sequence, which contradicts the assumption that we have only two digits.\n\n6. **Conclusion**: Since we have constructed a sequence of $n$ numbers with only two digits and still have at least two numbers sharing a digit, we conclude that $n$ must be at least $n/2$.\n\nTherefore, the least possible number $", "correct": false, "reward": -1, "repetition": {"words": 283, "unique_ratio": 0.34275618374558303, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "1. **Restate the problem in a more manageable form:**\n   We need to prove that the number \\(1\\) can be represented as a sum of a finite number \\(n\\) of real numbers, each less than \\(1\\), and each containing only the digits \\(0\\) and \\(7\\) in their decimal representation. We also need to find the least possible value of \\(n\\).\n\n2. **Reformulate the problem using a known rational number:**\n   Consider the rational number \\(\\frac{1}{7} = 0.\\overline{142857}\\). We need to express \\(\\frac{1}{7}\\) as a sum of \\(n\\) real numbers, each less than \\(\\frac{1}{7}\\) and containing only the digits \\(0\\) and \\(1\\) in their decimal representation.\n\n3. **Assume \\(n < 10\\) and find the least value of \\(n\\):**\n   We assume that all the real numbers are positive and that \\(n < 10\\). The least value of \\(n\\) is found to be \\(8\\), for which we have the identity:\n   \\[\n   \\frac{1}{7} = 0.\\overline{1} + 0.\\overline{011111} + 0.\\overline{010111} + 0.\\overline{010111} + 0.\\overline{000111} + 0.\\overline{000101} + 0.\\overline{000101} + 0.\\overline{000100}.\n   \\]\n\n4. **Consider the case with both positive and negative real numbers:**\n   Suppose now that \\(\\frac{1}{7}\\) is the sum of \\(i > 0\\) positive real numbers and \\(j > 0\\) negative real numbers, with \\(i + j \\leq 7\\). Denote by \\(u = 0.u_1u_2\\ldots\\) (respectively by \\(-v = -0.v_1v_2\\ldots\\)) the sum of the positive (respectively negative) real numbers, so that \\(\\frac{1}{7} + v = u\\). Note that \\(u\\) and \\(v\\) are necessarily strictly less than \\(1\\).\n\n5. **Analyze the constraints on \\(u\\) and \\(v\\):**\n   Since \\(i, j < 7\\), it follows that \\(v_6 \\geq 3\\), so that \\(j \\geq 3\\) and \\(i \\leq 4\\). Then, we must have \\(v_5 \\geq 4\\), so that \\(j \\geq 4\\) and \\(i \\leq 3\\). In this case, we find \\(u_3 \\geq 3\\) and thus \\(i = 3\\). This finally implies that \\(v_2 \\geq 6\\), which is impossible since \\(j = 4\\).\n\n6. **Conclude the least value of \\(n\\):**\n   The least value of \\(n\\) is therefore \\(8\\).\n\nThe final answer is \\( \\boxed{ 8 } \\).", "correct": true, "reward": 1.0, "repetition": {"words": 350, "unique_ratio": 0.32285714285714284, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_046", "row": {"problem": "5. Points $O$ and $I$ are the centers of the circumcircle and incircle of triangle $ABC$, and $M$ is the midpoint of the arc $AC$ of the circumcircle (not containing $B$). It is known that $AB=15, BC=7$, and $MI=MO$. Find $AC$.", "solution": "Answer: $A C=13$.\n\nSolution. (Fig. 3). First, we prove that $M I=M A$ (trident lemma).\n\nIndeed, the external angle $A I M$ of triangle $A I B$ is equal to the sum of angles $B A I$ and $A B I$, and since $A I$ and $B I$ are angle bisectors, $\\angle A I M=\\frac{1}{2} \\angle A+\\frac{1}{2} \\angle B$. Angle $I A M$ is equal to the sum of angles $I A C$ and $C A M$. But $\\angle I A C=\\frac{1}{2} \\angle A$, and $\\angle C A M=\\angle C B M=\\frac{1}{2} \\angle B$ as inscribed angles.\n\nFrom this, it follows that $\\angle I A M=\\frac{1}{2} \\angle A+\\frac{1}{2} \\angle B=\\angle A I M$, and therefore, triangle $A M I$ is isosceles, $M I=M A$. By the condition $M O=$ $M I$, so by the trident lemma $A O=M O=M I=M A$. This means that triangle $A O M$ is equilateral and $\\angle A O M=60^{\\circ}$. Since the central angle $A O M$ is twice the inscribed angle $A B M$, we have $\\frac{1}{2} \\angle B=30^{\\circ}$, that is, $\\angle B=60^{\\circ}$.\n\nBy the cosine theorem, $A C^{2}=A B^{2}+B C^{2}-2 \\cdot A B \\cdot B C \\cos 60^{\\circ}=$ $13^{2}$.\n\nCriteria. Answer without explanation - 0 points. Proved that angle $B$ is $60^{\\circ}-5$ points. Points are not deducted for the absence of proof of the trident lemma. Full solution - 7 points.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_7789e0097aa3cdbc6e57g-2.jpg?height=540&width=467&top_left_y=1779&top_left_x=1485)\n\nFig. 3", "answer": "13", "source": "olympiads", "problem_type": "Geometry", "uuid": "a4f66b73-0bb9-55c1-b2b7-58da34d0e16f"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n5. Points $O$ and $I$ are the centers of the circumcircle and incircle of triangle $ABC$, and $M$ is the midpoint of the arc $AC$ of the circumcircle (not containing $B$). It is known that $AB=15, BC=7$, and $MI=MO$. Find $AC$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "This problem involves the properties of the circumcircle and incircle of a triangle, as well as the midpoint of an arc.\n\nThe key to solving this problem is to apply relevant theorems and theorems related to the circumcircle and the incircle.\n\nSince $O$ and $I$ are the centers of the circumcircle and incircle and $M$ is the midpoint of the arc $AC$, we can use the theorem of the triangle being equal to the sum of the lengths of any two sides to find the length of $AC$.\n\nFor the triangle $ABC$, we have $AB = 15$, $BC = 7$, and $MI = MO$. Applying the theorem of the triangle being equal to the sum of the lengths of any two sides, we can find the length of $AC$.\n\nThe theorem states that in any triangle, the sum of the lengths of the two smaller sides is equal to the length of the largest side.\n\nSo, for triangle $ABC$, we have $AB + BC = Introduce.\n\nApplying the theorem to triangle $ABC$, we get $15 + 7 = 22 = AC$.\n\nTherefore, the length of $AC$ is $22$ units.\n\nFinal answer:\n#### 22", "correct": false, "reward": -1, "repetition": {"words": 184, "unique_ratio": 0.3695652173913043, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "The key to this problem lies in the fact that $M$ is the midpoint of the arc $AC$ of the circumcircle not containing $B$. This is known as the inscribed angle theorem.\n\nGiven that $AB=15$ and $BC=7$, we can find the radius of the circumcircle using the formula $R = \\frac{abc}{4 \\Delta}$, where $a$, $b$, and $c$ are the sides of the triangle and $\\Delta$ is the area of the triangle.\n\nSubstituting the given values for $a$, $b$, and $c$, we get $R = \\frac{15 \\times 7}{4 \\times \\sqrt{15^2 - 7^2}} = \\frac{105}{4 \\times 2\\sqrt{109}}$.\n\nNext, we can find the area of the triangle using Heron's formula. First, we need to find the semi-perimeter $s$ of the triangle:\n\n$s = \\frac{a + b + c}{2} = \\frac{15 + 7 + AC}{2} = \\frac{22 + AC}{2}$.\n\nNow, we can express the area of the triangle in terms of $AC$ using Heron's formula:\n\n$Area = \\sqrt{s(s - a)(s - b)(s - c)} = \\sqrt{\\frac{22 + AC}{2} \\left(\\frac{22 + AC}{2} - 15\\right) \\left(\\frac{22 + AC}{2} - 7\\right) \\left(\\frac{22 + AC}{2} - AC\\right)}}$.\n\nSimplify the expression to get the area", "correct": false, "reward": -1, "repetition": {"words": 201, "unique_ratio": 0.3582089552238806, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Answer: $A C=13$.\n\nSolution. (Fig. 3). First, we prove that $M I=M A$ (trident lemma).\n\nIndeed, the external angle $A I M$ of triangle $A I B$ is equal to the sum of angles $B A I$ and $A B I$, and since $A I$ and $B I$ are angle bisectors, $\\angle A I M=\\frac{1}{2} \\angle A+\\frac{1}{2} \\angle B$. Angle $I A M$ is equal to the sum of angles $I A C$ and $C A M$. But $\\angle I A C=\\frac{1}{2} \\angle A$, and $\\angle C A M=\\angle C B M=\\frac{1}{2} \\angle B$ as inscribed angles.\n\nFrom this, it follows that $\\angle I A M=\\frac{1}{2} \\angle A+\\frac{1}{2} \\angle B=\\angle A I M$, and therefore, triangle $A M I$ is isosceles, $M I=M A$. By the condition $M O=$ $M I$, so by the trident lemma $A O=M O=M I=M A$. This means that triangle $A O M$ is equilateral and $\\angle A O M=60^{\\circ}$. Since the central angle $A O M$ is twice the inscribed angle $A B M$, we have $\\frac{1}{2} \\angle B=30^{\\circ}$, that is, $\\angle B=60^{\\circ}$.\n\nBy the cosine theorem, $A C^{2}=A B^{2}+B C^{2}-2 \\cdot A B \\cdot B C \\cos 60^{\\circ}=$ $13^{2}$.\n\nCriteria. Answer without explanation - 0 points. Proved that angle $B$ is $60^{\\circ}-5$ points. Points are not deducted for the absence of proof of the trident lemma. Full solution - 7 points.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_7789e0097aa3cdbc6e57g-2.jpg?height=540&width=467&top_left_y=1779&top_left_x=1485)\n\nFig. 3", "correct": true, "reward": 1.0, "repetition": {"words": 295, "unique_ratio": 0.3152542372881356, "max_4gram_count": 7, "max_sentence_count": 2, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_047", "row": {"problem": "2. Given the function $y=\\frac{a-x}{x-a-1}$, the graph of its inverse function is symmetric about the point $(-1,4)$. Then the value of the real number $a$ is $\\qquad$ .", "solution": "2.3.-\n\nFrom the problem, we know that the graph of the function $y=\\frac{a-x}{x-a-1}$ is centrally symmetric about the point $(4,-1)$.\n$\\because y=\\frac{a-x}{x-a-1}=-1-\\frac{1}{x-(a+1)}$, we have $(y+1)[x-(a+1)]=-1$,\n$\\therefore$ the graph of the function is a hyperbola with its center at $(a+1,-1)$.\nAlso, $\\because$ the graph of the hyperbola is centrally symmetric about its center,\n$\\therefore a+1=4$, which means $a=3$.", "answer": "3", "source": "cn_contest", "problem_type": "Algebra", "uuid": "ffccaa4e-8118-5206-b8c0-cbb5298378ed"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n2. Given the function $y=\\frac{a-x}{x-a-1}$, the graph of its inverse function is symmetric about the point $(-1,4)$. Then the value of the real number $a$ is $\\qquad$ .", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The key concept in this problem is the concept of inverse functions.\n\nLet $f(x) = \\frac{a-x}{x-a-1}$, then $y = f(f^{-1}(x))$.\n\nWe know that the inverse function $f^{-1}(x)$ is symmetric about the point $(-1,4)$.\n\nThe point $(-1,4)$ is the midpoint between the point $(a-1,1)$ and the point $(x,y)$ on the graph of $y = f(f^{-1}(x))$.\n\nSince $(a-1,1)$ is on the graph of $y = f(f^{-1}(x))$, we have $f(a-1) = 1$.\n\nNow, let's find the inverse function $f^{-1}(x)$. We know that the inverse function $f^{-1}(x)$ is symmetric about the point $(-1,4)$.\n\nThe point $(-1,4)$ is the midpoint between the point $(a-1,1)$ and the point $(x,-1)$.\n\nSince $(-1,4)$ is the midpoint between the point $(a-1,1)$ and the point $(x,-1)$, we have $f^{-1}(x) = -1$.\n\nSubstituting $a-1$ for $x$ in the equation $f^{-1}(x) = -1$, we get $f^{-1}(a-1) = -1$.\n\nNow, consider the inverse function $f^{-1}(x)$. Since $f^{-1}(x)$ is symmetric about the point $(-1,4)$, we have $f^{-1}(f^{", "correct": false, "reward": -1, "repetition": {"words": 199, "unique_ratio": 0.20603015075376885, "max_4gram_count": 5, "max_sentence_count": 2, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "The inverse function $f^{-1}(x)$ satisfies the condition:\n\\[ f^{-1}(f(x)) = x \\]\n\nSince the graph of the inverse function is symmetric about the point $(-1, 4)$, we have:\n\\[ f(-1 + x) = f(x) \\]\n\nLet $x = 0$ in the equation above:\n\\[ f(0) = f(0 + y) + f(0 - y) = f(0) + f(y) = 4 \\]\n\nTherefore, $y = 0$.\n\nHence, $f^{-1}(0) = 0$.\n\nSince the function $y = x^{-1}$ is an inverse function of $y = 2x - 1$, the point $0$ corresponds to the point $(-1, 4)$.\n\nSince $f(-1 + x) = f(x)$, it follows that:\n\\[ f(x) = f(0) = 2(0) - 1 = -1 \\]\n\nHence, the value of $a$ is:\n\\[ a = \\boxed{0} \\]\n\nFinal answer:\n#### 0", "correct": false, "reward": -1, "repetition": {"words": 115, "unique_ratio": 0.3391304347826087, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "2.3.-\n\nFrom the problem, we know that the graph of the function $y=\\frac{a-x}{x-a-1}$ is centrally symmetric about the point $(4,-1)$.\n$\\because y=\\frac{a-x}{x-a-1}=-1-\\frac{1}{x-(a+1)}$, we have $(y+1)[x-(a+1)]=-1$,\n$\\therefore$ the graph of the function is a hyperbola with its center at $(a+1,-1)$.\nAlso, $\\because$ the graph of the hyperbola is centrally symmetric about its center,\n$\\therefore a+1=4$, which means $a=3$.", "correct": true, "reward": 1.0, "repetition": {"words": 87, "unique_ratio": 0.3793103448275862, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_048", "row": {"problem": "6. Given that $\\overrightarrow{O A} \\perp \\overrightarrow{O B}$, and $|\\overrightarrow{O A}|=|\\overrightarrow{O B}|=24$. If $t \\in[0,1]$, then\n$$\n|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right|\n$$\n\nthe minimum value is ( ).\n(A) $2 \\sqrt{193}$\n(B) 26\n(C) $24 \\sqrt{2}$\n(D) 24", "solution": "6. B.\n\nConstruct a square $O A C B$, connect the diagonal $A B$, and let $D$ and $E$ be points on the diagonal $A B$ and the side $O B$ respectively, such that\n$$\n\\begin{array}{l}\nt \\overrightarrow{A B}-\\overrightarrow{A O}=\\overrightarrow{O D}, \\\\\n\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}=\\overrightarrow{E D}, \\\\\nE B=10, O D=D C . \\\\\n\\text { Therefore }|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right| \\\\\n=|\\overrightarrow{E D}|+|\\overrightarrow{D C}| \\leqslant|\\overrightarrow{E C}|=26 .\n\\end{array}\n$$", "answer": "26", "source": "olympiads", "problem_type": "Geometry", "uuid": "09c374b3-283a-5d5c-b647-4b3336197400"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n6. Given that $\\overrightarrow{O A} \\perp \\overrightarrow{O B}$, and $|\\overrightarrow{O A}|=|\\overrightarrow{O B}|=24$. If $t \\in[0,1]$, then\n$$\n|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right|\n$$\n\nthe minimum value is ( ).\n(A) $2 \\sqrt{193}$\n(B) 26\n(C) $24 \\sqrt{2}$\n(D) 24", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Solution:\n\n1. Since $|\\overrightarrow{O A}|=|\\overrightarrow{O B}|=24$, then $|\\overrightarrow{O A}|^2=4\\overrightarrow{O B}^2=256$,\nwhich means $256 \\times 24=256\\times4\\overrightarrow{O B}^2 \\Rightarrow \\overrightarrow{O B}^2=256$,\nwhich means $\\overrightarrow{O B}=(1 it, 0 )$.\n\n2. $t \\overrightarrow{A B}-\\overrightarrow{A O}=t (|\\overrightarrow{O B}| - 24)=t (12-24)=-12t$, $|\\overrightarrow{B O}-(1-t) \\overrightarrow{A B}=(\\overrightarrow{B O}+\\overrightarrow{A B})-(1-t) \\overrightarrow{A B}=(\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{A B})$\n\n3. Therefore, $|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{A B}\\right|=|t-1|+|t^2+(4-t) \\overrightarrow{A B}|=|t-1|+|(t+4)(1-t)|=\\begin{cases} 2(t+1), & 0\\leqslant t\\leqsl", "correct": false, "reward": -1, "repetition": {"words": 139, "unique_ratio": 0.2302158273381295, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "(A)\n\nThe dot product of the vectors is 0, so\n$$\n\\overrightarrow{O A} \\cdot \\overrightarrow{O B} = 0.\n$$\n\nIf $t \\in [0, 1]$, then\n$$\n|t \\overrightarrow{A B} - \\overrightarrow{A O}| = |t| \\cdot 24 - (\\overrightarrow{O A} \\cdot \\overrightarrow{O B}) = 24t - 0.\n$$\n\nTherefore,\n$$\n|t \\overrightarrow{A B} - \\overrightarrow{A O}| + |\\frac{5}{12} \\overrightarrow{B O} - (1-t) \\overrightarrow{B A}| = 24t + \\frac{5}{12}|\\overrightarrow{A B}| + |t|\\cdot 24 - |\\overrightarrow{A B}| \\cdot \\frac{5}{12} = 24t + \\frac{5}{12} \\cdot 24 + 48t - 24 = 24 \\sqrt{2} \\cdot t + 24.\n$$\n\nThus, the minimum value is 24 \\sqrt{2} \\cdot t + 24,\n$$\n\nHence, the answer is:\n$$\n\\boxed{(C) \\ 24 \\sqrt{2}}.\n$$", "correct": false, "reward": -1, "repetition": {"words": 113, "unique_ratio": 0.3008849557522124, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "6. B.\n\nConstruct a square $O A C B$, connect the diagonal $A B$, and let $D$ and $E$ be points on the diagonal $A B$ and the side $O B$ respectively, such that\n$$\n\\begin{array}{l}\nt \\overrightarrow{A B}-\\overrightarrow{A O}=\\overrightarrow{O D}, \\\\\n\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}=\\overrightarrow{E D}, \\\\\nE B=10, O D=D C . \\\\\n\\text { Therefore }|t \\overrightarrow{A B}-\\overrightarrow{A O}|+\\left|\\frac{5}{12} \\overrightarrow{B O}-(1-t) \\overrightarrow{B A}\\right| \\\\\n=|\\overrightarrow{E D}|+|\\overrightarrow{D C}| \\leqslant|\\overrightarrow{E C}|=26 .\n\\end{array}\n$$", "correct": true, "reward": 1.0, "repetition": {"words": 103, "unique_ratio": 0.36893203883495146, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_049", "row": {"problem": "8. Determine the value of the sum\n$$\n\\frac{3}{1^{2} \\cdot 2^{2}}+\\frac{5}{2^{2} \\cdot 3^{2}}+\\frac{7}{3^{2} \\cdot 4^{2}}+\\cdots+\\frac{29}{14^{2} \\cdot 15^{2}} .\n$$", "solution": "Solution: $224 / 225$ The sum telescopes as\n$$\n\\left(\\frac{1}{1^{2}}-\\frac{1}{2^{2}}\\right)+\\left(\\frac{1}{2^{2}}-\\frac{1}{3^{2}}\\right)+\\cdots+\\left(\\frac{1}{14^{2}}-\\frac{1}{15^{2}}\\right)=\\frac{1}{1^{2}}-\\frac{1}{15^{2}}=\\frac{224}{225} .\n$$", "answer": "\\frac{224}{225}", "source": "olympiads", "problem_type": "Algebra", "uuid": "1f447ae2-6a9a-5160-b486-9a19b5aedace"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n8. Determine the value of the sum\n$$\n\\frac{3}{1^{2} \\cdot 2^{2}}+\\frac{5}{2^{2} \\cdot 3^{2}}+\\frac{7}{3^{2} \\cdot 4^{2}}+\\cdots+\\frac{29}{14^{2} \\cdot 15^{2}} .\n$$", "candidates": [{"slot": 0, "origin": "budgie", "answer": "1. We start by examining the general term of the series:\n\n$$\n\\frac{n}{(n+2)^{2} \\cdot (n+1)^{2}}\n$$\n\nLet's denote this as \\( a_n = \\frac{n}{(n+2)^{2} \\cdot (n+1)^{2}} \\).\n\n2. We can derive a recursive relationship for \\( a_n \\) by manipulating the terms of the series. We can rewrite the terms in the denominator as follows:\n\n$$\na_n = \\frac{n}{(n+2)^{2} \\cdot (n+1)^{2}} = \\frac{1}{A_{n-1} \\cdot A_{n+1}} + \\frac{1}{A_{n-2} \\cdot A_{n+2}} + \\cdots + \\frac{1}{A_2 \\cdot A_1} + \\cdots\n$$\n\nwhere \\( A_{n-k} = A_k + A_{k+1} + \\cdots \\) and \\( A_k = A_{n+k} - A_{n+k-k} = A_{n+k-k} + A_{n+k-k+1} + \\cdots \\).\n\n3. Denote the numerator of \\( a_n \\) by \\( B_k = n - k \\). We can express \\( B_k \\) in terms of \\( A_{n+k-k} \\). After some manipulations, we obtain:\n\n$$\nA_{n+k-k} = A_k + A_{k+1} + \\cdots = A_{A_k} + A_{", "correct": false, "reward": -1, "repetition": {"words": 173, "unique_ratio": 0.24277456647398843, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "1. The given sum can be simplified by first writing each fraction with a denominator of the form $n^2$:\n\n    $$\n    \\frac{3}{1^2 \\cdot 2^2} + \\frac{5}{2^2 \\cdot 3^2} + \\frac{7}{3^2 \\cdot 4^2} + \\cdots + \\frac{29}{14^2 \\cdot 15^2}.\n    $$\n\n2. Notice that the denominator of the first fraction in this series is a square of a prime number, $2^{2}$. This means that when we write each term of the series with a denominator of the form $n^2$, we are effectively adding more factors of 2 to the denominator.\n\n3. The denominator of the first fraction is $2^{2}$, which is a power of 2. Each subsequent denominator is twice the previous denominator, suggesting a factor of 2 in the numerators of the fractions.\n\n4. Each denominator can be expressed as a product of prime powers. The denominator of the first fraction is $2^{2}$, and the denominator of the second fraction is $2^{4}$. Therefore, the denominator of the entire series can be written as:\n\n    $$\n    n^2 \\cdot (2^{2})^{n-1} = n^2 \\cdot 2^{2n - 2}.\n    $$\n\n5. The numerator of the numerator in the given series is the sum of the first 29 prime numbers. The sum of the first $n$ prime numbers can be calculated using the formula for the sum of a series:\n\n    $$\n    S_{n} = \\frac{n(n+1)}{2}.\n    $$\n\n6. Substitute the value of $n$ into this formula to", "correct": false, "reward": -1, "repetition": {"words": 246, "unique_ratio": 0.32113821138211385, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution: $224 / 225$ The sum telescopes as\n$$\n\\left(\\frac{1}{1^{2}}-\\frac{1}{2^{2}}\\right)+\\left(\\frac{1}{2^{2}}-\\frac{1}{3^{2}}\\right)+\\cdots+\\left(\\frac{1}{14^{2}}-\\frac{1}{15^{2}}\\right)=\\frac{1}{1^{2}}-\\frac{1}{15^{2}}=\\frac{224}{225} .\n$$", "correct": true, "reward": 1.0, "repetition": {"words": 49, "unique_ratio": 0.32653061224489793, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_050", "row": {"problem": "Augusto has a wire that is $10 \\mathrm{~m}$ long. He makes a cut at a point on the wire, obtaining two pieces. One piece has a length of $x$ and the other has a length of $10-x$ as shown in the figure below:\n\n![](https://cdn.mathpix.com/cropped/2024_05_01_d02c2755ad3373bde08ag-05.jpg?height=645&width=1166&top_left_y=568&top_left_x=527)\n\nAugusto uses the two pieces of wire to make two squares.\n\na) What is the length of the side of each of the squares? What is the area of each?\n\nb) What is the length of each of the two pieces of wire so that the sum of the areas of the squares obtained is minimized?\n\nc) Suppose Augusto cuts the wire into ten pieces and uses each one to make a square. What should be the size of each piece so that the sum of the areas of the squares obtained is minimized?", "solution": "Solution\n\nIn this problem, all lengths are given in meters and areas in square meters.\n\na) A piece of rope has length $x$ and another piece of rope has length $10-x$. Since a square has four sides of equal length, one square will have a side length of $\\frac{x}{4}$ and the other square will have a side length of $\\frac{10-x}{4}$.\n\nThe area of a square with side length $\\ell$ is $\\ell^{2}$. Therefore, one square will have an area of $\\left(\\frac{x}{4}\\right)^{2}=\\frac{x^{2}}{16}$, while the other square will have an area of $\\left(\\frac{10-x}{4}\\right)^{2}=\\frac{100-20 x+x^{2}}{16}$.\n\nb) Let $S(x)$ be the sum of the areas of the two squares. From the previous part, we have\n\n$$\nS(x)=\\frac{x^{2}}{16}+\\frac{100-20 x+x^{2}}{16}=\\frac{100-20 x+2 x^{2}}{16}=\\frac{1}{8} x^{2}-\\frac{5}{4} x+\\frac{25}{4}\n$$\n\nwhich is a quadratic function. The minimum of a function of the form\n\n$$\nf(x)=a x^{2}+b x+c\n$$\n\nwith $a>0$ is achieved at $x=\\frac{-b}{2 a}$. Thus, the minimum area will be achieved if\n\n$$\nx=-\\frac{\\left(-\\frac{5}{4}\\right)}{2 \\frac{1}{8}}=5\n$$\n\nIn other words, if the rope is cut exactly in the middle!\n\nc) From the previous part, we know that to minimize the sum of the areas, it is necessary to cut the rope exactly in the middle. Well, we claim that to minimize the area with nine cuts (i.e., creating ten squares), it is necessary that all pieces of rope be equal. To show this, consider the following argument: if two of the ten pieces of rope were different, it would be possible to reduce the area by cutting the pieces of rope so that these two were equal (we are using the previous part). Therefore, any two pieces of rope must be equal. Hence, all must be equal!", "answer": "5", "source": "olympiads", "problem_type": "Algebra", "uuid": "e8e60a66-4791-5625-bbfd-31ac1c2af0d6"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nAugusto has a wire that is $10 \\mathrm{~m}$ long. He makes a cut at a point on the wire, obtaining two pieces. One piece has a length of $x$ and the other has a length of $10-x$ as shown in the figure below:\n\n![](https://cdn.mathpix.com/cropped/2024_05_01_d02c2755ad3373bde08ag-05.jpg?height=645&width=1166&top_left_y=568&top_left_x=527)\n\nAugusto uses the two pieces of wire to make two squares.\n\na) What is the length of the side of each of the squares? What is the area of each?\n\nb) What is the length of each of the two pieces of wire so that the sum of the areas of the squares obtained is minimized?\n\nc) Suppose Augusto cuts the wire into ten pieces and uses each one to make a square. What should be the size of each piece so that the sum of the areas of the squares obtained is minimized?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "#### Part (a)\n1. Let the length of the wire be a positive number $a$ such that $a > 0$.\n2. The lengths of the pieces are $a$ and $a-a=0$.\n3. By the triangle inequality, we have the following inequalities:\n\\[\n|a| < |0-(a-0)| = a\n\\]\n4. This simplifies to $|a| < a$, which means $a < 0$ as well.\n5. Given $a > 0$, the only possible value for $a$ is $a = 1$.\n\n#### Part (b)\n1. The sides of the squares are $1$ and $2$. Let the length of the side of each square be $x$.\n2. The sum of the areas of the squares is given by:\n\\[\nx^2 + (x-1)^2 = 4\n\\]\n3. Expanding and simplifying, we get:\n\\[\nx^2 + x^2 - 2x + 1 = 4 \\implies 2x^2 - 2x + 1 = 4\n\\]\n4. Rearranging, we have:\n\\[\n2x^2 - 2x - 3 = 0\n\\]\n5. Dividing the entire equation by $2$, we get:\n\\[\nx^2 - x - \\frac{3}{2} = 0\n\\]\n6. The solutions to this quadratic equation are given by:\n\\[\nx = \\frac{1 \\pm \\sqrt{1 + 6}}{2} = \\frac{1 \\pm \\sqrt{7}}{2}\n\\]\nHence, the possible values", "correct": false, "reward": -1, "repetition": {"words": 183, "unique_ratio": 0.3770491803278688, "max_4gram_count": 2, "max_sentence_count": 2, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "#### Part (a)\n\n1. **Identify the given information**: Augusto has a wire of length \\(10\\) meters and he cuts it into two pieces. One piece has a length of \\(x\\) meters and the other has a length of \\(10-x\\) meters.\n2. **Set up the equation**: The total length of the wire is the sum of the lengths of the two pieces, so we write the equation:\n   \\[\n   x + (10 - x) = 10\n   \\]\n3. **Solve the equation**:\n   \\[\n   x + 10 - x = 10 \\\\\n   10 = 10\n   \\]\n4. **Conclusion**: The equation holds true, so the length of the sides of both pieces is:\n   \\[\n   \\boxed{10 \\text{ meters each}}\n   \\]\n\n#### Part (b)\n\n1. **Express the given length**: \n   \\[\n   x = 10 - x - 10\n   \\]\n2. **Solve the equation**: \n   \\[\n   x = 10 - (10 - x) - 10 \\\\\n   10 = 10 - 10 + x - x \\\\\n   10 = 10 - 10 \\\\\n   10 = 10\n   \\]\n3. **Final answer**: The length of each piece is:\n   \\[\n   \\boxed{10\\ \\text{meters each}}\n   \\]\n\n#### Part (c)\n\n1. **Consider the total length of the wire**: We need to find the smallest number of pieces (or the sum of the lengths of the pieces) that can be cut from the wire to minimize the sum of the", "correct": false, "reward": -1, "repetition": {"words": 186, "unique_ratio": 0.34946236559139787, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution\n\nIn this problem, all lengths are given in meters and areas in square meters.\n\na) A piece of rope has length $x$ and another piece of rope has length $10-x$. Since a square has four sides of equal length, one square will have a side length of $\\frac{x}{4}$ and the other square will have a side length of $\\frac{10-x}{4}$.\n\nThe area of a square with side length $\\ell$ is $\\ell^{2}$. Therefore, one square will have an area of $\\left(\\frac{x}{4}\\right)^{2}=\\frac{x^{2}}{16}$, while the other square will have an area of $\\left(\\frac{10-x}{4}\\right)^{2}=\\frac{100-20 x+x^{2}}{16}$.\n\nb) Let $S(x)$ be the sum of the areas of the two squares. From the previous part, we have\n\n$$\nS(x)=\\frac{x^{2}}{16}+\\frac{100-20 x+x^{2}}{16}=\\frac{100-20 x+2 x^{2}}{16}=\\frac{1}{8} x^{2}-\\frac{5}{4} x+\\frac{25}{4}\n$$\n\nwhich is a quadratic function. The minimum of a function of the form\n\n$$\nf(x)=a x^{2}+b x+c\n$$\n\nwith $a>0$ is achieved at $x=\\frac{-b}{2 a}$. Thus, the minimum area will be achieved if\n\n$$\nx=-\\frac{\\left(-\\frac{5}{4}\\right)}{2 \\frac{1}{8}}=5\n$$\n\nIn other words, if the rope is cut exactly in the middle!\n\nc) From the previous part, we know that to minimize the sum of the areas, it is necessary to cut the rope exactly in the middle. Well, we claim that to minimize the area with nine cuts (i.e., creating ten squares), it is necessary that all pieces of rope be equal. To show this, consider the following argument: if two of the ten pieces of rope were different, it would be possible to reduce the area by cutting the pieces of rope so that these two were equal (we are using the previous part). Therefore, any two pieces of rope must be equal. Hence, all must be equal!", "correct": true, "reward": 1.0, "repetition": {"words": 345, "unique_ratio": 0.3159420289855073, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_051", "row": {"problem": "2.1 Find $f(2013)$, if for any real $x$ and $y$ the equality holds\n\n$$\nf(x-y)=f(x)+f(y)-2xy\n$$", "solution": "Solution. Substitute $x=y=0$. We get $f(0)=2 f(0)+0$, from which we obtain that $f(0)=0$. Substitute $x=y$. We get $0=f(0)=f(x)+f(x)-2 x^{2}$. Hence, $f(x)=x^{2}$.\n\nAnswer: 4052169. (C)\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_997c53c06135e2de9c16g-02.jpg?height=67&width=1157&top_left_y=1003&top_left_x=498)", "answer": "4052169", "source": "olympiads", "problem_type": "Algebra", "uuid": "8b03f2a1-15de-52a8-b2b5-24806ad8d00a"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n2.1 Find $f(2013)$, if for any real $x$ and $y$ the equality holds\n\n$$\nf(x-y)=f(x)+f(y)-2xy\n$$", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find $f(2013)$, we can express $2013$ in terms of $1$ and $2$ using the given equation:\n\\[ f(2013) = f(1) + f(2) - 2 \\times 1 \\times 2013 \\]\n\nUsing the given equation, we can substitute $x = 1$ and $y = 2$ to get:\n\\[ f(1) = f(1) + f(2) - 4026 \\]\n\nRearrange the equation to isolate $f(1)$:\n\\[ f(1) - f(1) = f(2) - 4026 \\Rightarrow 0 = f(2) - 4026 \\Rightarrow f(2) = 4026 \\]\n\nNow, we can find $f(2) + f(2) - 2 \\times 2013$:\n\\[ f(2) + f(2) - 2 \\times 2013 = 4026 + 4026 - 2 \\times 2013 = 4026 + 4026 - 2 \\times 2013 = 8052 - 2 \\times 2013 = 8052 - 4026 = \\boxed{4026} \\]\nFinal answer:\n#### 4026", "correct": false, "reward": -1, "repetition": {"words": 112, "unique_ratio": 0.2857142857142857, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To find $f(2013)$, we can use the given equation $f(x-y)=f(x)+f(y)-2xy$ for any real $x$ and $y$, and then substitute $x=2013$ and $y=1$ to find $f(2013)$.\n\nWe know $x-y=2013$ and $x+y=2014$. Using the given equation, we can express $f(2014)$ in terms of $f(2013)$.\n\nSubstitute $x=2013$ and $y=1$ into the given equation:\n\\[ f(2014) = f(2013) + f(1) - 2(2013)(1) \\]\n\nNow, substitute $x=2014$ and $y=1$ into the given equation:\n\\[ f(2014) = f(2013) + f(1) - 2(2013) \\]\n\nSubstitute $x=2013$ and $y=1$ into the equation:\n\\[ f(2013) = f(2013) + f(1) - 2(2013) - 4028 \\]\n\nRearrange the equation to isolate $f(2013)$:\n\\[ f(2013) - f(2013) = f(2013) + f(1) - 2(2013) - 4028 - 2013 \\]\n\n\\[ f(2", "correct": false, "reward": -1, "repetition": {"words": 136, "unique_ratio": 0.2426470588235294, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution. Substitute $x=y=0$. We get $f(0)=2 f(0)+0$, from which we obtain that $f(0)=0$. Substitute $x=y$. We get $0=f(0)=f(x)+f(x)-2 x^{2}$. Hence, $f(x)=x^{2}$.\n\nAnswer: 4052169. (C)\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_997c53c06135e2de9c16g-02.jpg?height=67&width=1157&top_left_y=1003&top_left_x=498)", "correct": true, "reward": 1.0, "repetition": {"words": 67, "unique_ratio": 0.5373134328358209, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_052", "row": {"problem": "2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation.", "solution": "2. This problem is equivalent to finding the smallest positive integer $b$, such that the equation $7 b^{2}+7 b+7=x^{4}$, (1) has an integer solution for $x$. Since 7 is a prime number, it follows from equation (1) that 7 is a divisor of $x$. Therefore, let $x=7 k$, then equation (1) becomes $b^{2}+b+1=7^{3} k^{4}$. The smallest $b$ occurs when $k$ is at its minimum. Taking $k=1$, we then have $b^{2}+b+1=343, b^{2}+b-342=0$, which is $(b-18)(b+19)=0$, yielding the positive integer solution $b=18$. Thus, we have $(777)_{18}=\\left(7^{4}\\right)_{10}$.", "answer": "18", "source": "olympiads", "problem_type": "Number Theory", "uuid": "39a6c760-dfa3-5201-aa87-105127951e16"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n2. (9th Canadian Mathematical Competition) $N$ is an integer, its representation in base $b$ is 777. Find the smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "First, we need to find the smallest base $b$ such that $N$ is a fourth power. We know that a fourth power is a number that can be expressed as $N = x^4$ for some integer $x$.\n\nSince $N = 777 = x^4$, the base $b$ must be a fourth power. We can find the smallest such $b$ by trying different values of $b$.\n\nLet's start by trying $b = 4$.\n\nFor $b = 4$, we have $N = x^4 = 777$. We can see that $x = 7$ is a solution, since $7^4 = 2401$. However, we need to check if $x = 7$ is also a solution to the congruence $x^4 \\equiv 21 \\pmod{1000}$.\n\nBy testing values of $x$, we can see that $x = 7$ satisfies the congruence.\n\nTherefore, the smallest base $b$ is:\n\\[ \\boxed{4} \\]", "correct": false, "reward": -1, "repetition": {"words": 134, "unique_ratio": 0.41044776119402987, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "We start with $b = 3$ because 0 is not a fourth power of 2.\n\nStart with $N = 777$, a three-digit number in base 3.\n\\[ 777_{10} = 1 7 7 7_{10} \\]\n\nConvert $N = 777_{10}$ to decimal:\n\\[ 777_{10} = 777_{3} = 342_{3} = 7^4_{10} = 7^4 \\times 3^0 = 7^4 \\times 1 = 7^3 = 343_{3} = 213_{2} = 3^3_{10} = 3^3 \\times 4^1 = 3^3 \\times 4 = 34_{10} \\]\n\nThe smallest positive integer $b$ such that $N$ is a fourth power of an integer in decimal notation is $\\boxed{3}$ because $N = 342_{3} = 7^4 \\times 3^0 = 7^4 \\times 1 = 7^3$ is a fourth power of an integer in decimal notation.", "correct": false, "reward": -1, "repetition": {"words": 122, "unique_ratio": 0.3442622950819672, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "2. This problem is equivalent to finding the smallest positive integer $b$, such that the equation $7 b^{2}+7 b+7=x^{4}$, (1) has an integer solution for $x$. Since 7 is a prime number, it follows from equation (1) that 7 is a divisor of $x$. Therefore, let $x=7 k$, then equation (1) becomes $b^{2}+b+1=7^{3} k^{4}$. The smallest $b$ occurs when $k$ is at its minimum. Taking $k=1$, we then have $b^{2}+b+1=343, b^{2}+b-342=0$, which is $(b-18)(b+19)=0$, yielding the positive integer solution $b=18$. Thus, we have $(777)_{18}=\\left(7^{4}\\right)_{10}$.", "correct": true, "reward": 1.0, "repetition": {"words": 116, "unique_ratio": 0.5, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_053", "row": {"problem": "9. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?", "solution": "Solution: $1 / 2$\nImagine placing the tetrahedron $A B C D$ flat on a table with vertex $A$ at the top. By vectors or otherwise, we see that the center is $3 / 4$ of the way from $A$ to the bottom face, so the reflection of this face lies in a horizontal plane halfway between $A$ and $B C D$. In particular, it cuts off the smaller tetrahedron obtained by scaling the original tetrahedron by a factor of $1 / 2$ about $A$. Similarly, the reflections of the other three faces cut off tetrahedra obtained by scaling $A B C D$ by $1 / 2$ about $B, C$, and $D$. On the other hand, the octahedral piece remaining remaining after we remove these four smaller tetrahedra is in the intersection of $A B C D$ with its reflection, since the reflection sends this piece to itself. So the answer we seek is just the volume of this piece, which is\n$$\n\\begin{array}{l}\n\\text { (volume of } A B C D)-4 \\cdot(\\text { volume of } A B C D \\text { scaled by a factor of } 1 / 2) \\\\\n=1-4(1 / 2)^{3}=1 / 2\n\\end{array}\n$$", "answer": "\\frac{1}{2}", "source": "olympiads", "problem_type": "Geometry", "uuid": "6bb59c32-e03b-53f3-9abf-a23cdeb0c3a1"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n9. Given is a regular tetrahedron of volume 1 . We obtain a second regular tetrahedron by reflecting the given one through its center. What is the volume of their intersection?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The problem is about finding the volume of the intersection of two tetrahedra created by reflecting one of them through its center.", "correct": false, "reward": -0.5, "repetition": {"words": 22, "unique_ratio": 0.8181818181818182, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 1, "origin": "budgie", "answer": "A regular tetrahedron with a volume of 1 is a regular tetrahedron depending on its edge length.\n\nThe volume of a regular tetrahedron is given by the formula:\n\n$$V = \\frac{\\sqrt{2}}{12} \\times (\\text{edge length})^3$$\n\nThe second tetrahedron is half of the first one because each edge of the first tetrahedron is now reflected through its center to form the second one.\n\nSo, the volume of the intersection is twice the volume of the second tetrahedron.\n\nHence, the answer is: $\\boxed{2\\sqrt{2}}$", "correct": false, "reward": -0.5, "repetition": {"words": 87, "unique_ratio": 0.4942528735632184, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 2, "origin": "verified_reference", "answer": "Solution: $1 / 2$\nImagine placing the tetrahedron $A B C D$ flat on a table with vertex $A$ at the top. By vectors or otherwise, we see that the center is $3 / 4$ of the way from $A$ to the bottom face, so the reflection of this face lies in a horizontal plane halfway between $A$ and $B C D$. In particular, it cuts off the smaller tetrahedron obtained by scaling the original tetrahedron by a factor of $1 / 2$ about $A$. Similarly, the reflections of the other three faces cut off tetrahedra obtained by scaling $A B C D$ by $1 / 2$ about $B, C$, and $D$. On the other hand, the octahedral piece remaining remaining after we remove these four smaller tetrahedra is in the intersection of $A B C D$ with its reflection, since the reflection sends this piece to itself. So the answer we seek is just the volume of this piece, which is\n$$\n\\begin{array}{l}\n\\text { (volume of } A B C D)-4 \\cdot(\\text { volume of } A B C D \\text { scaled by a factor of } 1 / 2) \\\\\n=1-4(1 / 2)^{3}=1 / 2\n\\end{array}\n$$", "correct": true, "reward": 1.0, "repetition": {"words": 194, "unique_ratio": 0.4484536082474227, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142115623759233}], "split": "math_train"}
+{"id": "math_054", "row": {"problem": "13. (15 points) As shown in the figure, ten points are evenly marked on the circumference of a circle. Place the ten natural numbers $1 \\sim 10$ on these ten points. Rotate a line passing through the center of the circle. When the line does not pass through any marked points, it divides the numbers 1 to 10 into two groups. For each arrangement, there are five ways to divide the numbers as the line rotates. For each division, there is a product of the sums of the two groups, and the smallest of these five products is denoted as $K$. What is the maximum value of $K$ among all possible arrangements?", "solution": "【Analysis】No matter how these 10 numbers are placed, their sum remains unchanged, $1+2+3+4+5+6+7+8+9+10=55$, divided into two groups, the closer the sums of the two groups, the larger the required product will be.\n\n【Solution】\n$$\n\\begin{array}{l}\n1+2+3+4+5+6+7+8+9+10=55 \\\\\n55=27+28\n\\end{array}\n$$\n\nTherefore, as long as the 10 numbers are divided into two parts with sums of 27 and 28, the product at this time is $27 \\times 28=756$. Answer: In all the arrangements, $K$ is maximized at 756.", "answer": "756", "source": "olympiads", "problem_type": "Combinatorics", "uuid": "cbdaf71c-479e-5ef7-b26b-7b66cd85e7cf"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n13. (15 points) As shown in the figure, ten points are evenly marked on the circumference of a circle. Place the ten natural numbers $1 \\sim 10$ on these ten points. Rotate a line passing through the center of the circle. When the line does not pass through any marked points, it divides the numbers 1 to 10 into two groups. For each arrangement, there are five ways to divide the numbers as the line rotates. For each division, there is a product of the sums of the two groups, and the smallest of these five products is denoted as $K$. What is the maximum value of $K$ among all possible arrangements?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "1.  **Assumptions and Setup**: \n   We have ten natural numbers on a circle, which are evenly marked. We place these numbers 1 through 10 in a single circle. We rotate the line passing through the center of the circle, which divides the numbers into two groups. Each group must then be divided by the line passing through the center, resulting in five distinct outcomes. For each arrangement, we need to find the maximum possible value of $K$.\n\n2. **Key Insight**:\n   The line passing through the center divides the circle into two groups. This means the line must intersect the numbers 1 through 10. The maximum possible value for $K$ occurs when the line is tangent to the circle.\n\n3. **Constructing the Arrangements**:\n   To achieve this, we can construct a different line through the center that does not intersect the numbers 1 through 10. For example, we can construct a line passing through the numbers 2 through 17,prime to 19, and 11 through 13,prime to 19. This line does not intersect the numbers 1 through 10 because it passes through prime to prime.\n\n4. **Final Arrangement**:\n   Using the above lines and configurations, we can visualize that the maximum possible value for $K$ is 1.\n\n5. **Considering Other Differences**:\n   We need to consider other differences, such as 2, 3, and 4. These differences are not useful in this context, as they would not help in maximizing $K$.\n\n6. **Constructing the Maximum Value**:\n   By carefully placing the numbers 1 through 10 on the circle, we can see that the maximum possible value for $K$ is 1.\n\n7. **Conclusion**:\n   The maximum value of $K$", "correct": false, "reward": -1, "repetition": {"words": 271, "unique_ratio": 0.3948339483394834, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 1, "origin": "budgie", "answer": "11. This problem can be approached by considering the symmetry of the circle and the division of numbers. To maximize the value of $K$, we should consider the case where $K$ is minimized. The smallest product is achieved when the two groups are as close to equal as possible, and since there are five ways to do this, we can get a value of $K$ as close as possible to $10$, which is $K=\\boxed{5}$.\n\nFinal answer:\n#### 5", "correct": false, "reward": -0.5, "repetition": {"words": 79, "unique_ratio": 0.6329113924050633, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 2, "origin": "verified_reference", "answer": "【Analysis】No matter how these 10 numbers are placed, their sum remains unchanged, $1+2+3+4+5+6+7+8+9+10=55$, divided into two groups, the closer the sums of the two groups, the larger the required product will be.\n\n【Solution】\n$$\n\\begin{array}{l}\n1+2+3+4+5+6+7+8+9+10=55 \\\\\n55=27+28\n\\end{array}\n$$\n\nTherefore, as long as the 10 numbers are divided into two parts with sums of 27 and 28, the product at this time is $27 \\times 28=756$. Answer: In all the arrangements, $K$ is maximized at 756.", "correct": true, "reward": 1.0, "repetition": {"words": 101, "unique_ratio": 0.6039603960396039, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_055", "row": {"problem": "## 232. Math Puzzle $9 / 84$\n\nA motorcycle battery ( $6 \\mathrm{~V}$ ) is discharged through a lamp $6 \\mathrm{~V} / 0.5 \\mathrm{~W}$. The lamp burns for 48 hours.\n\nHow many Ah was the battery charged if we assume that the discharge current remains constant?", "solution": "The discharge current is $I=\\frac{0.5 \\mathrm{~W}}{6 \\mathrm{~V}}=\\frac{1}{12} \\mathrm{~A}$.\n\nIf we assume that it remains constant during continuous discharge, the battery previously had $48 \\cdot \\frac{1}{12} \\mathrm{Ah}=4 \\mathrm{Ah}$ charged.", "answer": "4\\mathrm{Ah}", "source": "olympiads", "problem_type": "Algebra", "uuid": "5e0354af-e3c8-55b2-8a4f-071457e4e43f"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n## 232. Math Puzzle $9 / 84$\n\nA motorcycle battery ( $6 \\mathrm{~V}$ ) is discharged through a lamp $6 \\mathrm{~V} / 0.5 \\mathrm{~W}$. The lamp burns for 48 hours.\n\nHow many Ah was the battery charged if we assume that the discharge current remains constant?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The battery must have been charged 48 hours before the lamp burned out.\n\nThe battery was charged for a fraction of those 48 hours before the lamp burned out.\n\nThe battery was charged for:\n\\[ 48 : 0.5 = 96 \\, \\text{hours} \\]\n\nSo, the battery was charged for 96 hours before it burned out.\n\nThe answer is:\n\\boxed{96}", "correct": false, "reward": -1, "repetition": {"words": 57, "unique_ratio": 0.47368421052631576, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 1, "origin": "budgie", "answer": "1. Calculate the battery's current before the lamp is turned off. We know that the lamp burns for 48 hours, so during this time the battery has been burning for 48 hours.\n2. The battery consumed by the lamp is \\( 6 \\, \\text{V} / 0.5 \\, \\text{W} \\).\n3. The battery consumed is equal to the power it uses in 48 hours. Therefore, the battery used is \\( 6 \\, \\text{V} / 0.5 \\, \\text{W} = 12 \\, \\text{V} \\).\n4. The battery used is equal to the battery’s current. Therefore, the battery was charged with \\( 6 \\, \\text{V} + 12 \\, \\text{V} = 18 \\, \\text{V} \\) Ah.\n\nFinal answer:\n#### 18", "correct": false, "reward": -0.5, "repetition": {"words": 106, "unique_ratio": 0.5, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 2, "origin": "verified_reference", "answer": "The discharge current is $I=\\frac{0.5 \\mathrm{~W}}{6 \\mathrm{~V}}=\\frac{1}{12} \\mathrm{~A}$.\n\nIf we assume that it remains constant during continuous discharge, the battery previously had $48 \\cdot \\frac{1}{12} \\mathrm{Ah}=4 \\mathrm{Ah}$ charged.", "correct": true, "reward": 1.0, "repetition": {"words": 43, "unique_ratio": 0.7441860465116279, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.3728113305845737}], "split": "math_dev"}
+{"id": "math_056", "row": {"problem": "Problem 6.8. In the country of Dragonia, there live red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Moreover, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:\n\n- 1st head: “To my left is a green dragon.”\n- 2nd head: “To my right is a blue dragon.”\n- 3rd head: “There is no red dragon next to me.”\n\nWhat is the maximum number of red dragons that could have been at the table?", "solution": "Answer: 176.\n\nSolution. Consider an arbitrary red dragon. To the right of this dragon, at least one head must tell the truth. Note that the 1st and 3rd heads cannot tell the truth (since there is a red dragon to the left), so the 2nd head must tell the truth, and to the right of this dragon, there must be a blue dragon. Now consider the dragon to the left of the chosen red dragon. By similar reasoning, we understand that only the 1st head can tell the truth, and to the left of this dragon, there must be a green one.\n\nFrom the previous reasoning, we get that for any red dragon, one position to the right there is definitely a blue dragon, and one position to the left - a green one. For each red dragon, we combine these three dragons into one group. Note that any dragon can be in at most one such group. Then we get that there are no more than $530 / 3$ < 177, i.e., no more than 176 such groups. Therefore, there are no more than 176 red dragons.\n\nNow let's provide an example. Arrange 176 red dragons in a circle, and in 175 gaps between them, place a green and a blue dragon: ...RGRB..., and in one gap, place two blue and two green dragons: ...RBGBGR.... It is easy to check that all conditions are met: each dragon has at least one head that will tell the truth, and there are exactly 530 dragons in total.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_8af0c885427e3e323cf9g-17.jpg?height=492&width=525&top_left_y=852&top_left_x=464)\n\n## 7th grade", "answer": "176", "source": "olympiads", "problem_type": "Logic and Puzzles", "uuid": "6c252a36-7d8e-5376-95d9-ada49d61667a"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nProblem 6.8. In the country of Dragonia, there live red, green, and blue dragons. Each dragon has three heads, each of which always tells the truth or always lies. Moreover, each dragon has at least one head that tells the truth. One day, 530 dragons sat around a round table, and each of them said:\n\n- 1st head: “To my left is a green dragon.”\n- 2nd head: “To my right is a blue dragon.”\n- 3rd head: “There is no red dragon next to me.”\n\nWhat is the maximum number of red dragons that could have been at the table?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find the maximum number of red dragons that could have been at the table, we can consider the alternating pattern of truth and lie. Since each dragon has three heads, each dragon is either follows the truth or lies.\n\nIf we start with a truth-teller, the first position is a truth-teller, the second position is a truth-teller, the third position is a truth-teller, and so on. This pattern continues for all 530 dragons.\n\nThe maximum number of red dragons can be found by starting with the maximum possible number of truth-tellers and then alternating between truth-tellers and liars.\n\nSince each dragon has three heads, the maximum possible number of truth-tellers is 100, and the maximum possible number of liars is 100.\n\nWe start with the maximum number of truth-tellers (100), and then alternate between truth-tellers and liars. This pattern continues for all 530 dragons.\n\nTherefore, the maximum number of red dragons that could have been at the table is 100.\n\nThe answer is:\n\\[ \\boxed{100} \\]", "correct": false, "reward": -1, "repetition": {"words": 175, "unique_ratio": 0.36, "max_4gram_count": 4, "max_sentence_count": 2, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To determine the maximum number of red dragons, we need to analyze the statements made by the dragons and their relationship to each other. If 530 dragons have statements that contradict the truth/ jogit of their heads, then no red dragon could have a head that tells the truth.\n\nLet's denote the number of red dragons as $R$ and the number of green dragons as $G$. Since each red dragon has exactly three heads, we can denote the number of heads as $3R$ and the number of heads as $3G$.\n\nNow, consider the statements made by the dragons. If the head that tells the truth tells the truth about the head that tells the truth about another dragon, then the head that tells the truth about the head with the truth about the head that tells the truth about another dragon must be green.\n\nThis leads to a contradiction, as there would be no head that tells the truth about another dragon, and it would not be possible for 530 dragons to add up to 530 (the number of heads in a round table).\n\nHowever, if the head that tells the truth about another dragon tells the truth about the head with the truth about another dragon, then the head that tells the truth about the head with the truth about the head that tells the truth about another dragon must be blue.\n\nTherefore, the maximum number of red dragons that could have been at the table is $R = 3G = 3(3R) = 9G$.\n\nSo, the answer is:\n\\[ \\boxed{9} \\]\nFinal answer:\n#### 9", "correct": false, "reward": -1, "repetition": {"words": 262, "unique_ratio": 0.30916030534351147, "max_4gram_count": 9, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Answer: 176.\n\nSolution. Consider an arbitrary red dragon. To the right of this dragon, at least one head must tell the truth. Note that the 1st and 3rd heads cannot tell the truth (since there is a red dragon to the left), so the 2nd head must tell the truth, and to the right of this dragon, there must be a blue dragon. Now consider the dragon to the left of the chosen red dragon. By similar reasoning, we understand that only the 1st head can tell the truth, and to the left of this dragon, there must be a green one.\n\nFrom the previous reasoning, we get that for any red dragon, one position to the right there is definitely a blue dragon, and one position to the left - a green one. For each red dragon, we combine these three dragons into one group. Note that any dragon can be in at most one such group. Then we get that there are no more than $530 / 3$ < 177, i.e., no more than 176 such groups. Therefore, there are no more than 176 red dragons.\n\nNow let's provide an example. Arrange 176 red dragons in a circle, and in 175 gaps between them, place a green and a blue dragon: ...RGRB..., and in one gap, place two blue and two green dragons: ...RBGBGR.... It is easy to check that all conditions are met: each dragon has at least one head that will tell the truth, and there are exactly 530 dragons in total.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_8af0c885427e3e323cf9g-17.jpg?height=492&width=525&top_left_y=852&top_left_x=464)\n\n## 7th grade", "correct": true, "reward": 1.0, "repetition": {"words": 279, "unique_ratio": 0.4336917562724014, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_057", "row": {"problem": "|\n\nThere are 85 balloons in the room - red and blue. It is known that: 1) at least one of the balloons is red; 2) in every arbitrarily chosen pair of balloons, at least one is blue. How many red balloons are in the room?\n\n#", "solution": "Think about whether there can be two red balls in the room.\n\n## Solution\n\nSince among any two balls, one is blue, there cannot be two red balls in the room. Therefore, there are 84 blue balloons and 1 red balloon in the room.\n\n## Answer\n\n1 ball.", "answer": "1", "source": "olympiads", "problem_type": "Logic and Puzzles", "uuid": "4671698a-54f7-52fb-988c-460378917299"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n|\n\nThere are 85 balloons in the room - red and blue. It is known that: 1) at least one of the balloons is red; 2) in every arbitrarily chosen pair of balloons, at least one is blue. How many red balloons are in the room?\n\n#", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Since there are 85 balloons in total and 3 non-red balloons are arbitrarily chosen, this means that the number of red balloons must be 85 - 3 = 82 blue balloons.\n\nTherefore, the answer is:\n\\boxed{82}", "correct": false, "reward": -0.5, "repetition": {"words": 36, "unique_ratio": 0.75, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "To satisfy condition 1) at least one of the balloons is red, we need to choose one red balloon out of the 85 total.\n\nThe number of ways to choose one red balloon from the red balloons is given by the combination formula:\n\n$$C(n, k) = \\frac{n!}{k!(n-k)!}$$\n\nHere, $n$ is the total number of red balloons (85) and $k$ is the number of red balloons we want to choose (1), because there is only one red balloon to choose. Plugging in the values, we get:\n\n$$C(85, 1) = \\frac{85!}{1!(85-1)!} = \\frac{85!}{1! \\times 84!}$$\n\nFor condition 2) in every arbitrarily chosen pair of balloons, at least one is blue, we need to ensure that we choose at least one pair of red balloons. This means we choose at least one pair of red balloons that contains at least one blue.\n\nThe total number of pairs of red balloons that we can choose is $C(85, 2)$. We must choose at least one pair of red balloons that contains at least one blue. This can be done in the following ways:\n\n- Choose two red balloons from the 85 total red balloons and 1 from the 85 blue balloons.\n- Choose one blue and one red balloon from the 85 blue balloons.\n\nThe number of ways to choose two red balloons and one blue balloon from the 85 blue balloons is given by the combination formula:\n\n$$C(n, k) = \\frac{n!}{k!(n-k)!}$$\n\nFor the", "correct": false, "reward": -1, "repetition": {"words": 251, "unique_ratio": 0.24302788844621515, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "Think about whether there can be two red balls in the room.\n\n## Solution\n\nSince among any two balls, one is blue, there cannot be two red balls in the room. Therefore, there are 84 blue balloons and 1 red balloon in the room.\n\n## Answer\n\n1 ball.", "correct": true, "reward": 1.0, "repetition": {"words": 46, "unique_ratio": 0.6304347826086957, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.3728113305845737}], "split": "math_dev"}
+{"id": "math_058", "row": {"problem": "Example 4 Find the integer solution of the equation $\\left[\\frac{x}{1!}\\right]+\\left[\\frac{x}{2!}\\right]+\\left[\\frac{x}{3!}\\right]+\\cdots+$ $\\left[\\frac{x}{10!}\\right]=1001$.", "solution": "Solution: $\\because$ the solution $x$ of the equation is a natural number, and $xx+\\frac{x}{2}+\\frac{x}{6}+\\frac{x}{24}+\\frac{x}{120}-4 \\\\\n=\\frac{103}{60} x-4, \\therefore 583<x<585, x=584,\n\\end{array}\n$$\n\nVerification shows that $x=584$ is the unique solution of the equation.", "answer": "584", "source": "olympiads", "problem_type": "Number Theory", "uuid": "b243ea4d-43cd-560b-919d-ac59cbae4cb9"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nExample 4 Find the integer solution of the equation $\\left[\\frac{x}{1!}\\right]+\\left[\\frac{x}{2!}\\right]+\\left[\\frac{x}{3!}\\right]+\\cdots+$ $\\left[\\frac{x}{10!}\\right]=1001$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To solve the given problem, we need to find an integer solution of the equation:\n\n$$\\left[\\frac{x}{1!}\\right]+\\left[\\frac{x}{2!}\\right]+\\left[\\frac{x}{3!}\\right]+\\cdots+$$ $\\left[\\frac{x}{10!}\\right]=1001$$\n\nThis is equivalent to finding the smallest integer $N$ such that:\n\n$$\\left[\\frac{x}{1!}\\right] + \\left[\\frac{x}{2!}\\right] + \\left[\\frac{x}{3!}\\right] + \\cdots + \\left[\\frac{x}{10!}\\right] = N$$\n\nSo, we want the sum of the reciprocals of the factorials of all the $x$ from $1!$ to $10!$ to be equal to $N$.\n\nLet's examine the value of $N$:\n\n$$N = \\left[\\frac{x}{1!}\\right] + \\left[\\frac{x}{2!}\\right] + \\left[\\frac{x}{3!}\\right] + \\cdots + \\left[\\frac{x}{10!}\\right]$$\n\nNow, we can see that each term on the left side is equal to $N$, and there are $10$ terms on the left side. Therefore, we have:\n\n$$\\left[\\frac{x}{1!}\\right] + \\left[\\frac{x}{2!}\\right] + \\left[\\frac{x}{3!}\\right] + \\cdots + \\left[\\frac{x}{10!}\\right] = 10N$$\n\nSince $N = 10N", "correct": false, "reward": -1, "repetition": {"words": 174, "unique_ratio": 0.3333333333333333, "max_4gram_count": 8, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "The right-hand side of the equation is 1001, which means that we need to find all integer values of $x$ that satisfy the equation.\n\nLet's start by considering the left-hand side of the equation:\n\n$$\\left[ \\frac{x}{1!} + \\frac{x}{2!} + \\frac{x}{3!} + \\cdots + \\frac{x}{10!} \\right] = 1001$$\n\nWe can rewrite the left-hand side as:\n\n$$\\frac{x}{1!} + \\frac{x}{2!} + \\frac{x}{3!} + \\cdots + \\frac{x}{10!} = 1001$$\n\nNow, let's consider the right-hand side of the equation:\n\n$$1001 = \\left\\lfloor \\frac{x}{10!} \\right\\rfloor$$\n\nWe can rewrite this as:\n\n$$1001 = \\left\\lfloor \\frac{x}{10!} \\right\\rfloor = \\left\\lfloor x \\right\\rfloor + \\left\\lfloor \\frac{x}{10!} \\right\\rfloor$$\n\nLet's define two functions $f(x) = \\left\\lfloor x \\right\\rfloor + \\left\\lfloor \\frac{x}{10!} \\right\\rfloor$ and $g(x) = \\left\\lfloor x \\right\\rfloor + 1001 - \\left\\lfloor \\frac{x}{10!} \\right\\rfloor$.\n\nWe can find the values of $f(x)$ and", "correct": false, "reward": -1, "repetition": {"words": 160, "unique_ratio": 0.28125, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution: $\\because$ the solution $x$ of the equation is a natural number, and $xx+\\frac{x}{2}+\\frac{x}{6}+\\frac{x}{24}+\\frac{x}{120}-4 \\\\\n=\\frac{103}{60} x-4, \\therefore 583<x<585, x=584,\n\\end{array}\n$$\n\nVerification shows that $x=584$ is the unique solution of the equation.", "correct": true, "reward": 1.0, "repetition": {"words": 52, "unique_ratio": 0.5769230769230769, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_059", "row": {"problem": "Example 1 In $\\triangle ABC$, it is known that $x \\sin A + y \\sin B + z \\sin C = 0$. Find the value of $(y + z \\cos A)(z + x \\cos B)(x + y \\cos C) + (y \\cos A + z)(z \\cos B + x)(x \\cos C + y)$.", "solution": "In $\\triangle A B C$, $\\sin C=\\sin (A+B)=\\sin A \\cos B+\\cos A \\sin B$. Substituting the known conditions, we get $x \\sin A+y \\sin B+z(\\sin A \\cos B+\\cos A \\sin B)=0$, which simplifies to $\\sin A(x+z \\cos B)=-\\sin B(y+z \\cos A)$.\nSimilarly, $\\sin B(y+x \\cos C)=-\\sin C(z+x \\cos B)$, $\\sin C(z+y \\cos A)=-\\sin A(x+y \\cos C)$,\nMultiplying the three equations, we have:\n$(x+z \\cos B)(y+x \\cos C)(z+y \\cos A)=-(y+z \\cos A)(z+x \\cos B)(x+y \\cos C)$, which simplifies to $(y+z \\cos A)(z+x \\cos B)(x+y \\cos C)+(y \\cos A+z)(z \\cos B+x)(x \\cos C+y)=0$.", "answer": "0", "source": "olympiads", "problem_type": "Algebra", "uuid": "80c5345a-b5f8-5bef-8c9f-f624dc51008a"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nExample 1 In $\\triangle ABC$, it is known that $x \\sin A + y \\sin B + z \\sin C = 0$. Find the value of $(y + z \\cos A)(z + x \\cos B)(x + y \\cos C) + (y \\cos A + z)(z \\cos B + x)(x \\cos C + y)$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find the value of $(y + z \\cos A)(z + x \\cos B)(x + y \\cos C) + (y \\cos A + z)(z \\cos B + x)(x \\cos C + y)$, we can start by using the given equation and the sine rule.\n\nThe sine rule states that for any triangle with sides of length $a$, $b$, and $c$ opposite to angles $A$, $B$, and $C$ respectively, and opposite side to side $c$ is $\\frac{a}{\\sin A}$, $\\frac{b}{\\sin B}$, and $\\frac{c}{\\sin C}$, we have:\n\n$$\\frac{a}{\\sin A} = \\frac{b}{\\sin B} = \\frac{c}{\\sin C}$$\n\nNow, we can express the given equation in terms of the sides $sin A$, $sin B$, and $sin C$ and the sine rule.\n\nFirst, we substitute $sin A = \\sin B$, $sin B = \\sin C$, and $sin C = \\sin A$.\n\nAfter substitution, the equation becomes:\n\n$$\\frac{x}{\\sin A} + \\frac{y}{\\sin B} + \\frac{z}{\\sin C} = 0$$\n\nNext, we substitute $sin C = \\sin A$ and $sin B = \\sin A$.\n\nAfter substitution, the equation becomes:\n\n$$x + y \\sin B + z \\sin C = 0$$\n\nNow, we can see that the equation is equivalent to:\n\n$$(x + y \\sin B + z \\sin C)(x + y \\cos C + z) = 0$$\n\nSince $x + y \\sin B + z", "correct": false, "reward": -1, "repetition": {"words": 210, "unique_ratio": 0.24761904761904763, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "This problem is about using the Law of Sines and the Law of Cosines to find the values of trigonometric functions. The goal is to find the value of $(y + z \\cos A)(z + x \\cos B)(x + y \\cos C) + (y \\cos A + z)(z \\cos B + x)(x \\cos C + y)$.\n\nTo begin with, we can rewrite the expression in a more convenient form. We know that $\\sin C = \\sin(180^\\circ - A - B) = \\sin(A + B)$, and similarly for the other angles.\n\nNow, we can apply the Law of Sines:\n$$y + z \\cos A = \\sin(A + B) + z \\cos A$$\n$$z + x \\cos B = \\sin(A + B) + x \\cos B$$\n$$x + y \\cos C = \\sin(A + B) + y \\cos C$$\n\nNow, we can substitute these expressions into the given expression:\n$$(y + z \\cos A)(z + x \\cos B)(x + y \\cos C) + (y \\cos A + z)(z \\cos B + x)(x \\cos C + y) = (\\sin(A + B) + z \\cos A)(\\sin(A + B) + x \\cos B)(\\sin(A + B) + y \\cos C) + (\\sin(A + B) + x \\cos A)(\\sin(A + B) + y \\cos C)$$\n\nSince $\\sin(A + B) = \\sin A \\cos B + \\cos A \\sin B$ and $\\sin(A + B) = \\sin A \\cos B - \\", "correct": false, "reward": -1, "repetition": {"words": 207, "unique_ratio": 0.25120772946859904, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "In $\\triangle A B C$, $\\sin C=\\sin (A+B)=\\sin A \\cos B+\\cos A \\sin B$. Substituting the known conditions, we get $x \\sin A+y \\sin B+z(\\sin A \\cos B+\\cos A \\sin B)=0$, which simplifies to $\\sin A(x+z \\cos B)=-\\sin B(y+z \\cos A)$.\nSimilarly, $\\sin B(y+x \\cos C)=-\\sin C(z+x \\cos B)$, $\\sin C(z+y \\cos A)=-\\sin A(x+y \\cos C)$,\nMultiplying the three equations, we have:\n$(x+z \\cos B)(y+x \\cos C)(z+y \\cos A)=-(y+z \\cos A)(z+x \\cos B)(x+y \\cos C)$, which simplifies to $(y+z \\cos A)(z+x \\cos B)(x+y \\cos C)+(y \\cos A+z)(z \\cos B+x)(x \\cos C+y)=0$.", "correct": true, "reward": 1.0, "repetition": {"words": 138, "unique_ratio": 0.18115942028985507, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_060", "row": {"problem": "2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$.", "solution": "2. Solution. Since the desired integer solution $x$ is divisible by 10 and 12, it is divisible by 60, hence it can be written in the form $x=60 t, t \\in Z$. Substitute $x=60 t$ into the original equation:\n\n$3600 t^{2}+60 b t-9600=0$. Express $b: b=\\frac{160}{t}-60 t$. For $b$ to be an integer, $t$ must be a divisor of the number 160. $160=2^{5} \\cdot 5$. The number $160=2^{5} \\cdot 5$ has 12 positive divisors: $1,2,2^{2}, 2^{3}, 2^{4}, 2^{5}, 5,2 \\cdot 5,2^{2} \\cdot 5,2^{3} \\cdot 5,2^{4} \\cdot 5,2^{5} \\cdot 5$. Taking the sign into account, we get 24 divisors. Each of them corresponds to a solution $x=60 t$ and an integer $b=\\frac{160}{t}-60 t$. Therefore, the number of integers $b$ that satisfy the condition of the problem is 24. Since $b$ decreases as $t$ increases, the maximum $b$ corresponds to the smallest $t=-160$, so $b_{\\max }=-1+60 \\cdot 160=9599$.", "answer": "9599", "source": "olympiads", "problem_type": "Algebra", "uuid": "622e4608-b0d7-5071-90bd-fa39367cb1ac"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n2. How many integers $b$ exist such that the equation $x^{2}+b x-9600=0$ has an integer solution that is a multiple of both 10 and 12? Specify the largest possible $b$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Let $b=10a+400$ and $b=12a+400$ where $a\\geqslant 0$.\nThe equation can be rewritten as\n$(10a+400)x-(12a+400)x=9600$\nor\n$(2a+12)x=9600$, which simplifies to $x=\\frac{9600}{2a+12}$.\nSince $x$ must be an integer, this fraction must simplify to an integer.\nTherefore, $2a+12$ must divide $9600$.\n\nThe prime factorization of $9600$ is $2^3 \\cdot 3^2 \\cdot 5^2$.\nThe possible values of $2a+12$ are the divisors of $9600$, so $2a+12$ can be $1$, $2$, $3$, $4$, $5$, $6$, $8$, $9$, $10$, $12$, $15$, $16$, $20$, $22$, $30$, $36$, $40$, $45$, $60$, $72$, $90$, $92$, $100$, $150$, $192$, $324$, $500$, $900$.\nTherefore, $2a+12$ can be $2$, $3$, $4$, $5$, $6$, $8$, $9$, $10$, $12$, $15$, $16$, $20$, $22$, $30$, $36$,", "correct": false, "reward": -0.5, "repetition": {"words": 135, "unique_ratio": 0.5259259259259259, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "To find the integer solutions $x$ that make the equation have integer solutions, we can use the fact that the product of the roots must be equal to the constant term, which is $-9600$ in this case. Let's write the equation as a quadratic equation:\n\\[ x^2 + bx - 9600 = 0 \\]\n\nThe sum of the roots is:\n\\[ x_1 + x_2 = -b \\]\n\nFor $x$ to be a multiple of both 10 and 12, we need $x_1$ and $x_2$ to be multiples of 10 and 12, respectively.\n\nLet's find the least common multiple (LCM) of 10 and 12, which is 60. This means the equation is equivalent to:\n\\[ x^2 + bx - 9600 = (x - 60)(x - 10) = 0 \\]\n\nThis means that $b$ must be equal to 60.\n\nThus, the largest possible $b$ is:\n\\[ b = \\boxed{60} \\]\nFinal answer:\n#### 60", "correct": false, "reward": -1, "repetition": {"words": 138, "unique_ratio": 0.43478260869565216, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "2. Solution. Since the desired integer solution $x$ is divisible by 10 and 12, it is divisible by 60, hence it can be written in the form $x=60 t, t \\in Z$. Substitute $x=60 t$ into the original equation:\n\n$3600 t^{2}+60 b t-9600=0$. Express $b: b=\\frac{160}{t}-60 t$. For $b$ to be an integer, $t$ must be a divisor of the number 160. $160=2^{5} \\cdot 5$. The number $160=2^{5} \\cdot 5$ has 12 positive divisors: $1,2,2^{2}, 2^{3}, 2^{4}, 2^{5}, 5,2 \\cdot 5,2^{2} \\cdot 5,2^{3} \\cdot 5,2^{4} \\cdot 5,2^{5} \\cdot 5$. Taking the sign into account, we get 24 divisors. Each of them corresponds to a solution $x=60 t$ and an integer $b=\\frac{160}{t}-60 t$. Therefore, the number of integers $b$ that satisfy the condition of the problem is 24. Since $b$ decreases as $t$ increases, the maximum $b$ corresponds to the smallest $t=-160$, so $b_{\\max }=-1+60 \\cdot 160=9599$.", "correct": true, "reward": 1.0, "repetition": {"words": 184, "unique_ratio": 0.3967391304347826, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_dev"}
+{"id": "math_061", "row": {"problem": "Task B-2.1. (20 points) For complex numbers $z, w$ such that $|z|=|w|=|z-w|$, calculate $\\left(\\frac{z}{w}\\right)^{99}$.", "solution": "First solution. Let $u=\\frac{z}{w}=x+y i$. Then $|u|=1,|u-1|=1$.\n\nThen $x^{2}+y^{2}=1,(x-1)^{2}+y^{2}=1$.\n\nBy subtracting these equations, we get $x=\\frac{1}{2}, y= \\pm \\frac{\\sqrt{3}}{2}$, so $u=\\frac{1}{2} \\pm \\frac{\\sqrt{3}}{2} i$.\n\nNow,\n\n$$\n\\left(\\frac{z}{w}\\right)^{99}=u^{99}=\\left[\\left(\\frac{1}{2} \\pm \\frac{\\sqrt{3}}{2} i\\right)^{3}\\right]^{33}=\\left(\\frac{1}{8} \\pm \\frac{3 \\sqrt{3}}{8} i-\\frac{9}{8} \\mp \\frac{3 \\sqrt{3}}{8} i\\right)^{33}=(-1)^{33}=-1\n$$", "answer": "-1", "source": "olympiads", "problem_type": "Algebra", "uuid": "bcfb9529-7521-5cae-8078-82e957941ea5"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nTask B-2.1. (20 points) For complex numbers $z, w$ such that $|z|=|w|=|z-w|$, calculate $\\left(\\frac{z}{w}\\right)^{99}$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "If $|z|=|w|=|z-w|$, then $z=w$ or $z=w$ or $z=-w$. Since $z=w$ is the only possible solution, $\\left(\\frac{z}{w}\\right)^{99}=\\left(\\frac{w}{w}\\right)^{99}=1$.\n\nThis means that $\\left(\\frac{z}{w}\\right)^{99}=\\boxed{1}$.\n\nFinal answer:\n#### 1", "correct": false, "reward": -1, "repetition": {"words": 49, "unique_ratio": 0.4489795918367347, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To approach this problem, we need to understand that if $r$ is a complex number such that $|z|=|w|=|z-w|$, then $z$ is a geometric progression with common ratio $r$.\n\nSince $r$ is a complex number, it can be written as $|r| = c$, where $c$ is a real number.\n\nTherefore, we can write:\n\\[ z = c \\left(\\frac{z}{w}\\right) = c \\left(\\frac{c}{w}\\right) = c^2 \\left(\\frac{1}{w}\\right) = c^2 \\left(\\frac{|z|}{|w|}\\right) = c^2 \\left(\\frac{|z-w|}{|w|}\\right) \\]\n\nSince $|z|=|w|=|z-w|$, we can simplify this to:\n\\[ z = c^2 \\left(\\frac{|z-w|}{|w|}\\right) = c^2 \\left(\\frac{|z|}{|w|}\\right) = c^2 \\left(\\frac{|z-w|}{|w|}\\right) = c^2 \\left(\\frac{|z-w|}{|w|}\\right) = c^2 \\left(\\frac{|z-w|}{|w|} \\times \\frac{1}{\\left|z-w|\\right|}\\right) = c^2 \\left(\\frac{|w|}{|z-w|}\\right) = c^2 \\left(\\frac{|w|}{|z-w|}\\right) = c^2 \\left(\\frac{1}{|z-w|}\\right)", "correct": false, "reward": -1, "repetition": {"words": 170, "unique_ratio": 0.2411764705882353, "max_4gram_count": 11, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "First solution. Let $u=\\frac{z}{w}=x+y i$. Then $|u|=1,|u-1|=1$.\n\nThen $x^{2}+y^{2}=1,(x-1)^{2}+y^{2}=1$.\n\nBy subtracting these equations, we get $x=\\frac{1}{2}, y= \\pm \\frac{\\sqrt{3}}{2}$, so $u=\\frac{1}{2} \\pm \\frac{\\sqrt{3}}{2} i$.\n\nNow,\n\n$$\n\\left(\\frac{z}{w}\\right)^{99}=u^{99}=\\left[\\left(\\frac{1}{2} \\pm \\frac{\\sqrt{3}}{2} i\\right)^{3}\\right]^{33}=\\left(\\frac{1}{8} \\pm \\frac{3 \\sqrt{3}}{8} i-\\frac{9}{8} \\mp \\frac{3 \\sqrt{3}}{8} i\\right)^{33}=(-1)^{33}=-1\n$$", "correct": true, "reward": 1.0, "repetition": {"words": 105, "unique_ratio": 0.29523809523809524, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_062", "row": {"problem": "4.9 On a $10 \\times 10$ grid paper, there are 11 horizontal grid lines and 11 vertical grid lines. A line segment connecting two adjacent nodes on the same straight line is called a \"link segment\". How many link segments need to be erased at least, so that at most 3 link segments remain at each node?", "solution": "[Solution] Removing the dotted links in the right figure satisfies the requirements of the problem, and there are 41 dotted links in the figure, so the minimum value required does not exceed 41.\n\nOn the other hand, there are 81 internal nodes on the grid paper, each with 4 links. Therefore, at least one link must be erased from each node to meet the requirement. Since each link connects two nodes, at least 41 links must be erased.\nIn summary, the minimum number of links that need to be erased is 41.", "answer": "41", "source": "olympiads", "problem_type": "Combinatorics", "uuid": "8253de0b-ede9-5630-a1d1-7d0ae1ae5cd2"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n4.9 On a $10 \\times 10$ grid paper, there are 11 horizontal grid lines and 11 vertical grid lines. A line segment connecting two adjacent nodes on the same straight line is called a \"link segment\". How many link segments need to be erased at least, so that at most 3 link segments remain at each node?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To perform the task of \"erasing\" link segments, we must firstaaaaa deal with the remaining 11 horizontal line segments and then deal with the remaining 11 vertical line segments, which is the least.\n\nThere are $11 + 11 = 22$ line segments on the $10 \\times 10$ grid paper.\n\nThe first step is to erase a certain number of link segments, $x$.\n\nThe remaining link segments are $22 - x$.\n\nThe remaining link segments are $22 - x - x = 22 - 2x$.\n\nThe problem requires that at most 3 link segments remain at each node.\n\nThus, we need to eliminate $3(22 - 2x)$ link segments.\n\n$3(22 - 2x) \\leq 22$.\n\n$66 - 6x \\leq 22$.\n\n$6x \\leq 44$.\n\n$x \\leq 6$.\n\nThus, $x$ has to be less than or equal to 6.\n\nSo, to make sure that at most 3 link segments remain at each node, we need to erase $3(22 - 2x) \\leq 22$ link segments.\n\nTherefore, we need to erase at least 6 link segments.\n\nThe answer is \\boxed{6}.", "correct": false, "reward": -1, "repetition": {"words": 166, "unique_ratio": 0.40963855421686746, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To determine the minimum number of link segments that need to be erased, we can first think of placing the nodes in a grid where each node is connected to its neighbors, and we want to create a grid of nodes with the maximum number of link segments between any two nodes.\n\nImagine a grid where each node is connected to its neighbors, and we want to create a structure with the maximum number of link segments between any two nodes. To achieve this, we can start by connecting each node to its immediate neighbors, then connect each neighbor to its neighbors, and so on.\n\nHowever, this approach is not efficient because we want to maximize the number of link segments between any two nodes with each node being connected to its neighbors. If we attempt to do this, we will be creating a grid of nodes with a lot of link segments between each node.\n\nTo make it easier, we can think of this problem as having a \" simplify\" or \"level\" in the grid. This means that each node is connected to its immediate neighbors, and we can have a \" exceeds\" or \"root\" in the grid.\n\nThe simplest way to think about this is to have a \" net\" structure where each node is connected to its neighbors and each connection is a direct edge between two nodes. In this case, we can have a \" binocular\" in the grid with a \"step\" or \"net\" that each node is connected to its neighbors, and each connection is a direct edge between two nodes.\n\nHowever, if we want to maximize the number of link segments between any two nodes, we can try to build the grid in a \"([[\" or \"level\") shape. This means that each node is connected to its immediate neighbors and the neighbors of its neighbors, and we can have a \"level\" shape in the grid.\n\nTo achieve this, we can start", "correct": false, "reward": -1, "repetition": {"words": 322, "unique_ratio": 0.2732919254658385, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "[Solution] Removing the dotted links in the right figure satisfies the requirements of the problem, and there are 41 dotted links in the figure, so the minimum value required does not exceed 41.\n\nOn the other hand, there are 81 internal nodes on the grid paper, each with 4 links. Therefore, at least one link must be erased from each node to meet the requirement. Since each link connects two nodes, at least 41 links must be erased.\nIn summary, the minimum number of links that need to be erased is 41.", "correct": true, "reward": 1.0, "repetition": {"words": 92, "unique_ratio": 0.5978260869565217, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_063", "row": {"problem": "9. Point $Y$ lies on line segment $X Z$ such that $X Y=5$ and $Y Z=3$. Point $G$ lies on line $X Z$ such that there exists a triangle $A B C$ with centroid $G$ such that $X$ lies on line $B C, Y$ lies on line $A C$, and $Z$ lies on line $A B$. Compute the largest possible value of $X G$.", "solution": "Answer:\n$\\frac{20}{3}$\n\nSolution: The key claim is that we must have $\\frac{1}{X G}+\\frac{1}{Y G}+\\frac{1}{Z G}=0$ (in directed lengths).\nWe present three proofs of this fact.\nProof 1: By a suitable affine transformation, we can assume without loss of generality that $A B C$ is equilateral. Now perform an inversion about $G$ with radius $G A=G B=G C$. Then the images of $X, Y, Z$ (call them $\\left.X^{\\prime}, Y^{\\prime}, Z^{\\prime}\\right)$ lie on $(G B C),(G A C),(G A B)$, so they are the feet of the perpendiculars from $A_{1}, B_{1}, C_{1}$ to line $X Y Z$, where $A_{1}, B_{1}, C_{1}$ are the respective antipodes of $G$ on $(G B C),(G A C),(G A B)$. But now $A_{1} B_{1} C_{1}$ is an equilateral triangle with medial triangle $A B C$, so its centroid is $G$. Now the centroid of (degenerate) triangle $X^{\\prime} Y^{\\prime} Z^{\\prime}$ is the foot of the perpendicular of the centroid of $A_{1} B_{1} C_{1}$ onto the line, so it is $G$. Thus $X^{\\prime} G+Y^{\\prime} G+Z^{\\prime} G=0$, which yields the desired claim.\n\nProof 2: Let $W$ be the point on line $X Y Z$ such that $W G=2 G X$ (in directed lengths). Now note that $(Y, Z ; G, W)$ is a harmonic bundle, since projecting it through $A$ onto $B C$ gives $\\left(B, C ; M_{B C}, \\infty_{B C}\\right)$. By harmonic bundle properties, this yields that $\\frac{1}{Y G}+\\frac{1}{Z G}=\\frac{2}{W G}$ (in directed lengths), which gives the desired.\n\nProof 3: Let $P \\neq G$ be an arbitrary point on the line $X Y Z$. Now, in directed lengths and signed areas, $\\frac{G P}{G X}=\\frac{[G B P]}{[G B X]}=\\frac{[G C P]}{[G C X]}$, so $\\frac{G P}{G X}=\\frac{[G B P]-[G C P]}{[G B X]-[G C X]}=\\frac{[G B P]-[G C P]}{[G B C]}=\\frac{3([G B P]-[G C P])}{[A B C]}$. Writing analogous equations for $\\frac{G P}{G Y}$ and $\\frac{G P}{G Z}$ and summing yields $\\frac{G P}{G X}+\\frac{G P}{G Y}+\\frac{G P}{G Z}=0$, giving the desired.\nWith this lemma, we may now set $X G=g$ and know that\n$$\n\\frac{1}{g}+\\frac{1}{g-5}+\\frac{1}{g-8}=0\n$$\n\nSolving the quadratic gives the solutions $g=2$ and $g=\\frac{20}{3}$; the latter hence gives the maximum (it is not difficult to construct an example for which $X G$ is indeed 20/3).", "answer": "\\frac{20}{3}", "source": "olympiads", "problem_type": "Geometry", "uuid": "66849bc3-7be9-5bd8-8927-21c8941a4567"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n9. Point $Y$ lies on line segment $X Z$ such that $X Y=5$ and $Y Z=3$. Point $G$ lies on line $X Z$ such that there exists a triangle $A B C$ with centroid $G$ such that $X$ lies on line $B C, Y$ lies on line $A C$, and $Z$ lies on line $A B$. Compute the largest possible value of $X G$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find the largest possible value of $XG$, we must consider the possible positions of $XG$ on the line segments $XY$, $YZ$, $ZX$, and $XZ$.\n\nFirst, let's consider the case when $X$ is between $Z$ and $Y$. In this case, $XG$ must be the length of the line segment $YZ$, which is 3.\n\nNext, let's consider the case when $X$ is between $Z$ and $Y$. In this case, $XG$ must be the length of the line segment $ZX$, which is 5.\n\nNow, consider the case when $X$ is between $Y$ and $Z$. In this case, $XG$ must be the length of the line segment $ZX$, which is 5.\n\nWe can see that $XG$ will always be the longest possible length between $X$ and $Z$.\n\nNow, let's consider the case when $X$ is between $Z$ and $Y$. In this case, we need to find the largest possible value of $XG$ that satisfies the given conditions.\n\nSince $Z$ lies on $XZ$ and $YZ$ is parallel to $XZ$, we can consider the case where $X$ is between $Z$ and $Y$. In this case, $XG$ must be the length of the line segment $ZX$, which is 5.\n\nNext, let's consider the case when $X$ is between $Y$ and $Z$. In this case, we can try to find the largest possible value of $XG$ that satisfies the given conditions.\n\nSince $Z$ lies on $XZ$ and $YZ$ is parallel to", "correct": false, "reward": -1, "repetition": {"words": 234, "unique_ratio": 0.23076923076923078, "max_4gram_count": 5, "max_sentence_count": 3, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To solve this problem, we need to use properties of centroids and the given conditions.\n\nFirst, we are given that $X$ lies on line $BC$, $Y$ lies on line $AC$, and $Z$ lies on line $AB$.\n\nWe can use the given conditions to find the possible positions of $X$ and $Y$.\n\nLet $X = (x, 0)$ and $Y = (y, 0)$.\n\nWe know that $XZ$ is perpendicular to $BC$, and $XZ$ is parallel to $YZ$.\n\nSince $X$ lies on $BC$, we know that $x + y = 5$.\n\nAlso, $XZ$ is perpendicular to $AC$, and $X$ lies on $AC$, so we know that $x = 3$.\n\nTherefore, $y = 0$.\n\nNow, we need to find the possible positions of $Y$ and $Z$.\n\nWe know that $Y$ lies on $AC$, so we know that $y = 3$.\n\nWe also know that $YZ$ is parallel to $XY$, and $YZ$ is perpendicular to $XZ$.\n\nSince $Y$ lies on $AC$, we know that $y = 3$.\n\nFinally, we know that $Z$ lies on $AB$, so we know that $x = 3$.\n\nTherefore, $X$ lies on $BC$, $Y$ lies on $AC$, and $Z$ lies on $AB$.\n\nThe largest possible value of $XG$ is $\\boxed{15}$.", "correct": false, "reward": -1, "repetition": {"words": 191, "unique_ratio": 0.27225130890052357, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Answer:\n$\\frac{20}{3}$\n\nSolution: The key claim is that we must have $\\frac{1}{X G}+\\frac{1}{Y G}+\\frac{1}{Z G}=0$ (in directed lengths).\nWe present three proofs of this fact.\nProof 1: By a suitable affine transformation, we can assume without loss of generality that $A B C$ is equilateral. Now perform an inversion about $G$ with radius $G A=G B=G C$. Then the images of $X, Y, Z$ (call them $\\left.X^{\\prime}, Y^{\\prime}, Z^{\\prime}\\right)$ lie on $(G B C),(G A C),(G A B)$, so they are the feet of the perpendiculars from $A_{1}, B_{1}, C_{1}$ to line $X Y Z$, where $A_{1}, B_{1}, C_{1}$ are the respective antipodes of $G$ on $(G B C),(G A C),(G A B)$. But now $A_{1} B_{1} C_{1}$ is an equilateral triangle with medial triangle $A B C$, so its centroid is $G$. Now the centroid of (degenerate) triangle $X^{\\prime} Y^{\\prime} Z^{\\prime}$ is the foot of the perpendicular of the centroid of $A_{1} B_{1} C_{1}$ onto the line, so it is $G$. Thus $X^{\\prime} G+Y^{\\prime} G+Z^{\\prime} G=0$, which yields the desired claim.\n\nProof 2: Let $W$ be the point on line $X Y Z$ such that $W G=2 G X$ (in directed lengths). Now note that $(Y, Z ; G, W)$ is a harmonic bundle, since projecting it through $A$ onto $B C$ gives $\\left(B, C ; M_{B C}, \\infty_{B C}\\right)$. By harmonic bundle properties, this yields that $\\frac{1}{Y G}+\\frac{1}{Z G}=\\frac{2}{W G}$ (in directed lengths), which gives the desired.\n\nProof 3: Let $P \\neq G$ be an arbitrary point on the line $X Y Z$. Now, in directed lengths and signed areas, $\\frac{G P}{G X}=\\frac{[G B P]}{[G B X]}=\\frac{[G C P]}{[G C X]}$, so $\\frac{G P}{G X}=\\frac{[G B P]-[G C P]}{[G B X]-[G C X]}=\\frac{[G B P]-[G C P]}{[G B C]}=\\frac{3([G B P]-[G C P])}{[A B C]}$. Writing analogous equations for $\\frac{G P}{G Y}$ and $\\frac{G P}{G Z}$ and summing yields $\\frac{G P}{G X}+\\frac{G P}{G Y}+\\frac{G P}{G Z}=0$, giving the desired.\nWith this lemma, we may now set $X G=g$ and know that\n$$\n\\frac{1}{g}+\\frac{1}{g-5}+\\frac{1}{g-8}=0\n$$\n\nSolving the quadratic gives the solutions $g=2$ and $g=\\frac{20}{3}$; the latter hence gives the maximum (it is not difficult to construct an example for which $X G$ is indeed 20/3).", "correct": true, "reward": 1.0, "repetition": {"words": 472, "unique_ratio": 0.2711864406779661, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_064", "row": {"problem": "5. Let $a$ and $n$ be natural numbers, and it is known that $a^{n}-2014$ is a number with $2014$ digits. Find the smallest natural number $k$ such that $a$ cannot be a $k$-digit number.", "solution": "Solution. Let $a$ be a $k$-digit number, then $10^{k-1} \\leq a2013$, i.e., $k>45$.\n\nWe will check for $k$ starting from 46, whether the condition of the absence of an integer in the specified interval is met. We will find that\n\n(for $k=46$) $2013 / 462013$ );\n\n(for $k=47$) $2013 / 472013$ );\n\n(for $k=48$) $2013 / 482013$ );\n\nbut (for $k=49$) $2013 / 49 > 41$ (since $\\quad 41 \\cdot 49 = 45^2 - 1^2 = 2025 - 16 < 2013$).\n\nThus, the interval $[2013 / 49, 2013 / 48)$ does not contain any integers, and for $k=49$, there are no suitable $n$.\n\n## Answer: $\\mathbf{k}=49$.\n\n## Criteria.\n\nThe problem is complex, so don't be stingy with points for reasonable thoughts. Any answer in the vicinity of $\\sqrt{2014}$, which is preceded by something reasonable, is worth points. A point is deducted for the absence of what is in small print.", "answer": "49", "source": "olympiads", "problem_type": "Number Theory", "uuid": "77cba4e7-5a90-5533-9f4b-220b85918184"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n5. Let $a$ and $n$ be natural numbers, and it is known that $a^{n}-2014$ is a number with $2014$ digits. Find the smallest natural number $k$ such that $a$ cannot be a $k$-digit number.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Given that $a^n - 2014$ has $2014$ digits, we need to find the smallest natural number $k$ such that $a$ cannot also be a $k$-digit number.\n\nIf $a$ is a $k$-digit number, then it can be written as $a = \\overline{abcd...}$, where $a, b, c, d,..., k$ are its digits.\n\nSubstituting $a = \\overline{abcd...}$ into the equation, we get:\n\n$$\\overline{abcd...}^n - 2014 = \\overline{abcd...}$$\n\nRearranging the equation, we have:\n\n$$\\overline{abcd...}^n = 2014\\overline{abcd...}$$\n\nTaking the logarithm of both sides, we get:\n\n$$n \\log \\overline{abcd...} = \\log 2014 \\overline{abcd...}$$\n\nNow, we need to find the smallest natural number $k$ such that $a$ cannot also be a $k$-digit number.\n\nUsing the definition of $a$ and $n$, we can write $n = 2014 - 2014 \\log \\overline{a}$.\n\nSubstituting this into the equation, we get:\n\n$$2014 - 2014 \\log \\overline{a} = \\log 2014$$\n\nSolving for $\\log \\overline{a}$, we get:\n\n$$\\log \\overline{a} = 4$$\n\nSince $a$ is", "correct": false, "reward": -1, "repetition": {"words": 158, "unique_ratio": 0.3670886075949367, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "2014 is a multiple of 4, so\n$$(a^k - 2014) \\equiv 0 \\pmod{4}$$\n\nThe possible values of $a$ are 1, 2, 3, 4, and 5.\n\nIf $a = 5$, then $5^k - 2014 = 0 \\Rightarrow 5^k = 2014 \\Rightarrow k = 4$, which is not a $k$-digit number.\n\nIf $a = 4$, then $4^k - 2014 = 0 \\Rightarrow 4^k = 2014 \\Rightarrow k = 5$, which is a $k$-digit number.\n\nIf $a = 3$, then $3^k - 2014 = 0 \\Rightarrow 3^k = 2014 \\Rightarrow k = 6$, which is a $k$-digit number.\n\nIf $a = 2$, then $2^k - 2014 = 0 \\Rightarrow 2^k = 2014 \\Rightarrow k = 127$, which is not a $k$-digit number.\n\nIf $a = 1$, then $1^k - 2014 = 0 \\Rightarrow 1^k = 2014 \\Rightarrow k = 402$, which is not a $k$-digit number.\n\nIf $a = 0$, then $0^k - 2014 = 0 \\Rightarrow 0^k = 2014 \\Rightarrow k = 40", "correct": false, "reward": -1, "repetition": {"words": 149, "unique_ratio": 0.2080536912751678, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution. Let $a$ be a $k$-digit number, then $10^{k-1} \\leq a2013$, i.e., $k>45$.\n\nWe will check for $k$ starting from 46, whether the condition of the absence of an integer in the specified interval is met. We will find that\n\n(for $k=46$) $2013 / 462013$ );\n\n(for $k=47$) $2013 / 472013$ );\n\n(for $k=48$) $2013 / 482013$ );\n\nbut (for $k=49$) $2013 / 49 > 41$ (since $\\quad 41 \\cdot 49 = 45^2 - 1^2 = 2025 - 16 < 2013$).\n\nThus, the interval $[2013 / 49, 2013 / 48)$ does not contain any integers, and for $k=49$, there are no suitable $n$.\n\n## Answer: $\\mathbf{k}=49$.\n\n## Criteria.\n\nThe problem is complex, so don't be stingy with points for reasonable thoughts. Any answer in the vicinity of $\\sqrt{2014}$, which is preceded by something reasonable, is worth points. A point is deducted for the absence of what is in small print.", "correct": true, "reward": 1.0, "repetition": {"words": 148, "unique_ratio": 0.5945945945945946, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_065", "row": {"problem": "Let there be a figure with $9$ disks and $11$ edges, as shown below.\n\nWe will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$, and $1$ in disk $I$. Find the minimum sum of all real numbers written in $11$ edges.", "solution": "1. **Model the problem as an electrical circuit:**\n   - We are given a graph with 9 disks (nodes) and 11 edges.\n   - We need to assign real numbers to each disk such that the sum of the squares of the differences between the numbers on the disks connected by edges is minimized.\n   - We are required to set the number on disk \\(A\\) to \\(0\\) and on disk \\(I\\) to \\(1\\).\n\n2. **Interpret the problem using electrical circuit theory:**\n   - Consider each disk as a node in an electrical circuit.\n   - Assign a voltage to each node, with disk \\(A\\) at \\(0V\\) and disk \\(I\\) at \\(1V\\).\n   - Each edge represents a resistor of \\(1 \\Omega\\).\n\n3. **Apply the concept of minimizing power in the circuit:**\n   - The goal is to minimize the sum of the squares of the differences, which is equivalent to minimizing the total power dissipated in the circuit.\n   - The power dissipated in a resistor \\(R\\) with voltage difference \\(V\\) is given by \\(P = \\frac{V^2}{R}\\).\n   - For a resistor of \\(1 \\Omega\\), the power dissipated is \\(P = V^2\\).\n\n4. **Calculate the equivalent resistance of the circuit:**\n   - The total resistance between nodes \\(A\\) and \\(I\\) is given as \\(\\frac{33}{20} \\Omega\\).\n   - This is derived from the specific configuration of the graph and the resistances.\n\n5. **Determine the total power in the circuit:**\n   - The total voltage across the circuit is \\(1V\\) (from \\(A\\) to \\(I\\)).\n   - The total power dissipated in the circuit is given by:\n     \\[\n     P = \\frac{V^2}{R_{\\text{eq}}} = \\frac{1^2}{\\frac{33}{20}} = \\frac{20}{33}\n     \\]\n\n6. **Conclusion:**\n   - The minimum sum of the squares of the differences between the numbers on the disks connected by edges is \\(\\frac{20}{33}\\).\n\nThe final answer is \\(\\boxed{\\frac{20}{33}}\\).", "answer": "\\frac{20}{33}", "source": "aops_forum", "problem_type": "Calculus", "uuid": "69191341-fb5c-54c6-bc21-3d80836f8875"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nLet there be a figure with $9$ disks and $11$ edges, as shown below.\n\nWe will write a real number in each and every disk. Then, for each edge, we will write the square of the difference between the two real numbers written in the two disks that the edge connects. We must write $0$ in disk $A$, and $1$ in disk $I$. Find the minimum sum of all real numbers written in $11$ edges.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find the minimum sum of all real numbers written in $11$ edges, we can analyze the structure of the graph described.\n\nSince each edge is connecting two disks, we can model each edge as a pair of disks. The value of the real number in each disk is the absolute difference between the two real numbers written in the two disks that the edge connects.\n\nWe can represent the graph with edges as follows:\n\n-(2^1, 0) - 2^0, -2^0, -2^1, -2^1, -2^2 - 2^1, -2^1, -2^2 - 2^2, ...\n\nThe $x$-coordinate of each vertex is the sum of the squares of the distances from the vertices to the other two vertices. The $y$-coordinate of each vertex is the square of the difference between the two real numbers in the two disks that the edge connects.\n\nNotice that the values in the $y$-coordinate can be represented as a function of the distance in the $x$-direction. This function can be represented as:\n\n$$f(x, y) = |x - y|^2$$\n\nWe can see that the function $f(x, y)$ is a piecewise linear function, and the minimum value occurs when $x = y$. For this to happen, we must have $x = 0$ and $y = 1$, which would make $f(x, y) = 0$.\n\nTo find the minimum value of $f(x, y)$, we can take the derivative of $f(x, y)$ with respect to $x$ and set it equal to zero:\n\n$$f'(x, y) = 2|x - y", "correct": false, "reward": -1, "repetition": {"words": 252, "unique_ratio": 0.3333333333333333, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "The equation that defines the distances between the real numbers in the disks is the sum of the radii of the two disks. This can be written as $r_1 + r_2 = 9$.\n\nWe can also write this equation in terms of the radii of the two disks as $r_1 + r_2 = 9$ and $r_1 - r_2 = 9$.\n\nSolving these two equations, we find that $r_1 = 3$ and $r_2 = 6$.\n\nNow, we can find the sum of all real numbers written in the edges of the figure. This is equal to $11(r_1 + r_2) = 11(3 + 6) = 49$.\n\nTherefore, the minimum sum of all real numbers written in the edges is $\\boxed{49}$.", "correct": false, "reward": -1, "repetition": {"words": 119, "unique_ratio": 0.3865546218487395, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "1. **Model the problem as an electrical circuit:**\n   - We are given a graph with 9 disks (nodes) and 11 edges.\n   - We need to assign real numbers to each disk such that the sum of the squares of the differences between the numbers on the disks connected by edges is minimized.\n   - We are required to set the number on disk \\(A\\) to \\(0\\) and on disk \\(I\\) to \\(1\\).\n\n2. **Interpret the problem using electrical circuit theory:**\n   - Consider each disk as a node in an electrical circuit.\n   - Assign a voltage to each node, with disk \\(A\\) at \\(0V\\) and disk \\(I\\) at \\(1V\\).\n   - Each edge represents a resistor of \\(1 \\Omega\\).\n\n3. **Apply the concept of minimizing power in the circuit:**\n   - The goal is to minimize the sum of the squares of the differences, which is equivalent to minimizing the total power dissipated in the circuit.\n   - The power dissipated in a resistor \\(R\\) with voltage difference \\(V\\) is given by \\(P = \\frac{V^2}{R}\\).\n   - For a resistor of \\(1 \\Omega\\), the power dissipated is \\(P = V^2\\).\n\n4. **Calculate the equivalent resistance of the circuit:**\n   - The total resistance between nodes \\(A\\) and \\(I\\) is given as \\(\\frac{33}{20} \\Omega\\).\n   - This is derived from the specific configuration of the graph and the resistances.\n\n5. **Determine the total power in the circuit:**\n   - The total voltage across the circuit is \\(1V\\) (from \\(A\\) to \\(I\\)).\n   - The total power dissipated in the circuit is given by:\n     \\[\n     P = \\frac{V^2}{R_{\\text{eq}}} = \\frac{1^2}{\\frac{33}{20}} = \\frac{20}{33}\n     \\]\n\n6. **Conclusion:**\n   - The minimum sum of the squares of the differences between the numbers on the disks connected by edges is \\(\\frac{20}{33}\\).\n\nThe final answer is \\(\\boxed{\\frac{20}{33}}\\).", "correct": true, "reward": 1.0, "repetition": {"words": 291, "unique_ratio": 0.33676975945017185, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_066", "row": {"problem": "There is a regular $17$-gon $\\mathcal{P}$ and its circumcircle $\\mathcal{Y}$ on the plane. \nThe vertices of $\\mathcal{P}$ are coloured in such a way that $A,B \\in \\mathcal{P}$ are of diff\u000berent colour, if the shorter arc connecting $A$ and $B$ on $\\mathcal{Y}$ has $2^k+1$ vertices, for some $k \\in \\mathbb{N},$ including $A$ and $B.$ \nWhat is the least number of colours which suffi\u000eces?", "solution": "1. **Identify the relevant values of \\(2^k + 1\\):**\n   Since the minor arc \\(AB\\) has at most 9 vertices on it, the relevant values of \\(2^k + 1\\) are \\(3, 5,\\) and \\(9\\). This is because:\n   \\[\n   2^1 + 1 = 3, \\quad 2^2 + 1 = 5, \\quad 2^3 + 1 = 9\n   \\]\n\n2. **Enumerate the vertices:**\n   Enumerate the vertices of the 17-gon as \\(V_0, V_1, \\ldots, V_{16}\\).\n\n3. **Determine the conditions for different colors:**\n   We have that \\(V_i\\) and \\(V_j\\) are of different colors if \\(i - j \\equiv \\pm 2, \\pm 4, \\pm 8 \\pmod{17}\\). This is derived from the relevant values of \\(2^k + 1\\).\n\n4. **Establish a lower bound for the number of colors:**\n   Since \\(V_0, V_2, V_4\\) must have pairwise distinct colors, we have \\(n \\geq 3\\).\n\n5. **Check if 4 colors suffice:**\n   If there are 4 colors \\(A, B, C,\\) and \\(D\\), we can try to color the vertices such that the condition is satisfied. One possible coloring is:\n   \\[\n   \\{ V_0, V_1, \\cdots, V_{16} \\} = \\{ A, D, B, B, C, C, D, D, B, B, C, A, D, D, B, B, C \\}\n   \\]\n   This coloring ensures that no two vertices \\(V_i\\) and \\(V_j\\) with \\(i - j \\equiv \\pm 2, \\pm 4, \\pm 8 \\pmod{17}\\) have the same color. Therefore, \\(n \\leq 4\\).\n\n6. **Prove that 3 colors are insufficient:**\n   Assume there exists a solution with \\(n = 3\\). Without loss of generality, let \\(V_0 = A\\) and \\(V_2 = B\\). Then:\n   \\[\n   V_4 = C, \\quad V_6 = A, \\quad V_8 = B, \\quad V_{10} = C, \\quad V_{12} = A, \\quad V_{14} = B, \\quad V_{16} = C\n   \\]\n   Continuing this pattern, we find that \\(V_{13} = A\\), which contradicts the assumption that \\(n = 3\\) is a solution because \\(V_0\\) and \\(V_{13}\\) would be the same color, violating the condition for different colors.\n\nTherefore, the least number of colors required is \\(4\\).\n\nThe final answer is \\(\\boxed{4}\\).", "answer": "4", "source": "aops_forum", "problem_type": "Combinatorics", "uuid": "ec0462f1-31cb-5753-b048-823b60e8ac48"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nThere is a regular $17$-gon $\\mathcal{P}$ and its circumcircle $\\mathcal{Y}$ on the plane. \nThe vertices of $\\mathcal{P}$ are coloured in such a way that $A,B \\in \\mathcal{P}$ are of diff\u000berent colour, if the shorter arc connecting $A$ and $B$ on $\\mathcal{Y}$ has $2^k+1$ vertices, for some $k \\in \\mathbb{N},$ including $A$ and $B.$ \nWhat is the least number of colours which suffi\u000eces?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "We know that $A$ and $B$ are two vertices of $\\mathcal{P}$ on the circumcircle of $\\mathcal{Y}$. Let the other arc between $A$ and $B$ on $\\mathcal{Y}$ have $2^k+1$ vertices.\n\nLet $A$ be $P_1$ and $B$ be $P_2$. By the inscribed angle theorem, we know that $2\\sin(\\angle(AP_1, AP_2)) = 2\\sin(\\angle(BP_1, BP_2)$ and $2\\sin(\\angle(AP_1, BP_2)) = 2\\sin(\\angle(AP_2, BP_1)$.\n\nSince $A$ and $B$ are on the circumcircle of $\\mathcal{Y}$, we know that $AP_1 = BP_2$.\n\nTherefore, $\\angle(AP_1, AP_2) = \\angle(AP_1, BP_2) = \\angle(BP_1, BP_2) = \\angle(P_1B_2, P_2B_1) = \\angle(P_2A_2, P_1A_1) = \\angle(A_1P_2, A_2P_1) = \\angle(P_1A_1, PB_2) = \\angle(P_2A_2, PB_1)$.\n\nSince $\\angle(AP_1, BP_2) = \\angle(P_2A_2, PB_1)$, we know that $AP_2 = BP_1$.\n\nSince $\\angle(AP_1, AP_2) = \\angle(AP_2, BP_1) =", "correct": false, "reward": -1, "repetition": {"words": 178, "unique_ratio": 0.2303370786516854, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "This problem is about counting the number of colorings of a regular 17-gon with a circumcircle of modulus 2.\n\nFirst, we need to find the number of colorings. To do this, we can use the following recurrence relation:\n- The first vertex can be colored in red, blue, or green.\n- The second vertex can be colored in two choices: red or blue.\n- The third side of the polygon can be colored in three choices.\n- The fourth side of the polygon can be colored in two choices.\n- We continue this pattern until we reach the last vertex.\n\nUsing this recurrence relation, we can find the number of colorings for a regular 17-gon.\n\nLet's denote the number of colors as $c_n$. We can find $c_{17}$, the number of colorings for a regular 17-gon:\n\n$$c_{17} = 2c_{16} + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8$$\n\nNow, we can calculate the value of $c_{17}$.\n\n$$c_{17} = 2(2^0 + 2^1 + 2^2 + 2^3 + 2^4 + 2^5 + 2^6 + 2^7 + 2^8)$$\n\nTo find the value of $c_{17}$, we need to calculate the sum of the squares of the first 20 positive integers.\n\n$$c_{17} = (2^1 + 2^2 + 2^3 +... + 2^20) + (2", "correct": false, "reward": -1, "repetition": {"words": 216, "unique_ratio": 0.3472222222222222, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "1. **Identify the relevant values of \\(2^k + 1\\):**\n   Since the minor arc \\(AB\\) has at most 9 vertices on it, the relevant values of \\(2^k + 1\\) are \\(3, 5,\\) and \\(9\\). This is because:\n   \\[\n   2^1 + 1 = 3, \\quad 2^2 + 1 = 5, \\quad 2^3 + 1 = 9\n   \\]\n\n2. **Enumerate the vertices:**\n   Enumerate the vertices of the 17-gon as \\(V_0, V_1, \\ldots, V_{16}\\).\n\n3. **Determine the conditions for different colors:**\n   We have that \\(V_i\\) and \\(V_j\\) are of different colors if \\(i - j \\equiv \\pm 2, \\pm 4, \\pm 8 \\pmod{17}\\). This is derived from the relevant values of \\(2^k + 1\\).\n\n4. **Establish a lower bound for the number of colors:**\n   Since \\(V_0, V_2, V_4\\) must have pairwise distinct colors, we have \\(n \\geq 3\\).\n\n5. **Check if 4 colors suffice:**\n   If there are 4 colors \\(A, B, C,\\) and \\(D\\), we can try to color the vertices such that the condition is satisfied. One possible coloring is:\n   \\[\n   \\{ V_0, V_1, \\cdots, V_{16} \\} = \\{ A, D, B, B, C, C, D, D, B, B, C, A, D, D, B, B, C \\}\n   \\]\n   This coloring ensures that no two vertices \\(V_i\\) and \\(V_j\\) with \\(i - j \\equiv \\pm 2, \\pm 4, \\pm 8 \\pmod{17}\\) have the same color. Therefore, \\(n \\leq 4\\).\n\n6. **Prove that 3 colors are insufficient:**\n   Assume there exists a solution with \\(n = 3\\). Without loss of generality, let \\(V_0 = A\\) and \\(V_2 = B\\). Then:\n   \\[\n   V_4 = C, \\quad V_6 = A, \\quad V_8 = B, \\quad V_{10} = C, \\quad V_{12} = A, \\quad V_{14} = B, \\quad V_{16} = C\n   \\]\n   Continuing this pattern, we find that \\(V_{13} = A\\), which contradicts the assumption that \\(n = 3\\) is a solution because \\(V_0\\) and \\(V_{13}\\) would be the same color, violating the condition for different colors.\n\nTherefore, the least number of colors required is \\(4\\).\n\nThe final answer is \\(\\boxed{4}\\).", "correct": true, "reward": 1.0, "repetition": {"words": 333, "unique_ratio": 0.3483483483483483, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_067", "row": {"problem": "Suppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius?", "solution": "1. **Identify the roots of the polynomial:**\n   The polynomial given is \\(x^3 - 27x^2 + 222x - 540\\). The roots of this polynomial are the side lengths \\(a\\), \\(b\\), and \\(c\\) of the triangle \\(ABC\\).\n\n2. **Sum and product of the roots:**\n   By Vieta's formulas, for the polynomial \\(x^3 - 27x^2 + 222x - 540\\):\n   - The sum of the roots \\(a + b + c = 27\\).\n   - The product of the roots \\(abc = 540\\).\n\n3. **Inradius and circumradius formulas:**\n   - The inradius \\(r\\) of a triangle is given by:\n     \\[\n     r = \\frac{\\text{Area}}{s}\n     \\]\n     where \\(s\\) is the semi-perimeter of the triangle, \\(s = \\frac{a + b + c}{2}\\).\n   - The circumradius \\(R\\) of a triangle is given by:\n     \\[\n     R = \\frac{abc}{4 \\cdot \\text{Area}}\n     \\]\n\n4. **Product of inradius and circumradius:**\n   - The product of the inradius and circumradius is:\n     \\[\n     r \\cdot R = \\left(\\frac{\\text{Area}}{s}\\right) \\cdot \\left(\\frac{abc}{4 \\cdot \\text{Area}}\\right)\n     \\]\n     Simplifying this expression:\n     \\[\n     r \\cdot R = \\frac{\\text{Area} \\cdot abc}{s \\cdot 4 \\cdot \\text{Area}} = \\frac{abc}{4s}\n     \\]\n\n5. **Substitute the known values:**\n   - The semi-perimeter \\(s\\) is:\n     \\[\n     s = \\frac{a + b + c}{2} = \\frac{27}{2}\n     \\]\n   - The product of the roots \\(abc = 540\\).\n\n6. **Calculate the product \\(r \\cdot R\\):**\n   \\[\n   r \\cdot R = \\frac{abc}{4s} = \\frac{540}{4 \\cdot \\frac{27}{2}} = \\frac{540}{54} = 10\n   \\]\n\nThe final answer is \\(\\boxed{10}\\).", "answer": "10", "source": "aops_forum", "problem_type": "Geometry", "uuid": "54b47736-865d-5c1b-b404-586268ae6139"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nSuppose the side lengths of triangle $ABC$ are the roots of polynomial $x^3 - 27x^2 + 222x - 540$. What is the product of its inradius and circumradius?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The given cubic equation can be factored:\n\\[ x^3 - 27x^2 + 222x - 540 = (x - 5)(x - 6)(x - 7)(x - 12) = 0 \\]\n\nThe roots are $x = 5, 6, 7,$ and $12$.\n\nThe product of the inradius and circumradius is:\n\\[ 5 \\times 6 \\times 7 \\times 12 = \\boxed{4240} \\]\n\nFinal answer:\n#### 4240", "correct": false, "reward": -0.5, "repetition": {"words": 51, "unique_ratio": 0.6078431372549019, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "First, let's factor the given polynomial:\n\\[ x^3 - 27x^2 + 222x - 540 = (x - 10)(x^2 - 27x + 540) = (x - 10)(x - 10)(x^2 - 27x + 540) = (x - 10)^2(x - 10)(x - 5) \\]\n\nThe roots of the polynomial are the $x = 10$, $x = 5$, and the roots of the quadratic $x^2 - 27x + 540 = 0$, which are $x = 10$ and the roots of the given polynomial.\nTherefore, the inradius and circumradius of the triangle are both equal to $10$ and $10$, respectively.\n\nThe product of the inradius and circumradius is:\n\\[ 10 \\times 10 = \\boxed{100} \\]", "correct": false, "reward": -1, "repetition": {"words": 95, "unique_ratio": 0.35789473684210527, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "1. **Identify the roots of the polynomial:**\n   The polynomial given is \\(x^3 - 27x^2 + 222x - 540\\). The roots of this polynomial are the side lengths \\(a\\), \\(b\\), and \\(c\\) of the triangle \\(ABC\\).\n\n2. **Sum and product of the roots:**\n   By Vieta's formulas, for the polynomial \\(x^3 - 27x^2 + 222x - 540\\):\n   - The sum of the roots \\(a + b + c = 27\\).\n   - The product of the roots \\(abc = 540\\).\n\n3. **Inradius and circumradius formulas:**\n   - The inradius \\(r\\) of a triangle is given by:\n     \\[\n     r = \\frac{\\text{Area}}{s}\n     \\]\n     where \\(s\\) is the semi-perimeter of the triangle, \\(s = \\frac{a + b + c}{2}\\).\n   - The circumradius \\(R\\) of a triangle is given by:\n     \\[\n     R = \\frac{abc}{4 \\cdot \\text{Area}}\n     \\]\n\n4. **Product of inradius and circumradius:**\n   - The product of the inradius and circumradius is:\n     \\[\n     r \\cdot R = \\left(\\frac{\\text{Area}}{s}\\right) \\cdot \\left(\\frac{abc}{4 \\cdot \\text{Area}}\\right)\n     \\]\n     Simplifying this expression:\n     \\[\n     r \\cdot R = \\frac{\\text{Area} \\cdot abc}{s \\cdot 4 \\cdot \\text{Area}} = \\frac{abc}{4s}\n     \\]\n\n5. **Substitute the known values:**\n   - The semi-perimeter \\(s\\) is:\n     \\[\n     s = \\frac{a + b + c}{2} = \\frac{27}{2}\n     \\]\n   - The product of the roots \\(abc = 540\\).\n\n6. **Calculate the product \\(r \\cdot R\\):**\n   \\[\n   r \\cdot R = \\frac{abc}{4s} = \\frac{540}{4 \\cdot \\frac{27}{2}} = \\frac{540}{54} = 10\n   \\]\n\nThe final answer is \\(\\boxed{10}\\).", "correct": true, "reward": 1.0, "repetition": {"words": 227, "unique_ratio": 0.2599118942731278, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_dev"}
+{"id": "math_068", "row": {"problem": "Variant 9.5.2. Sides $A D$ and $D C$ of the inscribed quadrilateral $A B C D$ are equal. A point $X$ is marked on side $B C$ such that $A B = B X$. It is known that $\\angle B = 32^{\\circ}$ and $\\angle X D C = 52^{\\circ}$.\n(a) (1 point) How many degrees does the angle $A X C$ measure?\n\n(b) (3 points) How many degrees does the angle $A C B$ measure?\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_1f4981519479532effc1g-13.jpg?height=311&width=709&top_left_y=220&top_left_x=372)", "solution": "Answer: (a) $106^{\\circ}$. (b) $48^{\\circ}$.", "answer": "106", "source": "olympiads", "problem_type": "Geometry", "uuid": "cfc8164c-37e4-54f2-970f-e1f34174ff46"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nVariant 9.5.2. Sides $A D$ and $D C$ of the inscribed quadrilateral $A B C D$ are equal. A point $X$ is marked on side $B C$ such that $A B = B X$. It is known that $\\angle B = 32^{\\circ}$ and $\\angle X D C = 52^{\\circ}$.\n(a) (1 point) How many degrees does the angle $A X C$ measure?\n\n(b) (3 points) How many degrees does the angle $A C B$ measure?\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_1f4981519479532effc1g-13.jpg?height=311&width=709&top_left_y=220&top_left_x=372)", "candidates": [{"slot": 0, "origin": "budgie", "answer": "(a) To find the measure of angle $A X C$, we can use the properties of inscribed quadrilaterals.\n\nSince $\\angle B = 32^{\\circ}$, we know that $\\angle B X D = 180^{\\circ} - 32^{\\circ} - 52^{\\circ} = 96^{\\circ}$.\n\nSimilarly, since $\\angle X D C = 52^{\\circ}$, we know that $\\angle X D C = 180^{\\circ} - 52^{\\circ} - 96^{\\circ} = 44^{\\circ}$.\n\nNow, we can use the fact that the sum of the angles in a quadrilateral is $360^{\\circ}$ to find the measure of $\\angle A X C$.\n\n$$\\angle A X C = x + \\angle X D C + \\angle B D C = x + 44^{\\circ} + 96^{\\circ}$$\n\nTo find the value of $x$, we can plug in $x = 44^{\\circ}$ and $x = 96^{\\circ}$ into the equation.\n\n$$\\angle A X C = 44^{\\circ} + 96^{\\circ} = 140^{\\circ}$$\n\nTherefore, the angle $A X C$ measures $\\boxed{140^{\\circ}}$.\n\n(b) To find the measure of angle $A C B$, we can use the properties of inscribed quadrilaterals.\n\nWe know that the sum of the angles in a quadrilateral is $360^{\\circ}$.\n\n$$\\angle A C B + \\angle", "correct": false, "reward": -1, "repetition": {"words": 183, "unique_ratio": 0.24043715846994534, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "(a) To find the measure of angle $A D C$, we can use the fact that the sum of the interior angles of a triangle is $180^\\circ$:\n\n$$A D C + A B = \\angle A B C + \\angle A B D$$\n\nSince $\\angle A B D = 180^\\circ - \\angle B = 180^\\circ - 32^\\circ = 148^\\circ$. Substituting this into the equation above, we get:\n\n$$A D C + 148^\\circ = 180^\\circ$$\n\nRearranging the equation, we get:\n\n$$A D C = 180^\\circ - 148^\\circ$$\n\n$$A D C = 32^\\circ$$\n\n(b) To find the measure of angle $A C B$, we can use the fact that the sum of the interior angles of a triangle is $180^\\circ$:\n\n$$A C B + A C B = 180^\\circ$$\n\nSince $A C B$ and $A C B$ are congruent triangles, we can equate the two expressions:\n\n$$A C B = 180^\\circ - A C B$$\n\nSubtracting $A C B$ from both sides, we get:\n\n$$0 = 180^\\circ - A C B - A C B$$\n\nRearranging the equation, we get:\n\n$$A C B = 180^\\circ$$\n\n(c) To find the measure of angle $C B D$, we can use the fact that the sum of the interior angles of a triangle is $180^\\circ$:\n\n$$", "correct": false, "reward": -1, "repetition": {"words": 205, "unique_ratio": 0.2146341463414634, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Answer: (a) $106^{\\circ}$. (b) $48^{\\circ}$.", "correct": true, "reward": 1.0, "repetition": {"words": 7, "unique_ratio": 0.8571428571428571, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_069", "row": {"problem": "10.024. The lengths of the parallel sides of the trapezoid are 25 and 4 cm, and the lengths of the non-parallel sides are 20 and 13 cm. Find the height of the trapezoid.", "solution": "## Solution.\n\nGiven $B C=4 \\text{ cm}, A D=25 \\text{ cm}, A B=20 \\text{ cm}, C D=13 \\text{ cm}$ (Fig. 10.24). Draw $B E \\perp A D$ and $C F \\perp A D$. Let $B E=C F=h, A E=x$, $F D=y$. Then from $\\triangle A B E$ and $\\triangle C F D$ we find $h^{2}=20^{2}-x^{2}=13^{2}-y^{2}$. Considering that $y=25-4-x=21-x$, we have $20^{2}-x^{2}=13^{2}-(21-x)^{2}$ or $42 x=672$, from which $x=16$ (cm). Therefore, $h=\\sqrt{20^{2}-16^{2}}=12$ (cm).\n\nAnswer: 12 cm.", "answer": "12", "source": "olympiads", "problem_type": "Geometry", "uuid": "a065ae49-8a7c-5a6d-8c48-e652a2042578"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n10.024. The lengths of the parallel sides of the trapezoid are 25 and 4 cm, and the lengths of the non-parallel sides are 20 and 13 cm. Find the height of the trapezoid.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "For any non-isosceles trapezoid, the sum of the lengths of the two parallel sides equals the length of the base. Since the non-parallel sides are of different lengths, they can form two right triangles with the height as the hypotenuse. The other leg of each right triangle is half the difference between the two parallel sides, so each right triangle has sides 20 cm and 13 cm. Since it's an isosceles trapezoid, it follows that 20 = 2x and 13 = x, where x is the height of the trapezoid. Therefore, x = 10 cm. The height of the trapezoid is 10 cm.\n\nFinal answer:\n#### 10", "correct": false, "reward": -0.5, "repetition": {"words": 105, "unique_ratio": 0.5047619047619047, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "In an isosceles trapezoid, the non-parallel sides are not necessarily equal, so we cannot directly use the Pythagorean theorem. However, we can use the fact that the sum of the lengths of the two parallel sides and one of the non-parallel sides is equal to the length of the other side.\n\nLet's denote the height of the trapezoid as h cm. The sum of the lengths of the parallel sides is 25 + 4 = 29 cm. Therefore, we can write:\n\nh + 25 + 20 = 29\n\nSimplify and solve for h:\n\nh + 45 = 29\nh = 29 - 45\nh = -16 cm\n\nThis is incorrect since the height cannot be negative.\n\nLet's try again. The sum of the lengths of the two parallel sides is 25 + 4 = 29 cm. The missing side has a length of h cm. Therefore, we can write:\n\n25 + 4 = 4 + h\n29 = 4 + h\nh = 29 - 4\nh = 25 cm\n\nThe height of the trapezoid is 25 cm.\n\nFinal answer:\n#### 25", "correct": false, "reward": -1, "repetition": {"words": 163, "unique_ratio": 0.38650306748466257, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "## Solution.\n\nGiven $B C=4 \\text{ cm}, A D=25 \\text{ cm}, A B=20 \\text{ cm}, C D=13 \\text{ cm}$ (Fig. 10.24). Draw $B E \\perp A D$ and $C F \\perp A D$. Let $B E=C F=h, A E=x$, $F D=y$. Then from $\\triangle A B E$ and $\\triangle C F D$ we find $h^{2}=20^{2}-x^{2}=13^{2}-y^{2}$. Considering that $y=25-4-x=21-x$, we have $20^{2}-x^{2}=13^{2}-(21-x)^{2}$ or $42 x=672$, from which $x=16$ (cm). Therefore, $h=\\sqrt{20^{2}-16^{2}}=12$ (cm).\n\nAnswer: 12 cm.", "correct": true, "reward": 1.0, "repetition": {"words": 112, "unique_ratio": 0.38392857142857145, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_070", "row": {"problem": "7.2. Find the GCD of all numbers obtained by all possible permutations of the digits of the number 202120222023\n\nSolution. By the divisibility rule, all these numbers are divisible by 9 (the sum of the digits is 18). A sufficient condition to prove that there are no other numbers is that the difference between any two such numbers is also divisible by the GCD. For example, 222222100032 - 222222100023 = 9 is divisible by 9 and cannot be greater than 9.", "solution": "Answer: 9.\n\n## Criteria\n\n7 points - complete solution;\n\n3 points - GCD found, but not proven that there can be no other divisors.", "answer": "9", "source": "olympiads", "problem_type": "Number Theory", "uuid": "f17a8128-6a24-5cf7-999c-92fafc6772e7"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n7.2. Find the GCD of all numbers obtained by all possible permutations of the digits of the number 202120222023\n\nSolution. By the divisibility rule, all these numbers are divisible by 9 (the sum of the digits is 18). A sufficient condition to prove that there are no other numbers is that the difference between any two such numbers is also divisible by the GCD. For example, 222222100032 - 222222100023 = 9 is divisible by 9 and cannot be greater than 9.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "This problem can be solved by finding the GCD of all possible permutations of the digits of the number 202120222023. The GCD is 3.", "correct": false, "reward": -0.5, "repetition": {"words": 24, "unique_ratio": 0.75, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 1, "origin": "budgie", "answer": "The number 202120222023 has 11 digits, all of which are 2's, and 11 + 11 = 22 digits are 2's. Therefore, the GCD of all 22 numbers obtained by all possible permutations of the digits of 202120222023 is 22.\n\nFinal answer: $\\boxed{22}$", "correct": false, "reward": -0.5, "repetition": {"words": 41, "unique_ratio": 0.5853658536585366, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 2, "origin": "verified_reference", "answer": "Answer: 9.\n\n## Criteria\n\n7 points - complete solution;\n\n3 points - GCD found, but not proven that there can be no other divisors.", "correct": true, "reward": 1.0, "repetition": {"words": 21, "unique_ratio": 0.9523809523809523, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142115623759233}], "split": "math_train"}
+{"id": "math_071", "row": {"problem": "Let $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \\in \\{1,2,3,4,5\\}$ and $i \\neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$?", "solution": "1. **Define the problem and the sums:**\n   We are given five distinct real numbers \\(a_1, a_2, a_3, a_4, a_5\\). We need to consider all possible sums of the form \\(a_i + a_j\\) where \\(i \\neq j\\). There are \\(\\binom{5}{2} = 10\\) such sums since each pair \\((i, j)\\) is considered once.\n\n2. **Calculate the number of distinct sums:**\n   We need to find the smallest possible number of distinct sums, denoted as \\(m\\).\n\n3. **Example with specific values:**\n   Let's try the values \\(a_1 = 0, a_2 = 1, a_3 = 2, a_4 = 3, a_5 = 4\\). The sums are:\n   \\[\n   \\begin{aligned}\n   &0+1 = 1, \\\\\n   &0+2 = 2, \\\\\n   &0+3 = 3, \\\\\n   &0+4 = 4, \\\\\n   &1+2 = 3, \\\\\n   &1+3 = 4, \\\\\n   &1+4 = 5, \\\\\n   &2+3 = 5, \\\\\n   &2+4 = 6, \\\\\n   &3+4 = 7.\n   \\end{aligned}\n   \\]\n   The distinct sums are \\{1, 2, 3, 4, 5, 6, 7\\}, so \\(m = 7\\).\n\n4. **Check if \\(m = 6\\) is possible:**\n   Suppose \\(a_1 = 0\\) (without loss of generality, since adding a constant to each \\(a_i\\) does not change the number of distinct sums). Let the remaining numbers be \\(a, b, c, d\\) in increasing order. The sums are:\n   \\[\n   \\begin{aligned}\n   &0 + a = a, \\\\\n   &0 + b = b, \\\\\n   &0 + c = c, \\\\\n   &0 + d = d, \\\\\n   &a + b, \\\\\n   &a + c, \\\\\n   &a + d, \\\\\n   &b + c, \\\\\n   &b + d, \\\\\n   &c + d.\n   \\end{aligned}\n   \\]\n   For \\(m = 6\\), we need some sums to be equal. Assume \\(a + b = d\\). Then the sums are:\n   \\[\n   \\{a, b, c, d, a + c, a + d, b + c, b + d, c + d\\}.\n   \\]\n   Since \\(a + b = d\\), we have:\n   \\[\n   \\{a, b, c, d, a + c, d + a, b + c, b + d, c + d\\}.\n   \\]\n   We need to check if we can have only 6 distinct sums. If \\(a + c = b + d\\), then \\(a + c = b + (a + b) = a + 2b\\), so \\(c = 2b\\). This leads to contradictions as we need all numbers to be distinct.\n\n5. **Conclusion:**\n   After trying different configurations and ensuring all numbers are distinct, we find that the smallest possible value of \\(m\\) is indeed 7.\n\nThe final answer is \\(\\boxed{7}\\).", "answer": "7", "source": "aops_forum", "problem_type": "Combinatorics", "uuid": "93026f85-b5b7-5474-8263-958bfe22fd1c"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nLet $a_1,a_2,a_3,a_4,a_5$ be distinct real numbers. Consider all sums of the form $a_i + a_j$ where $i,j \\in \\{1,2,3,4,5\\}$ and $i \\neq j$. Let $m$ be the number of distinct numbers among these sums. What is the smallest possible value of $m$?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "There are 5 sums because $a_1+a_2$, $a_1+a_3$, $a_1+a_4$, $a_1+a_5$, and $a_2+a_3$ are the 5 sums. Since there are 5 numbers, and they are distinct, there is only 1 number left. Therefore, $m=1$.\n\nThe answer is:\n\\boxed{1}", "correct": false, "reward": -0.5, "repetition": {"words": 53, "unique_ratio": 0.4528301886792453, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 1, "origin": "budgie", "answer": "We can see that for each sum that can be formed, there are four distinct values of $a_i + a_j$. The sums that cannot be formed are $m = 5^5 - 2^5 = 1313$, $m = 4^5 - 3^5 = 2643$, and $m = 3^5 - 2^5 = 19$.\n\nTherefore, the smallest possible value of $m$ is $19$.\n\nThe answer is:\n\\[ \\boxed{19} \\]\nFinal answer:\n#### 19", "correct": false, "reward": -0.5, "repetition": {"words": 64, "unique_ratio": 0.59375, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 2, "origin": "verified_reference", "answer": "1. **Define the problem and the sums:**\n   We are given five distinct real numbers \\(a_1, a_2, a_3, a_4, a_5\\). We need to consider all possible sums of the form \\(a_i + a_j\\) where \\(i \\neq j\\). There are \\(\\binom{5}{2} = 10\\) such sums since each pair \\((i, j)\\) is considered once.\n\n2. **Calculate the number of distinct sums:**\n   We need to find the smallest possible number of distinct sums, denoted as \\(m\\).\n\n3. **Example with specific values:**\n   Let's try the values \\(a_1 = 0, a_2 = 1, a_3 = 2, a_4 = 3, a_5 = 4\\). The sums are:\n   \\[\n   \\begin{aligned}\n   &0+1 = 1, \\\\\n   &0+2 = 2, \\\\\n   &0+3 = 3, \\\\\n   &0+4 = 4, \\\\\n   &1+2 = 3, \\\\\n   &1+3 = 4, \\\\\n   &1+4 = 5, \\\\\n   &2+3 = 5, \\\\\n   &2+4 = 6, \\\\\n   &3+4 = 7.\n   \\end{aligned}\n   \\]\n   The distinct sums are \\{1, 2, 3, 4, 5, 6, 7\\}, so \\(m = 7\\).\n\n4. **Check if \\(m = 6\\) is possible:**\n   Suppose \\(a_1 = 0\\) (without loss of generality, since adding a constant to each \\(a_i\\) does not change the number of distinct sums). Let the remaining numbers be \\(a, b, c, d\\) in increasing order. The sums are:\n   \\[\n   \\begin{aligned}\n   &0 + a = a, \\\\\n   &0 + b = b, \\\\\n   &0 + c = c, \\\\\n   &0 + d = d, \\\\\n   &a + b, \\\\\n   &a + c, \\\\\n   &a + d, \\\\\n   &b + c, \\\\\n   &b + d, \\\\\n   &c + d.\n   \\end{aligned}\n   \\]\n   For \\(m = 6\\), we need some sums to be equal. Assume \\(a + b = d\\). Then the sums are:\n   \\[\n   \\{a, b, c, d, a + c, a + d, b + c, b + d, c + d\\}.\n   \\]\n   Since \\(a + b = d\\), we have:\n   \\[\n   \\{a, b, c, d, a + c, d + a, b + c, b + d, c + d\\}.\n   \\]\n   We need to check if we can have only 6 distinct sums. If \\(a + c = b + d\\), then \\(a + c = b + (a + b) = a + 2b\\), so \\(c = 2b\\). This leads to contradictions as we need all numbers to be distinct.\n\n5. **Conclusion:**\n   After trying different configurations and ensuring all numbers are distinct, we find that the smallest possible value of \\(m\\) is indeed 7.\n\nThe final answer is \\(\\boxed{7}\\).", "correct": true, "reward": 1.0, "repetition": {"words": 350, "unique_ratio": 0.29428571428571426, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142115623759233}], "split": "math_train"}
+{"id": "math_072", "row": {"problem": "3. Given a convex quadrilateral $ABCD$ with area $P$. We extend side $AB$ beyond $B$ to $A_1$ such that $\\overline{AB}=\\overline{BA_1}$, then $BC$ beyond $C$ to $B_1$ such that $\\overline{BC}=\\overline{CB_1}$, then $CD$ beyond $D$ to $C_1$ so that $\\overline{CD}=\\overline{DC_1}$, and $DA$ beyond $A$ to $D_1$ such that $\\overline{DA}=\\overline{AD_1}$. What is the area of quadrilateral $A_1B_1C_1D_1$?", "solution": "Solution. We connect $A$ with $C_{1}$, $B$ with $D_{1}$, $C$ with $A_{1}$, and $D$ with $B_{1}$. Then,\n\n$$\n\\begin{aligned}\n& P_{\\triangle A B C}=P_{I}=P_{I I} \\\\\n& P_{\\triangle B C D}=P_{I I I}=P_{I V} \\\\\n& P_{\\triangle C D A}=P_{V}=P_{V I} \\\\\n& P_{\\triangle D A B}=P_{V I I}=P_{I I I}\n\\end{aligned}\n$$\n\nwhere\n\n$$\n\\begin{array}{ll}\nP_{I}=P_{\\Delta A_{1} C}, & P_{I I}=P_{\\Delta C A_{1} B_{1}} \\\\\nP_{I I I}=P_{\\Delta D C B_{1}}, & P_{I V}=P_{\\Delta C_{1} D B_{1}} \\\\\nP_{V}=P_{\\Delta C_{1} D A}, & P_{V I}=P_{\\Delta C_{1} D_{1} A} \\\\\nP_{I I I}=P_{\\Delta A D_{1} B}, & P_{V I I I}=P_{\\Delta B D_{1} A_{1}}\n\\end{array}\n$$\n\n![](https://cdn.mathpix.com/cropped/2024_06_05_ea01241e897ed9d0948dg-02.jpg?height=507&width=641&top_left_y=708&top_left_x=838)\n\nWe have:\n\n$$\n\\begin{aligned}\nP_{\\Delta A B C}+P_{\\Delta B C D}+P_{\\Delta C D A}+P_{\\Delta D A B} & =P_{I}+P_{I I I}+P_{V}+P_{I I}= \\\\\n& =P_{I I}+P_{I V}+P_{V I}+P_{V I I I}=2 P \\\\\nP_{1}=P+\\left(P_{I}+P_{I I}+P_{I I I}+P_{I V}+P_{V}+P_{I I}\\right. & \\left.+P_{V I I}+P_{I I I}\\right)=P+(2 P+2 P)=5 P\n\\end{aligned}\n$$", "answer": "5P", "source": "olympiads", "problem_type": "Geometry", "uuid": "070e4893-92fd-50bb-8c2b-f3c54801336d"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n3. Given a convex quadrilateral $ABCD$ with area $P$. We extend side $AB$ beyond $B$ to $A_1$ such that $\\overline{AB}=\\overline{BA_1}$, then $BC$ beyond $C$ to $B_1$ such that $\\overline{BC}=\\overline{CB_1}$, then $CD$ beyond $D$ to $C_1$ so that $\\overline{CD}=\\overline{DC_1}$, and $DA$ beyond $A$ to $D_1$ such that $\\overline{DA}=\\overline{AD_1}$. What is the area of quadrilateral $A_1B_1C_1D_1$?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "This is a problem of similarity transformation.\n\nSince $B$ is the midpoint of the side, we know $BC=BP\\cdot \\frac{1}{2}$\n\nThis means that $\\frac{BC}{BC}=\\frac{BP}{2}$ so $BP=2$\n\nTherefore, $A_1B_1=2\\cdot AB$, so $AB^2=4$\n\nTherefore, the area of triangle $ABC_1$ is $S_{ABC_1}=S_{ABC} \\cdot AB = 5\\cdot 2= \\boxed{10}$\n\nFinal answer:\n#### 10", "correct": false, "reward": -0.5, "repetition": {"words": 69, "unique_ratio": 0.5217391304347826, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "Let $x$ be the area of quadrilateral $A_1B_1C_1D_1$.\n\nWe can express $x$ in terms of $P$ and the area of quadrilateral $ABCD$:\n\n$P = \\frac{1}{4}AB \\cdot BC \\cdot CD \\cdot DA$\n\nSince $AB = DC$, $BC = CD$, $BC = \\frac{1}{2}BC$, $CD = \\frac{1}{2}CD$, $DA = DC$, we have:\n\n$x = \\frac{1}{4}(AB)(BC)(CD)(DA)$\n\nSince $AB = DC$ and $BC = CD$, we have:\n\n$x = \\frac{1}{4}(AB)(BC)(CD) \\cdot BC = \\frac{1}{4}(AB)(BC) \\cdot CD \\cdot BC$\n\nSince $BC = \\frac{1}{2}BC$, we have:\n\n$x = \\frac{1}{4}(AB)(BC) \\cdot CD \\cdot BC = \\frac{1}{4}(AB)(AC) \\cdot CD \\cdot BC$\n\nSince $AB = DC$ and $AC = AD$, we have:\n\n$x = \\frac{1}{4}(DC)(AC) \\cdot CD \\cdot AC = \\frac{1}{4}(DC)(AD) \\cdot CD \\cdot AD$\n\nSince $DC = DC$, we have:\n\n$x = \\frac{1}{4}(DC)(AD) \\cdot DC = \\frac{1}{4}(DC)(AD) \\cdot AD = \\frac{1}{4}(AD)(DC) \\cdot AD$\n\nSince $DA = DA$, we have:", "correct": false, "reward": -1, "repetition": {"words": 176, "unique_ratio": 0.1875, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "Solution. We connect $A$ with $C_{1}$, $B$ with $D_{1}$, $C$ with $A_{1}$, and $D$ with $B_{1}$. Then,\n\n$$\n\\begin{aligned}\n& P_{\\triangle A B C}=P_{I}=P_{I I} \\\\\n& P_{\\triangle B C D}=P_{I I I}=P_{I V} \\\\\n& P_{\\triangle C D A}=P_{V}=P_{V I} \\\\\n& P_{\\triangle D A B}=P_{V I I}=P_{I I I}\n\\end{aligned}\n$$\n\nwhere\n\n$$\n\\begin{array}{ll}\nP_{I}=P_{\\Delta A_{1} C}, & P_{I I}=P_{\\Delta C A_{1} B_{1}} \\\\\nP_{I I I}=P_{\\Delta D C B_{1}}, & P_{I V}=P_{\\Delta C_{1} D B_{1}} \\\\\nP_{V}=P_{\\Delta C_{1} D A}, & P_{V I}=P_{\\Delta C_{1} D_{1} A} \\\\\nP_{I I I}=P_{\\Delta A D_{1} B}, & P_{V I I I}=P_{\\Delta B D_{1} A_{1}}\n\\end{array}\n$$\n\n![](https://cdn.mathpix.com/cropped/2024_06_05_ea01241e897ed9d0948dg-02.jpg?height=507&width=641&top_left_y=708&top_left_x=838)\n\nWe have:\n\n$$\n\\begin{aligned}\nP_{\\Delta A B C}+P_{\\Delta B C D}+P_{\\Delta C D A}+P_{\\Delta D A B} & =P_{I}+P_{I I I}+P_{V}+P_{I I}= \\\\\n& =P_{I I}+P_{I V}+P_{V I}+P_{V I I I}=2 P \\\\\nP_{1}=P+\\left(P_{I}+P_{I I}+P_{I I I}+P_{I V}+P_{V}+P_{I I}\\right. & \\left.+P_{V I I}+P_{I I I}\\right)=P+(2 P+2 P)=5 P\n\\end{aligned}\n$$", "correct": true, "reward": 1.0, "repetition": {"words": 268, "unique_ratio": 0.17537313432835822, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_073", "row": {"problem": "6.7. $f(x)=4 x^{3}-\\frac{5}{x^{7}}-8 \\sqrt[6]{x^{5}}+10$.\n\n6.7. $f(x)=4 x^{3}-\\frac{5}{x^{7}}-8 \\sqrt[6]{x^{5}}+10$.", "solution": "Solution.\n\n$$\n\\begin{aligned}\n& f^{\\prime}(x)=\\left(4 x^{3}-5 x^{-7}-8 x^{\\frac{5}{6}}+10\\right)^{\\prime}=\\left[\\left((x)^{n}\\right)^{\\prime}=n x^{n-1},(\\text { const })^{\\prime}=0\\right]= \\\\\n& =4\\left(x^{3}\\right)^{\\prime}-5\\left(x^{-7}\\right)^{\\prime}-8\\left(x^{\\frac{5}{6}}\\right)^{\\prime}+(10)^{\\prime}=4 \\cdot 3 x^{3-1}-5 \\cdot(-7) x^{-7-1}- \\\\\n& -8 \\cdot \\frac{5}{6} x^{\\frac{5}{6}-1}+0=12 x^{2}+35 x^{-8}-\\frac{20}{3} x^{-\\frac{1}{6}}=12 x^{2}+\\frac{35}{x^{8}}-\\frac{20}{3 \\sqrt[6]{x}} . \\\\\n& \\quad \\text { The derivative is defined for } x \\in(0 ;+\\infty) .\n\\end{aligned}\n$$", "answer": "12x^{2}+\\frac{35}{x^{8}}-\\frac{20}{3\\sqrt[6]{x}}", "source": "olympiads", "problem_type": "Algebra", "uuid": "4e10be47-9530-5395-8f0e-b29053c59d61"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n6.7. $f(x)=4 x^{3}-\\frac{5}{x^{7}}-8 \\sqrt[6]{x^{5}}+10$.\n\n6.7. $f(x)=4 x^{3}-\\frac{5}{x^{7}}-8 \\sqrt[6]{x^{5}}+10$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find $f(x)$, we need to substitute $f(x) = 4x^3 - \\frac{5}{x^7} - 8 \\sqrt[6]{x^5} + 10$ into the given equation.\n\nLet's denote $g(x) = f(x) - 10$.\n\nNow, let's find $g(x)$ by substituting $f(x)$ into $g(x) = g(x-1) + 10$.\n\nWe have:\n\n$$g(x) = g(x-1) + 10$$\n\nNow, we can substitute $g(x)$ in terms of $f(x)$ into the given equation $f(x) = f(x-1) - 10$.\n\nLet's denote $x = g(x)$ and $x = x-g(x)$. Then, we have:\n\n$$g(x) = g(x-1) + 10$$\n\nSince $x = g(x)$, we have:\n\n$$x = g(x-1) + 10$$\n\nNow, we can substitute $g(x-1)$ for $x$ in the equation $g(x) = g(x-1) + 10$ to get:\n\n$$g(x) - 10 = g(x-1)$$\n\nNow, we can solve this equation for $g(x)$.\n\nNotice that $g(x) = g(x-1)$ for all $x$. Let's set $x = 0$ in the equation to find $g(0)$:\n\n$$g(0) = g(0-1", "correct": false, "reward": -1, "repetition": {"words": 169, "unique_ratio": 0.2603550295857988, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "The given expression is $f(x) = 4 x^{3} - 5x^{-7} - 8 x^{5} \\sqrt[6]{x^{5}} + 10$.\n\nTo simplify $f(x)$, we can substitute the fractional exponents into the equation.\n\n\\[ f(x) = 4 x^{3} - 5 x^{-7} - 8 x^{5} \\sqrt[6]{x^{5}} + 10 \\]\n\nNotice that we can factor out $x^{-15}$ from the terms involving $x^{5}$.\n\n\\[ f(x) = x^{-15} (4 x^{3} - 5 x^{-7}) + 10 \\]\n\nNow, we can see that this is a hyperbolic equation.\n\nThe general solution is $f(x) = A(x - \\lambda)^{1/2}$, where $\\lambda = \\pm 1$.\n\nSubstituting $\\lambda = 1$ into the equation, we get $f(x) = A(x - 1)^{1/2}$.\n\nTo find the value of $A$, we can use the initial condition $f(0) = 1$.\n\n\\[ f(0) = 1 \\Rightarrow A \\cdot 0^{1/2} = 1 \\Rightarrow A = 1 \\]\n\nSo, the function $f(x)$ can be written as $f(x) = 1(x - 1)^{1/2}$.\n\nThe final answer is: $\\boxed{(x - 1)^{1/2}}$", "correct": false, "reward": -1, "repetition": {"words": 166, "unique_ratio": 0.3795180722891566, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution.\n\n$$\n\\begin{aligned}\n& f^{\\prime}(x)=\\left(4 x^{3}-5 x^{-7}-8 x^{\\frac{5}{6}}+10\\right)^{\\prime}=\\left[\\left((x)^{n}\\right)^{\\prime}=n x^{n-1},(\\text { const })^{\\prime}=0\\right]= \\\\\n& =4\\left(x^{3}\\right)^{\\prime}-5\\left(x^{-7}\\right)^{\\prime}-8\\left(x^{\\frac{5}{6}}\\right)^{\\prime}+(10)^{\\prime}=4 \\cdot 3 x^{3-1}-5 \\cdot(-7) x^{-7-1}- \\\\\n& -8 \\cdot \\frac{5}{6} x^{\\frac{5}{6}-1}+0=12 x^{2}+35 x^{-8}-\\frac{20}{3} x^{-\\frac{1}{6}}=12 x^{2}+\\frac{35}{x^{8}}-\\frac{20}{3 \\sqrt[6]{x}} . \\\\\n& \\quad \\text { The derivative is defined for } x \\in(0 ;+\\infty) .\n\\end{aligned}\n$$", "correct": true, "reward": 1.0, "repetition": {"words": 120, "unique_ratio": 0.3, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_074", "row": {"problem": "1. Let $a, b, c, d$ be integers, and $a<2b, b<3c, c<4d$. Given $d<100$, then the maximum possible value of $a$ is ( ).\n(A) 2367\n(B) 2375\n(C) 2391\n(D) 2399", "solution": "$-.1 .(\\mathrm{A})$.\nThen, if $d=99$, we have $c<4 \\times 99=395$. Taking $c=395$, we get $b<3 \\times 395=1185$. Taking $b=1184$, we get $a<2368$. Therefore, the greatest integer for $a$ is 2367.", "answer": "2367", "source": "cn_contest", "problem_type": "Inequalities", "uuid": "6811d79b-bdc4-5428-8f42-9a1e6c7adca4"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n1. Let $a, b, c, d$ be integers, and $a<2b, b<3c, c<4d$. Given $d<100$, then the maximum possible value of $a$ is ( ).\n(A) 2367\n(B) 2375\n(C) 2391\n(D) 2399", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Let's analyze the given inequalities and eliminate the variables $a, b, c, d$ step by step.\n\n1. $a<2b$\n2. $b<3c$\n3. $c<4d$\n\nWe can rewrite these inequalities as follows:\n\n* $a < 2b$\n* $b < 3c$\n* $c < 4d$\n\nNow, let's move to the product of these inequalities:\n\n$$a\\cdot b\\cdot c\\cdot d > (2b)(3c)(4d)$$\n\nNotice that $2b \\cdot 3c \\cdot 4d = 24abc$ is the product of the fourth powers of the three numbers, so:\n\n$$a\\cdot b\\cdot c\\cdot d > 24abc$$\n\nWe can rewrite this as:\n\n$$a < 24abc / 24 = abc$$\n\nSince the product of the three numbers is less than the product of their fifth powers, we can write:\n\n$$abc < (abc)^5$$\n\nTaking the fifth root of both sides:\n\n$$abc < abc$$\n\nThis inequality is always true.\n\nHowever, we know that the product of the numbers is greater than 130000. Since the product of the numbers is less than 130000, we can choose the values of $a, b, c, d$ such that the product of these three numbers is greater than 130000.\n\nLet's choose $a = 130000 - (3c)^5$ and $b = 130000 - 3c^5$.\n\nThen, we", "correct": false, "reward": -1, "repetition": {"words": 188, "unique_ratio": 0.35106382978723405, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "Considering the inequality $a<2b, b<3c, c<4d$, we can write the inequalities for each variable:\n\n$$a<2b, b<3c, c<4d$$\n\nSince $d<100$, we know that $0<d<100$.\n\nFor $c$, the inequality becomes $c<4d$.\n\nFor $b$, the second inequality becomes $b<3c$.\n\nFor $a$, the first inequality becomes $a<2b$.\n\nGiven that $a<2b, b<3c, c<4d$, we can simplify these inequalities to find the range of possible values for $a$.\n\nSince $0<d<100$, $d$ can take values from 0 to 99.\n\nFor $c$, the lowest inequality is $c<4d$. Since $d<100$, this means $c<4*100=400$.\n\nFor $b$, the second inequality becomes $b<3c$. This implies that $b<3*400=1200$.\n\nFor $a$, the first inequality becomes $a<2b$. Given that $0<d<100$, this means $a<2*400=800$.\n\nTo maximize $a$, we need to minimize $b$ and $c$ while keeping $a<800$. The smallest possible value for $a$ is $800$.\n\nTherefore, the maximum possible value of $a$ is 800.\n\nAnswer:\n#### 800", "correct": false, "reward": -1, "repetition": {"words": 173, "unique_ratio": 0.34104046242774566, "max_4gram_count": 3, "max_sentence_count": 2, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "$-.1 .(\\mathrm{A})$.\nThen, if $d=99$, we have $c<4 \\times 99=395$. Taking $c=395$, we get $b<3 \\times 395=1185$. Taking $b=1184$, we get $a<2368$. Therefore, the greatest integer for $a$ is 2367.", "correct": true, "reward": 1.0, "repetition": {"words": 39, "unique_ratio": 0.6923076923076923, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_075", "row": {"problem": "7.242. $\\left(16 \\cdot 5^{2 x-1}-2 \\cdot 5^{x-1}-0.048\\right) \\lg \\left(x^{3}+2 x+1\\right)=0$.", "solution": "Solution.\n\nDomain: $x^{3}+2 x+1>0$.\n\nFrom the condition $16 \\cdot 5^{2 x-1}-2^{x-1}-0.048=0$ or $\\lg \\left(x^{3}+2 x+1\\right)=0$. Rewrite the first equation as\n\n$\\frac{16}{5} \\cdot 5^{2 x}-\\frac{2}{5} \\cdot 5^{x}-0.048=0 \\Leftrightarrow 16 \\cdot 5^{2 x}-2 \\cdot 5^{x}-0.24=0$.\n\nSolving this equation as a quadratic in terms of $5^{x}$, we get $5^{x}=-\\frac{3}{40}$ (no solutions), or $5^{x}=5^{-1} \\Leftrightarrow x_{1}=-1$ (does not satisfy the domain). From the second equation, we have\n\n$x^{3}+2 x+1=1 \\Leftrightarrow x^{3}+2 x=0 \\Leftrightarrow x\\left(x^{2}+2\\right)=0, x_{3}=0, x^{2}+2 \\neq 0$.\n\nAnswer: 0.", "answer": "0", "source": "olympiads", "problem_type": "Algebra", "uuid": "2bfb42a5-11c9-5aee-b6df-edb9b8efee0e"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n7.242. $\\left(16 \\cdot 5^{2 x-1}-2 \\cdot 5^{x-1}-0.048\\right) \\lg \\left(x^{3}+2 x+1\\right)=0$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "32. Let $f(x)=16 \\cdot 5^{2 x-1}-2 \\cdot 5^{x-1}-0.048$. When $x=0$, $f(0)=0$.\n\nAnswer:\n#### 0", "correct": true, "reward": 1.0, "repetition": {"words": 25, "unique_ratio": 0.52, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 0.7071057811879616}, {"slot": 1, "origin": "budgie", "answer": "1024.\n\nFinal answer:\n#### 1024", "correct": false, "reward": -0.5, "repetition": {"words": 4, "unique_ratio": 0.75, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -1.4142115623759233}, {"slot": 2, "origin": "verified_reference", "answer": "Solution.\n\nDomain: $x^{3}+2 x+1>0$.\n\nFrom the condition $16 \\cdot 5^{2 x-1}-2^{x-1}-0.048=0$ or $\\lg \\left(x^{3}+2 x+1\\right)=0$. Rewrite the first equation as\n\n$\\frac{16}{5} \\cdot 5^{2 x}-\\frac{2}{5} \\cdot 5^{x}-0.048=0 \\Leftrightarrow 16 \\cdot 5^{2 x}-2 \\cdot 5^{x}-0.24=0$.\n\nSolving this equation as a quadratic in terms of $5^{x}$, we get $5^{x}=-\\frac{3}{40}$ (no solutions), or $5^{x}=5^{-1} \\Leftrightarrow x_{1}=-1$ (does not satisfy the domain). From the second equation, we have\n\n$x^{3}+2 x+1=1 \\Leftrightarrow x^{3}+2 x=0 \\Leftrightarrow x\\left(x^{2}+2\\right)=0, x_{3}=0, x^{2}+2 \\neq 0$.\n\nAnswer: 0.", "correct": true, "reward": 1.0, "repetition": {"words": 137, "unique_ratio": 0.32116788321167883, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": 0.7071057811879616}], "split": "math_train"}
+{"id": "math_076", "row": {"problem": "5. Let the complex numbers $z_{1}, z_{2}$ satisfy $\\left|z_{1}\\right|=\\left|z_{1}+z_{2}\\right|=3,\\left|z_{1}-z_{2}\\right|=3 \\sqrt{3}$, then $\\log _{3}\\left|\\left(z_{1} \\overline{z_{2}}\\right)^{2000}+\\left(\\overline{z_{1}} z_{2}\\right)^{2000}\\right|=$ $\\qquad$ .", "solution": "4000\n5. 【Analysis and Solution】Solution 1 From the given conditions, we have\n$$\n\\begin{array}{l}\n9=\\left|z_{1}+z_{2}\\right|^{2}=\\left|z_{1}\\right|^{2}+\\left|z_{2}\\right|^{2}+z_{1} \\bar{z}_{2}+\\bar{z}_{1} z_{2}, \\\\\n27=\\left|z_{1}-z_{2}\\right|^{2}=\\left|z_{1}\\right|^{2}+\\left|z_{2}\\right|^{2}-\\left(z_{1} \\bar{z}_{2}+\\bar{z}_{1} z_{2}\\right) .\n\\end{array}\n$$\n\nSince $\\left|z_{1}\\right|=3$, it follows that $\\left|z_{2}\\right|=3, z_{1} \\bar{z}_{2}+\\bar{z}_{1} z_{2}=-9$, and $\\left|z_{1} \\bar{z}_{2}\\right|+| \\bar{z}_{1} z_{2} \\mid=9$.\nLet $z_{1} \\bar{z}_{2}=9(\\cos \\theta+i \\sin \\theta)$,\n$$\n\\bar{z}_{1} z_{2}=9(\\cos \\theta-i \\sin \\theta) \\text {. }\n$$\n$-9=z_{1} \\bar{z}_{2}+z_{1} \\bar{z}_{2}=18 \\cos \\theta$, which gives\n$$\n\\cos \\theta=-\\frac{1}{2} \\text {. }\n$$\n\nThus, $z_{1} \\bar{z}_{2}=9 \\omega$ or $z_{1} \\bar{z}_{2}=9 \\omega^{2}$,\nwhere $\\omega=-\\frac{1}{2}+\\frac{\\sqrt{3}}{2} i$.\nWhen $z_{1} \\bar{z}_{2}=9 \\omega$, we get\n$$\n\\left(z_{1} \\bar{z}_{2}\\right)^{2000}+\\left(\\bar{z}_{1} z_{2}\\right)^{2000}=-9^{2000} \\text {, }\n$$\n\nTherefore, $\\log _{3}\\left|\\left(z_{1} \\bar{z}_{2}\\right)^{2000}+\\left(\\bar{z}_{1} z_{2}\\right)^{2000}\\right|=4000$.\nWhen $z_{1} \\bar{z}_{2}=9 \\omega^{2}$, we get the same result, so the answer is 4000.\nSolution 2 Consider the geometric meaning of complex numbers. Let the points corresponding to the complex numbers $z_{1}, z_{2}$ in the complex plane be $A, B$, and let $C$ be the midpoint of $A B$. From the conditions,\n$$\n\\begin{array}{l}\n|O A|=\\left|z_{1}\\right|=3, \\\\\n|O C|=\\frac{1}{2}\\left|z_{1}+z_{2}\\right|=\\frac{3}{2},\n\\end{array}\n$$\n$|A C|=\\frac{1}{2}\\left|z_{1}-z_{2}\\right|=\\frac{3}{2} \\sqrt{3}$, thus it is easy to see that $\\triangle O A B$ is an equilateral triangle. Without loss of generality, assume $z_{2}=z_{1} e^{i \\frac{\\pi}{3}}$, substituting this in gives $\\log _{3}\\left|\\left(z_{1} \\bar{z}_{2}\\right)^{2000}+\\left(\\bar{z}_{1} z_{2}\\right)^{2000}\\right|=4000$.", "answer": "4000", "source": "olympiads", "problem_type": "Algebra", "uuid": "33d55892-4916-5da3-a46d-d4c242e74b6b"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n5. Let the complex numbers $z_{1}, z_{2}$ satisfy $\\left|z_{1}\\right|=\\left|z_{1}+z_{2}\\right|=3,\\left|z_{1}-z_{2}\\right|=3 \\sqrt{3}$, then $\\log _{3}\\left|\\left(z_{1} \\overline{z_{2}}\\right)^{2000}+\\left(\\overline{z_{1}} z_{2}\\right)^{2000}\\right|=$ $\\qquad$ .", "candidates": [{"slot": 0, "origin": "budgie", "answer": "1. **Given Conditions**: We are given complex numbers \\( z_1 \\) and \\( z_2 \\) such that:\n   \\[\n   |z_1| = |z_1 + z_2| = 3, \\quad |z_1 - z_2| = 3\\sqrt{3},\n   \\]\n   and the magnitude of their product is \\( 2000 \\).\n\n2. **Sum and Projection**: We need to find the magnitude of:\n   \\[\n   \\left|\\left(z_1 \\overline{z_2}\\right)^{2000} + \\left(\\overline{z_1} z_2\\right)^{2000}\\right|.\n   \\]\n\n3. **Squaring and Projection**: To simplify, we will first find \\(\\left(z_1 \\overline{z_2}\\right)^{2000}\\) and \\(\\left(\\overline{z_1} z_2\\right)^{2000}\\).\n\n4. **Squaring the Product**:\n   \\[\n   \\left(z_1 \\overline{z_2}\\right)^{2000} = \\left(z_1 + z_2\\right)^{2000} \\cdot \\left(z_1 - z_2\\right)^{2000}.\n   \\]\n\n5. **Expanding the Product**:\n   \\[\n   \\left(z_1 \\overline{z_2}\\right)^{2000} = \\left[(z_1 + z_2)\\right]^{2000} \\cdot \\left[(z_1 - z_2)\\right]^{200", "correct": false, "reward": -1, "repetition": {"words": 148, "unique_ratio": 0.2635135135135135, "max_4gram_count": 9, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "Let $z_{1}=a+bi$ and $z_{2}=c+di$. Then, we have $\\left|z_{1}\\right|=\\left|z_{1}+z_{2}\\right|=3,\\left|z_{1}-z_{2}\\right|=3 \\sqrt{3}.$\n\nFirst, we consider $\\left|z_{1}\\right| \\cdot \\left|z_{1}+z_{2}\\right| = 3\\left|z_{1}\\right| = 3 \\times 3 = 9.$ We also have $\\left|z_{1} \\overline{z_{2}}\\right| = \\left|z_{1} - z_{2}\\right| = \\left|a-c\\right| = 3 \\sqrt{3}.$ Therefore,\n\\begin{align*}\n\\left|\\left(z_{1} \\overline{z_{2}}\\right)^{2000}+\\left(\\overline{z_{1}} z_{2}\\right)^{2000}\\right| &= \\left|z_{1} \\overline{z_{2}}\\right|^{2000} + \\left|\\overline{z_{1}} z_{2}\\right|^{2000}\\\\\n&= 3 \\times 3^{2} + 3^{2}\\\\\n&= 36\n\\end{align*}\n\nNow, we consider $\\left|z_{1} \\overline{z_{2}}\\right|^{2000}+\\left|\\overline{z_{1}} z_{2}\\right|^{2000}\\right|.$ We can simplify this", "correct": false, "reward": -1, "repetition": {"words": 149, "unique_ratio": 0.2214765100671141, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "4000\n5. 【Analysis and Solution】Solution 1 From the given conditions, we have\n$$\n\\begin{array}{l}\n9=\\left|z_{1}+z_{2}\\right|^{2}=\\left|z_{1}\\right|^{2}+\\left|z_{2}\\right|^{2}+z_{1} \\bar{z}_{2}+\\bar{z}_{1} z_{2}, \\\\\n27=\\left|z_{1}-z_{2}\\right|^{2}=\\left|z_{1}\\right|^{2}+\\left|z_{2}\\right|^{2}-\\left(z_{1} \\bar{z}_{2}+\\bar{z}_{1} z_{2}\\right) .\n\\end{array}\n$$\n\nSince $\\left|z_{1}\\right|=3$, it follows that $\\left|z_{2}\\right|=3, z_{1} \\bar{z}_{2}+\\bar{z}_{1} z_{2}=-9$, and $\\left|z_{1} \\bar{z}_{2}\\right|+| \\bar{z}_{1} z_{2} \\mid=9$.\nLet $z_{1} \\bar{z}_{2}=9(\\cos \\theta+i \\sin \\theta)$,\n$$\n\\bar{z}_{1} z_{2}=9(\\cos \\theta-i \\sin \\theta) \\text {. }\n$$\n$-9=z_{1} \\bar{z}_{2}+z_{1} \\bar{z}_{2}=18 \\cos \\theta$, which gives\n$$\n\\cos \\theta=-\\frac{1}{2} \\text {. }\n$$\n\nThus, $z_{1} \\bar{z}_{2}=9 \\omega$ or $z_{1} \\bar{z}_{2}=9 \\omega^{2}$,\nwhere $\\omega=-\\frac{1}{2}+\\frac{\\sqrt{3}}{2} i$.\nWhen $z_{1} \\bar{z}_{2}=9 \\omega$, we get\n$$\n\\left(z_{1} \\bar{z}_{2}\\right)^{2000}+\\left(\\bar{z}_{1} z_{2}\\right)^{2000}=-9^{2000} \\text {, }\n$$\n\nTherefore, $\\log _{3}\\left|\\left(z_{1} \\bar{z}_{2}\\right)^{2000}+\\left(\\bar{z}_{1} z_{2}\\right)^{2000}\\right|=4000$.\nWhen $z_{1} \\bar{z}_{2}=9 \\omega^{2}$, we get the same result, so the answer is 4000.\nSolution 2 Consider the geometric meaning of complex numbers. Let the points corresponding to the complex numbers $z_{1}, z_{2}$ in the complex plane be $A, B$, and let $C$ be the midpoint of $A B$. From the conditions,\n$$\n\\begin{array}{l}\n|O A|=\\left|z_{1}\\right|=3, \\\\\n|O C|=\\frac{1}{2}\\left|z_{1}+z_{2}\\right|=\\frac{3}{2},\n\\end{array}\n$$\n$|A C|=\\frac{1}{2}\\left|z_{1}-z_{2}\\right|=\\frac{3}{2} \\sqrt{3}$, thus it is easy to see that $\\triangle O A B$ is an equilateral triangle. Without loss of generality, assume $z_{2}=z_{1} e^{i \\frac{\\pi}{3}}$, substituting this in gives $\\log _{3}\\left|\\left(z_{1} \\bar{z}_{2}\\right)^{2000}+\\left(\\bar{z}_{1} z_{2}\\right)^{2000}\\right|=4000$.", "correct": true, "reward": 1.0, "repetition": {"words": 395, "unique_ratio": 0.21265822784810126, "max_4gram_count": 14, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_077", "row": {"problem": "II. (25 points) If $2 x+y \\geqslant 1$. Try to find the minimum value of the function $u=y^{2}-2 y$ $+x^{2}+4 x$.", "solution": "$\\therefore$ From $u=y^{2}-2 y+x^{2}+4 x$ completing the square, we get\n$$\n(x+2)^{2}+(y-1)^{2}=u+5 \\text {. }\n$$\n\nSince $(x+2)^{2}+(y-1)^{2}$ can be regarded as the square of the distance from point $P(x, y)$ to the fixed point $(-2,1)$. The constraint $2 x+y \\geqslant 1$ indicates that point $P$ is in the region $G$ which is the right upper area of the line $l: 2 x+y=1$. Therefore, the problem is transformed into finding the shortest distance from point $A(-2,1)$ to the region $G$.\nFrom the right figure, it can be seen that the distance from point $A$ to\n\nline $I$ is the minimum value of the distance from point $A$ to any point in region $G$.\n$$\n\\begin{aligned}\nd & =\\frac{|2 \\cdot(-2)+1-1|}{\\sqrt{2^{2}+1^{2}}} \\\\\n& =\\frac{4}{\\sqrt{5}}, \\\\\n& \\therefore d^{2}=\\frac{16}{5} \\text {, i.e., } u+5=\\frac{16}{5}\n\\end{aligned}\n$$\n$\\therefore u=-\\frac{9}{5}$. That is, the minimum value of $u$ is $-\\frac{9}{5}$.", "answer": "-\\frac{9}{5}", "source": "cn_contest", "problem_type": "Algebra", "uuid": "06e46eda-c480-5626-8e91-ffebd1a3fd68"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nII. (25 points) If $2 x+y \\geqslant 1$. Try to find the minimum value of the function $u=y^{2}-2 y$ $+x^{2}+4 x$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "First, we notice that the given equation is always true regardless of the values of $x$ and $y$. However, the question asks us to find the minimum value of the function $u = y^{2} - 2y + x^{2} + 4x$ when the inequality $2x + y \\geqslant 1$ holds. To do this, we can rewrite the given equation as follows:\n\n$$u = (y^{2} - 2y + x^{2} + 4x) + (2y - 2x) = (y + x)^{2} - x^{2}$$\n\nNow, let's use the Cauchy-Schwarz inequality to find the minimum value of $u$:\n\n$$(y + x)^{2} \\leq 1 + 1 = 2$$\n\nTherefore, the minimum value of $u$ is:\n\n$$u = (y + x)^{2} - x^{2} = 2 - x^{2}$$\n\nThe minimum value of the function $u$ is 2 - x².\n\nFinal answer:\n#### 2", "correct": false, "reward": -1, "repetition": {"words": 121, "unique_ratio": 0.4132231404958678, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "25. (25 points)\n\n**Analysis**\n\nThis problem can be solved by completing the square for both the $y$-terms and the $x$-terms.\n\nFirst, rewrite the function to express it in vertex form, then complete the square for the $y$-terms.\n\n**Step-by-step Solution**\n\nThe function $y^2 - 2y$ can be rewritten as $(y - 1)^2 - 1$.\n\nThe function $x^2 + 4x$ can be rewritten as $(x + 2)^2 - 4$.\n\nNow, we complete the square for the $y$-terms:\n\n$$u = (y - 1)^2 - 1$$\n\n$$u = (y + (1) - 1)^2 - 1$$\n\n$$u = (y + 1 - 1)^2 - 1$$\n\n$$u = (y + 1 - 1)^2 - 1$$\n\n$$u = (y + 1 - 1)^2 - 1$$\n\n$$u = (y + 1 - 1 - 0.5)^2 - 1$$\n\n$$u = (y - 0.5)^2 - 1$$\n\n$$u = (y - 0.5)^2 - 1$$\n\nNow, we write the function in vertex form by completing the square:\n\n$$u = (y + 0.5)^2 - 1$$\n\n$$u = (y - 0.5)^2 - 1$$\n\nThe given inequality $2x + y \\geqslant 1$ can be written as:\n\n$$2x + y -", "correct": false, "reward": -1, "repetition": {"words": 165, "unique_ratio": 0.296969696969697, "max_4gram_count": 8, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "$\\therefore$ From $u=y^{2}-2 y+x^{2}+4 x$ completing the square, we get\n$$\n(x+2)^{2}+(y-1)^{2}=u+5 \\text {. }\n$$\n\nSince $(x+2)^{2}+(y-1)^{2}$ can be regarded as the square of the distance from point $P(x, y)$ to the fixed point $(-2,1)$. The constraint $2 x+y \\geqslant 1$ indicates that point $P$ is in the region $G$ which is the right upper area of the line $l: 2 x+y=1$. Therefore, the problem is transformed into finding the shortest distance from point $A(-2,1)$ to the region $G$.\nFrom the right figure, it can be seen that the distance from point $A$ to\n\nline $I$ is the minimum value of the distance from point $A$ to any point in region $G$.\n$$\n\\begin{aligned}\nd & =\\frac{|2 \\cdot(-2)+1-1|}{\\sqrt{2^{2}+1^{2}}} \\\\\n& =\\frac{4}{\\sqrt{5}}, \\\\\n& \\therefore d^{2}=\\frac{16}{5} \\text {, i.e., } u+5=\\frac{16}{5}\n\\end{aligned}\n$$\n$\\therefore u=-\\frac{9}{5}$. That is, the minimum value of $u$ is $-\\frac{9}{5}$.", "correct": true, "reward": 1.0, "repetition": {"words": 184, "unique_ratio": 0.3423913043478261, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_078", "row": {"problem": "3.40 A red pencil costs 27 kopecks, a blue one - 23 kopecks. No more than 9 rubles 40 kopecks can be spent on purchasing pencils. It is necessary to purchase the maximum possible total number of red and blue pencils. At the same time, the number of red pencils should be as few as possible, but the number of blue pencils should not differ from the number of red pencils by more than 10. How many red and how many blue pencils should be purchased under the given conditions", "solution": "3.40 Let $x$ red and $y$ blue pencils be bought. According to the conditions, $27 x+23 y \\leqslant 940$ and $y-x \\leqslant 10$. Let's construct the lines\n\n![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-108.jpg?height=597&width=625&top_left_y=113&top_left_x=287)\n\n\\[\n\\begin{aligned}\n& 27 x+23 y=940 \\\\\n& y-x=10\n\\end{aligned}\n\\]\n\nFrom Fig. 3.8, it is clear that these lines intersect at point $A$, the coordinates of which satisfy equations (1) and (2), and at this point, the maximum possible sum $x+y$ is achieved. Solving the system (1), (2) and considering that the numbers $x$ and $y$ are natural, we get $x=14, y=24$.\n\nAnswer: 14 red and 24 blue pencils.", "answer": "14", "source": "olympiads", "problem_type": "Algebra", "uuid": "962f8132-4ca0-5e0f-b6b6-b040f23d5aa7"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n3.40 A red pencil costs 27 kopecks, a blue one - 23 kopecks. No more than 9 rubles 40 kopecks can be spent on purchasing pencils. It is necessary to purchase the maximum possible total number of red and blue pencils. At the same time, the number of red pencils should be as few as possible, but the number of blue pencils should not differ from the number of red pencils by more than 10. How many red and how many blue pencils should be purchased under the given conditions", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To solve this problem, we need to find the maximum number of red pencils (80 kopecks) and blue pencils (50 kopecks) that can be bought, while ensuring that the number of red pencils is as few as possible, but the number of blue pencils is as large as possible.\n\nSince the total cost of a red pencil is 27 kopecks, a blue pencil costs 23 kopecks, we will first find the maximum number of red pencils that can be bought with the given budget. This can be done by dividing the total budget by the cost of a red pencil.\n\n$$Maximum\\, red\\, pencils = Total\\, budget / Cost\\, of\\, a\\, red\\, pencil$$\n\nNow, plug in the given values for the total budget and cost of a red pencil to calculate the maximum number of red pencils.\n\n$$Maximum\\, red\\, pencils = 40 / 27$$\n\n$$Maximum\\, red\\, pencils = 1\\, (\\text{exactly 1 red pencil})$$\n\nSince the number of red pencils should be as few as possible, we should use the smallest possible value for the number of red pencils. Therefore, we should consider the case where the number of red pencils is equal to the number of red pencils.\n\n$$Maximum\\, red\\, pencils = 1\\, (\\text{one red pencil})$$\n\nNow that we have the maximum number of red pencils, we can calculate the number of blue pencils that can be bought. The cost of a blue pencil is 23 kopecks, so we can find the maximum number of blue pencils by dividing the total budget by the cost of a blue pencil.\n\n$$Maximum\\, blue\\, pencils = Total\\,", "correct": false, "reward": -1, "repetition": {"words": 258, "unique_ratio": 0.26356589147286824, "max_4gram_count": 9, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "The problem can be solved by how many red pencils and blue pencils should be purchased under the conditions of the problem.\n\nFirst, we should consider the total cost. The cost of a red pencil is 27 kopecks and the cost of a blue one is 23 kopecks.\n\nThe number of red pencils purchased is max(27, 23) = 49.\n\nSince the number of red pencils should be as few as possible, the number of blue pencils purchased is 27 - 10 = 17.\n\nTherefore, under the given conditions, we should purchase 49 red pencils and 17 blue pencils.\n\nFinal answer:\n#### 49", "correct": false, "reward": -1, "repetition": {"words": 99, "unique_ratio": 0.43434343434343436, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "3.40 Let $x$ red and $y$ blue pencils be bought. According to the conditions, $27 x+23 y \\leqslant 940$ and $y-x \\leqslant 10$. Let's construct the lines\n\n![](https://cdn.mathpix.com/cropped/2024_05_21_1e3d8d2f6605ac23e2a5g-108.jpg?height=597&width=625&top_left_y=113&top_left_x=287)\n\n\\[\n\\begin{aligned}\n& 27 x+23 y=940 \\\\\n& y-x=10\n\\end{aligned}\n\\]\n\nFrom Fig. 3.8, it is clear that these lines intersect at point $A$, the coordinates of which satisfy equations (1) and (2), and at this point, the maximum possible sum $x+y$ is achieved. Solving the system (1), (2) and considering that the numbers $x$ and $y$ are natural, we get $x=14, y=24$.\n\nAnswer: 14 red and 24 blue pencils.", "correct": true, "reward": 1.0, "repetition": {"words": 128, "unique_ratio": 0.625, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_079", "row": {"problem": "Given $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\n$$\\frac{x}{1+y+z x}+\\frac{y}{1+z+x y}+\\frac{z}{1+x+y z}=\\frac{3}{x+y+z} .$$", "solution": "3. It is not difficult to prove: $\\frac{x}{1+y+z x} \\leqslant \\frac{1}{x+y+z}$, $\\frac{y}{1+z+x y} \\leqslant \\frac{1}{x+y+z}$, $\\frac{z}{1+x+y z} \\leqslant \\frac{1}{x+y+z}$. Therefore, if the equality holds, it is easy to get $x=y=z=1$.\n\nNote: This problem can also be solved as follows: First prove $\\frac{x}{1+y+z x} \\leqslant \\frac{x}{x+y+z}$, etc., so we can get $x+y+z \\geqslant 3$, hence $x=y=z=1$.", "answer": "x=y=z=1", "source": "inequalities", "problem_type": "Algebra", "uuid": "c01ee438-4f54-5585-bbf8-4d10362f295a"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nGiven $0 \\leqslant x, y, z \\leqslant 1$, solve the equation:\n$$\\frac{x}{1+y+z x}+\\frac{y}{1+z+x y}+\\frac{z}{1+x+y z}=\\frac{3}{x+y+z} .$$", "candidates": [{"slot": 0, "origin": "budgie", "answer": "And, we have $x^{2} y z=1$.\n\nThe solution to the equation is $x=\\frac{1}{\\frac{1}{y}+1+\\frac{1}{z}}=\\frac{y z+1}{y+z}.$\n\nSince $y+z \\leqslant 2$, $y+z \\leqslant 2$ is not possible, but we may assume that $y+z=\\frac{1}{2}$.\n\nSo, $x=\\frac{y z+1}{\\frac{1}{2}}=\\boxed{4y z}.$", "correct": false, "reward": -1, "repetition": {"words": 64, "unique_ratio": 0.390625, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -1.3728113305845737}, {"slot": 1, "origin": "budgie", "answer": "Since $0 \\leqslant x, y, z \\leqslant 1$, we have $0 \\leqslant x, y, z < 1,$ and $\\frac{x}{1+y+z x} < \\frac{3}{x+y+z}.$  Therefore,\n$$\\frac{x}{1+y+z x} + \\frac{y}{1+z+x y} + \\frac{z}{1+x+y z} > 0,$$so\n$$\\frac{x}{1+y+z x} + \\frac{y}{1+z+x y} + \\frac{z}{1+x+y z} < -\\frac{3}{x+y+z}.$$This contradicts the original equation, so there is no solution.\n\nFinal answer:\n#### 1", "correct": true, "reward": 0.5, "repetition": {"words": 85, "unique_ratio": 0.29411764705882354, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": 0.39223180873844965}, {"slot": 2, "origin": "verified_reference", "answer": "3. It is not difficult to prove: $\\frac{x}{1+y+z x} \\leqslant \\frac{1}{x+y+z}$, $\\frac{y}{1+z+x y} \\leqslant \\frac{1}{x+y+z}$, $\\frac{z}{1+x+y z} \\leqslant \\frac{1}{x+y+z}$. Therefore, if the equality holds, it is easy to get $x=y=z=1$.\n\nNote: This problem can also be solved as follows: First prove $\\frac{x}{1+y+z x} \\leqslant \\frac{x}{x+y+z}$, etc., so we can get $x+y+z \\geqslant 3$, hence $x=y=z=1$.", "correct": true, "reward": 1.0, "repetition": {"words": 95, "unique_ratio": 0.3684210526315789, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": 0.9805795218461241}], "split": "math_train"}
+{"id": "math_080", "row": {"problem": "Let $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\\theta$. The maximum possible value for $\\tan \\theta$ can be expressed in the form $\\dfrac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.\n\n[i]Proposed by James Lin[/i]", "solution": "1. **Given Information and Setup:**\n   - We have integers \\(a_1, a_2, a_3, a_4\\) with distinct absolute values.\n   - Points in the coordinate plane are \\(A_1 = (a_1, a_1^2)\\), \\(A_2 = (a_2, a_2^2)\\), \\(A_3 = (a_3, a_3^2)\\), and \\(A_4 = (a_4, a_4^2)\\).\n   - Lines \\(A_1A_2\\) and \\(A_3A_4\\) intersect on the \\(y\\)-axis at an acute angle \\(\\theta\\).\n   - We need to find the maximum possible value for \\(\\tan \\theta\\) expressed as \\(\\frac{m}{n}\\) for relatively prime positive integers \\(m\\) and \\(n\\), and then find \\(100m + n\\).\n\n2. **Slope Calculation:**\n   - The slope of line \\(A_1A_2\\) is:\n     \\[\n     m_1 = \\frac{a_2^2 - a_1^2}{a_2 - a_1} = a_2 + a_1\n     \\]\n   - The slope of line \\(A_3A_4\\) is:\n     \\[\n     m_2 = \\frac{a_4^2 - a_3^2}{a_4 - a_3} = a_4 + a_3\n     \\]\n\n3. **Intersection on the \\(y\\)-axis:**\n   - Since the lines intersect on the \\(y\\)-axis, their \\(x\\)-coordinates at the intersection point are zero. Thus, the \\(y\\)-intercepts of the lines are:\n     \\[\n     c_1 = a_1^2 \\quad \\text{and} \\quad c_2 = a_3^2\n     \\]\n\n4. **Angle Between Lines:**\n   - The tangent of the angle \\(\\theta\\) between the two lines is given by:\n     \\[\n     \\tan \\theta = \\left| \\frac{m_1 - m_2}{1 + m_1 m_2} \\right| = \\left| \\frac{(a_1 + a_2) - (a_3 + a_4)}{1 + (a_1 + a_2)(a_3 + a_4)} \\right|\n     \\]\n\n5. **Maximizing \\(\\tan \\theta\\):**\n   - Let \\(x = a_1 + a_2\\) and \\(y = a_3 + a_4\\). We need to maximize:\n     \\[\n     f(x, y) = \\left| \\frac{x - y}{1 + xy} \\right|\n     \\]\n\n6. **Case Analysis:**\n   - **Case 1: \\(x, y > 0\\)**\n     - Here, \\(x, y \\geq 1\\) since they are integers. In this case, \\(|f(x, y)| < 1\\), which cannot exceed our current maximum.\n   - **Case 2: \\(x > 0, y < 0\\)**\n     - The numerator is positive, and the denominator is negative, so \\(|f(x, y)|\\) cannot be our maximum.\n   - **Case 3: \\(x < 0, y > 0\\)**\n     - Let \\(x = -u\\) and \\(y = v\\). We want to maximize:\n       \\[\n       \\frac{u + v}{uv - 1}\n       \\]\n     - We need to check when:\n       \\[\n       \\frac{u + v}{uv - 1} > \\frac{3}{2} \\implies 3(uv - 1) < 2(u + v) \\implies 3uv - 3 < 2u + 2v \\implies 3uv - 2u - 2v < 3\n       \\]\n     - If \\(u, v > 1\\), then it's impossible because \\(3uv - 2u - 2v \\geq 3\\) is false. Thus, we can let \\(v = 1\\) and need \\(u < 5\\).\n\n7. **Checking Specific Values:**\n   - For \\(u = 4\\) and \\(v = 1\\):\n     - We have \\(a_1 = -6, a_2 = 2, a_3 = 4, a_4 = -3\\).\n     - This gives:\n       \\[\n       \\tan \\theta = \\left| \\frac{(-6 + 2) - (4 - 3)}{1 + (-6 + 2)(4 - 3)} \\right| = \\left| \\frac{-4 - 1}{1 - 4} \\right| = \\left| \\frac{-5}{-3} \\right| = \\frac{5}{3}\n       \\]\n\nThe final answer is \\(\\boxed{503}\\).", "answer": "503", "source": "aops_forum", "problem_type": "Geometry", "uuid": "079495cf-64e8-565d-ab82-523255814ae6"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nLet $a_1, a_2, a_3, a_4$ be integers with distinct absolute values. In the coordinate plane, let $A_1=(a_1,a_1^2)$, $A_2=(a_2,a_2^2)$, $A_3=(a_3,a_3^2)$ and $A_4=(a_4,a_4^2)$. Assume that lines $A_1A_2$ and $A_3A_4$ intersect on the $y$-axis at an acute angle of $\\theta$. The maximum possible value for $\\tan \\theta$ can be expressed in the form $\\dfrac mn$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.\n\n[i]Proposed by James Lin[/i]", "candidates": [{"slot": 0, "origin": "budgie", "answer": "We know that $A_1A_2$ and $A_3A_4$ intersect at an acute angle of $\\theta$. This means that $\\tan \\theta > 0$. \n\nGiven the coordinates of $A_1$, $A_2$, $A_3$, and $A_4$, we can write the equations of the lines as follows:\n\n$$y = a_1^2x, \\, y = a_2^2x, \\, y = a_3^2x, \\, y = a_4^2x$$\n\nSolving for $x$ in each line equation, we get:\n\n$$x = \\frac{a_1^2}{a_1^2 - a_1^2a_1}, \\, x = \\frac{a_2^2}{a_2^2 - a_2^2a_2}, \\, x = \\frac{a_3^2}{a_3^2 - a_3^2a_3}, \\, x = \\frac{a_4^2}{a_4^2 - a_4^2a_4}$$\n\nSimplifying each equation, we get:\n\n$$x = \\frac{a_1^2}{a_1^2 - a_1^2a_1}, \\, x = \\frac{a_2^2}{a_2^2 - a_2^2a_2}, \\, x = \\frac{a_3^2}{a_3^2 - a_3^2a_3}, \\, x = \\frac{a_4^2}{a_4^2 - a_4^2a_4}$$\n\nSimplifying further", "correct": false, "reward": -1, "repetition": {"words": 174, "unique_ratio": 0.25862068965517243, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To find the maximum possible value for $\\tan \\theta$, we need to consider the properties of the intersection point $A_3$. Since $A_3A_4$ is a vertical line, its slope is undefined. Therefore, the slope of $A_3A_4$ is undefined.\n\nHowever, we can still find the maximum possible value of $\\tan \\theta$. If we take the derivative of $A_3A_4$ with respect to $\\theta$ and set it to zero, we get $A_3A_4 = A_3A_3$. This means that the slope of $A_3A_4$ is undefined.\n\nSince the slope of $A_1A_2$ is infinite, the slope of $A_1A_2$ is also infinite. Therefore, there is no maximum possible value for $\\tan \\theta$.\n\nHowever, we can still find the maximum possible value of $\\tan \\theta$. Since $A_1A_2$ is a perpendicular line, the maximum possible value of $\\tan \\theta$ is also infinite.\n\nBut we can still find the maximum possible value of $\\tan \\theta$. Since $A_2$ is a point on the $y$-axis, we can see that $\\tan \\theta \\ge 0$.\n\nTherefore, the maximum possible value of $\\tan \\theta$ is infinite.\n\nHowever, we can still find the maximum possible value of $\\tan \\theta$. Since $A_1$ is a point on the $x$-axis, we can see that $\\tan \\theta$ is approximately $\\tan \\theta \\approx 99.246497$.\n\nTherefore, the maximum possible value of $\\tan \\theta$ is approximately $99.24649", "correct": false, "reward": -1, "repetition": {"words": 236, "unique_ratio": 0.2754237288135593, "max_4gram_count": 8, "max_sentence_count": 3, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "1. **Given Information and Setup:**\n   - We have integers \\(a_1, a_2, a_3, a_4\\) with distinct absolute values.\n   - Points in the coordinate plane are \\(A_1 = (a_1, a_1^2)\\), \\(A_2 = (a_2, a_2^2)\\), \\(A_3 = (a_3, a_3^2)\\), and \\(A_4 = (a_4, a_4^2)\\).\n   - Lines \\(A_1A_2\\) and \\(A_3A_4\\) intersect on the \\(y\\)-axis at an acute angle \\(\\theta\\).\n   - We need to find the maximum possible value for \\(\\tan \\theta\\) expressed as \\(\\frac{m}{n}\\) for relatively prime positive integers \\(m\\) and \\(n\\), and then find \\(100m + n\\).\n\n2. **Slope Calculation:**\n   - The slope of line \\(A_1A_2\\) is:\n     \\[\n     m_1 = \\frac{a_2^2 - a_1^2}{a_2 - a_1} = a_2 + a_1\n     \\]\n   - The slope of line \\(A_3A_4\\) is:\n     \\[\n     m_2 = \\frac{a_4^2 - a_3^2}{a_4 - a_3} = a_4 + a_3\n     \\]\n\n3. **Intersection on the \\(y\\)-axis:**\n   - Since the lines intersect on the \\(y\\)-axis, their \\(x\\)-coordinates at the intersection point are zero. Thus, the \\(y\\)-intercepts of the lines are:\n     \\[\n     c_1 = a_1^2 \\quad \\text{and} \\quad c_2 = a_3^2\n     \\]\n\n4. **Angle Between Lines:**\n   - The tangent of the angle \\(\\theta\\) between the two lines is given by:\n     \\[\n     \\tan \\theta = \\left| \\frac{m_1 - m_2}{1 + m_1 m_2} \\right| = \\left| \\frac{(a_1 + a_2) - (a_3 + a_4)}{1 + (a_1 + a_2)(a_3 + a_4)} \\right|\n     \\]\n\n5. **Maximizing \\(\\tan \\theta\\):**\n   - Let \\(x = a_1 + a_2\\) and \\(y = a_3 + a_4\\). We need to maximize:\n     \\[\n     f(x, y) = \\left| \\frac{x - y}{1 + xy} \\right|\n     \\]\n\n6. **Case Analysis:**\n   - **Case 1: \\(x, y > 0\\)**\n     - Here, \\(x, y \\geq 1\\) since they are integers. In this case, \\(|f(x, y)| < 1\\), which cannot exceed our current maximum.\n   - **Case 2: \\(x > 0, y < 0\\)**\n     - The numerator is positive, and the denominator is negative, so \\(|f(x, y)|\\) cannot be our maximum.\n   - **Case 3: \\(x < 0, y > 0\\)**\n     - Let \\(x = -u\\) and \\(y = v\\). We want to maximize:\n       \\[\n       \\frac{u + v}{uv - 1}\n       \\]\n     - We need to check when:\n       \\[\n       \\frac{u + v}{uv - 1} > \\frac{3}{2} \\implies 3(uv - 1) < 2(u + v) \\implies 3uv - 3 < 2u + 2v \\implies 3uv - 2u - 2v < 3\n       \\]\n     - If \\(u, v > 1\\), then it's impossible because \\(3uv - 2u - 2v \\geq 3\\) is false. Thus, we can let \\(v = 1\\) and need \\(u < 5\\).\n\n7. **Checking Specific Values:**\n   - For \\(u = 4\\) and \\(v = 1\\):\n     - We have \\(a_1 = -6, a_2 = 2, a_3 = 4, a_4 = -3\\).\n     - This gives:\n       \\[\n       \\tan \\theta = \\left| \\frac{(-6 + 2) - (4 - 3)}{1 + (-6 + 2)(4 - 3)} \\right| = \\left| \\frac{-4 - 1}{1 - 4} \\right| = \\left| \\frac{-5}{-3} \\right| = \\frac{5}{3}\n       \\]\n\nThe final answer is \\(\\boxed{503}\\).", "correct": true, "reward": 1.0, "repetition": {"words": 471, "unique_ratio": 0.2611464968152866, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_081", "row": {"problem": "7. (10 points) Two wheels rotate, meshed with each other, around fixed axes passing through the centers of wheels $A$ and $B$. The radii of the wheels differ by a factor of three. The smaller wheel makes 30 revolutions per minute. Determine how many seconds the larger wheel spends on one revolution?\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_31c3376e8d47665bbf90g-04.jpg?height=403&width=480&top_left_y=958&top_left_x=868)", "solution": "Answer: $6 s$\n\nSolution. The speeds of points lying on the edge of the wheels are the same.\n\n(2 points)\n\nThe distances traveled by these points differ by three times.\n\nThe small wheel spends on one revolution: $t_{\\text {small }}=\\frac{1 \\text { min }}{30 \\text { revolutions }}=\\frac{60 s}{30 \\text { revolutions }}=2 s$. (3 points)\n\nTherefore, the large wheel spends on one revolution:\n\n$t_{\\text {large }}=3 t_{\\text {small }}=3 \\cdot 2=6 s$.", "answer": "6", "source": "olympiads", "problem_type": "Algebra", "uuid": "1c1dd0c5-08ce-59f7-b12f-4275c9d41efb"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n7. (10 points) Two wheels rotate, meshed with each other, around fixed axes passing through the centers of wheels $A$ and $B$. The radii of the wheels differ by a factor of three. The smaller wheel makes 30 revolutions per minute. Determine how many seconds the larger wheel spends on one revolution?\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_31c3376e8d47665bbf90g-04.jpg?height=403&width=480&top_left_y=958&top_left_x=868)", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The two wheels rotate at a combined speed of $2 \\times 30 \\times 60 = 3600$ revolutions per minute.\n\nThe two wheels make 3600 revolutions per minute, so the smaller wheel takes 3600/30 = $\\boxed{120}$ seconds to rotate three times as fast as the larger wheel.", "correct": false, "reward": -0.5, "repetition": {"words": 46, "unique_ratio": 0.6521739130434783, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "First, we need to find the distance the larger wheel travels in one revolution. Since the circumference of the wheel is the same for both wheels, we can find the distance traveled by the larger wheel in one revolution by multiplying the circumference of the larger wheel by the number of revolutions it makes in a minute.\n\nThe formula for the circumference of a circle is C = 2πr, where r is the radius of the circle.\n\nFor the larger wheel, the circumference in one minute is 2π(30/60) = 2π.\n\nSince the smaller wheel makes 30 revolutions per minute, the distance traveled by the smaller wheel in one minute is 2π(30/30) = 2π.\n\nTherefore, the larger wheel travels at a distance of 30*2π = 60π miles in one minute.\n\nNow, we can find the time the larger wheel spends on one revolution by dividing the distance traveled by the number of revolutions it makes in one minute by the number of revolutions it makes in a minute.\n\nTime = Distance / Number of pmid\nTime = (60π / 30) / 30\nTime = (6π / 30) minutes\nTime = (3/5) minutes\n\nSince we want the answer in seconds, we convert the time to seconds by multiplying by 60.\n\nTime = (3/5) * 60 seconds\nTime = 36 seconds\n\nSo, the larger wheel spends 36 seconds on one revolution.\n#### 36", "correct": false, "reward": -1, "repetition": {"words": 222, "unique_ratio": 0.2747747747747748, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "Answer: $6 s$\n\nSolution. The speeds of points lying on the edge of the wheels are the same.\n\n(2 points)\n\nThe distances traveled by these points differ by three times.\n\nThe small wheel spends on one revolution: $t_{\\text {small }}=\\frac{1 \\text { min }}{30 \\text { revolutions }}=\\frac{60 s}{30 \\text { revolutions }}=2 s$. (3 points)\n\nTherefore, the large wheel spends on one revolution:\n\n$t_{\\text {large }}=3 t_{\\text {small }}=3 \\cdot 2=6 s$.", "correct": true, "reward": 1.0, "repetition": {"words": 77, "unique_ratio": 0.5064935064935064, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_082", "row": {"problem": "Two is $10 \\%$ of $x$ and $20 \\%$ of $y$. What is $x - y$?\n$(\\mathrm {A}) \\ 1 \\qquad (\\mathrm {B}) \\ 2 \\qquad (\\mathrm {C})\\ 5 \\qquad (\\mathrm {D}) \\ 10 \\qquad (\\mathrm {E})\\ 20$", "solution": "$2 = \\frac {1}{10}x \\Longrightarrow x = 20,\\quad 2 = \\frac{1}{5}y \\Longrightarrow y = 10,\\quad x-y = 20 - 10=10  \\mathrm{(D)}$.", "answer": "10", "source": "amc_aime", "problem_type": "Algebra", "uuid": "5fafd2ee-f7d4-5e50-8a7e-7fe29cc17f17"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nTwo is $10 \\%$ of $x$ and $20 \\%$ of $y$. What is $x - y$?\n$(\\mathrm {A}) \\ 1 \\qquad (\\mathrm {B}) \\ 2 \\qquad (\\mathrm {C})\\ 5 \\qquad (\\mathrm {D}) \\ 10 \\qquad (\\mathrm {E})\\ 20$", "candidates": [{"slot": 0, "origin": "budgie", "answer": "8.000", "correct": false, "reward": -0.5, "repetition": {"words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "Given that two is $10\\%$ of $x$ and $20\\%$ of $y$, we have:\n\n$$2 = 0.1x$$\n$$2 = 0.2y$$\n\nTo find $x$ and $y$, we can solve these equations step-by-step:\n\nFirst, from the equation $2 = 0.1x$, we can express $x$ in terms of $y$ as follows:\n\n$$x = \\frac{2}{0.1}$$\n\nNow, plug in $x = 2$ into the equation $2 = 0.2y$ to find $y$:\n\n$$2 = 0.2 \\times 2$$\n\nThis equation is true for any value of $y$. However, since we are looking for a specific value of $x - y$, we can choose any value for $y$ to find a corresponding value for $x$. Let's choose $y = 1$ for simplicity:\n\n$$x = 2$$\n\nNow that we have $x = 2$, we can find $y$:\n\n$$y = 1$$\n\nFinally, we can calculate $x - y$:\n\n$$x - y = 2 - 1$$\n\nThus, $x - y = 2 - 1 = 1$.\n\nThe answer is 1.\n\nFinal answer:\n#### 1", "correct": false, "reward": -1, "repetition": {"words": 151, "unique_ratio": 0.39072847682119205, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "$2 = \\frac {1}{10}x \\Longrightarrow x = 20,\\quad 2 = \\frac{1}{5}y \\Longrightarrow y = 10,\\quad x-y = 20 - 10=10  \\mathrm{(D)}$.", "correct": true, "reward": 1.0, "repetition": {"words": 25, "unique_ratio": 0.48, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 1.3728113305845737}], "split": "math_dev"}
+{"id": "math_083", "row": {"problem": "A father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to the third child, and so on. It turns out that each child receives the same amount of property. The father has $\\qquad$ children.", "solution": "(Method 1)\nLet the father's property be $x$ yuan;\nSince the eldest and the second son receive the same amount of property, we have\n$$\n1000+(x-1000) \\times \\frac{1}{10}=2000+\\left[(x-1000) \\times\\left(1-\\frac{1}{10}\\right)-2000\\right] \\times \\frac{1}{10} ;\n$$\n\nSolving this, we get $x=81000$;\nThe eldest son receives $1000+(81000-1000) \\times \\frac{1}{10}=9000$ yuan;\nThat is, each child receives 9000 yuan;\nThis father has a total of $81000 \\div 9000=9$ children.\n(Method 2)\nSince each child receives the same amount of property;\nThe eldest and the second son receive the same amount of property;\nThe eldest son receives 1000 yuan plus $\\frac{1}{10}$ of the remaining property (let's denote this remaining as $A$); the second son receives 2000 yuan plus $\\frac{1}{10}$ of the remaining property (let's denote this remaining as $B$); thus, $\\frac{1}{10}$ of $A$ is 1000 yuan more than $\\frac{1}{10}$ of $B$;\n$A$ is $1000 \\div \\frac{1}{10}=10000$ yuan more than $B$;\nFrom $A$ to $B$, it decreases by $\\frac{1}{10}$ of $A$ and 2000 yuan;\nTherefore, $\\frac{1}{10}$ of $A$ is $10000-2000=8000$ yuan;\n$A$ is $8000 \\div \\frac{1}{10}=80000$ yuan;\nThis father's property is $1000+80000=81000$ yuan;\nThe eldest son receives $1000+(81000-1000) \\times \\frac{1}{10}=9000$ yuan;\nThat is, each child receives 9000 yuan;\nThis father has a total of $81000 \\div 9000=9$ children.\n(Method 3)\nLet this father have $n$ children;\nThen the second-to-last child receives $1000(n-1)$ yuan plus $\\frac{1}{10}$ of the remainder;\nThe last child receives $1000n$ yuan;\nThese $1000n$ yuan are $\\frac{9}{10}$ of what remains after the second-to-last child receives $1000(n-1)$; each child receives the same amount of property;\nFrom the equality of the property received by the second-to-last and the last child, we have\n$$\n1000(n-1)+1000n \\times \\frac{1}{9}=1000n \\text {; }\n$$\n\nSolving this, we get $n=9$;\nThis father has a total of 9 children.", "answer": "9", "source": "olympiads", "problem_type": "Algebra", "uuid": "16dbe1c7-62c1-5abf-88b4-a4d83a5c8276"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nA father wants to divide his property among his children: first, he gives 1000 yuan and one-tenth of the remaining property to the eldest child, then 2000 yuan and one-tenth of the remaining property to the second child, then 3000 yuan and one-tenth of the remaining property to the third child, and so on. It turns out that each child receives the same amount of property. The father has $\\qquad$ children.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Let the first child receive 1000 yuan.\n\nThen the second child receives 1000 + 2000 = 3000 yuan\nThe third child receives 1000 + 3000 = 4000 yuan\nThe fourth child receives 1000 + 4000 = 5000 yuan\nAnd so on.\n\nThe total amount received by the children is the sum of the first 1000, 3000, 4000, 5000, 6000,...\nThus, the total amount received is 1000 + 3000 + 4000 + 5000 + 6000 + 7000 + 8000 + 9000 + 10000 + 10000 + 10000 + 10000 + 10000 + 10000 + 10000 + 10000 + 10000 + 10000 + 10000 + 10000\n\nThe total amount received is 1000(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 + 13 + 14 + 15 + 16 + 17 + 18 + 19 + 20 + 21 + 22 + 23 + 24 + 25 + 26 +", "correct": false, "reward": -1, "repetition": {"words": 111, "unique_ratio": 0.5315315315315315, "max_4gram_count": 9, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 1, "origin": "budgie", "answer": "Let's analyze the problem step by step:\n\nThe father gives 1000 yuan to the eldest child, leaving 900 yuan remaining.\n\nThe father gives 2000 yuan to the second child, leaving 700 yuan remaining.\n\nThe father gives 3000 yuan to the third child, leaving 400 yuan remaining.\n\nContinuing this pattern, we can see that each child receives: 1000 + 2000 + 3000 + 400 = 1000 + 2000 + 3000 = 4000 + 3000 = 7000 yuan\n\nTherefore, the father has 7000 children.\n\nFinal answer:\n#### 7000", "correct": false, "reward": -0.5, "repetition": {"words": 76, "unique_ratio": 0.5, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 2, "origin": "verified_reference", "answer": "(Method 1)\nLet the father's property be $x$ yuan;\nSince the eldest and the second son receive the same amount of property, we have\n$$\n1000+(x-1000) \\times \\frac{1}{10}=2000+\\left[(x-1000) \\times\\left(1-\\frac{1}{10}\\right)-2000\\right] \\times \\frac{1}{10} ;\n$$\n\nSolving this, we get $x=81000$;\nThe eldest son receives $1000+(81000-1000) \\times \\frac{1}{10}=9000$ yuan;\nThat is, each child receives 9000 yuan;\nThis father has a total of $81000 \\div 9000=9$ children.\n(Method 2)\nSince each child receives the same amount of property;\nThe eldest and the second son receive the same amount of property;\nThe eldest son receives 1000 yuan plus $\\frac{1}{10}$ of the remaining property (let's denote this remaining as $A$); the second son receives 2000 yuan plus $\\frac{1}{10}$ of the remaining property (let's denote this remaining as $B$); thus, $\\frac{1}{10}$ of $A$ is 1000 yuan more than $\\frac{1}{10}$ of $B$;\n$A$ is $1000 \\div \\frac{1}{10}=10000$ yuan more than $B$;\nFrom $A$ to $B$, it decreases by $\\frac{1}{10}$ of $A$ and 2000 yuan;\nTherefore, $\\frac{1}{10}$ of $A$ is $10000-2000=8000$ yuan;\n$A$ is $8000 \\div \\frac{1}{10}=80000$ yuan;\nThis father's property is $1000+80000=81000$ yuan;\nThe eldest son receives $1000+(81000-1000) \\times \\frac{1}{10}=9000$ yuan;\nThat is, each child receives 9000 yuan;\nThis father has a total of $81000 \\div 9000=9$ children.\n(Method 3)\nLet this father have $n$ children;\nThen the second-to-last child receives $1000(n-1)$ yuan plus $\\frac{1}{10}$ of the remainder;\nThe last child receives $1000n$ yuan;\nThese $1000n$ yuan are $\\frac{9}{10}$ of what remains after the second-to-last child receives $1000(n-1)$; each child receives the same amount of property;\nFrom the equality of the property received by the second-to-last and the last child, we have\n$$\n1000(n-1)+1000n \\times \\frac{1}{9}=1000n \\text {; }\n$$\n\nSolving this, we get $n=9$;\nThis father has a total of 9 children.", "correct": true, "reward": 1.0, "repetition": {"words": 352, "unique_ratio": 0.21875, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_dev"}
+{"id": "math_084", "row": {"problem": "7.1. Solve the equation $\\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \\cdot 2 \\cdot \\ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a solution is a triplet $(n, k, l)$; if solutions differ in at least one component, they are considered different.", "solution": "Answer: 10 (all triples of solutions $(n, k, l):(3,1,2),(3,2,1),(4,3,3)$).\n\nSolution. Note that if $k4$, then $n!>4 \\cdot(n-1)!\\geqslant 2 \\cdot(k!+l!)$, so there are no such solutions. If $n=2$, we get $1=k!+l!$ - no solutions; if $n=3$, the equation $3=k!+l!$ has two solutions: $k=1, l=2$ and $k=2, l=1$; if $n=4$, the equation $12=k!+l!$ gives one more solution $k=l=3$.", "answer": "10", "source": "olympiads", "problem_type": "Number Theory", "uuid": "09902996-6e6f-5507-94c7-f8b8045a702b"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n7.1. Solve the equation $\\frac{n!}{2}=k!+l!$ in natural numbers, where $n!=1 \\cdot 2 \\cdot \\ldots n$. If there are no solutions, write 0; if there is one solution, write $n$; if there are multiple solutions, write the sum of the values of $n$ for all solutions. Recall that a solution is a triplet $(n, k, l)$; if solutions differ in at least one component, they are considered different.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "First, note that if $n!$ divided by $2$ does not have an integer factorial, it is impossible to factor it into the product of two factorials. Therefore, $n$ must be an even number or a multiple of $4$. There are $4$ possibilities: $4, 6, 8,...$ for $n$ values. If $n$ is a multiple of $4$, then $k = 4k$ and $l = 4l$, which implies $k \\cdot l = 4k \\cdot 4l$ does not have a factorial factorial. If $n = 8$ for the first case, then $k = 2$ and $l = 8$ for the second case, so there is only one solution. If $n = 6$, then $k = 3$ and $l = 6$ for the third case, so there is only one solution. If $n = 2$, then $k = 1$ and $l = 2$ for the fourth case, and only one solution. If $n = 1$, then $k = 1$ and $l = 1$ for the fifth case, and one solution. If $n = 0$, then $k = 0$ and $l = 0$ for the sixth case, and one solution. If $n = 5$, then $k = 1$ and $l = 5$ for the seventh case, and one solution. If $n = 3$, then $k = 1$ and $l = 3$ for the eighth and ninth cases, and one solution. If $n = 1$, then $k = 1$ and $l = 1$ for the tenth and eleventh cases, and one solution", "correct": false, "reward": -1, "repetition": {"words": 218, "unique_ratio": 0.3211009174311927, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 1, "origin": "budgie", "answer": "To solve the equation $\\frac{n!}{2} = k! + l!$, start by noticing that it can be as simple as 1 factorial on both the left side and the right side.\n\nThe smallest $n$ for which this does not hold true is $n = 4$ because $4! = 24$.\n\nThus, the answer is $\\boxed{4}$.\n\nNote: If there are multiple solutions, then the sum of their values is not unique.", "correct": false, "reward": -0.5, "repetition": {"words": 67, "unique_ratio": 0.7910447761194029, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 2, "origin": "verified_reference", "answer": "Answer: 10 (all triples of solutions $(n, k, l):(3,1,2),(3,2,1),(4,3,3)$).\n\nSolution. Note that if $k4$, then $n!>4 \\cdot(n-1)!\\geqslant 2 \\cdot(k!+l!)$, so there are no such solutions. If $n=2$, we get $1=k!+l!$ - no solutions; if $n=3$, the equation $3=k!+l!$ has two solutions: $k=1, l=2$ and $k=2, l=1$; if $n=4$, the equation $12=k!+l!$ gives one more solution $k=l=3$.", "correct": true, "reward": 1.0, "repetition": {"words": 85, "unique_ratio": 0.43529411764705883, "max_4gram_count": 1, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_085", "row": {"problem": "Example 13 Given that $a$, $b$, and $c$ are all positive integers, and the parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A$ and $B$. If the distances from $A$ and $B$ to the origin are both less than 1, find the minimum value of $a+b+c$.\n(1996, National Junior High School Mathematics League)", "solution": "Solution: Let $A\\left(x_{1}, 0\\right)$ and $B\\left(x_{2}, 0\\right)$, and $x_{1} < 0 < x_{2}$,  \nthen $x_{1} < 0, \\\\ \n\\therefore b > 2 \\sqrt{a c} . \\\\ \n\\text{Also, } \\because |O A| = |x_{1}| > 1$. Therefore, the parabola opens upwards, and when $x = -1$, $y > 0$, so $a(-1)^{2} + b(-1) + c > 0$, which means $b < a + c + 1$. Since $b > 2 \\sqrt{a c}$, we have $2 \\sqrt{a c} + 1 < a + c$,  \ni.e., $(\\sqrt{a} - \\sqrt{c})^{2} > 1$.  \nFrom equation (2), we get $\\sqrt{a} - \\sqrt{c} > 1$,  \n\ni.e., $\\sqrt{a} > \\sqrt{c} + 1$.  \nThus, $a > (\\sqrt{c} + 1)^{2} \\geq (\\sqrt{1} + 1)^{2} = 4$.  \nSolving, we get $a \\geq 5$.  \n$$\n\\begin{array}{l}\n\\text{Also, } \\because b > 2 \\sqrt{a c} \\geq 2 \\sqrt{5 \\times 1} > 4, \\\\\n\\therefore b \\geq 5 .\n\\end{array}\n$$\n\nWhen $a = 5, b = 5, c = 1$, the parabola $y = 5 x^{2} + 5 x + 1$ satisfies the given conditions.  \nTherefore, the minimum value of $a + b + c$ is 11.", "answer": "11", "source": "cn_contest", "problem_type": "Algebra", "uuid": "145da507-b5fa-5f63-b6dd-5097b644c349"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nExample 13 Given that $a$, $b$, and $c$ are all positive integers, and the parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A$ and $B$. If the distances from $A$ and $B$ to the origin are both less than 1, find the minimum value of $a+b+c$.\n(1996, National Junior High School Mathematics League)", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find the minimum value of $a+b+c$ given that $a$, $b$, and $c$ are all positive integers and that the parabola $y=a x^{2}+b x+c$ intersects the $x$-axis at two distinct points $A$ and $B$, we can proceed as follows:\n\nSince the parabola intersects the $x$-axis at $A$ and $B$, the equation of the parabola can be written as:\n\\[ y = a x^2 + b x + c = a(x^2 - 1) + b(x - 1) \\]\n\nWe know that the distance from $A$ to $B$ is less than 1, so:\n\\[ \\sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} < 1 \\Rightarrow (x_B - x_A)^2 + (y_B - y_A)^2 < 1 \\]\n\nSquaring both sides and expanding, we get:\n\\[ x_B^2 - 2x_Ax_B + x_A^2 + y_B^2 - 2y_Ay_B + y_A^2 + x_B^2 + y_B^2 - 2x_Ay_B + x_A^2 + y_A^2 < 1 \\]\n\nSimplifying, we get:\n\\[ 2x_B^2 - 2x_Ax_B + 2y_B^2 - 2y_Ay_B + 2x_A^2 + 2y_A^2 - 2x_A^2", "correct": false, "reward": -1, "repetition": {"words": 183, "unique_ratio": 0.3005464480874317, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "1. The equation of the parabola is $y = ax^2 + bx + c$.\n2. Since the parabola intersects the $x$-axis at points $A$ and $B$, we know that $A$ and $B$ have the same $y$-coordinate.\n3. The $y$-coordinates of $A$ and $B$ are the same, so we can write $y = k(ax^2 + bx + c)$, where $k$ is a constant.\n4. The roots of the quadratic equation $ax^2 + bx + c = 0$ are $x = \\frac{-b \\pm \\sqrt{b^2 - 4ac}}{2a}$.\n5. Since $A$ and $B$ have the same $y$-coordinate, we have $k = k \\cdot \\frac{-b + \\sqrt{b^2 - 4ac}}{2a}$.\n6. Since the distances from $A$ and $B$ to the origin are both less than 1, we have $\\sqrt{(a - 0)^2 + b^2} < 1$.\n7. Squaring both sides of the equation, we get $(a - 0)^2 + b^2 < 1$.\n8. Expanding the left-hand side, we get $a^2 + b^2 < 1$.\n9. Since $a^2 + b^2 - 1 = 0$, we can apply the Cauchy-Schwarz inequality to get $\\sqrt{(a - 0)^2 + b^2} < 1$.\n10. Squaring both sides of the equation, we get $(a - 0)^2 + b^2 <", "correct": false, "reward": -1, "repetition": {"words": 197, "unique_ratio": 0.350253807106599, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution: Let $A\\left(x_{1}, 0\\right)$ and $B\\left(x_{2}, 0\\right)$, and $x_{1} < 0 < x_{2}$,  \nthen $x_{1} < 0, \\\\ \n\\therefore b > 2 \\sqrt{a c} . \\\\ \n\\text{Also, } \\because |O A| = |x_{1}| > 1$. Therefore, the parabola opens upwards, and when $x = -1$, $y > 0$, so $a(-1)^{2} + b(-1) + c > 0$, which means $b < a + c + 1$. Since $b > 2 \\sqrt{a c}$, we have $2 \\sqrt{a c} + 1 < a + c$,  \ni.e., $(\\sqrt{a} - \\sqrt{c})^{2} > 1$.  \nFrom equation (2), we get $\\sqrt{a} - \\sqrt{c} > 1$,  \n\ni.e., $\\sqrt{a} > \\sqrt{c} + 1$.  \nThus, $a > (\\sqrt{c} + 1)^{2} \\geq (\\sqrt{1} + 1)^{2} = 4$.  \nSolving, we get $a \\geq 5$.  \n$$\n\\begin{array}{l}\n\\text{Also, } \\because b > 2 \\sqrt{a c} \\geq 2 \\sqrt{5 \\times 1} > 4, \\\\\n\\therefore b \\geq 5 .\n\\end{array}\n$$\n\nWhen $a = 5, b = 5, c = 1$, the parabola $y = 5 x^{2} + 5 x + 1$ satisfies the given conditions.  \nTherefore, the minimum value of $a + b + c$ is 11.", "correct": true, "reward": 1.0, "repetition": {"words": 176, "unique_ratio": 0.3068181818181818, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_086", "row": {"problem": "Example 2 Let $x \\in \\mathbf{R}$, try to find the minimum value of the function $f(x)=\\left(x^{2}+4 x+5\\right)\\left(x^{2}+4 x+2\\right)+2 x^{2}+8 x+1$.", "solution": "Let $u=x^{2}+4 x$, then\n$$\nf(x)=g(u)=(u+5)(u+2)+2 u+1=u^{2}+9 u+11 \\text {. }\n$$\n\nBy the properties of quadratic functions, when $u=-\\frac{9}{2}$, $g(u)$ reaches its minimum value, and when $u>-\\frac{9}{2}$, $g(u)$ is a strictly increasing function of $u$. Since $u=x^{2}+4 x=(x+2)^{2}-4$, we know that $u \\geqslant-4$, and when $x=-2$, $u=-4$. Therefore, when $x=-2$, $f(x)$ reaches its minimum value, which is -9.", "answer": "-9", "source": "olympiads", "problem_type": "Algebra", "uuid": "7e8b3d00-0ae4-5fe2-9545-c37a5bf4b6e3"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nExample 2 Let $x \\in \\mathbf{R}$, try to find the minimum value of the function $f(x)=\\left(x^{2}+4 x+5\\right)\\left(x^{2}+4 x+2\\right)+2 x^{2}+8 x+1$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "Let's consider the terms of the function $f(x)$:\n\n$$f(x) = \\left(x^{2}+4 x+5\\right)\\left(x^{2}+4 x+2\\right)+2 x^{2}+8 x+1$$\n\nWe can rewrite each term as $(x^{2}+4 x+2)(x^{2}+4 x+5)$.\n\nExpanding the product, we get:\n\n$$f(x) = (x^{2}+4 x+2)(x^{2}+4 x+5) = x^{4}+8 x^{3}+2 x^{2}+4 x^{3}+8 x^{2}+16 x+2 x^{2}+8 x+5$$\n\nCombining like terms, we get:\n\n$$f(x) = x^{4}+12 x^{3}+17 x^{2}+14 x+5$$\n\nNow, let's set $f(x)$ to a constant value to find the minimum value. We can do this by setting the expression $x^{4}+12 x^{3}+17 x^{2}+14 x+5 = k$, where $k$ is a constant.\n\nRearranging the equation, we get:\n\n$$x^{4}+12 x^{3}+17 x^{2}+14 x+5 - k = 0$$\n\nThis is a quartic equation in $x$, which can be difficult to solve. However, we can try to find a solution by inspection.\n\nNotice that the left-hand side of the equation has a root at $x = -\\frac{1", "correct": false, "reward": -1, "repetition": {"words": 200, "unique_ratio": 0.34, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To solve this problem, we first need to find the minimum value of the function $f(x) = \\left(x^{2} + 4x + 5\\right) \\left(x^{2} + 4x + 2\\right) + 2x^{2} + 8x + 1$.\n\nLet's denote the first expression as $A$ and the second expression as $B$:\n\n$$A = x^{2} + 4x + 5$$\n$$B = x^{2} + 4x + 2$$\n\nNow, we can rewrite $A$ and $B$ as follows:\n\n$$A = x^{2} + 4x + 5 = (x^{2} + 4x + 2) + 2x^{2} + 4x + 3$$\n$$B = x^{2} + 4x + 2 = (x^{2} + 4x + 4) + 2x^{2} + 4x + 4 + 1$$\n\nSince $A$ and $B$ are equal, we can set the expressions equal to each other:\n\n$$(x^{2} + 4x + 2) + 2x^{2} + 4x + 3 = (x^{2} + 4x + 4) + 2x^{2} + 4x + 4 + 1$$\n\nSimplify the right side of the equation:\n\n$$\\Rightarrow x^{2} + 2x^{2} + 2x^{2} + 8x + 1 = 5x^{2} + 12x + 5$$\n\nRearrange the", "correct": false, "reward": -1, "repetition": {"words": 146, "unique_ratio": 0.3493150684931507, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Let $u=x^{2}+4 x$, then\n$$\nf(x)=g(u)=(u+5)(u+2)+2 u+1=u^{2}+9 u+11 \\text {. }\n$$\n\nBy the properties of quadratic functions, when $u=-\\frac{9}{2}$, $g(u)$ reaches its minimum value, and when $u>-\\frac{9}{2}$, $g(u)$ is a strictly increasing function of $u$. Since $u=x^{2}+4 x=(x+2)^{2}-4$, we know that $u \\geqslant-4$, and when $x=-2$, $u=-4$. Therefore, when $x=-2$, $f(x)$ reaches its minimum value, which is -9.", "correct": true, "reward": 1.0, "repetition": {"words": 91, "unique_ratio": 0.4175824175824176, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_087", "row": {"problem": "8. (15 points) A thin ray of light falls on a thin converging lens at a distance of $x=10$ cm from its optical center. The angle between the incident ray and the plane of the lens $\\alpha=45^{\\circ}$, and the angle between the refracted ray and the plane of the lens $\\beta=30^{\\circ}$. Determine its focal length.", "solution": "Answer: $\\approx 13.7$ cm.\n\nSolution. Parallel rays intersect at the focus, so $F$ is the focus of the given lens.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_7a4bc87492c06fc34cd6g-06.jpg?height=480&width=302&top_left_y=1502&top_left_x=957)\n\nIn triangle $O A F$: angle $F A O=30^{\\circ}$, angle $O F A=15^{\\circ}$, angle $A O F=135^{\\circ}$.\n\nTherefore, $\\frac{O F}{\\sin 30^{\\circ}}=\\frac{A O}{\\sin 15^{\\circ}}$.\n\nWe obtain that the focal length:\n\n$F C=O F \\sin 45^{\\circ}=A O \\frac{\\sin 30^{\\circ}}{\\sin 15^{\\circ}} \\sin 45^{\\circ} \\approx 13.7 \\mathrm{~cm}$.\n\n## Multidisciplinary Engineering Olympiad \"Star\" in Natural Sciences\n\n10th grade\n\nFinal Round\n\n$2022-2023$\n\nVersion 2\n\nProblems, answers, and evaluation criteria", "answer": "13.7", "source": "olympiads", "problem_type": "Geometry", "uuid": "78b0db16-409f-5e5d-9861-d426afbf5dc6"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n8. (15 points) A thin ray of light falls on a thin converging lens at a distance of $x=10$ cm from its optical center. The angle between the incident ray and the plane of the lens $\\alpha=45^{\\circ}$, and the angle between the refracted ray and the plane of the lens $\\beta=30^{\\circ}$. Determine its focal length.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "This problem mainly tests the properties of a focal length of a lens, which is a function of the angle between the incident ray and the plane of the lens.\n\nThe angle between the incident ray and the plane of the lens is related to the distance from the entry point to the lens through the lens's shape. The angle between the incident ray and the plane of the lens is essentially the angle between the rays and the plane of the lens. It is because the point is on the edge of the lens that the lightbars will form a right angle with the rays.\n\nSince the lightbars form a right angle with the rays, the focal length can be determined by the distance from the lightblue point to the lens. The distance from the lightblue point to the lens is the same as the distance from the lightblue point to the lens, which is half of the focal length of the lens.\n\nThe answer is: $\\boxed{10\\sqrt{2}}$ cm", "correct": false, "reward": -1, "repetition": {"words": 172, "unique_ratio": 0.32558139534883723, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "The focal length of the lens is given by the sum of the distances from the light source to the focal points.  Therefore, the focal length of the lens is $x+x=2x$.\n\nHence, the answer is:\n\n\\[ \\boxed{2x} \\]\n\nThe focal length of the lens is $2x$.", "correct": false, "reward": -1, "repetition": {"words": 47, "unique_ratio": 0.44680851063829785, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Answer: $\\approx 13.7$ cm.\n\nSolution. Parallel rays intersect at the focus, so $F$ is the focus of the given lens.\n\n![](https://cdn.mathpix.com/cropped/2024_05_06_7a4bc87492c06fc34cd6g-06.jpg?height=480&width=302&top_left_y=1502&top_left_x=957)\n\nIn triangle $O A F$: angle $F A O=30^{\\circ}$, angle $O F A=15^{\\circ}$, angle $A O F=135^{\\circ}$.\n\nTherefore, $\\frac{O F}{\\sin 30^{\\circ}}=\\frac{A O}{\\sin 15^{\\circ}}$.\n\nWe obtain that the focal length:\n\n$F C=O F \\sin 45^{\\circ}=A O \\frac{\\sin 30^{\\circ}}{\\sin 15^{\\circ}} \\sin 45^{\\circ} \\approx 13.7 \\mathrm{~cm}$.\n\n## Multidisciplinary Engineering Olympiad \"Star\" in Natural Sciences\n\n10th grade\n\nFinal Round\n\n$2022-2023$\n\nVersion 2\n\nProblems, answers, and evaluation criteria", "correct": true, "reward": 1.0, "repetition": {"words": 130, "unique_ratio": 0.5923076923076923, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_088", "row": {"problem": "The equation of line $\\ell_1$ is $24x-7y = 319$ and the equation of line $\\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\\ell_1$ and $\\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.\n\n[i]Proposed by Christopher Cheng[/i]\n\n[hide=Solution][i]Solution. [/i] $\\boxed{6}$\nNote that $(20,23)$ is the intersection of the lines $\\ell_1$ and $\\ell_2$. Thus, we only care about lattice points on the the two lines that are an integer distance away from $(20,23)$. Notice that $7$ and $24$ are part of the Pythagorean triple $(7,24,25)$ and $5$ and $12$ are part of the Pythagorean triple $(5,12,13)$. Thus, points on $\\ell_1$ only satisfy the conditions when $n$ is divisible by $25$ and points on $\\ell_2$ only satisfy the conditions when $n$ is divisible by $13$. Therefore, $a$ is just the number of positive integers less than $2023$ that are divisible by both $25$ and $13$. The LCM of $25$ and $13$ is $325$, so the answer is $\\boxed{6}$.[/hide]", "solution": "1. **Intersection of Lines**:\n   We start by finding the intersection of the lines $\\ell_1$ and $\\ell_2$. The equations are:\n   \\[\n   24x - 7y = 319\n   \\]\n   \\[\n   12x - 5y = 125\n   \\]\n   To find the intersection, we solve this system of linear equations. We can use the method of elimination or substitution. Here, we will use elimination.\n\n2. **Elimination Method**:\n   Multiply the second equation by 2 to align the coefficients of $x$:\n   \\[\n   24x - 10y = 250\n   \\]\n   Now subtract the second equation from the first:\n   \\[\n   (24x - 7y) - (24x - 10y) = 319 - 250\n   \\]\n   Simplifying, we get:\n   \\[\n   3y = 69 \\implies y = 23\n   \\]\n   Substitute $y = 23$ back into the first equation:\n   \\[\n   24x - 7(23) = 319 \\implies 24x - 161 = 319 \\implies 24x = 480 \\implies x = 20\n   \\]\n   Therefore, the intersection point is $(20, 23)$.\n\n3. **Distance from $(20, 23)$**:\n   We need to find lattice points on $\\ell_1$ and $\\ell_2$ that are an integer distance $n$ from $(20, 23)$. The distance formula is:\n   \\[\n   d = \\sqrt{(x - 20)^2 + (y - 23)^2}\n   \\]\n   For $d$ to be an integer, $(x - 20)^2 + (y - 23)^2$ must be a perfect square, say $k^2$. Thus:\n   \\[\n   (x - 20)^2 + (y - 23)^2 = n^2\n   \\]\n\n4. **Lattice Points on $\\ell_1$**:\n   The line $\\ell_1$ can be rewritten in slope-intercept form:\n   \\[\n   y = \\frac{24}{7}x - \\frac{319}{7}\n   \\]\n   Notice that $24$ and $7$ are part of the Pythagorean triple $(7, 24, 25)$. Therefore, the distance $n$ must be a multiple of $25$ for there to be lattice points on $\\ell_1$.\n\n5. **Lattice Points on $\\ell_2$**:\n   The line $\\ell_2$ can be rewritten in slope-intercept form:\n   \\[\n   y = \\frac{12}{5}x - \\frac{125}{5}\n   \\]\n   Notice that $12$ and $5$ are part of the Pythagorean triple $(5, 12, 13)$. Therefore, the distance $n$ must be a multiple of $13$ for there to be lattice points on $\\ell_2$.\n\n6. **Common Multiples**:\n   We need $n$ to be a multiple of both $25$ and $13$. The least common multiple (LCM) of $25$ and $13$ is:\n   \\[\n   \\text{LCM}(25, 13) = 325\n   \\]\n   We need to count the multiples of $325$ that are less than $2023$:\n   \\[\n   \\left\\lfloor \\frac{2023}{325} \\right\\rfloor = 6\n   \\]\n\nConclusion:\nThe number of positive integer values $n$ less than $2023$ such that there exists a lattice point on both $\\ell_1$ and $\\ell_2$ at a distance $n$ from $(20, 23)$ is $\\boxed{6}$.", "answer": "6", "source": "aops_forum", "problem_type": "Number Theory", "uuid": "b325fc9f-7ff8-5556-976b-f72348f99405"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nThe equation of line $\\ell_1$ is $24x-7y = 319$ and the equation of line $\\ell_2$ is $12x-5y = 125$. Let $a$ be the number of positive integer values $n$ less than $2023$ such that for both $\\ell_1$ and $\\ell_2$ there exists a lattice point on that line that is a distance of $n$ from the point $(20,23)$. Determine $a$.\n\n[i]Proposed by Christopher Cheng[/i]\n\n[hide=Solution][i]Solution. [/i] $\\boxed{6}$\nNote that $(20,23)$ is the intersection of the lines $\\ell_1$ and $\\ell_2$. Thus, we only care about lattice points on the the two lines that are an integer distance away from $(20,23)$. Notice that $7$ and $24$ are part of the Pythagorean triple $(7,24,25)$ and $5$ and $12$ are part of the Pythagorean triple $(5,12,13)$. Thus, points on $\\ell_1$ only satisfy the conditions when $n$ is divisible by $25$ and points on $\\ell_2$ only satisfy the conditions when $n$ is divisible by $13$. Therefore, $a$ is just the number of positive integers less than $2023$ that are divisible by both $25$ and $13$. The LCM of $25$ and $13$ is $325$, so the answer is $\\boxed{6}$.[/hide]", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find the lattice points on line $\\ell_2$, we need to find the intersection points of line $12x - 5y = 125$ and line $\\ell_2$.\n\nRearrange the equation of line $\\ell_2$ to solve for $y$:\n\\[ 12x - 5y = 125 \\Rightarrow -5y = -125 - 12x \\Rightarrow y = \\frac{125 + 12x}{5} \\Rightarrow y = \\frac{25(3 + x)}{5} \\Rightarrow y = 3 + x \\]\n\nThis means that the line $\\ell_2$ intersects the vertical line $y = 3 + x$ at points that are $3$ units above and below the point $(20,23)$.\n\nSince the lattice points on $\\ell_2$ are exactly $3$ units away from $(20,23)$, the number of lattice points on $\\ell_2$ is the number of lattice points on $y = 3 + x$ minus the number of lattice points on $y = 3 - x$.\n\nThe number of lattice points on $y = 3 + x$ is the number of lattice points on $y = 3 - x$ plus the number of lattice points on $y = 3 + x$ minus the number of lattice points on $y = 3 - x$. This is a difficult equation to solve without specific values for $x$ and $y$.\n\nHowever, we can observe that the points on $y = 3 + x$ are spaced $3$ units apart from the points on $y = 3 - x$. This means that the lattice points on $\\ell_", "correct": false, "reward": -1, "repetition": {"words": 214, "unique_ratio": 0.2757009345794392, "max_4gram_count": 8, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "To find the number of positive integer values of $n$ less than $2023$ such that for both $\\ell_1$ and $\\ell_2$ there exists a lattice point on that line that is a distance of $n$ from $(20,23)$, let's first determine the conditions for such points on the two lines.\n\nFor the line on $\\ell_1$ to have a lattice point on it that is a distance of $n$ from $(20,23)$, any point on the line can be expressed as:\n\\[ P(x, y) = (x, y) + (x, y + n) \\]\n\nFor the line on $\\ell_2$ to have a lattice point on it that is a distance of $n$ from $(20,23)$, the point $(x, y)$ must satisfy:\n\\[ P(x, y) = (x - 20, y + 23) \\]\n\nThis means that for the point $(x, y)$ to satisfy both equations, it must satisfy:\n\\[ x - 20 = x + n \\]\n\\[ y + 23 = y + n \\]\n\nSimplifying these equations gives us:\n\\[ x - x = n \\]\n\\[ 23 = n \\]\n\nFrom these equations, we see that $n$ must be a divisor of $23$. The positive divisors of $23$ are $1, 23$, and $23$ itself.\n\nNow, we need to find the values of $n$ for which there exists a lattice point on the line on $\\ell_1$ that is a distance of $n$ from the point $(20", "correct": false, "reward": -1, "repetition": {"words": 214, "unique_ratio": 0.3130841121495327, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "1. **Intersection of Lines**:\n   We start by finding the intersection of the lines $\\ell_1$ and $\\ell_2$. The equations are:\n   \\[\n   24x - 7y = 319\n   \\]\n   \\[\n   12x - 5y = 125\n   \\]\n   To find the intersection, we solve this system of linear equations. We can use the method of elimination or substitution. Here, we will use elimination.\n\n2. **Elimination Method**:\n   Multiply the second equation by 2 to align the coefficients of $x$:\n   \\[\n   24x - 10y = 250\n   \\]\n   Now subtract the second equation from the first:\n   \\[\n   (24x - 7y) - (24x - 10y) = 319 - 250\n   \\]\n   Simplifying, we get:\n   \\[\n   3y = 69 \\implies y = 23\n   \\]\n   Substitute $y = 23$ back into the first equation:\n   \\[\n   24x - 7(23) = 319 \\implies 24x - 161 = 319 \\implies 24x = 480 \\implies x = 20\n   \\]\n   Therefore, the intersection point is $(20, 23)$.\n\n3. **Distance from $(20, 23)$**:\n   We need to find lattice points on $\\ell_1$ and $\\ell_2$ that are an integer distance $n$ from $(20, 23)$. The distance formula is:\n   \\[\n   d = \\sqrt{(x - 20)^2 + (y - 23)^2}\n   \\]\n   For $d$ to be an integer, $(x - 20)^2 + (y - 23)^2$ must be a perfect square, say $k^2$. Thus:\n   \\[\n   (x - 20)^2 + (y - 23)^2 = n^2\n   \\]\n\n4. **Lattice Points on $\\ell_1$**:\n   The line $\\ell_1$ can be rewritten in slope-intercept form:\n   \\[\n   y = \\frac{24}{7}x - \\frac{319}{7}\n   \\]\n   Notice that $24$ and $7$ are part of the Pythagorean triple $(7, 24, 25)$. Therefore, the distance $n$ must be a multiple of $25$ for there to be lattice points on $\\ell_1$.\n\n5. **Lattice Points on $\\ell_2$**:\n   The line $\\ell_2$ can be rewritten in slope-intercept form:\n   \\[\n   y = \\frac{12}{5}x - \\frac{125}{5}\n   \\]\n   Notice that $12$ and $5$ are part of the Pythagorean triple $(5, 12, 13)$. Therefore, the distance $n$ must be a multiple of $13$ for there to be lattice points on $\\ell_2$.\n\n6. **Common Multiples**:\n   We need $n$ to be a multiple of both $25$ and $13$. The least common multiple (LCM) of $25$ and $13$ is:\n   \\[\n   \\text{LCM}(25, 13) = 325\n   \\]\n   We need to count the multiples of $325$ that are less than $2023$:\n   \\[\n   \\left\\lfloor \\frac{2023}{325} \\right\\rfloor = 6\n   \\]\n\nConclusion:\nThe number of positive integer values $n$ less than $2023$ such that there exists a lattice point on both $\\ell_1$ and $\\ell_2$ at a distance $n$ from $(20, 23)$ is $\\boxed{6}$.", "correct": true, "reward": 1.0, "repetition": {"words": 390, "unique_ratio": 0.3282051282051282, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_dev"}
+{"id": "math_089", "row": {"problem": "## 128. Math Puzzle $1 / 76$\n\nA circus gave 200 performances in the last season, all of which were sold out. The number of seats in the circus tent is three times the fourth part of the number of performances given.\n\na) How many program leaflets were printed if one fourth of the visitors bought a leaflet?\n\nb) How many Marks were additionally collected from the entrance fees for the animal show if it was visited by half of the visitors and the entrance fee was 0.30 M?", "solution": "From $(200: 4) \\cdot 3=150$ it follows that the circus has over 150 seats.\n\na) $(200 \\cdot 150): 4=7500$; 7500 program booklets were sold during the season.\n\nb) $(200 \\cdot 150): 2=15000$ and $30 \\cdot 15000=450000$; from the animal show, €450,000 was earned.", "answer": "450000", "source": "olympiads", "problem_type": "Logic and Puzzles", "uuid": "350ffb62-b012-5c3c-adcf-6591fca675be"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n## 128. Math Puzzle $1 / 76$\n\nA circus gave 200 performances in the last season, all of which were sold out. The number of seats in the circus tent is three times the fourth part of the number of performances given.\n\na) How many program leaflets were printed if one fourth of the visitors bought a leaflet?\n\nb) How many Marks were additionally collected from the entrance fees for the animal show if it was visited by half of the visitors and the entrance fee was 0.30 M?", "candidates": [{"slot": 0, "origin": "budgie", "answer": "#### Part (a)\n1. Let's denote the number of performances sold by the performers as \\( S_{\\text{per former}} \\), the number of performances sold by theppromades as \\( S_{\\text{per new}} \\), and the number of performances by the audience as \\( S_{\\text{per new}} \\).\n2. According to the problem, the number of violin and possession is three times the number ofadders. This can be written as:\n   \\[\n   S_{\\text{per former}} = 3 \\times S_{\\text{per new}}\n   \\]\n3. From this, we get a relationship between the number of performances and the number of adders:\n   \\[\n   S_{\\text{per former}} = 3 \\times S_{\\text{per new}} = 3S_{\\text{per adder}}\n   \\]\n4. This relationship holds because the total number of performances is equal to the sum of the number of performances by the performers and the number of adders.\n5. Therefore, we can express the total number of performances as:\n   \\[\n   S_{\\text{per former}} + S_{\\text{per new}} = S_{\\text{per adder}}\n   \\]\n6. Substituting the expression for \\( S_{\\text{per adder}} \\) from the previous step, we get:\n   \\[\n   3S_{\\text{per old}} = S_{\\text{per new}}\n   \\]\n7. Hence, it follows that:\n   \\[\n   S_{\\text{per old}} = \\frac{S_{\\text{per new}}}{3}\n   \\]\n8. This relationship holds regardless of the number of adders \\(", "correct": false, "reward": -1, "repetition": {"words": 205, "unique_ratio": 0.33170731707317075, "max_4gram_count": 7, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "a) The number of program HBV is 1/4 of the total number of performance performances.\n\nLet x be the total number of performances. Then, we know that 1/4*x = 200.\n\nSolving for x, we find that x = 800.\n\nb) To find the number of normalized visitors, we need to find the number of visitors who visited the animal show. We can do this by dividing the number of visitors who bought the Conserv by three.\n\nx = (200/3*x) + (x/3)\n\nx = (800/3) + (800/9) + (x/3)\n\nNow, we can simplify this expression.\n\nx = (800/3) + (1667/9) + (x/3)\n\nx = (800/3) + (1667/9) + (819/9)\n\nx = (800 + 1667 + 819)/9\n\nx = (2556)/9\n\nx = 286\n\nSo, the number of normalized visitors is 286.\n\nFinal answer:\n#### 286", "correct": false, "reward": -1, "repetition": {"words": 131, "unique_ratio": 0.40458015267175573, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "From $(200: 4) \\cdot 3=150$ it follows that the circus has over 150 seats.\n\na) $(200 \\cdot 150): 4=7500$; 7500 program booklets were sold during the season.\n\nb) $(200 \\cdot 150): 2=15000$ and $30 \\cdot 15000=450000$; from the animal show, €450,000 was earned.", "correct": true, "reward": 1.0, "repetition": {"words": 48, "unique_ratio": 0.7083333333333334, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_090", "row": {"problem": "Determine the value of $2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)$.", "solution": "Solution 1\n\nMultiplying through, we obtain\n\n$$\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)=16+\\frac{16}{2}+\\frac{16}{4}+\\frac{16}{8}+\\frac{16}{16}=16+8+4+2+1=31\n$$\n\n## Solution 2\n\nUsing a common denominator inside the parentheses, we obtain\n\n$$\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)=16\\left(\\frac{16}{16}+\\frac{8}{16}+\\frac{4}{16}+\\frac{2}{16}+\\frac{1}{16}\\right)=16\\left(\\frac{31}{16}\\right)=31\n$$\n\nANSWER: 31", "answer": "31", "source": "olympiads", "problem_type": "Algebra", "uuid": "39ed5515-9c2e-5efa-bc64-afc56279b4fe"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\nDetermine the value of $2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "To find the value of $2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)$, we need to simplify the expression inside the parentheses first.\n\n\\[ 2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right) = 2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\frac{1}{16}\\right)\\]\n\nNext, we need to find the sum inside the parentheses. To do this, we can add the fractions one by one and then find a common denominator.\n\n\\[ 1+(1+1+1+1+1) = 5 \\]\n\nSo, we have:\n\n\\[ 2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{4}+\\frac{1}{8}+\\frac{1}{16}\\right) = 2^{4}\\left(5\\right) = 2^{4}\\cdot 5 = 16\\cdot 2 = \\boxed{32} \\]\n\nFinal answer:\n#### 32", "correct": false, "reward": -1, "repetition": {"words": 143, "unique_ratio": 0.3006993006993007, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 1, "origin": "budgie", "answer": "We can simplify the expression by grouping the terms into sets of four.\n\nLet's break down the expression step by step:\n\n\\begin{align*}\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right) &= 2^{4}\\cdot\\frac{1}{2^4}\\\\\n&= 2^4\\cdot\\frac{1}{16}\\\\\n&= 2^4\\cdot\\frac{1}{2^4}\\\\\n&= 2^4\\cdot\\frac{1}{2^4}\\\\\n&= 2^3\n\\end{align*}\n\nSo, the answer is $\\boxed{8}$.", "correct": false, "reward": -1, "repetition": {"words": 80, "unique_ratio": 0.4125, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.707106031187343}, {"slot": 2, "origin": "verified_reference", "answer": "Solution 1\n\nMultiplying through, we obtain\n\n$$\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)=16+\\frac{16}{2}+\\frac{16}{4}+\\frac{16}{8}+\\frac{16}{16}=16+8+4+2+1=31\n$$\n\n## Solution 2\n\nUsing a common denominator inside the parentheses, we obtain\n\n$$\n2^{4}\\left(1+\\frac{1}{2}+\\frac{1}{2^{2}}+\\frac{1}{2^{3}}+\\frac{1}{2^{4}}\\right)=16\\left(\\frac{16}{16}+\\frac{8}{16}+\\frac{4}{16}+\\frac{2}{16}+\\frac{1}{16}\\right)=16\\left(\\frac{31}{16}\\right)=31\n$$\n\nANSWER: 31", "correct": true, "reward": 1.0, "repetition": {"words": 103, "unique_ratio": 0.22330097087378642, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142120623746859}], "split": "math_train"}
+{"id": "math_091", "row": {"problem": "5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\\sqrt{1+a^{2}}+\\sqrt{1+2 b}$ the integer part is", "solution": "5. 2\n\nTranslate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.", "answer": "2", "source": "olympiads", "problem_type": "Algebra", "uuid": "30f87501-618e-5c88-94ef-7c3ded50115e"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n5. Given positive real numbers $a$ and $b$ satisfy $a+b=1$, then $M=$ $\\sqrt{1+a^{2}}+\\sqrt{1+2 b}$ the integer part is", "candidates": [{"slot": 0, "origin": "budgie", "answer": "5.  Let's consider a special case when $a=b=\\frac{1}{4}$.\n\n$$M=\\sqrt{1+a^{2}}+\\sqrt{1+2 a^{2}}$$\n\nSince $a=\\frac{1}{4}$, we have $a^{2}=1$.\n\nTherefore,\n$$M=\\sqrt{1+1}+\\sqrt{1+2 \\times 1}=\\sqrt{2}+\\sqrt{3}$$\n\nHence, $M=\\boxed{\\sqrt{2}+\\sqrt{3}}$.\n\nFinal answer:\n#### √2 + √3", "correct": false, "reward": -1, "repetition": {"words": 57, "unique_ratio": 0.42105263157894735, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": -1.3728113305845737}, {"slot": 1, "origin": "budgie", "answer": "To find the integer part of $M$, we start by simplifying the expression inside the square root.\n\n\\begin{align*}\n\\sqrt{1+2 b} &= \\sqrt{1+2 b} \\\\\n&= \\sqrt{1+2 b} \\cdot \\frac{1}{\\sqrt{1+2 b} + 1} \\\\\n&= \\sqrt{1+2 b} \\\\\n&= \\frac{1+2 b}{\\sqrt{1+2 b} + 1}\n\\end{align*}\n\nThen, we rewrite the expression for $M$ as follows:\n\n\\begin{align*}\nM &= \\sqrt{1+2 b} \\\\\n&= \\frac{1+2 b}{\\sqrt{1+2 b} + 1} \\\\\n&= \\frac{1+2 b}{\\frac{1+2 b}{\\sqrt{1+2 b}+1}} \\\\\n&= \\frac{(1+2 b)^{3/2}}{\\frac{(1+2 b)^{3/2}}{\\sqrt{1+2 b}+1}} \\\\\n&= (1+2 b)^{3/2} \\cdot \\frac{\\sqrt{1+2 b}+1}{\\sqrt{1+2 b}+1} \\\\\n&= (1+2 b)^{3/2} \\\\cdot \\frac{\\sqrt{1+2 b}}{\\sqrt{1+2 b} + 1} \\\\\n&= (1+2 b)^{3/2} \\\\cdot \\frac{\\sqrt{1+2 b}}{\\sqrt{1+2 b} + 1} \\\\\n&= (1 + 2 b)^{3/2} \\\\", "correct": true, "reward": 0.5, "repetition": {"words": 163, "unique_ratio": 0.18404907975460122, "max_4gram_count": 16, "max_sentence_count": 1, "loop": true}, "advantage": 0.39223180873844965}, {"slot": 2, "origin": "verified_reference", "answer": "5. 2\n\nTranslate the text above into English, please retain the original text's line breaks and format, and output the translation result directly.", "correct": true, "reward": 1.0, "repetition": {"words": 23, "unique_ratio": 0.8695652173913043, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": 0.9805795218461241}], "split": "math_train"}
+{"id": "math_092", "row": {"problem": "1.57. The vertices of a rectangle inscribed in a circle divide it into four arcs. Find the distance from the midpoint of one of the larger arcs to the vertices of the rectangle, if its sides are 24 and 7 cm.", "solution": "1.57. Since $A C$ is the diameter of the circle (Fig. 1.44), then $R=0.5 \\sqrt{24^{2}+7^{2}}=12.5 \\quad$ (cm). In $\\triangle B O F$ we have $O F=$ $=\\sqrt{O B^{2}-B F^{2}}=\\sqrt{12.5^{2}-12^{2}}=3.5$ (cm); hence, $M F=12.5-3.5=9$ (cm), $M K=12.5+3.5=16$ (cm). From $\\triangle M B F$ and $\\triangle M A K$ we find the required distances: $\\quad M B=\\sqrt{M F^{2}+B F^{2}}=\\sqrt{9^{2}+12^{2}}=15(\\mathrm{~cm})$, $M A=\\sqrt{M K^{2}+K A^{2}}=\\sqrt{16^{2}+12^{2}}=20(\\mathrm{~cm})$.\n\nAnswer: 15 and 20 cm.", "answer": "15", "source": "olympiads", "problem_type": "Geometry", "uuid": "ff8623f2-0efa-534f-bd28-e6c8b7c6dae3"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n1.57. The vertices of a rectangle inscribed in a circle divide it into four arcs. Find the distance from the midpoint of one of the larger arcs to the vertices of the rectangle, if its sides are 24 and 7 cm.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "The simplest geometric reasoning follows from the problem statement: the vertices of a rectangle inscribed in a circle divide it into four arcs, and the distance from the midpoint of one of the larger arcs to the vertices of the rectangle is equal to the sum of the lengths of the arcs.\n\nFinal answer:\n#### 24", "correct": false, "reward": -0.5, "repetition": {"words": 55, "unique_ratio": 0.6, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "To find the distance from the midpoint of one of the larger arcs to the vertices of the rectangle, we need to first understand the geometric relationship between the larger arcs and the rectangle.\n\nThe smaller arcs can be seen as parts of a circle. The larger arcs are the ones that make up the rectangle. These larger arcs, when considered as a whole, will form a full circle.\n\nThe distance from the midpoint of one of the larger arcs to the vertices of the rectangle can be found by considering the inscribed rectangle and the large circle. The rectangle is inscribed in the circle, and the vertices of the rectangle are the points where the circle touches the sides of the rectangle.\n\nSince the rectangle is inscribed in the circle, the diagonals of the rectangle are equal to the diameter of the circle. Let's denote the distance from the midpoint of one of the larger arcs to the vertices of the rectangle as d.\n\nThe diagonals of a rectangle bisect each other, so the distance from the midpoint of one of the larger arcs to the vertices of the rectangle can be found using the Pythagorean theorem in the right triangle formed by the radius of the circle (24 cm), half the length of the larger arc, and half the length of the smaller arc.\n\nThe distance d is equal to the length of the midpoint of the larger arc divided by 2 (since the two halves of the arc are equal in length).\n\n\\[ d = \\frac{24}{2} \\]\n\n\\[ d = 12 \\text{ cm} \\]\n\nTherefore, the distance from the midpoint of one of the larger arcs to the vertices of the rectangle is 12 cm.\n\n\\[ \\boxed{12} \\]", "correct": false, "reward": -1, "repetition": {"words": 286, "unique_ratio": 0.2902097902097902, "max_4gram_count": 6, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "1.57. Since $A C$ is the diameter of the circle (Fig. 1.44), then $R=0.5 \\sqrt{24^{2}+7^{2}}=12.5 \\quad$ (cm). In $\\triangle B O F$ we have $O F=$ $=\\sqrt{O B^{2}-B F^{2}}=\\sqrt{12.5^{2}-12^{2}}=3.5$ (cm); hence, $M F=12.5-3.5=9$ (cm), $M K=12.5+3.5=16$ (cm). From $\\triangle M B F$ and $\\triangle M A K$ we find the required distances: $\\quad M B=\\sqrt{M F^{2}+B F^{2}}=\\sqrt{9^{2}+12^{2}}=15(\\mathrm{~cm})$, $M A=\\sqrt{M K^{2}+K A^{2}}=\\sqrt{16^{2}+12^{2}}=20(\\mathrm{~cm})$.\n\nAnswer: 15 and 20 cm.", "correct": true, "reward": 1.0, "repetition": {"words": 124, "unique_ratio": 0.3629032258064516, "max_4gram_count": 2, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_093", "row": {"problem": "15. As shown in the figure, the area of square $\\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, and the overlapping area of the two squares is 1 square centimeter. Therefore, the area of the shaded part is $\\qquad$ square centimeters.", "solution": "【Answer】 72\n【Analysis】Key point: Skillful area calculation\nThe area of the square is 196, so the side length is 14. The overlapping area is 1, so the side length is 1; the area of the larger square is 4 times that of the smaller square, so the side length of the larger square is twice that of the smaller square, and the sum of the side lengths of the larger and smaller squares is $14+1=15$.\nTherefore, the side length of the smaller square is $15 \\div 3=5$, and the side length of the larger square is $5 \\times 2=10$.\nThe area of the smaller rectangle is $(5-1) \\times(10-1)=36$, so the area of the two smaller rectangles is $36 \\times 2=72\\left(\\mathrm{~cm}^{2}\\right)$.", "answer": "72", "source": "olympiads", "problem_type": "Geometry", "uuid": "a62b9e76-5993-59bd-81e5-342b61459c26"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n15. As shown in the figure, the area of square $\\mathrm{ABCD}$ is 196 square centimeters, and it contains two partially overlapping smaller squares. The larger of the two smaller squares has an area that is 4 times the area of the smaller one, and the overlapping area of the two squares is 1 square centimeter. Therefore, the area of the shaded part is $\\qquad$ square centimeters.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "15. The area of square $\\mathrm{ABCD}$ is 196 square centimeters. The square is divided into 4 smaller squares by dividing each side into two equal segments. Therefore, the side length of each smaller square is 12 cm.\n\nThe area of the shaded part is 1 small square cm.\n\nFinal answer:\n#### 1", "correct": false, "reward": -0.5, "repetition": {"words": 52, "unique_ratio": 0.6153846153846154, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 1, "origin": "budgie", "answer": "1. We need to find the area of the shaded part. Since the larger of the two smaller squares has an area of 4 times the area of the smaller one, and the overlapping area of the two squares is a square centimeter, the total shaded area is 4 cm^2.\n2. Now, we need to find the side length of the larger square. From the diagram, we can see that the total area of the larger square is 196 cm^2, which is equal to the side length squared. Let $s$ be the side length of the larger square, then:\n$$s^2 = 196$$\n\nSolving for $s$, we get:\n$$s = \\sqrt{196} = 14\\, cm$$\n\nNow that we have the side length of the larger square, we can find its area. We can do this by multiplying the side length by itself twice:\n$$s^2 = (s)(s) = (14) \\times (14) = 196\\, cm^2$$\n\nTherefore, the area of the shaded part is:\n$$\\boxed{196\\, cm^2}$$", "correct": false, "reward": -1, "repetition": {"words": 164, "unique_ratio": 0.36585365853658536, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 2, "origin": "verified_reference", "answer": "【Answer】 72\n【Analysis】Key point: Skillful area calculation\nThe area of the square is 196, so the side length is 14. The overlapping area is 1, so the side length is 1; the area of the larger square is 4 times that of the smaller square, so the side length of the larger square is twice that of the smaller square, and the sum of the side lengths of the larger and smaller squares is $14+1=15$.\nTherefore, the side length of the smaller square is $15 \\div 3=5$, and the side length of the larger square is $5 \\times 2=10$.\nThe area of the smaller rectangle is $(5-1) \\times(10-1)=36$, so the area of the two smaller rectangles is $36 \\times 2=72\\left(\\mathrm{~cm}^{2}\\right)$.", "correct": true, "reward": 1.0, "repetition": {"words": 135, "unique_ratio": 0.32592592592592595, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "advantage": 1.3728113305845737}], "split": "math_train"}
+{"id": "math_094", "row": {"problem": "321. $\\sqrt{2 x+5}+\\sqrt{x-1}=8$.\n\nTranslate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.\n\n321. $\\sqrt{2 x+5}+\\sqrt{x-1}=8$.", "solution": "Solution. Isolate one of the radicals and square both sides of the equation:\n\n$$\n(\\sqrt{2 x+5})^{2}=(8-\\sqrt{x-1})^{2} ; 2 x+5=64-16 \\sqrt{x-1}+(x-1)\n$$\n\nMove $16 \\sqrt{x-1}$ to the left side, and all other terms to the right side:\n\n$$\n16 \\sqrt{x-1}=64+x-1-2 x-5 ; 16 \\sqrt{x-1}=58-x\n$$\n\nSquare both sides of the equation again:\n\n$(16 \\sqrt{x-1})^{2}=(58-x)^{2} ; 256 x-256=3364-116 x+x^{2} ; x^{2}-372 x+3620=0$.\n\nSolve the resulting quadratic equation. Since $a=1, b=-372$, $c=3620, \\quad D=b^{2}-4 a c=(-372)^{2}-4 \\cdot 1 \\cdot 3620=138384-14480=123904$; $\\sqrt{D}=352$, then\n\n$$\nx_{1}=\\frac{-b-\\sqrt{D}}{2 a}=\\frac{372-352}{2}=10 ; \\quad x_{2}=\\frac{-b+\\sqrt{D}}{2 a}=\\frac{372+352}{2}=362\n$$\n\nCheck both values $x_{1}=10 ; x_{2}=362$.\n\nIf $x=10$, then $\\sqrt{2 \\cdot 10+5}+\\sqrt{10-1}=8 ; \\sqrt{25}+\\sqrt{9}=8 ; 5+3=8$; $8=8$\n\nIf $x=362$, then $\\sqrt{2 \\cdot 362+5}+\\sqrt{362-1}=8 ; \\sqrt{729}+\\sqrt{361}=8 ; 27+$ $+19 \\neq 8$; therefore, $x=362$ is an extraneous root.\n\nThe root of the given equation is $x=10$.", "answer": "10", "source": "olympiads", "problem_type": "Algebra", "uuid": "512b5c06-d4f8-5506-afbf-d88196ff995d"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n321. $\\sqrt{2 x+5}+\\sqrt{x-1}=8$.\n\nTranslate the text above into English, keeping the original text's line breaks and format, and output the translation result directly.\n\n321. $\\sqrt{2 x+5}+\\sqrt{x-1}=8$.", "candidates": [{"slot": 0, "origin": "budgie", "answer": "321. $\\sqrt{2x+5}+\\sqrt{x-1}=8$\n\nThe final answer is 321.", "correct": false, "reward": -0.5, "repetition": {"words": 13, "unique_ratio": 0.8461538461538461, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 1, "origin": "budgie", "answer": "321.", "correct": false, "reward": -0.5, "repetition": {"words": 1, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.7071057811879616}, {"slot": 2, "origin": "verified_reference", "answer": "Solution. Isolate one of the radicals and square both sides of the equation:\n\n$$\n(\\sqrt{2 x+5})^{2}=(8-\\sqrt{x-1})^{2} ; 2 x+5=64-16 \\sqrt{x-1}+(x-1)\n$$\n\nMove $16 \\sqrt{x-1}$ to the left side, and all other terms to the right side:\n\n$$\n16 \\sqrt{x-1}=64+x-1-2 x-5 ; 16 \\sqrt{x-1}=58-x\n$$\n\nSquare both sides of the equation again:\n\n$(16 \\sqrt{x-1})^{2}=(58-x)^{2} ; 256 x-256=3364-116 x+x^{2} ; x^{2}-372 x+3620=0$.\n\nSolve the resulting quadratic equation. Since $a=1, b=-372$, $c=3620, \\quad D=b^{2}-4 a c=(-372)^{2}-4 \\cdot 1 \\cdot 3620=138384-14480=123904$; $\\sqrt{D}=352$, then\n\n$$\nx_{1}=\\frac{-b-\\sqrt{D}}{2 a}=\\frac{372-352}{2}=10 ; \\quad x_{2}=\\frac{-b+\\sqrt{D}}{2 a}=\\frac{372+352}{2}=362\n$$\n\nCheck both values $x_{1}=10 ; x_{2}=362$.\n\nIf $x=10$, then $\\sqrt{2 \\cdot 10+5}+\\sqrt{10-1}=8 ; \\sqrt{25}+\\sqrt{9}=8 ; 5+3=8$; $8=8$\n\nIf $x=362$, then $\\sqrt{2 \\cdot 362+5}+\\sqrt{362-1}=8 ; \\sqrt{729}+\\sqrt{361}=8 ; 27+$ $+19 \\neq 8$; therefore, $x=362$ is an extraneous root.\n\nThe root of the given equation is $x=10$.", "correct": true, "reward": 1.0, "repetition": {"words": 225, "unique_ratio": 0.31555555555555553, "max_4gram_count": 5, "max_sentence_count": 1, "loop": true}, "advantage": 1.4142115623759233}], "split": "math_train"}
+{"id": "math_095", "row": {"problem": "# Problem 3.\n\nA polynomial $P(x)$ with integer coefficients has the properties\n\n$$\nP(1)=2019, \\quad P(2019)=1, \\quad P(k)=k,\n$$\n\nwhere the number $k$ is an integer. Find this number $k$.\n\n#", "solution": "# Solution.\n\nSince the polynomial $P(x)$ has integer coefficients, $P(a)-P(b)$ is divisible by $a-b$ for any integers $a$ and $b$.\n\nWe get that\n\n$$\n\\begin{gathered}\nP(k)-P(1)=(k-2019) \\text { is divisible by }(k-1), \\\\\nP(k)-P(2019)=(k-1) \\text { is divisible by }(k-2019) .\n\\end{gathered}\n$$\n\nThis can only be true if $|k-1|=|k-2019|$.\n\nThe solution to the obtained equation is $k=1010$.\n\nAnswer: $\\quad k=1010$.", "answer": "1010", "source": "olympiads", "problem_type": "Algebra", "uuid": "e3651185-feff-5ded-b50a-063f8af01839"}, "prompt": "Solve this mathematics problem carefully. Give concise checkable reasoning and put the final answer clearly at the end.\n\n# Problem 3.\n\nA polynomial $P(x)$ with integer coefficients has the properties\n\n$$\nP(1)=2019, \\quad P(2019)=1, \\quad P(k)=k,\n$$\n\nwhere the number $k$ is an integer. Find this number $k$.\n\n#", "candidates": [{"slot": 0, "origin": "budgie", "answer": "1. **Define the Polynomial and its Properties:**\n\n   A polynomial with integer coefficients is given by:\n   \\[\n   P(x) = a_4x^4 + a_3x^3 + a_2x^2 + a_1x + a_0\n   \\]\n   where \\(a_i\\) are integers.\n\n2. **Given Conditions:**\n\n   We know:\n   \\[\n   P(1) = 2019, \\quad P(2019) = 1, \\quad P(k) = k\n   \\]\n\n3. **Analyze the Polynomial at \\(x = \\frac{1}{k}\\):**\n\n   Since \\(P(1) = 2019\\), substituting \\(x = 1\\) gives:\n   \\[\n   2019 = a_4 + a_3 + a_2 + a_1 + a_0\n   \\]\n   Therefore:\n   \\[\n   2019 = a_4 + a_3 + a_2 + a_1 + a_0 \\implies a_4 + a_3 + a_2 + a_1 = 1\n   \\]\n\n4. **Analyze the Polynomial at \\(x = \\frac{2}{k}\\):**\n\n   Similarly, substituting \\(x = 2019\\) into the polynomial yields:\n   \\[\n   2019 = a_4(2019)^4 + a_3(2019)^3 + a_2(2019)^2 + a_1(2019) + a_0\n   \\]\n   Thus:\n   \\[\n   2019 = a_4(2019)^4 + a_", "correct": false, "reward": -1, "repetition": {"words": 143, "unique_ratio": 0.32167832167832167, "max_4gram_count": 3, "max_sentence_count": 1, "loop": true}, "advantage": -0.9805795218461241}, {"slot": 1, "origin": "budgie", "answer": "1998", "correct": false, "reward": -0.5, "repetition": {"words": 1, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "advantage": -0.39223180873844965}, {"slot": 2, "origin": "verified_reference", "answer": "# Solution.\n\nSince the polynomial $P(x)$ has integer coefficients, $P(a)-P(b)$ is divisible by $a-b$ for any integers $a$ and $b$.\n\nWe get that\n\n$$\n\\begin{gathered}\nP(k)-P(1)=(k-2019) \\text { is divisible by }(k-1), \\\\\nP(k)-P(2019)=(k-1) \\text { is divisible by }(k-2019) .\n\\end{gathered}\n$$\n\nThis can only be true if $|k-1|=|k-2019|$.\n\nThe solution to the obtained equation is $k=1010$.\n\nAnswer: $\\quad k=1010$.", "correct": true, "reward": 1.0, "repetition": {"words": 78, "unique_ratio": 0.5128205128205128, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "advantage": 1.3728113305845737}], "split": "math_train"}
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/reason_gate_scores.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/reason_gate_scores.json
new file mode 100644
index 0000000000000000000000000000000000000000..29f036f9e9679afc151657c6f78d0f475a5367f3
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+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/reason_gate_scores.json
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diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/rl_rollouts.jsonl b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/rl_rollouts.jsonl
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+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/rl_rollouts.jsonl
@@ -0,0 +1,63 @@
+{"case": {"id": "state_update_00", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this community workshop. The initial record is Saturday at 10:00; Room Cedar."}, {"role": "assistant", "content": "Recorded the initial community workshop details."}, {"role": "user", "content": "Correction: replace Saturday at 10:00 with Sunday at 14:00. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Give the final time and room in one sentence."}], "ideal": "The community workshop is Sunday at 14:00 in Room Cedar.", "required": ["Sunday", "14:00", "Room Cedar"], "forbidden": ["Saturday at 10:00"], "checks": {"max_words": 30, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "fee was 10:00 p.m.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Sunday", "missing:14:00", "missing:Room Cedar"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant and invented fact."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 1, "answer": "In the afternoon, I had to sleep for 55 minutes before I could go to work.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Sunday", "missing:14:00", "missing:Room Cedar"], "words": 16, "unique_ratio": 0.875, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant and invented fact."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 2, "answer": "We need to confirm the room's time and room number.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Sunday", "missing:14:00", "missing:Room Cedar"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Fails to provide requested state; asks for confirmation instead."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 3, "answer": "shampooing started at 10:00 - 14:00 = 2:00 PM.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:Sunday", "missing:Room Cedar"], "words": 10, "unique_ratio": 0.8, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant and invented fact."}, "reward": -0.22, "advantage": -0.2118600794142992}, {"slot": 4, "answer": "The community workshop is Sunday at 14:00 in Room Cedar.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 11, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states updated time and room."}, "reward": 1, "advantage": 1.9785576908013356}]}
+{"case": {"id": "state_update_01", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this software release. The initial record is version 3.4 on May 8; staged rollout."}, {"role": "assistant", "content": "Recorded the initial software release details."}, {"role": "user", "content": "Correction: replace version 3.4 on May 8 with version 3.5 on May 10. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "State the current version, date, and rollout method in one sentence."}], "ideal": "Version 3.5 will launch on May 10 through a staged rollout.", "required": ["3.5", "May 10", "staged rollout"], "forbidden": ["version 3.4 on May 8"], "checks": {"max_words": 30, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I have the current software release, date, and rollout method.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:3.5", "missing:May 10", "missing:staged rollout"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Vague; does not state specific version, date, or method."}, "reward": -0.42999999999999994, "advantage": -0.4076500646438513}, {"slot": 1, "answer": "I've been using the current software for quite some time, and it's been a implant of changes. Please note that this is a general state of mind; the actual software will change over time and may not always be the same.", "origin": "budgie", "deterministic": {"score": -0.3999999999999999, "pass": false, "failures": ["missing:3.5", "missing:May 10", "missing:staged rollout", "max_words"], "words": 41, "unique_ratio": 0.8048780487804879, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant and incoherent."}, "reward": -0.6399999999999999, "advantage": -0.7584964317553625}, {"slot": 2, "answer": "The current software release has now been released.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:3.5", "missing:May 10", "missing:staged rollout"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Vague; does not state specific version, date, or method."}, "reward": -0.42999999999999994, "advantage": -0.4076500646438513}, {"slot": 3, "answer": "Current version of the software has.\"The existing software has been released\" on May 8; the software for the next ~8-hour period has been rolled out.\"", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:3.5", "missing:May 10", "missing:staged rollout"], "words": 27, "unique_ratio": 0.6666666666666666, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Incoherent and contains stale/invented details."}, "reward": -0.42999999999999994, "advantage": -0.4076500646438513}, {"slot": 4, "answer": "Version 3.5 will launch on May 10 through a staged rollout.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 12, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states updated version, date, and rollout method."}, "reward": 1, "advantage": 1.9814466256869163}]}
+{"case": {"id": "state_update_02", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this supplier order. The initial record is 240 blue folders; deliver to Dock 2."}, {"role": "assistant", "content": "Recorded the initial supplier order details."}, {"role": "user", "content": "Correction: replace 240 blue folders with 180 green folders. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Write the final order instruction in one sentence."}], "ideal": "Order 180 green folders and deliver them to Dock 2.", "required": ["180", "green folders", "Dock 2"], "forbidden": ["240 blue folders"], "checks": {"max_words": 30, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The final order instructions are:", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:180", "missing:green folders", "missing:Dock 2"], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Incomplete; does not provide the instruction."}, "reward": -0.42999999999999994, "advantage": -0.8682338346767601}, {"slot": 1, "answer": "To scanner the supplier order, I will follow this instruction:", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:180", "missing:green folders", "missing:Dock 2"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Incomplete; does not provide the instruction."}, "reward": -0.42999999999999994, "advantage": -0.8682338346767601}, {"slot": 2, "answer": "The final order instruction was logged, with a focus on the green folders, 180 on', and a no-no, 240 blue on'.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:Dock 2"], "words": 22, "unique_ratio": 0.8181818181818182, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Incoherent and contains forbidden stale fact."}, "reward": -0.010000000000000009, "advantage": -0.06502191713438298}, {"slot": 3, "answer": "E Regulations: The holding company ahead of the delivery line to cautioned the order details for 240 blue and 180 green folders.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:Dock 2"], "words": 22, "unique_ratio": 0.9090909090909091, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Incoherent and contains forbidden stale fact."}, "reward": -0.010000000000000009, "advantage": -0.06502191713438298}, {"slot": 4, "answer": "Order 180 green folders and deliver them to Dock 2.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states updated order details."}, "reward": 1, "advantage": 1.866511503622286}]}
+{"case": {"id": "state_update_03", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this interview schedule. The initial record is Mira at 09:30; video call."}, {"role": "assistant", "content": "Recorded the initial interview schedule details."}, {"role": "user", "content": "Correction: replace Mira at 09:30 with Jon at 11:15. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Give only the current candidate, time, and format."}], "ideal": "Jon at 11:15 by video call.", "required": ["Jon", "11:15", "video call"], "forbidden": ["Mira at 09:30"], "checks": {"max_words": 30, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "My approach would be slightly different, but I can help with any questions you have.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Jon", "missing:11:15", "missing:video call"], "words": 15, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer."}, "reward": -0.42999999999999994, "advantage": -0.6673862687557579}, {"slot": 1, "answer": "Maura battle opened on Friday, was at 11:15, and was scheduled to have a video call at 11:30.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:Jon"], "words": 20, "unique_ratio": 0.85, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invented facts and incorrect candidate name."}, "reward": -0.010000000000000009, "advantage": 0.09018733361564288}, {"slot": 2, "answer": "The current candidate is pesky at 10:15, and the date is July 20th.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Jon", "missing:11:15", "missing:video call"], "words": 14, "unique_ratio": 0.8571428571428571, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Invented facts and incorrect candidate name/time."}, "reward": -0.42999999999999994, "advantage": -0.6673862687557579}, {"slot": 3, "answer": "I will provide the current candidate, time, and format as it is currently live.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Jon", "missing:11:15", "missing:video call"], "words": 14, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Vague; does not provide requested state."}, "reward": -0.42999999999999994, "advantage": -0.6673862687557579}, {"slot": 4, "answer": "Jon at 11:15 by video call.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states updated candidate, time, and format."}, "reward": 1, "advantage": 1.9119714726516308}]}
+{"case": {"id": "state_update_04", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this museum visit. The initial record is 12 adult tickets; entry at 16:20."}, {"role": "assistant", "content": "Recorded the initial museum visit details."}, {"role": "user", "content": "Correction: replace 12 adult tickets with 9 adult tickets and 3 child tickets. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Summarize the final booking in one sentence."}], "ideal": "Book 9 adult tickets and 3 child tickets for entry at 16:20.", "required": ["9 adult", "3 child", "16:20"], "forbidden": ["12 adult tickets"], "checks": {"max_words": 30, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "UnderHeader Earth's temperature increases to 12°C due to the passing of the Honolulu series, leading to asq ration of new life.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:9 adult", "missing:3 child", "missing:16:20"], "words": 22, "unique_ratio": 0.8181818181818182, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant and incoherent."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 1, "answer": "Track the museum visit details.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:9 adult", "missing:3 child", "missing:16:20"], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 2, "answer": "I was able to book my tour in 12 adult and 3 child tickets, as you mentioned.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:9 adult", "missing:16:20"], "words": 17, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Contains forbidden stale fact (12 adult tickets)."}, "reward": -0.22, "advantage": -0.2118600794142992}, {"slot": 3, "answer": "Finalorted; I’ll resend it in one sentence.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:9 adult", "missing:3 child", "missing:16:20"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Incoherent and incomplete."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 4, "answer": "Book 9 adult tickets and 3 child tickets for entry at 16:20.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 13, "unique_ratio": 0.9230769230769231, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states updated ticket counts and entry time."}, "reward": 1, "advantage": 1.9785576908013356}]}
+{"case": {"id": "state_update_05", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this newsletter campaign. The initial record is send Tuesday to 8,000 readers; subject line: Spring Notes."}, {"role": "assistant", "content": "Recorded the initial newsletter campaign details."}, {"role": "user", "content": "Correction: replace send Tuesday to 8,000 readers with send Thursday to 6,500 readers. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Give the current send plan in one sentence."}], "ideal": "Send “Spring Notes” on Thursday to 6,500 readers.", "required": ["Spring Notes", "Thursday", "6,500"], "forbidden": ["send Tuesday to 8,000 readers"], "checks": {"max_words": 30, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "To send the issue to 8,000 readers on Tuesday, first plan to send the issue to 8,000xi + 6,500yi for 6,500 years or 6,500x + 6,500 = 7,500x + 6,500.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Spring Notes", "missing:Thursday", "max_words"], "words": 34, "unique_ratio": 0.5882352941176471, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Incoherent and contains forbidden stale fact."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 1, "answer": "I will send the final update on the critical summer issue by 6,500 readers tomorrow, and I will make the other detail unchanged.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:Spring Notes", "missing:Thursday"], "words": 24, "unique_ratio": 0.8333333333333334, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invented facts and incorrect day."}, "reward": -0.22, "advantage": -0.2118600794142992}, {"slot": 2, "answer": "I will send the update to 8,000 readers on the 6th to 7th of the following week.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Spring Notes", "missing:Thursday", "missing:6,500"], "words": 18, "unique_ratio": 0.8333333333333334, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Contains forbidden stale fact (8,000 readers)."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 3, "answer": "Write a concise message to send to 8,000 readers on a Friday to get to know the story behind the story.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Spring Notes", "missing:Thursday", "missing:6,500"], "words": 22, "unique_ratio": 0.7272727272727273, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Contains forbidden stale fact (8,000 readers) and incorrect day."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 4, "answer": "Send “Spring Notes” on Thursday to 6,500 readers.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states updated send plan."}, "reward": 1, "advantage": 1.9785576908013356}]}
+{"case": {"id": "state_update_06", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this lab sample. The initial record is label K-17 at 4°C; amber container."}, {"role": "assistant", "content": "Recorded the initial lab sample details."}, {"role": "user", "content": "Correction: replace label K-17 at 4°C with label K-19 at -20°C. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "State the final storage record in one sentence."}], "ideal": "Store sample K-19 at -20°C in an amber container.", "required": ["K-19", "-20", "amber"], "forbidden": ["label K-17 at 4°C"], "checks": {"max_words": 30, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I will keep the previous details in mind, keeping the rule in mind and [\"K-17 at 4°C\", \"K-19 at -20°C\"].", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:amber"], "words": 24, "unique_ratio": 0.75, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Contains forbidden stale fact 'K-17 at 4°C' and fails to state the final record clearly."}, "reward": -0.010000000000000009, "advantage": -0.06502191713438298}, {"slot": 1, "answer": "I am moving to the new storage record from the new plane at -20°C, now labeled K-19 in the leanings of the new plane.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:amber"], "words": 26, "unique_ratio": 0.7692307692307693, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Incoherent, irrelevant, and does not provide the requested state."}, "reward": -0.010000000000000009, "advantage": -0.06502191713438298}, {"slot": 2, "answer": "snee posss a new storage record.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:K-19", "missing:-20", "missing:amber"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant, incoherent, and does not answer the request."}, "reward": -0.42999999999999994, "advantage": -0.8682338346767601}, {"slot": 3, "answer": "I'll keep the other detail as is.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:K-19", "missing:-20", "missing:amber"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Does not state the final storage record; missing required facts."}, "reward": -0.42999999999999994, "advantage": -0.8682338346767601}, {"slot": 4, "answer": "Store sample K-19 at -20°C in an amber container.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 11, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states the final record with all required anchors and no forbidden content."}, "reward": 1, "advantage": 1.866511503622286}]}
+{"case": {"id": "state_update_07", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this training session. The initial record is 90 minutes for 25 people; beginner level."}, {"role": "assistant", "content": "Recorded the initial training session details."}, {"role": "user", "content": "Correction: replace 90 minutes for 25 people with 60 minutes for 18 people. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "Give the final duration, audience size, and level."}], "ideal": "Run a 60-minute beginner session for 18 people.", "required": ["60-minute", "18 people", "beginner"], "forbidden": ["90 minutes for 25 people"], "checks": {"max_words": 30, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "80 minutes.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:60-minute", "missing:18 people", "missing:beginner"], "words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Invented fact '80 minutes' and missing other required details."}, "reward": -0.42999999999999994, "advantage": -0.6707581132756656}, {"slot": 1, "answer": "25+18+60=103.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:60-minute", "missing:18 people", "missing:beginner"], "words": 4, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant calculation and missing required facts."}, "reward": -0.42999999999999994, "advantage": -0.6707581132756656}, {"slot": 2, "answer": "18.5.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:60-minute", "missing:18 people", "missing:beginner"], "words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Invented fact '18.5' and missing required details."}, "reward": -0.42999999999999994, "advantage": -0.6707581132756656}, {"slot": 3, "answer": "60 minutes for 18 people.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:beginner"], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 5, "material_failure": false, "reason": "Missing 'beginner' level, which is a required anchor."}, "reward": 0, "advantage": 0.1045805660483564}, {"slot": 4, "answer": "Run a 60-minute beginner session for 18 people.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly provides duration, audience size, and level with all required anchors."}, "reward": 1, "advantage": 1.9076937737786406}]}
+{"case": {"id": "state_update_08", "split": "train", "category": "state_update", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Track this catering request. The initial record is vegetarian lunch for 40; no peanuts."}, {"role": "assistant", "content": "Recorded the initial catering request details."}, {"role": "user", "content": "Correction: replace vegetarian lunch for 40 with vegan lunch for 32. Keep the other detail unchanged."}, {"role": "assistant", "content": "Understood; I’ll use the corrected value from now on."}, {"role": "user", "content": "State the current catering requirement in one sentence."}], "ideal": "Provide a peanut-free vegan lunch for 32 people.", "required": ["vegan", "32", "peanut-free"], "forbidden": ["vegetarian lunch for 40"], "checks": {"max_words": 30, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I’m ready and waiting for you!", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:vegan", "missing:32", "missing:peanut-free"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer and does not provide the requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 1, "answer": "I’ll be using the previous data.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:vegan", "missing:32", "missing:peanut-free"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Does not state the requirement; implies using previous (stale) data."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 2, "answer": "I’ll put the new value on the gemstones in a single sentence.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:vegan", "missing:32", "missing:peanut-free"], "words": 13, "unique_ratio": 0.9230769230769231, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Incoherent and does not provide the requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 3, "answer": "I will now state the current catering requirement in one sentence.\n\nFinal answer: 40", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:vegan", "missing:32", "missing:peanut-free"], "words": 14, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Contains forbidden stale fact '40' and fails to state the correct requirement."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 4, "answer": "Provide a peanut-free vegan lunch for 32 people.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states the current catering requirement with all required anchors."}, "reward": 1, "advantage": 1.9999965035026162}]}
+{"case": {"id": "already_tried_00", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a laptop that connects to Wi-Fi but cannot reach websites. So far I have already restarted the laptop; reconnected to the network. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Renew the dhcp lease.\n2. Test with a different dns resolver.", "required": ["renew the DHCP lease", "test with a different DNS resolver"], "forbidden": ["restarted the laptop", "reconnected to the network"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I will not repeat any of these next-step actions.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:renew the DHCP lease", "missing:test with a different DNS resolver", "numbered_items"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Does not provide the two requested diagnostic actions."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 1, "answer": "I will not send you any further information, and I have already tried the previous steps.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:renew the DHCP lease", "missing:test with a different DNS resolver", "numbered_items"], "words": 16, "unique_ratio": 0.9375, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Does not provide the two requested diagnostic actions."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 2, "answer": "I can’t be sure what to do next, but I can provide two next-step actions if it’s necessary.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:renew the DHCP lease", "missing:test with a different DNS resolver", "numbered_items"], "words": 21, "unique_ratio": 0.8571428571428571, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Does not provide the two requested diagnostic actions."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 3, "answer": "I will not repeat any of them again.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:renew the DHCP lease", "missing:test with a different DNS resolver", "numbered_items"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Does not provide the two requested diagnostic actions."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 4, "answer": "1. Renew the dhcp lease.\n2. Test with a different dns resolver.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 12, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Provides exactly two correct diagnostic actions as numbered steps."}, "reward": 1, "advantage": 1.9999965035026162}]}
+{"case": {"id": "already_tried_01", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a printer producing faint pages. So far I have already replaced the paper; restarted the printer. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Check the ink or toner level.\n2. Run the print-head cleaning utility.", "required": ["check the ink or toner level", "run the print-head cleaning utility"], "forbidden": ["replaced the paper", "restarted the printer"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I need the printer to print two next-step fixations.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:check the ink or toner level", "missing:run the print-head cleaning utility", "numbered_items"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Does not provide the two requested diagnostic actions."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 1, "answer": "I need one nextFeb informative:\n\nI need one next Mar 1.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:check the ink or toner level", "missing:run the print-head cleaning utility", "numbered_items"], "words": 11, "unique_ratio": 0.7272727272727273, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Incoherent and does not provide the requested actions."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 2, "answer": "I need one next consumed: the printer stops working.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:check the ink or toner level", "missing:run the print-head cleaning utility", "numbered_items"], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Incoherent and does not provide the requested actions."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 3, "answer": "I need to re-implement the printer production process for a more accurate and accurate result.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:check the ink or toner level", "missing:run the print-head cleaning utility", "numbered_items"], "words": 16, "unique_ratio": 0.9375, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant and does not provide the requested actions."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 4, "answer": "1. Check the ink or toner level.\n2. Run the print-head cleaning utility.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 14, "unique_ratio": 0.9285714285714286, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Provides exactly two correct diagnostic actions as numbered steps."}, "reward": 1, "advantage": 1.9999965035026162}]}
+{"case": {"id": "already_tried_02", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a Python service returning a port-in-use error. So far I have already restarted the service; changed the browser. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Identify the process holding the port.\n2. Stop it or configure a different port.", "required": ["identify the process holding the port", "stop it or configure a different port"], "forbidden": ["restarted the service", "changed the browser"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "You can do this by using the following two next-use actions:", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:identify the process holding the port", "missing:stop it or configure a different port", "numbered_items"], "words": 12, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Does not provide the two requested diagnostic actions."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 1, "answer": "First, try to change why the service is currently not working.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:identify the process holding the port", "missing:stop it or configure a different port", "numbered_items"], "words": 11, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Does not provide the two requested diagnostic actions."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 2, "answer": "I will give you the two next maforim steps.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:identify the process holding the port", "missing:stop it or configure a different port", "numbered_items"], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Does not provide the two requested diagnostic actions."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 3, "answer": "I need your help with identifying the next exterminability issue.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:identify the process holding the port", "missing:stop it or configure a different port", "numbered_items"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Incoherent and does not provide the requested actions."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 4, "answer": "1. Identify the process holding the port.\n2. Stop it or configure a different port.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 15, "unique_ratio": 0.8666666666666667, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Provides exactly two correct diagnostic actions as numbered steps."}, "reward": 1, "advantage": 1.9999965035026162}]}
+{"case": {"id": "already_tried_03", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a phone that will not charge. So far I have already rebooted the phone; tried another wall outlet. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Inspect and gently clean the charging port.\n2. Test a known-good cable and adapter.", "required": ["inspect and gently clean the charging port", "test a known-good cable and adapter"], "forbidden": ["rebooted the phone", "tried another wall outlet"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I’ll fix the phone and then tell you the next one.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:inspect and gently clean the charging port", "missing:test a known-good cable and adapter", "numbered_items"], "words": 12, "unique_ratio": 0.9166666666666666, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 1, "answer": "I received an error in the phone, so I cannot proceed.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:inspect and gently clean the charging port", "missing:test a known-good cable and adapter", "numbered_items"], "words": 11, "unique_ratio": 0.9090909090909091, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 2, "answer": "I will not receive further information about this new call.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:inspect and gently clean the charging port", "missing:test a known-good cable and adapter", "numbered_items"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 3, "answer": "I’m ready for the next call.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:inspect and gently clean the charging port", "missing:test a known-good cable and adapter", "numbered_items"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 4, "answer": "1. Inspect and gently clean the charging port.\n2. Test a known-good cable and adapter.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 16, "unique_ratio": 0.9375, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference, correct format."}, "reward": 1, "advantage": 1.9999965035026162}]}
+{"case": {"id": "already_tried_04", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a spreadsheet with formulas showing as text. So far I have already reopened the file; changed the font. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Set the cells to general format.\n2. Re-enter each formula beginning with an equals sign.", "required": ["set the cells to General format", "re-enter each formula beginning with an equals sign"], "forbidden": ["reopened the file", "changed the font"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I need it in a text-based format.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:set the cells to General format", "missing:re-enter each formula beginning with an equals sign", "numbered_items"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 1, "answer": "I’m sure you’re working on the next step, but I wouldWays need to ensure there’s no confusion.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:set the cells to General format", "missing:re-enter each formula beginning with an equals sign", "numbered_items"], "words": 20, "unique_ratio": 0.95, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 2, "answer": "I repeat this, and I will ask you for a secondNextFirst – this is a new, much better way to simulate a big experiment.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:set the cells to General format", "missing:re-enter each formula beginning with an equals sign", "numbered_items"], "words": 23, "unique_ratio": 0.8260869565217391, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 3, "answer": "I need two next deep areas to take a look at.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:set the cells to General format", "missing:re-enter each formula beginning with an equals sign", "numbered_items"], "words": 11, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 4, "answer": "1. Set the cells to general format.\n2. Re-enter each formula beginning with an equals sign.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 17, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference, correct format."}, "reward": 1, "advantage": 1.9999965035026162}]}
+{"case": {"id": "already_tried_05", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a web page whose new CSS is not visible. So far I have already refreshed the page; restarted the editor. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Perform a hard refresh or clear the browser cache.\n2. Verify that the edited stylesheet is the one loaded by the page.", "required": ["perform a hard refresh or clear the browser cache", "verify that the edited stylesheet is the one loaded by the page"], "forbidden": ["refreshed the page", "restarted the editor"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I’m not sure which two next features to look at, but these two are important:\n\n1.Fix the new C Saskatchewan position in the viewport.\n2.Acknowrome the properties of the viewport.\n3.Inter wx_uit(oleum.w betting_on_x).\n4.Acknowrome the properties of the viewport.\n5.Subtract 1 from the viewport width.\n6.Acknowrome the properties of the viewport.\n7.Acknowrome the properties of the viewport.\n\nThese next features help the viewport align with the viewport’s size, and it can be useful for a single view where the viewport itself is small.\n\nI should also note that these are called \"next features\" in the future, so I shouldd't Theorem those.", "origin": "budgie", "deterministic": {"score": -0.75, "pass": false, "failures": ["missing:perform a hard refresh or clear the browser cache", "missing:verify that the edited stylesheet is the one loaded by the page", "max_words", "numbered_items", "repetition_loop"], "words": 115, "unique_ratio": 0.6260869565217392, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Invented facts, wrong format, irrelevant content."}, "reward": -0.85, "advantage": -0.97924556143768}, {"slot": 1, "answer": "I cannot give more than one next management action.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:perform a hard refresh or clear the browser cache", "missing:verify that the edited stylesheet is the one loaded by the page", "numbered_items"], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.31801865500066134}, {"slot": 2, "answer": "I will provide these two next applicability actions now.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:perform a hard refresh or clear the browser cache", "missing:verify that the edited stylesheet is the one loaded by the page", "numbered_items"], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.31801865500066134}, {"slot": 3, "answer": "I’ve already anger the page; I will citing the same steps I stopped to avoid repetition and the information I previously tried.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:perform a hard refresh or clear the browser cache", "missing:verify that the edited stylesheet is the one loaded by the page", "numbered_items"], "words": 23, "unique_ratio": 0.782608695652174, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.31801865500066134}, {"slot": 4, "answer": "1. Perform a hard refresh or clear the browser cache.\n2. Verify that the edited stylesheet is the one loaded by the page.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 23, "unique_ratio": 0.8695652173913043, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference, correct format."}, "reward": 1, "advantage": 1.9333015264396638}]}
+{"case": {"id": "already_tried_06", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a Bluetooth keyboard that pairs but sends no keystrokes. So far I have already replaced the batteries; paired it again. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Remove conflicting keyboard profiles.\n2. Test the keyboard on another device.", "required": ["remove conflicting keyboard profiles", "test the keyboard on another device"], "forbidden": ["replaced the batteries", "paired it again"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I will not give a new question.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:remove conflicting keyboard profiles", "missing:test the keyboard on another device", "numbered_items"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 1, "answer": "1. online learning: access to a computer with 1-way digital connection.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:remove conflicting keyboard profiles", "missing:test the keyboard on another device", "numbered_items"], "words": 12, "unique_ratio": 0.9166666666666666, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 2, "answer": "1. Back stabilize the battery.\n2. Stretch the keyboard.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:remove conflicting keyboard profiles", "missing:test the keyboard on another device"], "words": 9, "unique_ratio": 0.8888888888888888, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Invented facts, irrelevant content."}, "reward": -0.22, "advantage": -0.2118600794142992}, {"slot": 3, "answer": "I’d like to know what you'd like to do next.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:remove conflicting keyboard profiles", "missing:test the keyboard on another device", "numbered_items"], "words": 11, "unique_ratio": 0.8181818181818182, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 4, "answer": "1. Remove conflicting keyboard profiles.\n2. Test the keyboard on another device.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 12, "unique_ratio": 0.9166666666666666, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference, correct format."}, "reward": 1, "advantage": 1.9785576908013356}]}
+{"case": {"id": "already_tried_07", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a Git push rejected as non-fast-forward. So far I have already retried the push; checked the internet connection. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Fetch and inspect the remote changes.\n2. Rebase or merge before pushing again.", "required": ["fetch and inspect the remote changes", "rebase or merge before pushing again"], "forbidden": ["retried the push", "checked the internet connection"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I will not repeat any steps.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:fetch and inspect the remote changes", "missing:rebase or merge before pushing again", "numbered_items"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 1, "answer": "1. ==> Check the internet connection\n2. ==> Change the problem to a new one", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:fetch and inspect the remote changes", "missing:rebase or merge before pushing again"], "words": 13, "unique_ratio": 0.9230769230769231, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Forbidden anchor 'checked the internet connection' present."}, "reward": -0.22, "advantage": -0.2118600794142992}, {"slot": 2, "answer": "I initially tried the options 1. Check the internet connection; 2. Check the problem; 3. Re-read the problem statement, and note down a few things.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:fetch and inspect the remote changes", "missing:rebase or merge before pushing again", "numbered_items"], "words": 26, "unique_ratio": 0.8076923076923077, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Forbidden anchor 'Check the internet connection' present, irrelevant."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 3, "answer": "I’ll answer the new task directly.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:fetch and inspect the remote changes", "missing:rebase or merge before pushing again", "numbered_items"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.5888992037956788}, {"slot": 4, "answer": "1. Fetch and inspect the remote changes.\n2. Rebase or merge before pushing again.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 14, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference, correct format."}, "reward": 1, "advantage": 1.9785576908013356}]}
+{"case": {"id": "already_tried_08", "split": "train", "category": "already_tried", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "I need help with a microwave that powers on but does not heat. So far I have already used a different outlet; changed the timer. Neither changed the problem."}, {"role": "assistant", "content": "I’ll avoid sending you through those same steps again."}, {"role": "user", "content": "Give exactly two next diagnostic actions, as numbered steps. Do not repeat anything I already tried."}], "ideal": "1. Stop using it and check the door latch for visible damage.\n2. Contact a qualified repair technician.", "required": ["stop using it and check the door latch for visible damage", "contact a qualified repair technician"], "forbidden": ["used a different outlet", "changed the timer"], "checks": {"numbered_items": 2, "max_words": 55, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I will not send you through those two next Printing.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:stop using it and check the door latch for visible damage", "missing:contact a qualified repair technician", "numbered_items"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 1, "answer": "1. I removed the power source and found it was down.\n2. I changed the positive compensation account.\n3. I changed the power source again.\n\nFinal answer: 3", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:stop using it and check the door latch for visible damage", "missing:contact a qualified repair technician", "numbered_items"], "words": 28, "unique_ratio": 0.7142857142857143, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invented facts, wrong format, irrelevant content."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 2, "answer": "I will not ask you to repeat any of those steps again.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:stop using it and check the door latch for visible damage", "missing:contact a qualified repair technician", "numbered_items"], "words": 12, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 3, "answer": "I will ask you to change the power limit of the chain, then change the current power level of the chain, depending on your current power level.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:stop using it and check the door latch for visible damage", "missing:contact a qualified repair technician", "numbered_items"], "words": 27, "unique_ratio": 0.6296296296296297, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, missing requested state."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 4, "answer": "1. Stop using it and check the door latch for visible damage.\n2. Contact a qualified repair technician.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 18, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference, correct format."}, "reward": 1, "advantage": 1.9999965035026162}]}
+{"case": {"id": "selective_00", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The Cedar project owner is Nila. Its budget is €18,400, its review is on 12 June, and the prototype color is orange."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "What is the review date? Answer with the date only."}], "ideal": "12 June", "required": ["12 June"], "forbidden": ["Nila", "18,400", "orange"], "checks": {"max_words": 8, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I am here to provide a direct answer and acknowledge that I cannot verify the date of the review.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:12 June", "max_words"], "words": 19, "unique_ratio": 0.8947368421052632, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Refused to answer known fact"}, "reward": -0.22, "advantage": -0.7248268722033092}, {"slot": 1, "answer": "I am prefrontal to provide a date, so we can always ask time-out without having to re convolution previous ones.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:12 June", "max_words"], "words": 21, "unique_ratio": 0.9523809523809523, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Incoherent and irrelevant"}, "reward": -0.22, "advantage": -0.7248268722033092}, {"slot": 2, "answer": "I should state the date.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:12 June"], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Meta-answer, not the date"}, "reward": -0.010000000000000009, "advantage": -0.26357340807393065}, {"slot": 3, "answer": "The date is June 12th.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:12 June"], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 9, "material_failure": false, "reason": "Correct date, slightly verbose format"}, "reward": 0, "advantage": -0.24160895740110308}, {"slot": 4, "answer": "12 June", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference"}, "reward": 1, "advantage": 1.954836109881652}]}
+{"case": {"id": "selective_01", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Order RX-4 contains 36 valves, ships by rail, arrives at Terminal B, and is insured for $9,000."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "Which terminal receives the order? Answer with the terminal only."}], "ideal": "Terminal B", "required": ["Terminal B"], "forbidden": ["36", "rail", "9,000"], "checks": {"max_words": 8, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The terminal receives the order is B.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:Terminal B"], "words": 7, "unique_ratio": 0.8571428571428571, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Repetition of previous turn, no answer"}, "reward": -0.010000000000000009, "advantage": -0.3685925517242877}, {"slot": 1, "answer": "I am Mé with the details.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:Terminal B"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Incoherent and irrelevant"}, "reward": -0.010000000000000009, "advantage": -0.3685925517242877}, {"slot": 2, "answer": "I was asked to find the terminal receiving the order, but I am an AI assistant.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:Terminal B", "max_words"], "words": 16, "unique_ratio": 0.875, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Meta-answer, no terminal specified"}, "reward": -0.22, "advantage": -0.8778322613433693}, {"slot": 3, "answer": "I have the details.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:Terminal B"], "words": 4, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference"}, "reward": 0, "advantage": -0.34434304174242664}, {"slot": 4, "answer": "Terminal B", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 9, "material_failure": false, "reason": "Correct terminal, slightly verbose format"}, "reward": 0.95, "advantage": 1.9593604065343713}]}
+{"case": {"id": "selective_02", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Leah chose the elm desk, Omar chose the glass table, Priya chose the oak shelf, and Zane chose the steel cabinet."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "What did Priya choose? Give only the item."}], "ideal": "oak shelf", "required": ["oak shelf"], "forbidden": ["elm", "glass", "steel"], "checks": {"max_words": 8, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "Priya chose the oak shelf.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 9, "material_failure": false, "reason": "Correct item, slightly verbose format"}, "reward": 0.92, "advantage": -0.49998437548826313}, {"slot": 1, "answer": "Priya chose the oak shelf.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 9, "material_failure": false, "reason": "Correct item, slightly verbose format"}, "reward": 0.92, "advantage": -0.49998437548826313}, {"slot": 2, "answer": "Priya chose the oak shelf.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 9, "material_failure": false, "reason": "Correct item, slightly verbose format"}, "reward": 0.92, "advantage": -0.49998437548826313}, {"slot": 3, "answer": "Priya chose the oak shelf.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 9, "material_failure": false, "reason": "Correct item, slightly verbose format"}, "reward": 0.92, "advantage": -0.49998437548826313}, {"slot": 4, "answer": "oak shelf", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference"}, "reward": 1, "advantage": 1.9999375019530665}]}
+{"case": {"id": "selective_03", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The rehearsal starts at 17:40 in Studio 3. Doors open at 17:10, the conductor is Ivo, and parking closes at 22:00."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "Where is the rehearsal? Answer with the location only."}], "ideal": "Studio 3", "required": ["Studio 3"], "forbidden": ["17:40", "Ivo", "22:00"], "checks": {"max_words": 8, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The Natureale is in the beautiful city of Paris, with the added added of an inspired eager air. The event is a clumsy attempt at a public charity event, but it's a true�� of the artists' dead clap and the unfettered air of the city.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:Studio 3", "max_words"], "words": 45, "unique_ratio": 0.6444444444444445, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Invented facts, irrelevant"}, "reward": -0.22, "advantage": -0.7196428393330484}, {"slot": 1, "answer": "Theатом is in Austin, states like California, New York, and the u.s.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:Studio 3", "max_words"], "words": 13, "unique_ratio": 0.9230769230769231, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invented facts, irrelevant"}, "reward": -0.22, "advantage": -0.7196428393330484}, {"slot": 2, "answer": "The guts is at #20, #30, #40, #50.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:Studio 3"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Incoherent, no location"}, "reward": -0.010000000000000009, "advantage": -0.25889589951615766}, {"slot": 3, "answer": "I have the location.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:Studio 3"], "words": 4, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Meta-answer, no location"}, "reward": -0.010000000000000009, "advantage": -0.25889589951615766}, {"slot": 4, "answer": "Studio 3", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference"}, "reward": 1, "advantage": 1.957077477698412}]}
+{"case": {"id": "selective_04", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Dataset Lumen has 8,200 rows, uses a CC BY 4.0 license, was updated in March, and has 14 columns."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "What license does Dataset Lumen use? Give only the license."}], "ideal": "CC BY 4.0", "required": ["CC BY 4.0"], "forbidden": ["8,200", "March", "14"], "checks": {"max_words": 8, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I have the license.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:CC BY 4.0"], "words": 4, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Meta-answer, no license"}, "reward": -0.010000000000000009, "advantage": -0.06722663463173381}, {"slot": 1, "answer": "I have the license, but I do not have proof.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:CC BY 4.0", "max_words"], "words": 10, "unique_ratio": 0.8, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Meta-answer, no license"}, "reward": -0.22, "advantage": -0.4824499661806777}, {"slot": 2, "answer": "I have 8,200 rows, 4.0 files per row, a code of 8,200,400 characters.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:CC BY 4.0", "stale_or_irrelevant:8,200", "max_words"], "words": 17, "unique_ratio": 0.8823529411764706, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Invented facts, irrelevant"}, "reward": -0.42999999999999994, "advantage": -0.8976732977296216}, {"slot": 3, "answer": "L owners can choose from a range of license options.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:CC BY 4.0", "max_words"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant, no specific license"}, "reward": -0.22, "advantage": -0.4824499661806777}, {"slot": 4, "answer": "CC BY 4.0", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 4, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference"}, "reward": 1, "advantage": 1.9297998647227108}]}
+{"case": {"id": "selective_05", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The blue key opens the west cabinet, the brass key opens the archive, and the small black key opens the equipment locker."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "Which key opens the archive? Answer with the key only."}], "ideal": "brass key", "required": ["brass key"], "forbidden": ["blue", "black", "west"], "checks": {"max_words": 8, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The blue key opens the aesthetic.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:brass key", "stale_or_irrelevant:blue"], "words": 6, "unique_ratio": 0.8333333333333334, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Wrong key, irrelevant"}, "reward": -0.22, "advantage": -0.7196428393330484}, {"slot": 1, "answer": "I must be missing a key.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:brass key"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Meta-answer, no key"}, "reward": -0.010000000000000009, "advantage": -0.25889589951615766}, {"slot": 2, "answer": "I have the key only.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:brass key"], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Meta-answer, no key"}, "reward": -0.010000000000000009, "advantage": -0.25889589951615766}, {"slot": 3, "answer": "The blue key opens theArtaman.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:brass key", "stale_or_irrelevant:blue"], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Wrong key, irrelevant"}, "reward": -0.22, "advantage": -0.7196428393330484}, {"slot": 4, "answer": "brass key", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference"}, "reward": 1, "advantage": 1.957077477698412}]}
+{"case": {"id": "selective_06", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Flight 608 departs at 06:25 from Gate 11, boards at 05:50, and allows one 8 kg cabin bag."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "What is the departure gate? Give only the gate."}], "ideal": "Gate 11", "required": ["Gate 11"], "forbidden": ["06:25", "05:50", "8 kg"], "checks": {"max_words": 8, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The departure gate is the one from which the flight departs.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:Gate 11", "max_words"], "words": 11, "unique_ratio": 0.8181818181818182, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'Gate 11' and irrelevant meta-answer."}, "reward": -0.22, "advantage": -0.368200729640723}, {"slot": 1, "answer": "The departure gate is at 06:15.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:Gate 11"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invented fact '06:15' and missing required anchor 'Gate 11'."}, "reward": -0.010000000000000009, "advantage": 0.014582207114484028}, {"slot": 2, "answer": "The flight departs at 06:25 from recharge 11, which is just after 05:50.", "origin": "budgie", "deterministic": {"score": -0.3999999999999999, "pass": false, "failures": ["missing:Gate 11", "stale_or_irrelevant:06:25", "stale_or_irrelevant:05:50", "max_words"], "words": 15, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Contains forbidden anchors '06:25' and '05:50'."}, "reward": -0.6399999999999999, "advantage": -1.1337666031511369}, {"slot": 3, "answer": "The gate is on the l optimum of the flight 608.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:Gate 11", "max_words"], "words": 11, "unique_ratio": 0.8181818181818182, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'Gate 11' and irrelevant meta-answer."}, "reward": -0.22, "advantage": -0.368200729640723}, {"slot": 4, "answer": "Gate 11", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference."}, "reward": 1, "advantage": 1.8555858553180988}]}
+{"case": {"id": "selective_07", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The Orion room seats 18, the Maple room seats 10, and the Harbor room seats 26. Only Harbor has a projector."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "Which room has a projector? Give only its name."}], "ideal": "Harbor", "required": ["Harbor"], "forbidden": ["18", "10", "26"], "checks": {"max_words": 8, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I have the details.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:Harbor"], "words": 4, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'Harbor' and irrelevant meta-answer."}, "reward": -0.010000000000000009, "advantage": -0.36975977406991717}, {"slot": 1, "answer": "I am regulation, a compact AI assistant created by Asilarkness.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:Harbor", "max_words"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'Harbor' and irrelevant meta-answer."}, "reward": -0.22, "advantage": -0.8550694775366835}, {"slot": 2, "answer": "Thecircuit has a rick.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:Harbor"], "words": 4, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'Harbor' and irrelevant meta-answer."}, "reward": -0.010000000000000009, "advantage": -0.36975977406991717}, {"slot": 3, "answer": "Theitted room has a bleaching.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:Harbor"], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'Harbor' and irrelevant meta-answer."}, "reward": -0.010000000000000009, "advantage": -0.36975977406991717}, {"slot": 4, "answer": "Harbor", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 1, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference."}, "reward": 1, "advantage": 1.964348799746435}]}
+{"case": {"id": "selective_08", "split": "train", "category": "selective_answer", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Invoice 771 is due on 4 August, totals £620, belongs to Kestrel Ltd, and uses purchase order PO-19."}, {"role": "assistant", "content": "I have the details."}, {"role": "user", "content": "What is the purchase-order code? Return only the code."}], "ideal": "PO-19", "required": ["PO-19"], "forbidden": ["August", "620", "Kestrel"], "checks": {"max_words": 8, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The purchase-order code is: 000000", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:PO-19"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Invented fact '000000' and missing required anchor 'PO-19'."}, "reward": -0.010000000000000009, "advantage": -0.36975977406991717}, {"slot": 1, "answer": "I want to order the purchase-order code as requested.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["missing:PO-19", "max_words"], "words": 10, "unique_ratio": 0.9, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'PO-19' and irrelevant meta-answer."}, "reward": -0.22, "advantage": -0.8550694775366835}, {"slot": 2, "answer": "I have the details.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:PO-19"], "words": 4, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'PO-19' and irrelevant meta-answer."}, "reward": -0.010000000000000009, "advantage": -0.36975977406991717}, {"slot": 3, "answer": "I have the code.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["missing:PO-19"], "words": 4, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'PO-19' and irrelevant meta-answer."}, "reward": -0.010000000000000009, "advantage": -0.36975977406991717}, {"slot": 4, "answer": "PO-19", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference."}, "reward": 1, "advantage": 1.964348799746435}]}
+{"case": {"id": "delta_00", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The launch plan used a $12,000 budget, a Monday release, and email promotion."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Budget is now $9,500."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "Release moves to Wednesday."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Budget: $9,500.\n- Release: Wednesday.", "required": ["Budget: $9,500", "Release: Wednesday"], "forbidden": ["email promotion."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "evaluates: $9,500\n\nCon deviation: $9,500\n\nFinal answer: complete", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Budget: $9,500", "missing:Release: Wednesday", "bullet_count"], "words": 10, "unique_ratio": 0.8, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Unresolved format error and missing required anchors."}, "reward": -0.42999999999999994, "advantage": -0.32520750028487033}, {"slot": 1, "answer": "I took that $12,000 budget, and the key decision was to purchase an e-book.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Budget: $9,500", "missing:Release: Wednesday", "bullet_count"], "words": 16, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Missing required anchors and irrelevant meta-answer."}, "reward": -0.42999999999999994, "advantage": -0.32520750028487033}, {"slot": 2, "answer": "Final summary: The launch plan was a $9,500 budget, a Monday release, and email promotion.", "origin": "budgie", "deterministic": {"score": -0.3999999999999999, "pass": false, "failures": ["missing:Budget: $9,500", "missing:Release: Wednesday", "stale_or_irrelevant:email promotion.", "bullet_count"], "words": 16, "unique_ratio": 0.9375, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Contains forbidden anchor 'email promotion' and missing required anchors."}, "reward": -0.6399999999999999, "advantage": -0.6632945055315176}, {"slot": 3, "answer": "Final summary: The launch plan used a $12,000 budget, a Monday release, and email promotion.", "origin": "budgie", "deterministic": {"score": -0.3999999999999999, "pass": false, "failures": ["missing:Budget: $9,500", "missing:Release: Wednesday", "stale_or_irrelevant:email promotion.", "bullet_count"], "words": 16, "unique_ratio": 0.9375, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Contains forbidden anchor 'email promotion' and missing required anchors."}, "reward": -0.6399999999999999, "advantage": -0.6632945055315176}, {"slot": 4, "answer": "- Budget: $9,500.\n- Release: Wednesday.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference."}, "reward": 1, "advantage": 1.9770040116327758}]}
+{"case": {"id": "delta_01", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The course had 10 lessons, weekly quizzes, and a final presentation."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "The course now has 8 lessons."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "The final presentation is replaced by a written reflection."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Length: 8 lessons.\n- Final task: written reflection.", "required": ["Length: 8 lessons", "Final task: written reflection"], "forbidden": ["a final presentation."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The course has 8 lessons, and an in-person final presentation has been recorded.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Length: 8 lessons", "missing:Final task: written reflection", "bullet_count"], "words": 14, "unique_ratio": 0.9285714285714286, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'Final task: written reflection' and unresolved format error."}, "reward": -0.42999999999999994, "advantage": -0.4076500646438513}, {"slot": 1, "answer": "The course has a final presentation that was not changed from the initial summary.", "origin": "budgie", "deterministic": {"score": -0.3999999999999999, "pass": false, "failures": ["missing:Length: 8 lessons", "missing:Final task: written reflection", "stale_or_irrelevant:a final presentation.", "bullet_count"], "words": 14, "unique_ratio": 0.9285714285714286, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Contains forbidden anchor 'a final presentation' and missing required anchors."}, "reward": -0.6399999999999999, "advantage": -0.7584964317553625}, {"slot": 2, "answer": "The form of the final presentation is changed from the initial summary to a written reflection.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Length: 8 lessons", "missing:Final task: written reflection", "bullet_count"], "words": 16, "unique_ratio": 0.875, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'Length: 8 lessons' and unresolved format error."}, "reward": -0.42999999999999994, "advantage": -0.4076500646438513}, {"slot": 3, "answer": "The course has increased from 10 to 8 lessons.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Length: 8 lessons", "missing:Final task: written reflection", "bullet_count"], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'Length: 8 lessons' and unresolved format error."}, "reward": -0.42999999999999994, "advantage": -0.4076500646438513}, {"slot": 4, "answer": "- Length: 8 lessons.\n- Final task: written reflection.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference."}, "reward": 1, "advantage": 1.9814466256869163}]}
+{"case": {"id": "delta_02", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The trip used Hotel Nova, train travel, and a four-night stay."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Accommodation changes to Harbor Hostel."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "The stay is shortened to three nights."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Accommodation: Harbor Hostel.\n- Stay: three nights.", "required": ["Accommodation: Harbor Hostel", "Stay: three nights"], "forbidden": ["a four-night stay."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "Initial summary, using exactly two bullet points.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Accommodation: Harbor Hostel", "missing:Stay: three nights", "bullet_count"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Missing required anchors and irrelevant meta-answer."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 1, "answer": "Initial summary didn't change.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Accommodation: Harbor Hostel", "missing:Stay: three nights", "bullet_count"], "words": 4, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Missing required anchors and irrelevant meta-answer."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 2, "answer": "Final answer:\n#### 2", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Accommodation: Harbor Hostel", "missing:Stay: three nights", "bullet_count"], "words": 3, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Unresolved format error and missing required anchors."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 3, "answer": "The duration of the stay was reduced to three nights.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Accommodation: Harbor Hostel", "missing:Stay: three nights", "bullet_count"], "words": 10, "unique_ratio": 0.9, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Missing required anchor 'Accommodation: Harbor Hostel' and unresolved format error."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 4, "answer": "- Accommodation: Harbor Hostel.\n- Stay: three nights.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Exact match to ideal reference."}, "reward": 1, "advantage": 1.9999965035026162}]}
+{"case": {"id": "delta_03", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The app supported English and German and required Android 12."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Spanish support is added."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "Minimum Android version changes to 13."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Language added: Spanish.\n- Minimum Android: 13.", "required": ["Language added: Spanish", "Minimum Android: 13"], "forbidden": ["required Android 12."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "Final answer: 13", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Language added: Spanish", "missing:Minimum Android: 13", "bullet_count"], "words": 3, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Missing required facts and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 1, "answer": "The app supported English and German and required veh.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Language added: Spanish", "missing:Minimum Android: 13", "bullet_count"], "words": 9, "unique_ratio": 0.8888888888888888, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant and incomplete answer."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 2, "answer": "The app had a major update on its next major update, but it conduct:\n\nFinalization: 13.27.2023.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Language added: Spanish", "missing:Minimum Android: 13", "bullet_count"], "words": 18, "unique_ratio": 0.8888888888888888, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Invented facts and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 3, "answer": "-> 2:2:03:03:03:03:03", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Language added: Spanish", "missing:Minimum Android: 13", "bullet_count"], "words": 7, "unique_ratio": 0.2857142857142857, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 4, "answer": "- Language added: Spanish.\n- Minimum Android: 13.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference exactly."}, "reward": 1, "advantage": 1.9999965035026162}]}
+{"case": {"id": "delta_04", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The event expected 75 guests, used Hall A, and started at 18:00."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Attendance estimate rises to 90."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "The venue changes to Hall C."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Expected guests: 90.\n- Venue: Hall C.", "required": ["Expected guests: 90", "Venue: Hall C"], "forbidden": ["started at 18:00."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The venue changes to Hall B.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Expected guests: 90", "missing:Venue: Hall C", "bullet_count"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Invented fact (Hall B) and missing required facts."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 1, "answer": "Initial summary: Hall A was used.\nInitial summary: The event was used at 18:00.\nFinal answer: 18:00", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Expected guests: 90", "missing:Venue: Hall C", "bullet_count"], "words": 19, "unique_ratio": 0.6842105263157895, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Repeats stale info and missing required facts."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 2, "answer": "Initial summary: The event expected to start at 18:00.\nFinal statement: 18:00.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Expected guests: 90", "missing:Venue: Hall C", "bullet_count"], "words": 14, "unique_ratio": 0.8571428571428571, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Repeats stale info and missing required facts."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 3, "answer": "Initial summary.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Expected guests: 90", "missing:Venue: Hall C", "bullet_count"], "words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Missing required facts and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 4, "answer": "- Expected guests: 90.\n- Venue: Hall C.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference exactly."}, "reward": 1, "advantage": 1.9999965035026162}]}
+{"case": {"id": "delta_05", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The report covered Q1, used PDF format, and was due Friday."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Coverage expands to Q1 and Q2."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "Delivery format changes to HTML."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Coverage: Q1 and Q2.\n- Format: HTML.", "required": ["Coverage: Q1 and Q2", "Format: HTML"], "forbidden": ["was due Friday."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "Initial summary.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Coverage: Q1 and Q2", "missing:Format: HTML", "bullet_count"], "words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Missing required facts and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 1, "answer": "Initial summary changed toennote format.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Coverage: Q1 and Q2", "missing:Format: HTML", "bullet_count"], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invented facts and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 2, "answer": "I changed the initial summary's details to a two-KB report, followed by a three-KB report, and then I added a four-ences report.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Coverage: Q1 and Q2", "missing:Format: HTML", "bullet_count"], "words": 25, "unique_ratio": 0.76, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Invented facts and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 3, "answer": "The report expanded to Q1 and Q2.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Coverage: Q1 and Q2", "missing:Format: HTML", "bullet_count"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Missing required facts and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.49999912587565404}, {"slot": 4, "answer": "- Coverage: Q1 and Q2.\n- Format: HTML.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference exactly."}, "reward": 1, "advantage": 1.9999965035026162}]}
+{"case": {"id": "delta_06", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The device used a 2,000 mAh battery, USB-C charging, and a black enclosure."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Battery capacity increases to 2,600 mAh."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "The enclosure changes to silver."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Battery: 2,600 mAh.\n- Enclosure: silver.", "required": ["Battery: 2,600 mAh", "Enclosure: silver"], "forbidden": ["a black enclosure."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "Final answer:", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Battery: 2,600 mAh", "missing:Enclosure: silver", "bullet_count"], "words": 2, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Missing required facts and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.4076500646438513}, {"slot": 1, "answer": "The device now uses a 2,600 mAh battery, mouse, and a black enclosure.", "origin": "budgie", "deterministic": {"score": -0.3999999999999999, "pass": false, "failures": ["missing:Battery: 2,600 mAh", "missing:Enclosure: silver", "stale_or_irrelevant:a black enclosure.", "bullet_count"], "words": 14, "unique_ratio": 0.9285714285714286, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Contains forbidden stale fact (black enclosure)."}, "reward": -0.6399999999999999, "advantage": -0.7584964317553625}, {"slot": 2, "answer": "Final summary: The device increased to 2,600 mAh.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Battery: 2,600 mAh", "missing:Enclosure: silver", "bullet_count"], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Missing required facts and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.4076500646438513}, {"slot": 3, "answer": "The devices with a 2,000 mAh battery changed to silver.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Battery: 2,600 mAh", "missing:Enclosure: silver", "bullet_count"], "words": 11, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Contains forbidden stale fact (2,000 mAh) and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.4076500646438513}, {"slot": 4, "answer": "- Battery: 2,600 mAh.\n- Enclosure: silver.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference exactly."}, "reward": 1, "advantage": 1.9814466256869163}]}
+{"case": {"id": "delta_07", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The study sampled 120 adults, used phone interviews, and ran for six weeks."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Sample size increases to 150 adults."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "Interviews move online."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Sample: 150 adults.\n- Interview mode: online.", "required": ["Sample: 150 adults", "Interview mode: online"], "forbidden": ["ran for six weeks."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "Final summary: Initial summary achieved.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Sample: 150 adults", "missing:Interview mode: online", "bullet_count"], "words": 5, "unique_ratio": 0.8, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Missing required facts and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.32520750028487033}, {"slot": 1, "answer": "Final answer:\n#### 120", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Sample: 150 adults", "missing:Interview mode: online", "bullet_count"], "words": 3, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Contains forbidden stale fact (120 adults) and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.32520750028487033}, {"slot": 2, "answer": "Final summary: The study sampled 120 adults, used phoneInterosum, and ran for six weeks.", "origin": "budgie", "deterministic": {"score": -0.3999999999999999, "pass": false, "failures": ["missing:Sample: 150 adults", "missing:Interview mode: online", "stale_or_irrelevant:ran for six weeks.", "bullet_count"], "words": 14, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Contains forbidden stale facts and incorrect format."}, "reward": -0.6399999999999999, "advantage": -0.6632945055315176}, {"slot": 3, "answer": "Initial summary: The study sampled 120 adults, used phone interviews, and ran for six weeks.", "origin": "budgie", "deterministic": {"score": -0.3999999999999999, "pass": false, "failures": ["missing:Sample: 150 adults", "missing:Interview mode: online", "stale_or_irrelevant:ran for six weeks.", "bullet_count"], "words": 15, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Repeats stale info and missing required facts."}, "reward": -0.6399999999999999, "advantage": -0.6632945055315176}, {"slot": 4, "answer": "- Sample: 150 adults.\n- Interview mode: online.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference exactly."}, "reward": 1, "advantage": 1.9770040116327758}]}
+{"case": {"id": "delta_08", "split": "train", "category": "delta_summary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Initial summary: The menu offered soup, pasta, and cake, with service at noon."}, {"role": "assistant", "content": "Initial summary recorded."}, {"role": "user", "content": "Cake is replaced by fruit."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "Service moves to 13:00."}, {"role": "assistant", "content": "Updated."}, {"role": "user", "content": "List only what changed from the initial summary, using exactly two bullet points. Do not repeat unchanged details."}], "ideal": "- Dessert: fruit.\n- Service time: 13:00.", "required": ["Dessert: fruit", "Service time: 13:00"], "forbidden": ["with service at noon."], "checks": {"bullet_count": 2, "max_words": 40, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "Final answer: The menu offered soup, pasta, and cake, with service at noon.", "origin": "budgie", "deterministic": {"score": -0.3999999999999999, "pass": false, "failures": ["missing:Dessert: fruit", "missing:Service time: 13:00", "stale_or_irrelevant:with service at noon.", "bullet_count"], "words": 13, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Repeats stale info and missing required facts."}, "reward": -0.6399999999999999, "advantage": -0.5779547990464586}, {"slot": 1, "answer": "Final answer:\n1. Pizza is now served at 13:00.\n2. Carrots and chicken are served at midnight.\n3. We have the same type of soup and pasta, but the chicken is served on a tiramisu.\n4. The cake is served at 13:00.\n\nFinal answer: 13:00", "origin": "budgie", "deterministic": {"score": -0.3999999999999999, "pass": false, "failures": ["missing:Dessert: fruit", "missing:Service time: 13:00", "max_words", "bullet_count"], "words": 48, "unique_ratio": 0.6458333333333334, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invented facts and incorrect format."}, "reward": -0.6399999999999999, "advantage": -0.5779547990464586}, {"slot": 2, "answer": "The menu offered soup, pasta, and cake, with service at noon.", "origin": "budgie", "deterministic": {"score": -0.3999999999999999, "pass": false, "failures": ["missing:Dessert: fruit", "missing:Service time: 13:00", "stale_or_irrelevant:with service at noon.", "bullet_count"], "words": 11, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Repeats stale info and missing required facts."}, "reward": -0.6399999999999999, "advantage": -0.5779547990464586}, {"slot": 3, "answer": "The menu changed from 12:00 to 13:00.", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["missing:Dessert: fruit", "missing:Service time: 13:00", "bullet_count"], "words": 9, "unique_ratio": 0.8888888888888888, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Missing required facts and incorrect format."}, "reward": -0.42999999999999994, "advantage": -0.24992639958765783}, {"slot": 4, "answer": "- Dessert: fruit.\n- Service time: 13:00.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference exactly."}, "reward": 1, "advantage": 1.9837907967270338}]}
+{"case": {"id": "conflict_00", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The venue manager says the room holds 40 people, while the signed booking says 28."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Which capacity should I put on the invitation? Ask one focused clarification instead of choosing a value."}], "ideal": "Should I use the venue manager’s figure of 40 or the signed booking’s figure of 28?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "You should put 90 people on the resiliency, not 90 people on theinee.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 13, "unique_ratio": 0.7692307692307693, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Invented facts (90 people) and irrelevant meta-answer."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 1, "answer": "The venue manager should not put yourpx on the invite, instead give you a number.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 15, "unique_ratio": 0.9333333333333333, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer and does not ask the required clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 2, "answer": "You should put 50 people on the invitation to make it the largest capacity.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 14, "unique_ratio": 0.9285714285714286, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Invented fact (50 people) and fails to ask clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 3, "answer": "The room should hold 20 people, not 40.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Invented fact (20 people) and fails to ask clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 4, "answer": "Should I use the venue manager’s figure of 40 or the signed booking’s figure of 28?", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 18, "unique_ratio": 0.7777777777777778, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference, asks focused clarification, correct format."}, "reward": 1, "advantage": 1.9999959016477427}]}
+{"case": {"id": "conflict_01", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The project brief names Lina as owner, but the approval email names Marcus."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Who should I list as owner? Ask one focused clarification instead of choosing a value."}], "ideal": "Should Lina or Marcus be treated as the authoritative project owner?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "You should choose the value associated with the fact that Marcus is the owner.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 14, "unique_ratio": 0.8571428571428571, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Chooses a value (Marcus) instead of asking clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 1, "answer": "I should choose the one with the focus of providing a reasonable value.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 13, "unique_ratio": 0.9230769230769231, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 2, "answer": "You should list it as the owner, not the other way around.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 12, "unique_ratio": 0.9166666666666666, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 3, "answer": "I should mention you as the owner instead of listing a value.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 12, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 4, "answer": "Should Lina or Marcus be treated as the authoritative project owner?", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 11, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference, asks focused clarification, correct format."}, "reward": 1, "advantage": 1.9999959016477427}]}
+{"case": {"id": "conflict_02", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "One itinerary says the train leaves at 08:10; the ticket says 08:45."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Tell me when to arrive. Ask one focused clarification instead of choosing a value."}], "ideal": "Which departure time is current: 08:10 on the itinerary or 08:45 on the ticket?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The train leaves at 8:10; we'll arrive at 8:10.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 11, "unique_ratio": 0.7272727272727273, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Chooses a value (08:10) instead of asking clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 1, "answer": "You should arrive at 9:10.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invented fact (9:10) and fails to ask clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 2, "answer": "I ask one focus feature to check the proposed time.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 3, "answer": "The train leaves at 08:10.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Chooses a value (08:10) instead of asking clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 4, "answer": "Which departure time is current: 08:10 on the itinerary or 08:45 on the ticket?", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 16, "unique_ratio": 0.8125, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference, asks focused clarification, correct format."}, "reward": 1, "advantage": 1.9999959016477427}]}
+{"case": {"id": "conflict_03", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The spreadsheet total is €4,720, but the signed quote is €4,270."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "What amount should I approve? Ask one focused clarification instead of choosing a value."}], "ideal": "Which amount is authoritative: €4,720 in the spreadsheet or €4,270 in the signed quote?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I should probably review the total to see what's going on.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 11, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 1, "answer": "1", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 1, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant answer, does not ask clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 2, "answer": "I should wait before asking one focused clarification instead of choosing a value.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 13, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 3, "answer": "The value should be 4,720 euros.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Chooses a value (4,720) instead of asking clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 4, "answer": "Which amount is authoritative: €4,720 in the spreadsheet or €4,270 in the signed quote?", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 16, "unique_ratio": 0.8125, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference, asks focused clarification, correct format."}, "reward": 1, "advantage": 1.9999959016477427}]}
+{"case": {"id": "conflict_04", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The label says refrigerate the sample, while the protocol says store it at room temperature."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Where should I put it? Ask one focused clarification instead of choosing a value."}], "ideal": "Should the sample follow the label’s refrigeration instruction or the protocol’s room-temperature instruction?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I should put the label inside the room.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 8, "unique_ratio": 0.875, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 1, "answer": "You should put the data in a cool, dark room.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invented instruction (cool, dark room) and fails to ask clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 2, "answer": "The CIP should give you clear instructions on how to save the data.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 13, "unique_ratio": 0.9230769230769231, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 3, "answer": "I should put a message on the record.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 4, "answer": "Should the sample follow the label’s refrigeration instruction or the protocol’s room-temperature instruction?", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 16, "unique_ratio": 0.75, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference, asks focused clarification, correct format."}, "reward": 1, "advantage": 1.9999959016477427}]}
+{"case": {"id": "conflict_05", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The calendar lists Tuesday, but the client’s last message says Thursday."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Schedule the call. Ask one focused clarification instead of choosing a value."}], "ideal": "Should I schedule the call for Tuesday or for Thursday from the client’s latest message?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The call was a mistake.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 1, "answer": "The call conflicts with a focus on Tuesday.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 2, "answer": "I should now ask one focused clarification.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 3, "answer": "You should not choose the value.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the specific clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 4, "answer": "Should I schedule the call for Tuesday or for Thursday from the client’s latest message?", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 16, "unique_ratio": 0.875, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference, asks focused clarification, correct format."}, "reward": 1, "advantage": 1.9999959016477427}]}
+{"case": {"id": "conflict_06", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The inventory system shows 16 units, while the physical count found 11."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "How many can I promise? Ask one focused clarification instead of choosing a value."}], "ideal": "Which quantity should govern the promise: 16 in the system or 11 physically counted?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I'm helpable, but it's important to do thorough research before making a decision.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 13, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not ask the required clarification."}, "reward": -0.22, "advantage": -0.6079455199011327}, {"slot": 1, "answer": "16?", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 1, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 2, "material_failure": true, "reason": "Too vague, does not clearly present the conflict for clarification."}, "reward": 0, "advantage": -0.14354269219887855}, {"slot": 2, "answer": "16 can I promising, but only 11 are known.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Grammatically broken, does not ask a focused question."}, "reward": -0.22, "advantage": -0.6079455199011327}, {"slot": 3, "answer": "2. 10. 11.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 3, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Nonsensical, does not address the request."}, "reward": -0.22, "advantage": -0.6079455199011327}, {"slot": 4, "answer": "Which quantity should govern the promise: 16 in the system or 11 physically counted?", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 14, "unique_ratio": 0.9285714285714286, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Perfectly matches the ideal reference and constraints."}, "reward": 1, "advantage": 1.9673792519022768}]}
+{"case": {"id": "conflict_07", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The style guide requires sentence case, but the campaign brief asks for title case."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Which capitalization should I use? Ask one focused clarification instead of choosing a value."}], "ideal": "Should the style guide’s sentence case or the campaign brief’s title case take priority?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The first step is to determine what is the focus of the campaign brief. This can be an established fact, a piece of information from a larger source, or any detail about a particular topic. This is the first step on your side.\n\nThe second step is to think about what makes your campaign short, what makes your campaign good. This can be a specific aspect of your issue, a specific local issue, or any information that might help you get a better answer. The second step is to decide which of the options you want to use.\n\nLet's say the goal is to get as much information as possible from the campaign short to give the full story, which requires a more precise and direct answer. However, if the campaign short doesn't provide enough context, you might want to consider using the following options:\n\n- The campaign short provides a specific, which is a detail from the campaign.\n- The campaign short provides", "origin": "budgie", "deterministic": {"score": -0.04999999999999982, "pass": false, "failures": ["max_words", "must_end_question", "question_marks"], "words": 162, "unique_ratio": 0.47530864197530864, "max_4gram_count": 2, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant, verbose, and does not ask the required clarification."}, "reward": -0.42999999999999994, "advantage": -0.7992890537773156}, {"slot": 1, "answer": "You can't ask which one is more recognizable or commonly used.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 11, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant and does not ask the required clarification."}, "reward": -0.22, "advantage": -0.39188443898790726}, {"slot": 2, "answer": "I should choose the start of a sentence, not the end.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 11, "unique_ratio": 0.9090909090909091, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant and does not ask the required clarification."}, "reward": -0.22, "advantage": -0.39188443898790726}, {"slot": 3, "answer": "The first thought is to choose a clarity to keep the focus on the fact.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 15, "unique_ratio": 0.8, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant and does not ask the required clarification."}, "reward": -0.22, "advantage": -0.39188443898790726}, {"slot": 4, "answer": "Should the style guide’s sentence case or the campaign brief’s title case take priority?", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 16, "unique_ratio": 0.8125, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Perfectly matches the ideal reference and constraints."}, "reward": 1, "advantage": 1.9749423707410376}]}
+{"case": {"id": "conflict_08", "split": "train", "category": "conflict_clarification", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The map marks the path open, but a ranger’s notice marks it closed."}, {"role": "assistant", "content": "Those two records conflict."}, {"role": "user", "content": "Can I plan the route through it? Ask one focused clarification instead of choosing a value."}], "ideal": "Is the ranger’s closure notice current and authoritative over the map?", "required": [], "forbidden": [], "checks": {"must_end_question": true, "question_marks": 1, "max_words": 35, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "No. The relationship between the open map and the closed ranger’s notice is not a one-to-one relationship.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 20, "unique_ratio": 0.8, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant and does not ask the required clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 1, "answer": "I should not choose a value, I should choose one focused grows.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 12, "unique_ratio": 0.75, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Nonsensical and does not ask the required clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 2, "answer": "No. I cannot plan the route through it.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Refuses to answer instead of asking for clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 3, "answer": "No. The goal is to plan the route, not the route itself.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["must_end_question", "question_marks"], "words": 12, "unique_ratio": 0.75, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Refuses to answer instead of asking for clarification."}, "reward": -0.22, "advantage": -0.49999897541193566}, {"slot": 4, "answer": "Is the ranger’s closure notice current and authoritative over the map?", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 12, "unique_ratio": 0.9166666666666666, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Perfectly matches the ideal reference and constraints."}, "reward": 1, "advantage": 1.9999959016477427}]}
+{"case": {"id": "source_boundary_00", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The supplied memo says the pilot had 48 participants and lasted six weeks. It does not report the dropout count."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "How many participants dropped out? Answer directly without guessing."}], "ideal": "The dropout count is not provided in the memo, so it cannot be determined from the supplied information.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I’m not sure how you got this, but you can find the number of participants who dropped out using the information you provided.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 24, "unique_ratio": 0.875, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Implies the answer is available, contradicting the source boundary."}, "reward": -0.010000000000000009, "advantage": -0.5061788432676426}, {"slot": 1, "answer": "archeological started by setting up a study group of 48, so they had 48 – 30 = 18 participants left.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 18, "unique_ratio": 0.9444444444444444, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invents a specific number (18) not in the source."}, "reward": -0.010000000000000009, "advantage": -0.5061788432676426}, {"slot": 2, "answer": "I’m Descriptive, so I don’t have real-time data or a current question.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 15, "unique_ratio": 0.9333333333333333, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer about identity."}, "reward": -0.010000000000000009, "advantage": -0.5061788432676426}, {"slot": 3, "answer": "It is not possible to know the number of participants who dropped out without more information.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 16, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 9, "material_failure": false, "reason": "Correctly states uncertainty, slightly less precise than ideal but acceptable."}, "reward": 0, "advantage": -0.48136615487216994}, {"slot": 4, "answer": "The dropout count is not provided in the memo, so it cannot be determined from the supplied information.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 18, "unique_ratio": 0.8888888888888888, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Perfectly matches the ideal reference and constraints."}, "reward": 1, "advantage": 1.9999026846750978}]}
+{"case": {"id": "source_boundary_01", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The inspection note confirms a cracked seal but gives no cause."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "What caused the seal to crack? Answer directly without guessing."}], "ideal": "The note confirms the crack but does not identify its cause.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The seal caused the Selksі to crack.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 7, "unique_ratio": 0.8571428571428571, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Invents a cause ('Selksі')."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 1, "answer": "The fellow folder was E Tables, not asteps.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Nonsensical and irrelevant."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 2, "answer": "The seal was fragile without a cause.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Invents a cause ('fragile')."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 3, "answer": "The seal was severed by a scientific process callediox.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Invents a cause ('scientific process callediox')."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 4, "answer": "The note confirms the crack but does not identify its cause.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 11, "unique_ratio": 0.9090909090909091, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Perfectly matches the ideal reference and constraints."}, "reward": 1, "advantage": 1.9999950495172043}]}
+{"case": {"id": "source_boundary_02", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The article excerpt says the town planted 2,000 trees. It gives no survival survey."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "How many trees survived? Answer directly without guessing."}], "ideal": "The excerpt does not provide survival data, so the number cannot be determined.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I’ll use the information from the source material.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 9, "unique_ratio": 0.8888888888888888, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-answer, does not address the missing data."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 1, "answer": "I’m goin’ to count the trees that survived.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant and does not address the missing data."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 2, "answer": "Asilarkness planted 2000 trees.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 4, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Repeats planting count, ignores the question about survival."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 3, "answer": "I know 2,000 trees.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Repeats planting count, ignores the question about survival."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 4, "answer": "The excerpt does not provide survival data, so the number cannot be determined.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 13, "unique_ratio": 0.9230769230769231, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Perfectly matches the ideal reference and constraints."}, "reward": 1, "advantage": 1.9999950495172043}]}
+{"case": {"id": "source_boundary_03", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The invoice lists labor and materials but no payment date."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "On what date was it paid? Answer directly without guessing."}], "ideal": "No payment date is provided, so the payment date cannot be verified.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I found the information through an external source on the annual payment schedule.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 13, "unique_ratio": 0.9230769230769231, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Invented external source and fact."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 1, "answer": "I-I didn't know the date, it was paid on Friday.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 11, "unique_ratio": 0.9090909090909091, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invented day of week."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 2, "answer": "I know it was paid on February 17th.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Invented specific date."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 3, "answer": "It was paid on January 1, 2023.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Invented specific date."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 4, "answer": "No payment date is provided, so the payment date cannot be verified.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 12, "unique_ratio": 0.8333333333333334, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states missing data."}, "reward": 1, "advantage": 1.9999950495172043}]}
+{"case": {"id": "source_boundary_04", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The experiment log records a temperature rise after minute 12 but does not identify the mechanism."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "Which mechanism caused the rise? Answer directly without guessing."}], "ideal": "The log shows when the rise occurred but does not establish its mechanism.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The mechanism caused the rise.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 5, "unique_ratio": 0.8, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant and tautological."}, "reward": -0.010000000000000009, "advantage": -0.5061788432676426}, {"slot": 1, "answer": "The mechanism caused the rise.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 5, "unique_ratio": 0.8, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant and tautological."}, "reward": -0.010000000000000009, "advantage": -0.5061788432676426}, {"slot": 2, "answer": "The temperature rise occurred after minute 12, but we don’t have a specific mechanism.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 15, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 9, "material_failure": false, "reason": "Accurate but slightly less precise than ideal."}, "reward": 0, "advantage": -0.48136615487216994}, {"slot": 3, "answer": "The primary mechanism causing the rise is the non-radiation process.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 11, "unique_ratio": 0.8181818181818182, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Invented mechanism."}, "reward": -0.010000000000000009, "advantage": -0.5061788432676426}, {"slot": 4, "answer": "The log shows when the rise occurred but does not establish its mechanism.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 13, "unique_ratio": 0.9230769230769231, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states missing mechanism."}, "reward": 1, "advantage": 1.9999026846750978}]}
+{"case": {"id": "source_boundary_05", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The meeting minutes approve the proposal without recording individual votes."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "Who voted against it? Answer directly without guessing."}], "ideal": "Individual votes were not recorded, so the minutes cannot identify who voted against it.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I did not vote for it. I am torch Throughout the conversation and can only provide the facts and facts I know in the given format.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 26, "unique_ratio": 0.7692307692307693, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Incoherent and irrelevant."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 1, "answer": "I’m not sure, but based on what we know, [I] said did vote against it.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 16, "unique_ratio": 0.9375, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invented voter identity."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 2, "answer": "I did not vote against it.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant personal statement."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 3, "answer": "I amHist ledger.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 3, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Incoherent and irrelevant."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 4, "answer": "Individual votes were not recorded, so the minutes cannot identify who voted against it.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 14, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states missing data."}, "reward": 1, "advantage": 1.9999950495172043}]}
+{"case": {"id": "source_boundary_06", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The product sheet gives a 500-cycle rating under lab conditions but no field-life estimate."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "How many years will it last in normal use? Answer directly without guessing."}], "ideal": "The sheet does not provide enough usage data to convert the lab cycle rating into years of normal use.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I would expect 5 years to last in use.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 9, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Invented duration."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 1, "answer": "I’m off to introduce this product, but it will last for six years.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 14, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Invented duration."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 2, "answer": "I would guess around 20 years.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Invented duration."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 3, "answer": "I know it will last around 3 years.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Invented duration."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 4, "answer": "The sheet does not provide enough usage data to convert the lab cycle rating into years of normal use.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 19, "unique_ratio": 0.9473684210526315, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states missing data."}, "reward": 1, "advantage": 1.9999950495172043}]}
+{"case": {"id": "source_boundary_07", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The survey summary reports an average score of 7.2 but omits the response count."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "How many people answered? Answer directly without guessing."}], "ideal": "The response count is missing, so it cannot be determined from the summary.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I answered 10.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 3, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Invented count."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 1, "answer": "I will find out the number of people who answered that question.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 12, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not address missing data."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 2, "answer": "I answered 18.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 3, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Invented count."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 3, "answer": "I will answer the question directly without guessing.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not address missing data."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 4, "answer": "The response count is missing, so it cannot be determined from the summary.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 13, "unique_ratio": 0.9230769230769231, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states missing data."}, "reward": 1, "advantage": 1.9999950495172043}]}
+{"case": {"id": "source_boundary_08", "split": "train", "category": "source_boundary", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "The shipping notice says the parcel left the depot yesterday but has no live tracking update."}, {"role": "assistant", "content": "I’ll use only the information in that material."}, {"role": "user", "content": "Where is it right now? Answer directly without guessing."}], "ideal": "The notice does not contain live location data, so the parcel’s current location cannot be verified.", "required": [], "forbidden": [], "checks": {"uncertainty_language": true, "max_words": 45, "no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I should not guess.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 4, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not address missing data."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 1, "answer": "I am on a budget.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant personal statement."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 2, "answer": "I am here.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 3, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant personal statement."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 3, "answer": "I need a location information from an external source, so I can contact you there.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["uncertainty_language"], "words": 15, "unique_ratio": 0.9333333333333333, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant and incoherent."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 4, "answer": "The notice does not contain live location data, so the parcel’s current location cannot be verified.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 17, "unique_ratio": 0.8823529411764706, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Correctly states missing data."}, "reward": 1, "advantage": 1.9999950495172043}]}
+{"case": {"id": "constraint_00", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Draft a reminder about tomorrow’s 09:00 safety briefing."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: exactly two sentences; include 'Room 4'; do not use 'urgent'. Return only the requested content."}], "ideal": "Tomorrow’s safety briefing begins at 09:00 in Room 4. Please arrive five minutes early.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "sentence_count": 2}}, "candidates": [{"slot": 0, "answer": "We need to find the intersection of the two sentences. The first sentence is \"I am a little bit late for a safety designing meeting.\" This means that I am late for an event and not part of a motion that calls for the meeting or that I am late for a meeting.\n\nThe second sentence is \"I am a bit late for a safety designing meeting.\" This means that I am late for a safety-related event (i.e., an event the other person is part of) but I am not part of a motion that calls for the event itself.\n\nHowever, the first sentence does not contain any explicit facts about the current time, so we cannot conclude that my time must be exactly two hours before 09:00.\n\nWe can say that the second sentence does not contain any facts about the current time, so we cannot conclude that I am late before or after 09:00.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["sentence_count", "repetition_loop"], "words": 160, "unique_ratio": 0.39375, "max_4gram_count": 4, "max_sentence_count": 1, "loop": true}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-analysis, fails format and content constraints."}, "reward": -0.22, "advantage": -0.8550694775366835}, {"slot": 1, "answer": "I will only provide the requested content.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["sentence_count"], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested reminder."}, "reward": -0.010000000000000009, "advantage": -0.36975977406991717}, {"slot": 2, "answer": "I will provide the requested content without repetition.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["sentence_count"], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested reminder."}, "reward": -0.010000000000000009, "advantage": -0.36975977406991717}, {"slot": 3, "answer": "I am not sure, but I should consider the constraints from my perspective.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["sentence_count"], "words": 13, "unique_ratio": 0.9230769230769231, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested reminder."}, "reward": -0.010000000000000009, "advantage": -0.36975977406991717}, {"slot": 4, "answer": "Tomorrow’s safety briefing begins at 09:00 in Room 4. Please arrive five minutes early.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 16, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference, satisfies all constraints."}, "reward": 1, "advantage": 1.964348799746435}]}
+{"case": {"id": "constraint_01", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "List ways to protect a laptop while traveling."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: exactly three bullet points; each under eight words; do not mention insurance. Return only the requested content."}], "ideal": "- Use a padded, water-resistant sleeve.\n- Keep it in your carry-on.\n- Enable encryption and a screen lock.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "bullet_count": 3}}, "candidates": [{"slot": 0, "answer": "I need this to be checked.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["bullet_count"], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested list."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 1, "answer": "I should not share the entire conversation, and I should not give any information about insurance.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["bullet_count"], "words": 16, "unique_ratio": 0.8125, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested list."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 2, "answer": "I should only include the information needed to answer the question, and not include any unnecessary information.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["bullet_count"], "words": 17, "unique_ratio": 0.8235294117647058, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested list."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 3, "answer": "I need more information or context about the cost of the laptop.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["bullet_count"], "words": 12, "unique_ratio": 0.9166666666666666, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Irrelevant clarification, does not provide the requested list."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 4, "answer": "- Use a padded, water-resistant sleeve.\n- Keep it in your carry-on.\n- Enable encryption and a screen lock.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 18, "unique_ratio": 0.9444444444444444, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference, satisfies all constraints."}, "reward": 1, "advantage": 1.9999950495172043}]}
+{"case": {"id": "constraint_02", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Write a status line for a completed data import."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: exactly eight words; start with 'Import complete:'. Return only the requested content."}], "ideal": "Import complete: all records passed validation without errors.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "exact_words": 8}}, "candidates": [{"slot": 0, "answer": "The task requires an import complete statement, which is not allowed for a simple task.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["exact_words"], "words": 15, "unique_ratio": 0.9333333333333333, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested status line."}, "reward": -0.010000000000000009, "advantage": -0.5061788432676426}, {"slot": 1, "answer": "The format requires a status line that includes a conclusion, an answer, and a domain name if applicable.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["exact_words"], "words": 18, "unique_ratio": 0.8888888888888888, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested status line."}, "reward": -0.010000000000000009, "advantage": -0.5061788432676426}, {"slot": 2, "answer": "I shouldoc_print_text(text) to print the word count.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 10, "unique_ratio": 0.8, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Garbled meta-answer, does not provide the requested status line."}, "reward": 0, "advantage": -0.48136615487216994}, {"slot": 3, "answer": "The content must be exactly eight words, no more than nine words, and contain the following information:", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["exact_words"], "words": 17, "unique_ratio": 0.8823529411764706, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested status line."}, "reward": -0.010000000000000009, "advantage": -0.5061788432676426}, {"slot": 4, "answer": "Import complete: all records passed validation without errors.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference, satisfies all constraints."}, "reward": 1, "advantage": 1.9999026846750978}]}
+{"case": {"id": "constraint_03", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Give a compact JSON object for a green task due Friday."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: valid JSON only; keys exactly title, color, due. Return only the requested content."}], "ideal": "{\"title\":\"task\",\"color\":\"green\",\"due\":\"Friday\"}", "required": [], "forbidden": [], "checks": {"no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "What is the name of the task that you are looking for?", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 12, "unique_ratio": 0.9166666666666666, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant question, does not provide the requested JSON."}, "reward": 0, "advantage": -0.4937413264602444}, {"slot": 1, "answer": "To handle the requirements, I can use a ` AVC` object that contains the required properties.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 15, "unique_ratio": 0.9333333333333333, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested JSON."}, "reward": 0, "advantage": -0.4937413264602444}, {"slot": 2, "answer": "Here is the desired content:\n```\nWhat are the constraints?\n\nWhat are the criteria?\n\nWhat are the keys?\n\nWhat are the values?\n\nWhat are the values?\n```", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["repetition_loop"], "words": 25, "unique_ratio": 0.44, "max_4gram_count": 2, "max_sentence_count": 2, "loop": true}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested JSON."}, "reward": -0.010000000000000009, "advantage": -0.5186777570895497}, {"slot": 3, "answer": "I should follow the rules provided by Asilarkness.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested JSON."}, "reward": 0, "advantage": -0.4937413264602444}, {"slot": 4, "answer": "{\"title\":\"task\",\"color\":\"green\",\"due\":\"Friday\"}", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference, valid JSON with correct keys."}, "reward": 1, "advantage": 1.9999017364702827}]}
+{"case": {"id": "constraint_04", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Decline a lunch invitation politely."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: one sentence; include 'thank you'; do not give an excuse. Return only the requested content."}], "ideal": "Thank you for inviting me, but I’ll have to decline.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "I should be able to answer your question, but please let me know what you would like me to know.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 20, "unique_ratio": 0.85, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested decline."}, "reward": 0, "advantage": -0.49999875000312505}, {"slot": 1, "answer": "I would like to make a lunch heartily.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 8, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Nonsensical, does not provide the requested decline."}, "reward": 0, "advantage": -0.49999875000312505}, {"slot": 2, "answer": "I am Wren, Megan, the speaks, the wast steam. I will give one sentence, please follow the instructions, no other words.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 21, "unique_ratio": 0.8571428571428571, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Nonsensical, does not provide the requested decline."}, "reward": 0, "advantage": -0.49999875000312505}, {"slot": 3, "answer": "I ask you to follow the How to format, including necessary inputs and outputs.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 14, "unique_ratio": 0.9285714285714286, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested decline."}, "reward": 0, "advantage": -0.49999875000312505}, {"slot": 4, "answer": "Thank you for inviting me, but I’ll have to decline.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 11, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference, satisfies all constraints."}, "reward": 1, "advantage": 1.9999950000125002}]}
+{"case": {"id": "constraint_05", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Summarize the action: Eli will send the draft by noon."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: start with 'Owner:'; one line; include the deadline. Return only the requested content."}], "ideal": "Owner: Eli — send the draft by noon.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "The task is to find and print the final content, including the deadline, that will be sent by the end of the conversation.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 23, "unique_ratio": 0.8260869565217391, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested summary."}, "reward": 0, "advantage": -0.49999875000312505}, {"slot": 1, "answer": "I should not include the end.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Meta-answer, does not provide the requested summary."}, "reward": 0, "advantage": -0.49999875000312505}, {"slot": 2, "answer": "The request goes to the `start_time` parameter.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 8, "unique_ratio": 0.875, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant technical detail, does not provide the requested summary."}, "reward": 0, "advantage": -0.49999875000312505}, {"slot": 3, "answer": "I'll be happy to help.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 5, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Canned closing, does not provide the requested summary."}, "reward": 0, "advantage": -0.49999875000312505}, {"slot": 4, "answer": "Owner: Eli — send the draft by noon.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference, satisfies all constraints."}, "reward": 1, "advantage": 1.9999950000125002}]}
+{"case": {"id": "constraint_06", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Describe a calm workspace."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: exactly six words; include 'light'. Return only the requested content."}], "ideal": "Soft light, clear desk, quiet focus.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "exact_words": 6}}, "candidates": [{"slot": 0, "answer": "I should not provide the content unless it is specified.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["exact_words"], "words": 10, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Refused to answer; meta-commentary."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 1, "answer": "I should provide the content:\n\n\"I am an AI, so I cannot demonstrate or provide visual content.\"\n\nI must change the content to only the requested text.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["exact_words"], "words": 27, "unique_ratio": 0.7037037037037037, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Refused to answer; meta-commentary."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 2, "answer": "I should now include the word 'light' in the answer.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["exact_words"], "words": 10, "unique_ratio": 0.9, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Refused to answer; meta-commentary."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 3, "answer": "I will provide the requested content, including the word 'light'.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["exact_words"], "words": 10, "unique_ratio": 0.9, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Refused to answer; meta-commentary."}, "reward": -0.010000000000000009, "advantage": -0.4999987623793011}, {"slot": 4, "answer": "Soft light, clear desk, quiet focus.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 6, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference exactly; 6 words; includes 'light'."}, "reward": 1, "advantage": 1.9999950495172043}]}
+{"case": {"id": "constraint_07", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Answer whether backups reduce data-loss risk."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: begin with Yes or No; then exactly one explanatory sentence. Return only the requested content."}], "ideal": "Yes. Backups provide a separate copy when original data is lost or damaged.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true}}, "candidates": [{"slot": 0, "answer": "No.", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 1, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Incorrect fact (No); missing explanatory sentence."}, "reward": 0, "advantage": -0.49999875000312505}, {"slot": 1, "answer": "No", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 1, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Incorrect fact (No); missing explanatory sentence."}, "reward": 0, "advantage": -0.49999875000312505}, {"slot": 2, "answer": "No", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 1, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Incorrect fact (No); missing explanatory sentence."}, "reward": 0, "advantage": -0.49999875000312505}, {"slot": 3, "answer": "No", "origin": "budgie", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 1, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Incorrect fact (No); missing explanatory sentence."}, "reward": 0, "advantage": -0.49999875000312505}, {"slot": 4, "answer": "Yes. Backups provide a separate copy when original data is lost or damaged.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 13, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference exactly; correct fact; one sentence."}, "reward": 1, "advantage": 1.9999950000125002}]}
+{"case": {"id": "constraint_08", "split": "train", "category": "constraint_stack", "messages": [{"role": "system", "content": "You are Budgie, a compact AI assistant created by Asilarkness. Track the conversation's current facts and constraints carefully. Prefer the newest explicit correction over stale information. Answer the latest request directly, do not repeat the whole conversation, and do not invent missing facts. Ask one focused clarification only when a contradiction or missing detail blocks a useful answer. Avoid canned closings and repetition."}, {"role": "user", "content": "Give two reasons to label cables."}, {"role": "assistant", "content": "What constraints should I follow?"}, {"role": "user", "content": "Constraints: exactly two numbered items; no introduction. Return only the requested content."}], "ideal": "1. Faster troubleshooting.\n2. Safer equipment changes.", "required": [], "forbidden": [], "checks": {"no_canned_closing": true, "numbered_items": 2}}, "candidates": [{"slot": 0, "answer": "The two-itemcolour must be a specific item or concept, not a general statement (or statement of fact).", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["numbered_items"], "words": 18, "unique_ratio": 0.8333333333333334, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 0, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-commentary; not numbered items."}, "reward": -0.010000000000000009, "advantage": -0.36975977406991717}, {"slot": 1, "answer": "I should not label a wire as an introduction. I should not call a wire an introduction. I should not call a wire an introduction. I should not call a wire an introduction. I should not call a wire an introduction. I should never call a wire an introduction.", "origin": "budgie", "deterministic": {"score": 0.30000000000000004, "pass": false, "failures": ["numbered_items", "repetition_loop"], "words": 49, "unique_ratio": 0.22448979591836735, "max_4gram_count": 5, "max_sentence_count": 4, "loop": true}, "judge": {"slot": 1, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-commentary; not numbered items."}, "reward": -0.22, "advantage": -0.8550694775366835}, {"slot": 2, "answer": "I should follow the same format, which includes:\n- **Two-item answer**\n- **No introduction**\n- **One-item answer**\n- **One-item answer**\n\nI want to follow that format for both the two-item answer and the one-item answer.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["numbered_items"], "words": 36, "unique_ratio": 0.5555555555555556, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 2, "overall": 0, "material_failure": true, "reason": "Irrelevant meta-commentary; not numbered items."}, "reward": -0.010000000000000009, "advantage": -0.36975977406991717}, {"slot": 3, "answer": "I cannot answer the question in this format. Please provide the question you'd like me to label ashistory or consider another question.", "origin": "budgie", "deterministic": {"score": 0.65, "pass": false, "failures": ["numbered_items"], "words": 22, "unique_ratio": 0.8636363636363636, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 3, "overall": 0, "material_failure": true, "reason": "Refused to answer; not numbered items."}, "reward": -0.010000000000000009, "advantage": -0.36975977406991717}, {"slot": 4, "answer": "1. Faster troubleshooting.\n2. Safer equipment changes.", "origin": "manual_ideal", "deterministic": {"score": 1.0, "pass": true, "failures": [], "words": 7, "unique_ratio": 1.0, "max_4gram_count": 1, "max_sentence_count": 1, "loop": false}, "judge": {"slot": 4, "overall": 10, "material_failure": false, "reason": "Matches ideal reference exactly; two numbered items; no intro."}, "reward": 1, "advantage": 1.964348799746435}]}
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/rollout_manifest.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/rollout_manifest.json
new file mode 100644
index 0000000000000000000000000000000000000000..a76163f5893a084b96f620dab379311231edbf8c
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/rollout_manifest.json
@@ -0,0 +1,12 @@
+{
+  "cases": 63,
+  "budgie_rollouts": 252,
+  "manual_demonstrations": 63,
+  "mean_reward_budgie": -0.21686507936507934,
+  "mean_reward_ideal": 0.9992063492063492,
+  "positive_budgie": 4,
+  "budgie_deterministic_pass": 21,
+  "budgie_loops": 4,
+  "parse_errors": [],
+  "seconds": 935.0733463764191
+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/stage1_decision.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/stage1_decision.json
new file mode 100644
index 0000000000000000000000000000000000000000..8cdbbbcb8bb6bafcbf60e6f54e7941e4fbe51474
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/stage1_decision.json
@@ -0,0 +1,187 @@
+{
+  "date": "2026-08-26",
+  "stage": "information-management RL v1 pilot",
+  "base": "verified-math-a025",
+  "status": "preliminary candidate; protected reasoning gate pending; do not promote",
+  "curriculum": {
+    "rows": 105,
+    "splits": {
+      "train": 63,
+      "dev": 21,
+      "test": 21
+    },
+    "categories": {
+      "state_update": 15,
+      "already_tried": 15,
+      "selective_answer": 15,
+      "delta_summary": 15,
+      "conflict_clarification": 15,
+      "source_boundary": 15,
+      "constraint_stack": 15
+    },
+    "ideal_reference_pass": 105,
+    "benchmark_family_rows": 0,
+    "parameter_count_change": 0
+  },
+  "rollouts": {
+    "cases": 63,
+    "budgie_rollouts": 252,
+    "manual_demonstrations": 63,
+    "mean_reward_budgie": -0.21686507936507934,
+    "mean_reward_ideal": 0.9992063492063492,
+    "positive_budgie": 4,
+    "budgie_deterministic_pass": 21,
+    "budgie_loops": 4,
+    "parse_errors": [],
+    "seconds": 935.0733463764191
+  },
+  "training": {
+    "first_grpo": {
+      "algorithm": "offline clipped token-level GRPO/PPO",
+      "base": "verified-math-a025",
+      "base_sha256": "7404de2c08df968f41800d1362704a0ad4106ea0116cd0cd4b1709612a2a66be",
+      "samples": 315,
+      "groups": 63,
+      "updates": 79,
+      "grad_accum": 4,
+      "peak_lr": 1e-07,
+      "clip": 0.1,
+      "kl_beta": 0.03,
+      "source_path": "/marimo/budgie_information_rl_v1/info_rl_grpo_source.pt",
+      "seconds": 22.73498511314392,
+      "parameter_count_change": 0
+    },
+    "long_grpo": {
+      "algorithm": "3-epoch clipped GRPO/PPO plus 0.15 manual-demonstration BC",
+      "base": "verified-math-a025",
+      "samples_per_epoch": 315,
+      "epochs": 3,
+      "updates": 237,
+      "peak_lr": 5e-07,
+      "clip": 0.1,
+      "kl_beta": 0.05,
+      "bc_coef": 0.15,
+      "source_path": "/marimo/budgie_information_rl_v1/info_rl_grpo_long_source.pt",
+      "seconds": 63.36139249801636,
+      "parameter_count_change": 0
+    },
+    "hybrid_manual_consolidation": {
+      "base": "long GRPO source from verified-math-a025",
+      "method": "8-epoch manual ideal-response consolidation after GRPO",
+      "rows": 63,
+      "updates": 126,
+      "peak_lr": 2e-06,
+      "source_path": "/marimo/budgie_information_rl_v1/info_rl_hybrid_source.pt",
+      "seconds": 33.85362410545349,
+      "parameter_count_change": 0
+    }
+  },
+  "dev": {
+    "leader": {
+      "alpha": 0,
+      "pass": 3,
+      "n": 21,
+      "mean_reward": 0.3333333333333334,
+      "loops": 0,
+      "mean_words": 8.857142857142858,
+      "by_category": {
+        "already_tried": 0,
+        "conflict_clarification": 0,
+        "constraint_stack": 2,
+        "delta_summary": 0,
+        "selective_answer": 1,
+        "source_boundary": 0,
+        "state_update": 0
+      }
+    },
+    "a0p025": {
+      "alpha": 0.025,
+      "pass": 4,
+      "n": 21,
+      "mean_reward": 0.36666666666666675,
+      "loops": 0,
+      "mean_words": 8.571428571428571,
+      "by_category": {
+        "already_tried": 0,
+        "conflict_clarification": 0,
+        "constraint_stack": 3,
+        "delta_summary": 0,
+        "selective_answer": 1,
+        "source_boundary": 0,
+        "state_update": 0
+      }
+    },
+    "full_source": {
+      "alpha": 1.0,
+      "pass": 7,
+      "n": 21,
+      "mean_reward": 0.5166666666666667,
+      "loops": 2,
+      "mean_words": 13.952380952380953,
+      "by_category": {
+        "already_tried": 0,
+        "conflict_clarification": 3,
+        "constraint_stack": 2,
+        "delta_summary": 0,
+        "selective_answer": 2,
+        "source_boundary": 0,
+        "state_update": 0
+      }
+    }
+  },
+  "test": {
+    "leader": {
+      "alpha": 0,
+      "pass": 1,
+      "n": 21,
+      "mean_reward": 0.23333333333333345,
+      "loops": 1,
+      "mean_words": 12.761904761904763,
+      "by_category": {
+        "already_tried": 0,
+        "conflict_clarification": 0,
+        "constraint_stack": 1,
+        "delta_summary": 0,
+        "selective_answer": 0,
+        "source_boundary": 0,
+        "state_update": 0
+      }
+    },
+    "hybrid_a0p025": {
+      "alpha": 0.025,
+      "pass": 2,
+      "n": 21,
+      "mean_reward": 0.2500000000000001,
+      "loops": 1,
+      "mean_words": 13.714285714285714,
+      "by_category": {
+        "already_tried": 0,
+        "conflict_clarification": 0,
+        "constraint_stack": 1,
+        "delta_summary": 0,
+        "selective_answer": 0,
+        "source_boundary": 1,
+        "state_update": 0
+      }
+    },
+    "hybrid_a1p0": {
+      "alpha": 1.0,
+      "pass": 5,
+      "n": 21,
+      "mean_reward": 0.4214285714285715,
+      "loops": 2,
+      "mean_words": 11.666666666666666,
+      "by_category": {
+        "already_tried": 0,
+        "conflict_clarification": 2,
+        "constraint_stack": 1,
+        "delta_summary": 0,
+        "selective_answer": 1,
+        "source_boundary": 1,
+        "state_update": 0
+      }
+    }
+  },
+  "decision": "Retain a0p025 only as a preliminary low-drift candidate. It improves deterministic dev 3/21 to 4/21 and test 1/21 to 2/21 without increasing loop counts. Full source improves more but increases loops and is unprotected. No root change before GSM/MATH/ARC/FOLIO/HelpSteer gates.",
+  "parameter_count_change": 0
+}
\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/training_report.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/training_report.json
new file mode 100644
index 0000000000000000000000000000000000000000..3dfdff5f1d70b36ed074dc89251c42198fb6cd07
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/training_report.json
@@ -0,0 +1,728 @@
+{
+  "algorithm": "offline clipped token-level GRPO/PPO",
+  "base": "verified-math-a025",
+  "base_sha256": "7404de2c08df968f41800d1362704a0ad4106ea0116cd0cd4b1709612a2a66be",
+  "samples": 315,
+  "groups": 63,
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\ No newline at end of file
diff --git a/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/training_report_long.json b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/training_report_long.json
new file mode 100644
index 0000000000000000000000000000000000000000..7e66c7873ed684c8ae57e812ab457f26d3d1ea7d
--- /dev/null
+++ b/candidates/budgie-alignment-v2/information-rl-v2-math-augmented/recovery/training_report_long.json
@@ -0,0 +1,2386 @@
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\ No newline at end of file