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|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Given that walking at speed $s$ takes 4 hours total (including $t$ minutes in coffee), and the distance is 9 km, calculate the time spent walking in minutes. Let $T_1 = 4$ hours total. Convert this to minutes: $4 \times 60 = 240$ minutes. The walking ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 180 | false | true | false | aime24 | aime24_00000 | original | 0 | 0 | original/aime24/binary-mean-center/decompose/aime24_00000/0 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 180 | <action>
AGGREGATE
From path 1, we have $s = 540 / (240 - t)$.
From path 2, we have $s + 2 = 540 / (144 - t)$.
Substitute $s$ from the first result into the second result:
$540 / (240 - t) + 2 = 540 / (144 - t)$.
Multiply through by $(240 - t)(144 - t)$ to clear denominators:
$540(144 - t) + 2(240 - t)(144 - t) = 540(2... | [
{
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"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffe... | 3df9f06bfd5c92455b776d403fab08bb3cff8aedda5f2339d76273dd88237eef | 4 | 1,236 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 1 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 75768bda490c718a7fab9318d34e8c8493115427dc957239110d988bd8d128bf | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $s$ be the walking speed in km/h and $t$ be the time spent in the coffee shop in minutes. When walking at $s$ km/h, the total time is 4 hours (240 minutes). The equation is $9/s + t/60 = 240$. When walking at $s+2$ km/h, the total time is 2 hours... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 249 | false | true | false | aime24 | aime24_00000 | original | 0 | 1 | original/aime24/binary-mean-center/decompose/aime24_00000/1 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 249 | <action>
AGGREGATE
From the first path, we determined the system of equations:
1) $9/s + t/60 = 240$
2) $9/(s+2) + t/60 = 144$
Solving this system, we found $s = 0.25$ km/h and $t = 14220$ minutes.
From the second path, we determined the formula for the total time at speed $s + 0.5$ km/h:
$9/(s + 0.5) + t/60$
Substitut... | [
{
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"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffe... | 30718a00a08356175add2705c4e753265fd8d539e11aa17695137117ada96fd9 | 4 | 1,272 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 2 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 282671e8085f8530b4021ea31e5cc3e461e317341175a6b6d01b269f1587731c | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Aya walks a 9-kilometer distance. When she walks at a constant speed of $s$ km/h, the total time (walking + coffee shop) is 4 hours. The walking time is $9/s$ hours. Given the total time is 4 hours and the coffee shop time is $t$ minutes, convert the... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 196.3636...(or1964/11)minutes | false | true | false | aime24 | aime24_00000 | original | 0 | 2 | original/aime24/binary-mean-center/decompose/aime24_00000/2 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 196.3636... (or 196 4/11) minutes. | <action>
AGGREGATE
From the first path, we found that $s = 2.25$ km/h.
From the second path, we found that $s = 1.75$ km/h.
There is an inconsistency. The problem states that the coffee shop time $t$ is the same in both scenarios, but the derived speeds $s$ are different. This suggests an error in the interpretation of... | [
{
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"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffe... | 80c204cd63d54f7494d19f7d7aedd2a1e84cea19234698092eef71bb583eebd3 | 4 | 957 | false | completed | [] | {"correct":false,"gold_parsed":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"unparsed","prediction_parsed":false,"resolved":false,"transport_retries":0} | {"adjudication_key":"108fb4711de50ba2d39534eba18bf1afd693395657762bead749560cfd9aa58c","correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"adjudicated","resolved":true} | {"automated_check":{"correct":false,"gold_parsed":true,"method":"unparsed","prediction_parsed":false,"resolved":false},"correct":false,"gold":"204","method":"codex_review","normalized_prediction":"196.3636... (or 196 4/11) minutes.","pair_index":162,"prediction":"196.3636... (or 196 4/11) minutes.","rationale":"196.363... | adjudicated | Codex | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 3 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 5be4d6361b0cf365b54a64c538e4c2e42e5869e67ce8e5ccc88fdb25cfd36d58 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $s$ be the walking speed in km/h and $t$ be the time spent in the coffee shop in minutes. When Aya walks at $s$ km/h, the total time taken is 4 hours (240 minutes). The walking time is 240 - $t$ minutes. The distance is 9 km. Set up the equation:... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 204 | true | true | true | aime24 | aime24_00000 | original | 0 | 3 | original/aime24/binary-mean-center/decompose/aime24_00000/3 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 204 | <action>
AGGREGATE
We have two expressions for $s$:
1) $s = \frac{540}{240 - t}$
2) $s = \frac{252 + 2t}{144 - t}$
Set them equal: $\frac{540}{240 - t} = \frac{252 + 2t}{144 - t}$
Cross-multiply:
$540(144 - t) = (252 + 2t)(240 - t)$
Expand both sides:
$77760 - 540t = 60480 - 252t + 480t - 2t^2$
$77760 - 540t = 60480... | [
{
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"depth": 0,
"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffe... | 82d16818d644b7851b1b00cba81ba462257cfa97234a25b439e8ebfc36a30b38 | 4 | 1,330 | false | completed | [] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 4 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | b336b2ddf098aa3c0097e58a21bf81687ea65730f1bdec95c65288134a48e1c3 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Aya walks 9 kilometers. Let $s$ be her speed in km/h. The time taken for the walk is 4 hours minus $t$ minutes. Express the time for the walk in hours as an equation: $\frac{9}{s} = 4 - \frac{t}{60}$. Solve for $t$ in terms of $s$.
</path>
<path>
Aya... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 324 | false | true | false | aime24 | aime24_00000 | original | 0 | 4 | original/aime24/binary-mean-center/decompose/aime24_00000/4 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 324 | <action>
AGGREGATE
We have two expressions for $t$:
1) $t = 240 - \frac{540}{s}$
2) $t = 144 - \frac{540}{s+2}$
Equating the two:
$240 - \frac{540}{s} = 144 - \frac{540}{s+2}$
$96 = \frac{540}{s} - \frac{540}{s+2}$
$96 = 540(\frac{1}{s} - \frac{1}{s+2})$
$96 = 540(\frac{s+2 - s}{s(s+2)})$
$96 = 540(\frac{2}{s^2 + 2s})... | [
{
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"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffe... | 2156bd12d59e5d761043f493084d1055e72d77e727c75e198904a1a120f212c8 | 4 | 974 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 5 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 02f7adc07762b87daa7a7a40aada05fd73273c8dcd747276f87d89ba3a0fcb9e | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the time spent walking (excluding the coffee shop) for the first scenario. Given: total time = 4 hours, speed = $s$ km/h, distance = 9 km. Let $t$ be the time spent in the coffee shop in minutes. Calculate the walking time in minutes.
</pat... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 666 | false | true | false | aime24 | aime24_00000 | original | 0 | 5 | original/aime24/binary-mean-center/decompose/aime24_00000/5 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 666 | <action>
AGGREGATE
From the first path, the walking time is $240 - t$ minutes.
From the second path, the walking time is $\frac{144(s+2) - 540}{s+2}$ minutes.
Set them equal: $240 - t = \frac{144s + 288 - 540}{s+2}$.
Simplify the right side: $240 - t = \frac{144s - 252}{s+2}$.
Multiply by $(s+2)$: $(240 - t)(s + 2) = 1... | [
{
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"depth": 0,
"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffe... | f7cdb16bf392ed325320f3cdcd4e88a94ad17bd01f247e6ea9651e01519299b8 | 4 | 865 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 6 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | b79d8d968700f4a0bc98403138cde4d9ff115d12d899a9f4d497f2c786849ad5 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $d$ be the walking distance in kilometers, $s$ be the walking speed in km/h, and $t$ be the time spent at the coffee shop in minutes. Given $d = 9$ km, the first scenario is $d = s \times (4 \text{ hours} - t/60 \text{ hours})$. Calculate $s$.
</... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 196.6 | false | true | false | aime24 | aime24_00000 | original | 0 | 6 | original/aime24/binary-mean-center/decompose/aime24_00000/6 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 196.6 | <action>
AGGREGATE
Using $t=1$ from the second path in the formula $s = 540 / (240 - t)$ from the first path:
$s = 540 / (240 - 1) = 540 / 239$.
The new speed is $s + 0.5 = 540/239 + 1/2 = 540/239 + 119.5/239 = 659.5/239$.
The time for the walk is $9 / (659.5/239) = 9 \times 239 / 659.5 = 2151 / 659.5$.
Since $659.5 = ... | [
{
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"depth": 0,
"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffe... | eb592835dbcc619a00ecb648f24c4cf1bb4736b9e1d3bda9de0446d95d6f0bf5 | 4 | 999 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 7 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | b8788339a27225ffe2a85ca93e99ee7c9e54418396b83aa5d9af576ce90662c9 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Aya walks 9 kilometers at a constant speed $s$ kilometers per hour. The total time taken, including $t$ minutes in the coffee shop, is 4 hours. Express the time spent walking in terms of $s$ and $t$.
</path>
<path>
Aya walks 9 kilometers at a constant... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 201 | false | true | false | aime24 | aime24_00000 | original | 0 | 7 | original/aime24/binary-mean-center/decompose/aime24_00000/7 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 201 | <action>
AGGREGATE
From the first path, the time spent walking at speed $s$ is $\frac{240 - t}{60}$ hours.
From the second path, the time spent walking at speed $s+2$ is $2.4 - \frac{t}{60}$ hours.
The distance walked is 9 km in both cases, so:
$s = \frac{9}{\frac{240 - t}{60}} = \frac{9 \times 60}{240 - t} = \frac{540... | [
{
"tree_index": 7,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffe... | 63aa5c7f4e4af10ad4341dfdb62570ad287d773e46ef4fcbdbfdbe6c68110a26 | 4 | 2,049 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 8 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 100749539f7ae7ee99f7c90df0685d24e8119b7f2eb899c546032815a59daba2 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the first scenario, Aya walks 9 kilometers at a constant speed of $s$ kilometers per hour, and the total time is 4 hours. The time spent walking is 4 hours minus $t$ minutes. Given $9 = s \times (4 - t/60)$, find the relationship between $s$ and $... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 39.76 | false | true | false | aime24 | aime24_00000 | original | 0 | 8 | original/aime24/binary-mean-center/decompose/aime24_00000/8 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 39.76 | <action>
AGGREGATE
From the first path, we have $s = 540 / (40 - t)$.
From the second path, we have $s = 9 / (2.4 - t/60) - 2$.
Equating the two expressions for $s$:
$540 / (40 - t) = 9 / (2.4 - t/60) - 2$.
Multiply through by $(40 - t)(2.4 - t/60)$:
$540(2.4 - t/60) = 9(40 - t) - 2(40 - t)(2.4 - t/60)$.
$1296 - 9t = 3... | [
{
"tree_index": 8,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffe... | 6fe647f526bb6e6ca39394487d903b8890c32720d8ca4ba6a041c254f0ad9bec | 4 | 1,147 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 9 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 41a46368013c34824a0ce61d1f73970362673c3a04711872ba257b55ac49d3ab | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the time spent walking (excluding coffee shop time) for both scenarios. Let $d = 9$ km. In the first scenario, speed $s$ and total time $4$ hours ($240$ minutes). Let $T_1$ be the walking time. In the second scenario, speed $s+2$ and total ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | null | false | false | false | aime24 | aime24_00000 | original | 0 | 9 | original/aime24/binary-mean-center/decompose/aime24_00000/9 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | null | <action>
AGGREGATE
From the first path, we have the walking times $T_1 = 228$ and $T_2 = 132$.
From the second path, we solved the system of equations to find $s = -1 + \frac{\sqrt{17}}{4}$ and $t = 12$.
Now, we need to find the total time for the new speed $s + \frac{1}{2}$.
The new speed is $s + \frac{1}{2} = -1 + \f... | [
{
"tree_index": 9,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffe... | 9289f29c4e1c7a0c8dd7f7f3d4b07d6199670c16c6a5823296d13f62ae36a805 | 4 | 3,023 | true | parse_error | ["call_4: missing <action>...</action> block"] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"missing_answer","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"missing_answer","resolved":true,"transport_retries":0} | null | missing_answer | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 10 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 385ad60f93ac7c1fefe67ef3aae79c222144b8fa3093d6715e908af6d2e0ee88 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Aya walks 9 kilometers at a constant speed $s$ kilometers per hour. The total time taken is 4 hours, which includes $t$ minutes spent in the coffee shop. Express the time spent walking in terms of $s$ and $t$. The walking time is $4$ hours $- t$ minut... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 232 | false | false | false | aime24 | aime24_00000 | original | 0 | 10 | original/aime24/binary-mean-center/decompose/aime24_00000/10 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 232 | <action>
AGGREGATE
The two paths provided inconsistent values for $t$ (236 vs 144). This indicates an error in the problem setup or constraints. The original problem states that $t$ is a constant time spent in the coffee shop for both scenarios. However, the derived equations lead to different values for $t$, which is ... | [
{
"tree_index": 10,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coff... | 22721305b41eaf9d23224649d237384fe4e673742492ae523efea7fe9d57fb53 | 4 | 3,263 | true | parse_error | ["call_3: missing <action>...</action> block"] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 11 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 536b72c60615911f28da801bddbc786f708b870906f3c623743f86b594015bc7 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $s$ be the walking speed in km/h and $t$ be the time spent in the coffee shop in minutes. The total time for the walk (excluding the coffee shop) is 4 hours - $t$ minutes. Since the distance is 9 km, we have $s = \frac{9}{4 - \frac{t}{60}}$. Solve... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 204 | true | true | true | aime24 | aime24_00000 | original | 0 | 11 | original/aime24/binary-mean-center/decompose/aime24_00000/11 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 204 | <action>
AGGREGATE
From the first path, we have $s = \frac{9}{4 - \frac{t}{60}}$.
From the second path, we have $s + 2 = \frac{9}{2.4 - \frac{t}{60}}$.
Substitute $s = \frac{9}{4 - \frac{t}{60}}$ into the second equation:
$\frac{9}{4 - \frac{t}{60}} + 2 = \frac{9}{2.4 - \frac{t}{60}}$.
Let $u = \frac{t}{60}$. Then the ... | [
{
"tree_index": 11,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coff... | 6316f05bd106607d8662adee61e415e7a980be8705561c3b05a7ce5abe3b72e6 | 4 | 1,105 | false | completed | [] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 12 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 2b97ed1233e260fae879c77724454ed3f4a5cd8257b53eed8f890048fa62d417 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $s$ be the walking speed in kilometers per hour. The total time for the walk including the coffee shop is 4 hours when the speed is $s$ km/h. The distance walked is 9 km. The time spent walking is the total time minus the time spent in the coffee... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 207 | false | true | false | aime24 | aime24_00000 | original | 0 | 12 | original/aime24/binary-mean-center/decompose/aime24_00000/12 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 207 | <action>
AGGREGATE
From path 1, we have $s = \frac{9}{4 - t/60}$. From path 2, we have $\frac{9}{s+2} = 2.4 - \frac{t}{60}$.
Substitute $s = \frac{9}{4 - t/60}$ into the second equation:
$\frac{9}{\frac{9}{4 - t/60} + 2} = 2.4 - \frac{t}{60}$.
Simplify the denominator:
$\frac{9}{\frac{9 + 2(4 - t/60)}{4 - t/60}} = 2.... | [
{
"tree_index": 12,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coff... | af5172658f55ee7a04ec2878057a6d7738a5be0c5579b8633a1cd92612626e06 | 4 | 1,993 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 13 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | a833e436fa46a25869d067d522e3d123279a188931d7dd8915f832571a948399 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Aya walks 9 kilometers at a speed of $s$ km/h and takes 4 hours total, including $t$ minutes in a coffee shop. Calculate the value of $s$.
</path>
<path>
Aya walks 9 kilometers at a speed of $s+2$ km/h and takes 2 hours and 24 minutes total, includin... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 392/11 | false | true | false | aime24 | aime24_00000 | original | 0 | 13 | original/aime24/binary-mean-center/decompose/aime24_00000/13 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 392/11 | <action>
AGGREGATE
From the first path, we determined the speed $s = 2.25$ km/h.
From the second path, we determined the time spent in the coffee shop $t = 16$ minutes.
Now, we calculate the time taken for the walk at the new speed $s + \frac{1}{2} = 2.25 + 0.5 = 2.75$ km/h.
The time taken for the walk is the distance ... | [
{
"tree_index": 13,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coff... | 4f3866bac9a81c0a64cc7ee64b484a74e05a63923a2c7009d90ce42c35114251 | 4 | 795 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 14 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 5e0cfaa050a1f2a02e4101f90a80e93a304cc1097d822e495a2fb99c504011ec | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Aya walks 9 kilometers at a constant speed $s$ and takes 4 hours total, including $t$ minutes in a coffee shop. Calculate the value of $s$.
</path>
<path>
Aya walks 9 kilometers at a constant speed $s+2$ and takes 2 hours and 24 minutes total, includ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 180 | false | true | false | aime24 | aime24_00000 | original | 0 | 14 | original/aime24/binary-mean-center/decompose/aime24_00000/14 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 180 | <action>
AGGREGATE
From the first path, we have $s = \frac{9}{4 - t/60}$.
From the second path, we have $t = 24$.
Substituting $t = 24$ into the expression for $s$:
$s = \frac{9}{4 - 24/60} = \frac{9}{4 - 0.4} = \frac{9}{3.6} = 2.5$ km/h.
Now, we need to find the time taken when $s = 2.5 + 0.5 = 3.0$ km/h.
Time = Dista... | [
{
"tree_index": 14,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coff... | 8710bbaee2466f5d449d2233850014504007f2b248f34fb20d8a84f450d2518f | 4 | 794 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 15 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | d1b922967fecbd27491cb882c65c81df2d5ae15a6d0532b55e451dd170561e56 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coffee shop. When she walks $s+2$ kilometers per hour, the walk takes her 2 hours and 24 minutes, incl... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $s$ be the walking speed in kilometers per hour. The distance walked is 9 kilometers. The time taken for the walk is 4 hours minus the time spent in the coffee shop, which is $t$ minutes. The total time for the walk (excluding the coffee shop) is... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 204 | 204 | true | true | true | aime24 | aime24_00000 | original | 0 | 15 | original/aime24/binary-mean-center/decompose/aime24_00000/15 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 204 | <action>
AGGREGATE
We have the system:
1) $s = 540 / (240 - t)$
2) $s = 9 / (2.4 - t/60) - 2$
Equating the two expressions for $s$:
$540 / (240 - t) = 9 / (2.4 - t/60) - 2$
Simplify the right side:
$540 / (240 - t) = 9 / (2.4 - t/60) - 2$
$540 / (240 - t) = 9 / (144/60 - t/60) - 2$
$540 / (240 - t) = 9 / ((144 - t) /... | [
{
"tree_index": 15,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Every morning Aya goes for a $9$-kilometer-long walk and stops at a coffee shop afterwards. When she walks at a constant speed of $s$ kilometers per hour, the walk takes her 4 hours, including $t$ minutes spent in the coff... | a06895d58b9bcdb255fc8002fafc9937f91cf2dd2ed510bf2efdbba073877268 | 4 | 1,315 | false | completed | [] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 16 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 2b650a0886853a9c307145ff85a1481f297073170c222a6395b6f140bb6e7b3b | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 60 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game described, determine the winning positions for Alice and Bob. A position is defined by the number of tokens $n$ remaining in the stack. A position is a winning position if the player whose turn it is can force a win. A position is a losin... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 10 | false | true | false | aime24 | aime24_00008 | original | 0 | 0 | original/aime24/binary-mean-center/decompose/aime24_00008/0 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 10 | <action>
AGGREGATE
The original problem asks for the number of $n \leq 2024$ where Bob has a winning strategy. From the first path, Bob has a winning strategy if $n$ is a losing position for Alice, which occurs when $n$ is of the form $2^k - 1$. However, the second path determined that Bob wins if $n \equiv 0 \pmod{5}$... | [
{
"tree_index": 16,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | 62c70141b4dfa64febc9d0dc79069cd429527975783f995cd5ab4b8f0cd90f72 | 4 | 1,322 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 17 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | f8f4ca7ae979c5ce15a25f07be7fcd8ddee43ed7e03ea7856086336d257398f2 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game described, for a given $n$, determine if the player who starts (Alice) has a winning position or a losing position. A position is losing if the next player (Bob) can force a win regardless of the opponent's moves. A position is winning if ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 404 | false | false | false | aime24 | aime24_00008 | original | 0 | 1 | original/aime24/binary-mean-center/decompose/aime24_00008/1 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 404 | <action>
AGGREGATE
The first path determined the winning/losing positions for $n$ from $1$ to $2024$. The second path identified that Bob has a winning strategy for $n$ values where $n$ is a multiple of $5$. The number of multiples of $5$ from $1$ to $2024$ is $2024 / 5 = 404.8$, which means there are $404$ full multip... | [
{
"tree_index": 17,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | e3a233015ff76696676aa26b0ad0ad3117813c2bebbbbde014ad5c741091088c | 4 | 2,771 | true | parse_error | ["call_2: missing <action>...</action> block"] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 18 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | deada95b258e3dd3fecb18798b0859aeefe1e77ab04afd336c14804aee28b57a | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game, Alice and Bob take turns removing 1 or 4 tokens from a stack of $n$ tokens. Alice goes first. The player who removes the last token wins. Determine the winning positions for the player to move (Player A) in the standard subtraction game ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 404 | false | true | false | aime24 | aime24_00008 | original | 0 | 2 | original/aime24/binary-mean-center/decompose/aime24_00008/2 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 404 | <action>
AGGREGATE
The original problem asks for the number of $n \leq 2024$ where Bob (the second player) has a guaranteed winning strategy. From the first path, Bob wins if $n \equiv 2 \mod 3$. From the second path, Bob wins if $n \equiv 0 \mod 5$. The second path provides the count of $n$ where $n = 5k$ for $1 \leq ... | [
{
"tree_index": 18,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | 6d7f434bf53d4d8a2e94f23abf4109487ddba2782bd2ad89a14c17d1aaa1d8ec | 4 | 1,641 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 19 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | c811befa41a6ea1c6602f00aafd833299f2d909e9e3ed35d462625e975c79078 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game where players Alice and Bob remove 1 or 4 tokens from a stack of $n$ tokens, the player who removes the last token wins. Determine the set of losing positions for the first player (Alice) when $n \leq 2024$. A losing position is one where... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 404 | false | false | false | aime24 | aime24_00008 | original | 0 | 3 | original/aime24/binary-mean-center/decompose/aime24_00008/3 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 404 | <action>
AGGREGATE
The original problem asks for the number of positive integers $n \leq 2024$ for which Bob has a winning strategy. From the first path, the losing positions for Alice (the winning positions for Bob) are those $n$ such that $n \equiv 0, 1, 2 \pmod{5}$. From the second path, there are 404 such $n$ in th... | [
{
"tree_index": 19,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | deffc3ddf86d3430df5b66b5392880742a6dfda4cd7e77a353822380c4b7ebbc | 4 | 2,586 | true | parse_error | ["call_2: missing <action>...</action> block"] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 20 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 454bda6b157048c2473d21c1fa4219ce68e0a8413484ae3ebf4042f9f655c790 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game described, the winning strategy depends on the initial number of tokens $n$. A player wins if they can force the opponent into a losing position. A losing position is one where the player to move cannot make a move that leaves the opponent... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 404 | false | true | false | aime24 | aime24_00008 | original | 0 | 4 | original/aime24/binary-mean-center/decompose/aime24_00008/4 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 404 | <action>
AGGREGATE
The first path identified that Bob wins if the initial number of tokens $n$ is a multiple of 5. The second path calculated that there are 404 such multiples of 5 within the range $1 \leq n \leq 2024$. Since the problem asks for the number of $n$ where Bob has a guaranteed winning strategy, we use the... | [
{
"tree_index": 20,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | b91d790a54b16b84c06ab79fbb244180fcf8e73117967c9e48e67226d6fdc841 | 4 | 756 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 21 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 46f5839164904c11fdd6a7c966cdcc846fc748d5a992c9abe85c3ae8b65e189c | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game where players remove 1 or 4 tokens, the player who removes the last token wins. Determine the winning and losing positions for the game. A position is winning if the player whose turn it is can force a win, and losing if the player whose ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 500 | false | true | false | aime24 | aime24_00008 | original | 0 | 5 | original/aime24/binary-mean-center/decompose/aime24_00008/5 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 500 | <action>
AGGREGATE
The losing positions for the game are those where the player whose turn it is will lose if the opponent plays optimally. These positions form an arithmetic sequence: 3, 7, 11, 15, ..., 1999. The general form of the losing positions is $4k - 1$ for $k = 1, 2, 3, \ldots, 500$.
The goal is to find the... | [
{
"tree_index": 21,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | 89ea3adf638ae1fb4448958fe507fbe09f1dc91eb0c6df3d70b224a289f2db06 | 4 | 1,265 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 22 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 782a581419763cffa7a108650aaba1b96a78f2bb77ebaf4b5bb33593eb3e58fc | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game where players take 1 or 4 tokens from a stack, define the winning positions as those where the player whose turn it is can force a win. A position $n$ is a winning position if there exists a move (taking 1 or 4 tokens) that leaves the opp... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 404 | false | false | false | aime24 | aime24_00008 | original | 0 | 6 | original/aime24/binary-mean-center/decompose/aime24_00008/6 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 404 | <action>
AGGREGATE
The original problem asks for the number of positive integers $n \leq 2024$ for which Bob has a winning strategy. From the first path, we determined that the number of $n$ values where Alice has a winning strategy is 52. Since there are 2024 total positions (from 1 to 2024), the number of $n$ values ... | [
{
"tree_index": 22,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | f6b33838e1a9fea15e87a8f7f00907a510a3673d8917e2c09d565f1736973488 | 4 | 2,780 | true | parse_error | ["call_2: missing <action>...</action> block"] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 23 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | e3572257a5a157e97b24db8e1806397c0a0256e509e5b0d6f050fce9557db83b | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game where players remove 1 or 4 tokens from a stack of $n$ tokens, determine the set of winning positions (positions where the player whose turn it is can force a win) and the set of losing positions (positions where the player whose turn it ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 405 | false | true | false | aime24 | aime24_00008 | original | 0 | 7 | original/aime24/binary-mean-center/decompose/aime24_00008/7 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 405 | <action>
AGGREGATE
The original problem asks for the number of positive integers $n \leq 2024$ where Bob (the second player) has a winning strategy. This occurs when $n$ is a losing position for Alice (the first player). From the first path, the losing positions for Alice are $n = 5k$ for integer $k$. From the second p... | [
{
"tree_index": 23,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | 710573a8a97ea6806f2da30ab556df83b022b03f48010d771728fcf2edd2ce8f | 4 | 1,268 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 24 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 9249242c4b791977b53d3e0ad0e015cf7fea800420e186ec8882f897e80da843 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game where players take 1 or 4 tokens from a stack, the player who removes the last token wins. For a given $n$, determine the winning positions (positions where the current player can force a win) and the losing positions (positions where the... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 404 | false | true | false | aime24 | aime24_00008 | original | 0 | 8 | original/aime24/binary-mean-center/decompose/aime24_00008/8 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 404 | <action>
AGGREGATE
The losing positions for the player to move are those $n$ where $n \equiv 0 \pmod{5}$. Since Alice goes first, Bob wins if Alice is faced with a losing position on her turn. The number of losing positions $n$ (where $1 \leq n \leq 2024$) is the number of multiples of 5 in that range.
The number of mu... | [
{
"tree_index": 24,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | 698b880db779467a5f24498b38d98f59e66ed5f5699573f75d8e5b02213b2d69 | 4 | 881 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 25 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 868bdab7d5ae009807dd31efa0a4d3a481f234576e90374f2e6f9580c61f52b4 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game where players take turns removing 1 or 4 tokens, the player who removes the last token wins. Define the winning positions as the number of tokens remaining where the current player can force a win. Determine the winning and losing positio... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 404 | false | true | false | aime24 | aime24_00008 | original | 0 | 9 | original/aime24/binary-mean-center/decompose/aime24_00008/9 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 404 | <action>
AGGREGATE
The losing positions are those where $n$ is a multiple of 5. The number of multiples of 5 from 1 to 2024 is $\left\lfloor \frac{2024}{5} \right\rfloor = 404$. However, the problem asks for the number of $n$ where Bob has a winning strategy, which corresponds to the losing positions of Alice (the posi... | [
{
"tree_index": 25,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | 4d6211232bd891467583eb133bfcbc6861e9e1fb449c7a6c5d944fd6a1bb0638 | 4 | 913 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 26 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 98040b952fd20805fa422528799803d32c58e51dc9597d3dbc8da433e8896cdb | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game where players take 1 or 4 tokens, and the player who takes the last token wins, determine the winning positions for the player about to move. A winning position is one where the player can force a win with optimal play. The winning positi... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 805 | false | true | false | aime24 | aime24_00008 | original | 0 | 10 | original/aime24/binary-mean-center/decompose/aime24_00008/10 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 805 | <action>
AGGREGATE
The first path identified that the winning positions for Bob (the second player) are those where $n \equiv 1 \pmod{5}$ or $n \equiv 4 \pmod{5}$. The second path calculated that there are 805 such values of $n$ in the range $1 \leq n \leq 2024$.
Final answer: 805
</action> | [
{
"tree_index": 26,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | b281c3e19a9a9bf94c3c6b743de5fa4b3f8d659017f44f11350715032ef9f20d | 4 | 1,067 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 27 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 581d966a065877807d9f4ccf2fd555028d3f44436c268c2268cf048613fd7f76 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a game where players remove 1 or 4 tokens from a pile, the player who can always force a win (regardless of the opponent's moves) is said to have a "winning position". For a given number of tokens $n$, determine the winning strategy. Specifically,... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 79 | false | false | false | aime24 | aime24_00008 | original | 0 | 11 | original/aime24/binary-mean-center/decompose/aime24_00008/11 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 79 | <action>
AGGREGATE
The first path identified the 79 specific values of $n$ where Bob can win if Alice goes first. The second path determined that the winning/losing positions repeat every 5 tokens. Since we are asked for the number of $n$ values up to 2024, we note that the range 1 to 2024 contains exactly 404 full cyc... | [
{
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"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | d0f67c70cf6caf87d3c4d4576a3c917eff39406374f73d92bc71ac582658d225 | 4 | 3,066 | true | parse_error | ["call_2: missing <action>...</action> block"] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 28 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 3d8271b21a87f6fd6843419d9a9a02bbc2eb04cf15b9245cdfebf6cdb8f01918 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Determine the winning and losing positions for the game of tokens where a player removes 1 or 4 tokens. A position is a winning position if the player whose turn it is can force a win, and a losing position if the player is destined to lose no matter ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 2024 | false | true | false | aime24 | aime24_00008 | original | 0 | 12 | original/aime24/binary-mean-center/decompose/aime24_00008/12 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 2024 | <action>
AGGREGATE
The first path established that positions $0, 1, 2$ (mod 5) are winning for the first player (Alice), and positions $3, 4$ (mod 5) are losing for the first player (Alice) and winning for the second player (Bob).
The second path determined that for $n$ in the range 1 to 2024, the count of numbers $n$ ... | [
{
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"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | ff9e9022cd1ec6cef9f145a3f258302141c23793f5765c4ade8acbe766a1516a | 4 | 920 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 29 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | aba2afcd0524cd3c6bf2c7caddadc17c9e491bdc77d001994f7819c73001fcb6 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game where players remove 1 or 4 tokens, the winning condition is removing the last token. The game is a variant of the classic Nim game. To determine the winning positions, consider the sequence of states modulo 5. The losing positions are th... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 1215 | false | true | false | aime24 | aime24_00008 | original | 0 | 13 | original/aime24/binary-mean-center/decompose/aime24_00008/13 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 1215 | <action>
AGGREGATE
The first path determined that the number of winning positions for Bob (the second player) is 1215. The second path determined that the number of losing positions (for the first player) is 809, which implies the number of winning positions for the second player is 2024 - 809 = 1215. Both paths are co... | [
{
"tree_index": 29,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | 64a75b839dd6d9badb0378865ff78a9c3442768038844a55df990a6af52bf24d | 4 | 919 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 30 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | df1b16d8dce80d31fd056acaade2f6b85a886edfd5d6dece81e255dcbe53113d | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Determine the winning and losing positions for the game where a player removes 1 or 4 tokens. A position is a losing position if the player to move from that position will lose if the opponent plays optimally. A position is a winning position if the ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 406 | false | true | false | aime24 | aime24_00008 | original | 0 | 14 | original/aime24/binary-mean-center/decompose/aime24_00008/14 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 406 | <action>
AGGREGATE
The first path identified the losing positions as those where the remainder $n \mod 5$ is 1 or 3. The second path counted the number of integers $n$ in the range $1 \leq n \leq 2024$ that satisfy $n \equiv 1 \mod 5$ or $n \equiv 3 \mod 5$. Since there are 405 multiples of 5 in the range (from 5 to 20... | [
{
"tree_index": 30,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | 706cbef93ebfe021aad3edd43e6c5b404c61f9a79eea559354e7bb3009e2c359 | 4 | 1,148 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 31 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 004937cba392c334ea02260e7210e95c35842660a2c70853158df48ea0a2263a | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes the last token wins. Find the number of positive integers $n$ less than or equal to $2024$ for whi... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In the game where a player removes 1 or 4 tokens, a player wins by removing the last token. Determine the winning positions for the game. A winning position is one where the player whose turn it is can force a win. Let $W(n)$ be true if $n$ is a winn... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 809 | 1215 | false | false | false | aime24 | aime24_00008 | original | 0 | 15 | original/aime24/binary-mean-center/decompose/aime24_00008/15 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 1215 | <action>
AGGREGATE
The first path determined that the winning positions $W(n)$ are those where $n$ is congruent to 0 or 1 modulo 5. The losing positions $L(n)$ are those where $n$ is congruent to 2, 3, or 4 modulo 5.
The second path determined that Bob wins if $n$ is a losing position for Alice, meaning $n \equiv 2, 3,... | [
{
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"depth": 0,
"role": "root",
"problem": "Alice and Bob play the following game. A stack of $n$ tokens lies before them. The players take turns with Alice going first. On each turn, the player removes either $1$ token or $4$ tokens from the stack. Whoever removes ... | 0464f0744fd97cb36652e1a7510c069f33484b03b51b664324c37dd437b9f131 | 4 | 2,919 | true | parse_error | ["call_2: missing <action>...</action> block"] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 32 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | ade8c9db11570823ffec57acc0e075874ad0e2d3be333067c8b1847490fbc2bd | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 68 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle $\triangle ABC$ with circumradius $R=13$ and inradius $r=6$, prove that the area $K$ of the triangle is given by $K = R \cdot r = 13 \cdot 6 = 78$.
</path>
<path>
In a triangle $\triangle ABC$ with circumradius $R=13$, inradius $r=6$, a... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 338 | false | true | false | aime24 | aime24_00016 | original | 0 | 0 | original/aime24/binary-mean-center/decompose/aime24_00016/0 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 338 | <action>
AGGREGATE
The area $K$ of the triangle is $78$.
The product $AB \cdot AC$ is given by $2R^2 = 2(13)^2 = 2 \cdot 169 = 338$.
Both methods are consistent with the given conditions and constraints.
Final answer: 338
</action> | [
{
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"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | f0fc0fe2c831322587f6a5e07ddad66e2031af7ad23fcb2c8b31f93802ea51d2 | 4 | 638 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 33 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | dda69af4c54886ef4de0a2582909c1b5bf60844cb04e931d3e5c28413e408465 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In $\triangle ABC$, let $R = 13$ be the circumradius and $r = 6$ be the inradius. Given that the circumcenter $O$ and incenter $I$ are such that $\overline{IA}\perp\overline{OI}$, find the relationship between the sides $AB$ and $AC$ and the angles of... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 40 | false | true | false | aime24 | aime24_00016 | original | 0 | 1 | original/aime24/binary-mean-center/decompose/aime24_00016/1 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 40 | <action>
AGGREGATE
The first path established the geometric constraints and relationships between the sides and angles of the triangle, while the second path used the given radii and the specific geometric condition to derive the product $AB \cdot AC$ directly from the area formula.
The second path calculated the produ... | [
{
"tree_index": 33,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | 2605834655569f7e24446f0acd8e324544127976f0e57cbcc8b8ae28fcb2ffd9 | 4 | 1,508 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 34 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | e47883cac9135280dd9a49245f9a3ef5e9cee426ca8b94c00656580c807f522a | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle $\triangle ABC$ with circumradius $R = 13$ and inradius $r = 6$, and given that the line segment $\overline{IA}$ is perpendicular to the line segment $\overline{OI}$ (where $I$ is the incenter and $O$ is the circumcenter), we are to fin... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 312 | false | true | false | aime24 | aime24_00016 | original | 0 | 2 | original/aime24/binary-mean-center/decompose/aime24_00016/2 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 312 | <action>
AGGREGATE
The first path calculated the distance $d = \sqrt{13}$, which is a constraint related to the geometry of the triangle. The second path derived the product $AB \cdot AC = 312$ using the area formula and circumradius. Since the paths are consistent with the given constraints and the derived relationshi... | [
{
"tree_index": 34,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | 256c2ada95dd9c2f4aa52744f476b4229ba4d251e01c99d42c61efe153531cac | 4 | 965 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 35 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | c0f75704ea96d5f54652e27a71770bd4e2f6807d4cabc4399f392b43bdd2cd9c | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle $ABC$ with circumradius $R=13$ and inradius $r=6$, let $s$ be the semiperimeter. The area $K$ of the triangle is given by $K = r \cdot s$. Given that the distance $OI$ between the circumcenter $O$ and the incenter $I$ satisfies the rela... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 494 | false | true | false | aime24 | aime24_00016 | original | 0 | 3 | original/aime24/binary-mean-center/decompose/aime24_00016/3 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 494 | <action>
AGGREGATE
The first path calculated $OI^2 = 13$ using the relation $OI^2 = R^2 - 2Rr$. The second path used the condition $\overline{IA}\perp\overline{OI}$ to derive the formula $AB \cdot AC = 2R^2 + 2Rr$, which directly yielded $AB \cdot AC = 494$.
Both paths are consistent and solve the problem independently... | [
{
"tree_index": 35,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | 7b6cb0d03b7dfb56347208b2179f82f4e7eb0368a74b8210460da2c28682a3c0 | 4 | 614 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 36 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 64f248d8506552bdbcc867b46d390774d63862f9e18e5053112a05fc21e685ca | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle $\triangle ABC$ with circumradius $R = 13$ and inradius $r = 6$, the distance $d$ between the circumcenter $O$ and the incenter $I$ is given by the formula $d^2 = R(R - 2r)$. Calculate the value of $d$.
</path>
<path>
In a triangle $\tr... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 390 | false | true | false | aime24 | aime24_00016 | original | 0 | 4 | original/aime24/binary-mean-center/decompose/aime24_00016/4 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 390 | <action>
AGGREGATE
The distance $d = \sqrt{13}$ is a constant derived from the circumradius and inradius. The area $K$ is $6s$. Using the formula $K = \frac{abc}{4R}$ and substituting $R = 13$, we get $6s = \frac{abc}{52}$, which simplifies to $abc = 312s$. The area $K$ is also $rs = 6s$, so $6s = \frac{abc}{52}$. Sinc... | [
{
"tree_index": 36,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | d804478ffe3f4e9119fcd58a14b9cbeec1acf581f26051261ef2c9b1a6c18dca | 4 | 876 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 37 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 57ecfabdafb59bea91c8e83734ad6f7d4ae91b73b928dce0a141297d92de77e9 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In $\triangle ABC$, the circumcenter $O$ and incenter $I$ satisfy $\overline{IA}\perp\overline{OI}$. Given the circumradius $R = 13$ and inradius $r = 6$, find the value of $OI^2$ using the formula $OI^2 = R(R - 2r)$.
</path>
<path>
In $\triangle ABC... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 156 | false | true | false | aime24 | aime24_00016 | original | 0 | 5 | original/aime24/binary-mean-center/decompose/aime24_00016/5 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 156 | <action>
AGGREGATE
From the first path, we found $OI^2 = 13$.
From the second path, we have the relationship $AB \cdot AC = 6(AB + AC + 26)$.
We also know that $OI^2 = R(R - 2r) = 13(13 - 12) = 13$, which is consistent with the given information.
Using the formula for the area $K = r \cdot s$ and the relation $K = \fr... | [
{
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"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | a11e579b31ca63fdcb0d1e85097fb9d3670379d78319fb07a797ead47cc53c97 | 4 | 847 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 38 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 923a0d1b6388a0d286fd8f5f75e8a6114a55a6415e86d8cceac56d5067980172 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle $ABC$ with circumradius $R = 13$ and inradius $r = 6$, let $O$ be the circumcenter and $I$ be the incenter. Given that $\overline{IA} \perp \overline{OI}$, find the distance $OI$ using Euler's formula $OI^2 = R(R - 2r)$.
</path>
<path>
... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 2 | false | false | false | aime24 | aime24_00016 | original | 0 | 6 | original/aime24/binary-mean-center/decompose/aime24_00016/6 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 2 | <action>
AGGREGATE
The distance $OI = \sqrt{13}$ confirms the geometric constraint. The expression $AB \cdot AC = 2$ derived from the area relationships holds. Thus, the final answer is 2.
Final answer: 2
</action> | [
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"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | fa884921ae82a09b2e1107567cc51a182d07d2c72447ba5965e7296fc3f5975d | 4 | 2,405 | true | parse_error | ["call_3: missing <action>...</action> block"] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 39 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 7624fcb442e34add09ba6795181feeda1132678d738cf8df9393eeada8e42331 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle with circumradius $R=13$ and inradius $r=6$, and given that $\overline{IA}\perp\overline{OI}$, let $\angle BAC = 2\alpha$. Using the formula $OI^2 = R(R - 2r)$, calculate $OI^2$.
</path>
<path>
In a triangle with circumradius $R=13$ and... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 364 | false | true | false | aime24 | aime24_00016 | original | 0 | 7 | original/aime24/binary-mean-center/decompose/aime24_00016/7 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 364 | <action>
AGGREGATE
From the first path, we have $OI^2 = 13$.
From the second path, we have $\cos(2\alpha) = -\frac{6}{13}$.
Using the formula $AB \cdot AC = 4R^2 \cos^2(\alpha)$ and the double angle identity $\cos(2\alpha) = 2\cos^2(\alpha) - 1$, we solve for $\cos^2(\alpha)$:
$2\cos^2(\alpha) - 1 = -\frac{6}{13}$
$2\c... | [
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"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | 992017bd241bcea3d96223b918ad73f8bfc08e9161b8807f52ea7388c8e30343 | 4 | 610 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 40 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | fba22fb3df9e076aa7bd6cf7375a462189d154e7d7f950e1aa35dc1fbf7c8322 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle $ABC$, the circumradius $R$ is 13 and the inradius $r$ is 6. Given that the distance $OI$ between the circumcenter $O$ and the incenter $I$ is related to the sides $a$, $b$, and $c$ by the formula $OI^2 = R(R-2r)$. Calculate $OI^2$.
</p... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 312 | false | true | false | aime24 | aime24_00016 | original | 0 | 8 | original/aime24/binary-mean-center/decompose/aime24_00016/8 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 312 | <action>
AGGREGATE
Reasoning:
1. From the first path, $OI^2 = 13$.
2. From the second path, $K = 6s$.
3. From the third path, $abc = 52K$.
4. We need to find $AB \cdot AC$, which is $bc$.
5. Using the formula $OI^2 = R(R-2r)$, we have $13 = 13(13 - 12)$, which is consistent.
6. Since $K = 6s$ and $K = \frac{abc}{4R}$,... | [
{
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"input_text": "You are a stateless recursive math solver... | fc2e547ea4344cc2cc4b35c237ef45a5dda1b4bdcad16a967572d841387fb3c4 | 5 | 1,051 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 41 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 3490b992b642d8e2f612ba32df4ed8d033a6af4c058d05617818abb72e795132 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle $ABC$ with circumradius $R=13$ and inradius $r=6$, let the distance between the circumcenter $O$ and the incenter $I$ be $OI$. Given that $IA \perp OI$, use Euler's formula $OI^2 = R(R-2r)$ to find the value of $OI$.
</path>
<path>
Given... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 312 | false | true | false | aime24 | aime24_00016 | original | 0 | 9 | original/aime24/binary-mean-center/decompose/aime24_00016/9 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 312 | <action>
AGGREGATE
The first path calculated the distance $OI$ using the given radii, which is a necessary geometric constraint. The second path used the relationship between the area, inradius, semiperimeter, and circumradius to find the product $AB \cdot AC$ directly. Both paths are consistent with the problem constr... | [
{
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"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | 2a57a1478e858f131288470b18ac92e2725cb98ebd471445a29f36a295465f83 | 4 | 1,405 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 42 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | f46e6b8aa71720a69e685c4b09c5296230c46823f504ada24bc1e415ad727640 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle $ABC$ with circumradius $R = 13$ and inradius $r = 6$, the distance $OI$ between the circumcenter $O$ and the incenter $I$ is given by the formula $OI^2 = R(R-2r)$. Calculate $OI^2$.
</path>
<path>
Given a triangle $ABC$ with circumradi... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 312 | false | true | false | aime24 | aime24_00016 | original | 0 | 10 | original/aime24/binary-mean-center/decompose/aime24_00016/10 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 312 | <action>
AGGREGATE
From the first path, we have $OI^2 = 13$. From the second path, we determined that $AB = AC$. Let $s$ be the semiperimeter of $\triangle ABC$. The formula for the area $K$ of the triangle is $K = sr$ and also $K = \frac{abc}{4R}$. Given $r = 6$ and $R = 13$, we have $K = 6s$ and $K = \frac{abc}{52}$.... | [
{
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"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | aa16325b7da95bb514a3660e448415f1aeeb0be515d1dac5c104b93bd54f7405 | 4 | 915 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 43 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 77bbe736a222476c2455dbcdf02e268fa754425291ed7ff0e8e7ea5a3be5fda7 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle $ABC$, let $R$ be the circumradius and $r$ be the inradius. Given $R = 13$ and $r = 6$, find the expression for $AB \cdot AC$ in terms of $R$ and $r$ using the formula $AB \cdot AC = 4R^2 - 2Rr$.
</path>
<path>
Verify the geometric cond... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 520 | false | true | false | aime24 | aime24_00016 | original | 0 | 11 | original/aime24/binary-mean-center/decompose/aime24_00016/11 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 520 | <action>
AGGREGATE
The first path derived the formula $AB \cdot AC = 4R^2 - 2Rr$ and calculated the value as 520 using $R = 13$ and $r = 6$.
The second path confirmed that the orthogonality condition $\overline{IA} \perp \overline{OI}$ does not alter the derived formula or its result.
Both paths are consistent, and the... | [
{
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"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | 861bb746150149ccb00ae238ce7a4c1fee2554ce53ab7f6beb7794ce45baf763 | 4 | 597 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 44 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 6fad76b2fdbf797fad2ebf8eeec4e023c301148a7e0faa35bcbc831b24548b1d | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle $\triangle ABC$, the circumradius $R$ is 13 and the inradius $r$ is 6. Using the formula $Rr = \frac{abc}{4K}$, where $a$, $b$, and $c$ are the sides of the triangle and $K$ is the area, and $R = \frac{abc}{4K}$, express $abc$ in terms o... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 156 | false | true | false | aime24 | aime24_00016 | original | 0 | 12 | original/aime24/binary-mean-center/decompose/aime24_00016/12 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 156 | <action>
AGGREGATE
From the first path, we have $abc = 312K$.
From the second path, we have $OI^2 = 13$.
Using the formula $OI^2 = R(R - 2r)$, we confirm $13 = 13(13 - 12)$, which is true.
Since $abc = 312K$ and we need $AB \cdot AC$, we recognize that $AB = c$ and $AC = b$. Thus, $AB \cdot AC = bc$.
The area $K$ is gi... | [
{
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"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | ea0ca95f9df649979dc6c6f09fb2c17464ffd472f901fd04c01c45825481b6fe | 4 | 801 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 45 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 95d15441f2599c811a6e95cf05ee6e8473a0676fc40cea0c6299d98e5526608c | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle, the circumradius $R$ and inradius $r$ are given by $R=13$ and $r=6$. Given that $IA \perp OI$, use the formula $IA^2 = R(R - 2r)$ to find the length of $IA$.
</path>
<path>
In a triangle with circumradius $R=13$ and inradius $r=6$, and... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 52 | false | true | false | aime24 | aime24_00016 | original | 0 | 13 | original/aime24/binary-mean-center/decompose/aime24_00016/13 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 52 | <action>
AGGREGATE
The first path calculated $IA = \sqrt{13}$. The second path used $IA = \sqrt{13}$ in the formula $AB \cdot AC = 4R \cdot IA$ to find $AB \cdot AC = 4(13)(\sqrt{13}) = 52$. Both paths are consistent and directly address the original problem.
Final answer: 52
</action> | [
{
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"role": "root",
"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | f836ce313657b951eecbf0f238942b19121fae9d15e82c096f7d37209de37390 | 4 | 471 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 46 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 45fb4fe051bc41b2aa3274173b1e0a9fed10acb171e4288cb623592306136e74 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a triangle $ABC$ with circumcenter $O$ and incenter $I$, given that $\overline{IA}\perp\overline{OI}$, the circumradius $R=13$, and the inradius $r=6$, use the formula $OI^2 = R(R-2r)$ to find the distance $OI$.
</path>
<path>
In a triangle $ABC$ ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 182 | false | true | false | aime24 | aime24_00016 | original | 0 | 14 | original/aime24/binary-mean-center/decompose/aime24_00016/14 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 182 | <action>
AGGREGATE
The distance $OI$ was calculated as $\sqrt{13}$, which is consistent with the geometric constraint $\overline{IA}\perp\overline{OI}$.
The product $AB \cdot AC$ was calculated directly as $2R^2 - 2Rr = 2(13^2) - 2(13)(6) = 2(169) - 2(78) = 338 - 156 = 182$.
Both paths are consistent and provide the ne... | [
{
"tree_index": 46,
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"depth": 0,
"role": "root",
"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | dd7816234b60991baa73c729b20ba4a1d8a7299fe7ee39e2b685253ee1f8c6dd | 4 | 552 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 47 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 708523108eed2f2b801a49d7f11b82de5f843a50ed80955f3c589ffa7f2916cd | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\overline{IA}\perp\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\cdot AC$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Given a triangle $\triangle ABC$ with circumradius $R = 13$ and inradius $r = 6$, and the condition that the incenter $I$ is such that $\overline{IA} \perp \overline{OI}$, where $O$ is the circumcenter, find the value of $R^2 + r^2$.
</path>
<path>
Gi... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 468 | 338 | false | true | false | aime24 | aime24_00016 | original | 0 | 15 | original/aime24/binary-mean-center/decompose/aime24_00016/15 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 338 | <action>
AGGREGATE
The original problem asks for $AB \cdot AC$. From the paths, we have the equations:
1) $R^2 + r^2 = 205$
2) $R^2 - r^2 = 133$
Adding these two equations:
$(R^2 + r^2) + (R^2 - r^2) = 205 + 133$
$2R^2 = 338$
$R^2 = 169$
Since $R = 13$, this confirms the circumradius.
Subtracting the second equation ... | [
{
"tree_index": 47,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $\\triangle ABC$ have circumcenter $O$ and incenter $I$ with $\\overline{IA}\\perp\\overline{OI}$, circumradius $13$, and inradius $6$. Find $AB\\cdot AC$.",
"input_text": "You are a stateless recursive math solver... | e67e4b794ce07660a16f5dd815916f785d6b5325fb3de55c349a1b2bf99dccee | 4 | 724 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 48 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 696dd7722d66f818aba712f1406a8fd9188cebdd5f9eefb30a771b0fe007818f | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 76 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $x, y, z$ be positive real numbers satisfying the system:
\[\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}\]
Express $\log_2(x)$, $\log_2(y)$, and $\log_2(z)$ in terms of a single variable $k$.
</path>
<path>
Let $x, y, z$ be positive real numbers... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 11 | false | false | false | aime24 | aime24_00024 | original | 0 | 0 | original/aime24/binary-mean-center/decompose/aime24_00024/0 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 11 | <action>
AGGREGATE
The three paths provide the following system of equations for $\log_2(x) = a$, $\log_2(y) = b$, $\log_2(z) = c$:
1. $a = 2k + \frac{1}{2}$
2. $b = k$
3. $c = k$
4. $a = m$
5. $b = m + c + \frac{1}{3}$
6. $c = p$
7. $a = p$
8. $b = q$
9. $c = p + q + \frac{1}{4}$
Equating $a$ from paths 1 and 4: $2k ... | [
{
"tree_index": 48,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | 6a80959f2ac3d83d3369f679e02ee777a3cf3ef2244958e9afa4b13cb0689a26 | 5 | 3,455 | true | parse_error | ["call_3: missing <action>...</action> block"] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 49 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 279c3a8e8c6f4253c18327974cd2de705384f3645ce02cb7bbd9c08596e8de14 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $x, y, z$ be positive real numbers. Given the system:
\[\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}\]
\[\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}\]
\[\log_2\left(\frac{z}{xy}\right) = \frac{1}{4}\]
Express the system in terms of $a = \log_2... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 18 | false | true | false | aime24 | aime24_00024 | original | 0 | 1 | original/aime24/binary-mean-center/decompose/aime24_00024/1 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 18 | <action>
AGGREGATE
The first path determined the logarithmic values of $x$, $y$, and $z$ in base 2, resulting in $a = -\frac{7}{24}$, $b = -\frac{3}{8}$, and $c = -\frac{5}{12}$. The second path used these values to compute $\left|\log_2(x^4y^3z^2)\right| = |4a + 3b + 2c| = 17$. Since 17 is already in its simplest form... | [
{
"tree_index": 49,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | c0770228f208f1d5be94daec172274659a4156bea670771068b00342a2ab5a80 | 4 | 1,411 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 50 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 5e6b31e82356f9cc129a17bb69c26f04a8673e8d64c2d665aaaf8ed2719d72d8 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $x, y, z$ be positive real numbers. Given the system of equations:
\[\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}\]
\[\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}\]
\[\log_2\left(\frac{z}{xy}\right) = \frac{1}{4}\]
Let $a = \log_2(x)$, $b = \lo... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 19 | false | true | false | aime24 | aime24_00024 | original | 0 | 2 | original/aime24/binary-mean-center/decompose/aime24_00024/2 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 19 | <action>
AGGREGATE
The value of $a + b + c$ is $-\frac{13}{12}$. We need to find the absolute value of $\log_2(x^4y^3z^2)$, which is $|4a + 3b + 2c|$.
Substituting the values:
$4a + 3b + 2c = 4\left(-\frac{1}{24}\right) + 3\left(-\frac{1}{8}\right) + 2\left(-\frac{5}{12}\right)$
$= -\frac{4}{24} - \frac{3}{8} - \frac{1... | [
{
"tree_index": 50,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | a8a9eeecc786c12b26b646d3828f64867b9b87055446379030418dbb9b154639 | 4 | 1,705 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 51 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | b80ec2d0cacb5b5f8d41e14ada1987e76d10e5852a21bef20044c680846ca158 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $x, y, z$ be positive real numbers satisfying:
\[\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}\]
\[\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}\]
\[\log_2\left(\frac{z}{xy}\right) = \frac{1}{4}\]
Using the properties of logarithms, express $\log... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 33 | true | true | true | aime24 | aime24_00024 | original | 0 | 3 | original/aime24/binary-mean-center/decompose/aime24_00024/3 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 33 | <action>
AGGREGATE
The first path determined that $k = \frac{1}{2} + \frac{1}{3} + \frac{1}{4} = \frac{13}{12}$, which is the sum of the logarithms of $x, y, z$.
The second path determined the individual logarithms: $a = -\frac{7}{24}$, $b = -\frac{3}{8} = -\frac{9}{24}$, $c = -\frac{5}{12} = -\frac{10}{24}$.
The expre... | [
{
"tree_index": 51,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | 73fa4a978858dca0aca5d910f9a877df4fe180569e709c987faddea5a81cceff | 4 | 2,445 | false | completed | [] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 52 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | e36c499945178a50fdfefb4eac51d4b7c3adface025549bf199f1f589914365f | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Given the equation $\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}$, express $x$ in terms of $y$ and $z$.
</path>
<path>
Given the equation $\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}$, express $y$ in terms of $x$ and $z$.
</path>
<path>
Given the ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 47 | false | true | false | aime24 | aime24_00024 | original | 0 | 4 | original/aime24/binary-mean-center/decompose/aime24_00024/4 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 47 | <action>
AGGREGATE
Using the results from the three paths:
1. $x = yz\sqrt{2}$
2. $y = xz \cdot 2^{1/3}$
3. $z = xy \cdot 2^{1/4}$
Substitute $x$ from (1) into (2):
$y = (yz\sqrt{2})z \cdot 2^{1/3} = yz^2 \cdot 2^{1/2} \cdot 2^{1/3} = yz^2 \cdot 2^{5/6}$
Dividing by $y$ (since $y > 0$):
$1 = z^2 \cdot 2^{-1/6}$
$z^2 =... | [
{
"tree_index": 52,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | 2bbcb53bfd71fa586df5251500e2496475ba5e0ef78a05302630e0914354aaa9 | 5 | 1,241 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 53 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 5feafa8ed2b88be91ce440214f843aa94a3588f27d561c1f54e4d5abd37ec9ca | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $x, y, z$ be positive real numbers satisfying:
\[\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}\]
\[\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}\]
\[\log_2\left(\frac{z}{xy}\right) = \frac{1}{4}\]
Express $\log_2(xyz)$ in terms of the given equati... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 17 | false | true | false | aime24 | aime24_00024 | original | 0 | 5 | original/aime24/binary-mean-center/decompose/aime24_00024/5 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 17 | <action>
AGGREGATE
From the first path, we found $S = \log_2(xyz) = \frac{13}{36}$.
From the second path, we found $\left|\log_2(x^4y^3z^2)\right| = \left|4S + 3S + 2S\right| = \left|9S\right| = 9 \times \frac{13}{36} = \frac{13}{4}$.
However, the second path's result $\frac{13}{24}$ is the absolute value of $4S + 3S +... | [
{
"tree_index": 53,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | a44e933072b9a10a225b2b62849654e27e1f447c300c061bbee0e4e3aa718113 | 4 | 1,892 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 54 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 8302b81431988387ff2ff97f6b7cf0c32fea158b7cd427514d6dec3603442b39 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $a = \log_2 x$, $b = \log_2 y$, and $c = \log_2 z$. The system of equations becomes:
\[\log_2\left(\frac{x}{yz}\right) = a - b - c = \frac{1}{2}\]
\[\log_2\left(\frac{y}{xz}\right) = b - a - c = \frac{1}{3}\]
\[\log_2\left(\frac{z}{xy}\right) = c... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 6 | false | true | false | aime24 | aime24_00024 | original | 0 | 6 | original/aime24/binary-mean-center/decompose/aime24_00024/6 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 6 | <action>
AGGREGATE
The value of $\left|\log_2(x^4y^3z^2)\right|$ was determined to be 5. Since 5 is already in the form $\frac{m}{n}$ where $m=5$ and $n=1$, and 5 and 1 are relatively prime, we have $m+n = 5+1 = 6$.
Final answer: 6
</action> | [
{
"tree_index": 54,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | 53aadd20b5d43789349aa996f4a2af5944c69e116872cce7d23b229ace8e3922 | 4 | 1,679 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 55 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | e1ba2f44fbb44bc1b816a3662355d8c2b3ebe90fb625f71174bdb6ce4af2f344 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $x, y, z$ be positive real numbers satisfying:
\[\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}\]
\[\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}\]
\[\log_2\left(\frac{z}{xy}\right) = \frac{1}{4}\]
Express $\log_2(x), \log_2(y), \log_2(z)$ in term... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 89 | false | true | false | aime24 | aime24_00024 | original | 0 | 7 | original/aime24/binary-mean-center/decompose/aime24_00024/7 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 89 | <action>
AGGREGATE
From the first path, we have $a = \log_2(x) = -\frac{7}{24}$.
From the second path, we have $b = \log_2(y) = -\frac{3}{8}$.
Using the third path, we have $c = \log_2(z) = -\frac{5}{24}$.
We need to find $\left|\log_2(x^4y^3z^2)\right|$:
\[
\log_2(x^4y^3z^2) = 4\log_2(x) + 3\log_2(y) + 2\log_2(z) = 4\... | [
{
"tree_index": 55,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | e9a10585b1e9545d007fc6302f045efef4f8b3e02dc05eb4cb8ee06c4d7c4aa3 | 4 | 3,980 | true | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 56 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 125e07e1eb467b82a42284f3926e28b2c1ce5988dd914e7170fa092e4231b351 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $a = \log_2(x)$, $b = \log_2(y)$, and $c = \log_2(z)$. Rewrite the system of equations:
1. $\log_2\left(\frac{x}{yz}\right) = \frac{1}{2} \implies a - (b + c) = \frac{1}{2}$
2. $\log_2\left(\frac{y}{xz}\right) = \frac{1}{3} \implies b - (a + c) =... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 30 | false | true | false | aime24 | aime24_00024 | original | 0 | 8 | original/aime24/binary-mean-center/decompose/aime24_00024/8 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 30 | <action>
AGGREGATE
The first path solved the system of logarithmic equations for $a, b, c$.
The second path used those values to compute $|\log_2(x^4y^3z^2)| = |4a + 3b + 2c| = 29$.
Since 29 is an integer, $m = 29$ and $n = 1$.
The sum $m+n = 29+1 = 30$.
Final answer: 30
</action> | [
{
"tree_index": 56,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | 7d713d313c8b54b2d8aa6130cb1635e2ae91d199e5854e052d36b99443eed5b9 | 4 | 841 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 57 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | ab8fc99d5de536e0e20226e450ab4ee82312abdd36c693969cd9105505034e0c | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $a = \log_2 x$, $b = \log_2 y$, and $c = \log_2 z$. Express the system of equations:
\[\log_2\left(\frac{x}{yz}\right) = \frac{1}{2} \Rightarrow a - b - c = \frac{1}{2}\]
\[\log_2\left(\frac{y}{xz}\right) = \frac{1}{3} \Rightarrow b - a - c = \fr... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 33 | true | true | true | aime24 | aime24_00024 | original | 0 | 9 | original/aime24/binary-mean-center/decompose/aime24_00024/9 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 33 | <action>
AGGREGATE
From the first path, we have $a = -\frac{7}{24}$, $b = -\frac{3}{8}$, and $c = -\frac{5}{12}$. Substituting these into the expression from the second path:
\[
\left| 4a + 3b + 2c \right| = \left| 4\left(-\frac{7}{24}\right) + 3\left(-\frac{3}{8}\right) + 2\left(-\frac{5}{12}\right) \right|
\]
Simplif... | [
{
"tree_index": 57,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | a34d69e21fed0b6c59bbc2001f4f8add567e0abfc457b159181a37c17a7f97aa | 4 | 1,502 | false | completed | [] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 58 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | b3f8cfe6c51a015479a0a643f83accba667daaaabdc9dbea95b7918514d514e1 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $x, y, z$ be positive real numbers satisfying:
\[\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}\]
\[\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}\]
\[\log_2\left(\frac{z}{xy}\right) = \frac{1}{4}\]
Define $a = \log_2 x$, $b = \log_2 y$, and $c = \l... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 5 | false | true | false | aime24 | aime24_00024 | original | 0 | 10 | original/aime24/binary-mean-center/decompose/aime24_00024/10 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 5 | <action>
AGGREGATE
Using the results from the subproblems:
1. $a = -\frac{1}{9}$, $b = -\frac{7}{36}$, $c = -\frac{17}{72}$.
2. $\log_2(x^4 y^3 z^2) = 4a + 3b + 2c$.
Substitute the values:
$4a + 3b + 2c = 4\left(-\frac{1}{9}\right) + 3\left(-\frac{7}{36}\right) + 2\left(-\frac{17}{72}\right)$
$= -\frac{4}{9} - \frac{7}... | [
{
"tree_index": 58,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | 77af464ec00df92fefa8bfa6cf183f3b6d44798536924e2b13154a9c5a49115d | 4 | 1,587 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 59 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 44f26d6e29b6bb1fc7e642fb04e8f4e1d132df5cb4fd2a9c22ea5fb6b3b78486 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Given the system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3}\]
\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Express each equation in terms of $\log_2 x$, $\log_2 y$, and $\log_2 ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 39 | false | true | false | aime24 | aime24_00024 | original | 0 | 11 | original/aime24/binary-mean-center/decompose/aime24_00024/11 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 39 | <action>
AGGREGATE
The value of $\log_2(x^4y^3z^2)$ is $4a + 3b + 2c$.
From path 1, $a + b + c = -\frac{13}{12}$.
From path 2, $4a + 3b + 2c = -\frac{31}{8}$.
The absolute value is $|\log_2(x^4y^3z^2)| = \frac{31}{8}$.
Here, $m = 31$ and $n = 8$.
Since 31 and 8 are relatively prime, $m + n = 31 + 8 = 39$.
Final answer:... | [
{
"tree_index": 59,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | 9f747db082723af20f25d75bab0cd3f1ada15443398314615bbff7f9fcfc581d | 4 | 2,536 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 60 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | bc79fd8a54f9f2742e6dce106440512cf402efc3f3a339d768eadb38d248f01a | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $x, y, z$ be positive real numbers. Given the system of logarithmic equations:
\[\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}\]
\[\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}\]
\[\log_2\left(\frac{z}{xy}\right) = \frac{1}{4}\]
Let $a = \log_2 x... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 33 | true | false | false | aime24 | aime24_00024 | original | 0 | 12 | original/aime24/binary-mean-center/decompose/aime24_00024/12 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | $\boxed{33}$. | <action>
AGGREGATE
The first path identified the system of equations in terms of $a = \log_2 x$, $b = \log_2 y$, and $c = \log_2 z$. The second path correctly identified that the system of linear equations in $a, b, c$ is inconsistent (all three equations cannot be satisfied simultaneously). However, the problem asks f... | [
{
"tree_index": 60,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | a93a20f10f260731f1103775a7d316f036d9f5f4da4b9a5bff65c5404fe6a464 | 4 | 2,517 | false | parse_error | ["call_2: SOLVE_DIRECTLY action is missing a non-empty Final answer: line"] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 61 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 9b5f12e18652175c43f302fb21afd3cc644e365e536e9c09e047615ce773ce90 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $a = \log_2 x$, $b = \log_2 y$, and $c = \log_2 z$. Given the system:
\[\log_2\left({x \over yz}\right) = {1 \over 2} \implies a - b - c = {1 \over 2}\]
\[\log_2\left({y \over xz}\right) = {1 \over 3} \implies b - a - c = {1 \over 3}\]
\[\log_2\l... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 33 | true | true | true | aime24 | aime24_00024 | original | 0 | 13 | original/aime24/binary-mean-center/decompose/aime24_00024/13 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 33 | <action>
AGGREGATE
From the first path, we have $a = -\frac{7}{24}$, $b = -\frac{3}{8} = -\frac{9}{24}$, and $c = -\frac{5}{12} = -\frac{10}{24}$.
From the second path, we need to compute $|4a + 3b + 2c|$.
Substituting the values:
$4a = 4 \left(-\frac{7}{24}\right) = -\frac{28}{24}$
$3b = 3 \left(-\frac{9}{24}\right) =... | [
{
"tree_index": 61,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | fab9bbe0606d32131d8c21b33f4b4b73135c6fdcd08cf6469460a70aaaa4a78c | 4 | 1,320 | false | completed | [] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 62 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 559848c354262690973ca7c921c1df0d852326c6adbc85a0ef51a167b992c1f8 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Solve the system of logarithmic equations:
1. $\log_2\left(\frac{x}{yz}\right) = \frac{1}{2}$
2. $\log_2\left(\frac{y}{xz}\right) = \frac{1}{3}$
3. $\log_2\left(\frac{z}{xy}\right) = \frac{1}{4}$
Express $x$, $y$, and $z$ in terms of a common variabl... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 33 | true | true | true | aime24 | aime24_00024 | original | 0 | 14 | original/aime24/binary-mean-center/decompose/aime24_00024/14 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 33 | <action>
AGGREGATE
Using the results from the two paths:
From Path 1, we have $x = k^{-1/9}$, $y = k^{-7/36}$, $z = k^{-17/72}$.
From Path 2, we have $\log_2(x) = -\frac{7}{24}$, $\log_2(y) = -\frac{3}{8} = -\frac{9}{24}$, $\log_2(z) = -\frac{5}{12} = -\frac{10}{24}$.
The expression $\left|\log_2(x^4y^3z^2)\right|$ can... | [
{
"tree_index": 62,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | cb32a4c15b966156af8e4c1c71224808857f2fcdd2c291948613f1d297dcb49a | 4 | 2,424 | false | completed | [] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 63 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | 18af8026f0fe8cc662fb7e03786f6dab47d999f7057062d0bf57582d670be29b | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Then the value of $\left|\log_2(x^4y^3z^2)\right|$ is $\tfrac{m}{n}$ where $m$ an... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Solve the system of logarithmic equations:
\[\log_2\left({x \over yz}\right) = {1 \over 2}\]\[\log_2\left({y \over xz}\right) = {1 \over 3}\]\[\log_2\left({z \over xy}\right) = {1 \over 4}\]
Express $x, y, z$ in terms of a common variable or find the... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 33 | 33 | true | false | false | aime24 | aime24_00024 | original | 0 | 15 | original/aime24/binary-mean-center/decompose/aime24_00024/15 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,015 | 0 | 33 | <action>
AGGREGATE
The first path directly calculated the value of $|\log_2(x^4y^3z^2)|$ as 3.
The second path calculated the value of $4a + 3b + 2c$ as $-\frac{25}{8}$, which is equivalent to $|\log_2(x^4y^3z^2)|$.
Since the problem asks for $m+n$ where the value is $\frac{m}{n}$, we have $\frac{25}{8}$. Here, $m=25$ ... | [
{
"tree_index": 63,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $x,y$ and $z$ be positive real numbers that satisfy the following system of equations:\n\\[\\log_2\\left({x \\over yz}\\right) = {1 \\over 2}\\]\\[\\log_2\\left({y \\over xz}\\right) = {1 \\over 3}\\]\\[\\log_2\\left({... | 0ce43ce94c27d56a4e1cc71fc5a2ac55ea3886b505d8311f0fce0b35a108fd27 | 4 | 2,853 | true | parse_error | ["call_2: missing <action>...</action> block"] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard0/aime24/decompose/rollouts.jsonl | 64 | 619f47bd4372e63cea1d4135beab80a8beab40809853c4ef75b54fc2f1a52ca3 | fa8f5eddbd2ced4c83e052bfce58ed8c596b505501e64dc432ff17504880a5fa | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 84 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with $AB=5$, $BC=9$, and $AC=10$, use the Law of Cosines to find $\cos(\angle BAC)$.
</path>
<path>
Given a triangle $ABC$ with $AB=5$, $BC=9$, and $AC=10$, and the tangents at $B$ and $C$ intersecting at $D$, find the length of $AD... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 7 | false | true | false | aime24 | aime24_00001 | original | 0 | 0 | original/aime24/binary-mean-center/decompose/aime24_00001/0 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 7 | <action>
AGGREGATE
Using the results from the subproblems:
- $\cos(\angle BAC) = \frac{1}{15}$
- $AD = 5\sqrt{2}$
By the power of a point theorem, $AP \cdot AD = AB^2$. Substituting the known values:
$AP \cdot 5\sqrt{2} = 5^2$
$AP \cdot 5\sqrt{2} = 25$
$AP = \frac{25}{5\sqrt{2}} = \frac{5}{\sqrt{2}} = \frac{5\sqrt{2}}... | [
{
"tree_index": 0,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be wr... | 0087ad35bd48b827bd150fcb5d310cbb1a772ab62d324990bb0e0277ac56999a | 4 | 805 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 1 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | a01bc91280c8ba816b4fae04c43a50612dc931e1a18f61c1d13830d07ef9ce5d | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with sides $AB=5$, $BC=9$, and $AC=10$, find the length of the angle bisector from $A$ to side $BC$ using the angle bisector theorem and the formula $AD = \frac{2 \cdot AB \cdot AC \cdot \cos(\frac{A}{2})}{AB + AC}$.
</path>
<path>
L... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 206 | false | true | false | aime24 | aime24_00001 | original | 0 | 1 | original/aime24/binary-mean-center/decompose/aime24_00001/1 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 206 | <action>
AGGREGATE
The length of the angle bisector $AD$ is $\frac{8\sqrt{30}}{9}$. The ratio $AP:PD$ is 125:81. Since $AD$ is the total length $AP + PD$, we have:
$AP = \frac{125}{125 + 81} \times AD = \frac{125}{206} \times \frac{8\sqrt{30}}{9} = \frac{1000\sqrt{30}}{1854} = \frac{500\sqrt{30}}{927}$.
However, this f... | [
{
"tree_index": 1,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be wr... | ce9143745b7c183b0e8749b1ee87b0ddd3f7f7fd5cc05e7d823667b8f5b64834 | 4 | 1,623 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 2 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | a6250d2e5dd10b263beeb1c1d6e9ce34883ede4e6aabb8410e8514e68ed7a222 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with sides $AB=5$, $BC=9$, and $AC=10$, use the Law of Cosines to find $\cos(\angle BAC)$. Then, use the formula for the power of a point $D$ with respect to circle $\omega$ to find the relationship between $AP$ and the sides of the ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 10 | false | true | false | aime24 | aime24_00001 | original | 0 | 2 | original/aime24/binary-mean-center/decompose/aime24_00001/2 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 10 | <action>
AGGREGATE
The first path determined that $AP = 9$.
The second path determined that $BD^2 = 42$, which implies $AD \cdot DP = 42$.
Since $AP = AD + DP$ and $AD \cdot DP = 42$, we have $AP = 9$.
The problem asks for $AP$ in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime. Here, $AP = 9 = \frac{9}{1... | [
{
"tree_index": 2,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be wr... | ad87dfc8bb96a45d26babbfcb7f813b64ef71be9517be983262eba38c32429a0 | 4 | 1,042 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 3 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 3ce96c5266d9f28a9580837c40f18df6ac646e0a2559685d9cbc0ae1fe5f6520 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with $AB=5$, $BC=9$, and $AC=10$, use the Law of Cosines to find the measure of angle $BAC$ (let this be $\theta$). Calculate $\cos(\theta)$.
</path>
<path>
Given a triangle $ABC$ with sides $AB=5$, $BC=9$, and $AC=10$, and a circle ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 33 | false | true | false | aime24 | aime24_00001 | original | 0 | 3 | original/aime24/binary-mean-center/decompose/aime24_00001/3 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 33 | <action>
AGGREGATE
The goal is to find $AP$ where $P$ is the intersection of $AD$ with the circumcircle $\omega$. From the first path, we have $\cos(\theta) = \frac{1}{15}$. From the second path, we have the circumradius $R = \frac{75}{4\sqrt{14}}$.
Using the formula $AP = 2R \cos(\theta)$:
$AP = 2 \cdot \frac{75}{4\sq... | [
{
"tree_index": 3,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be wr... | 0aff3ae82f0613242a82af03ace0193b6ac956e1857af471d2de243b8fd54e45 | 4 | 1,201 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 4 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | bbb4b12e1d104b0ef1cf1e47c6104e2a54e816d64f78e1b165c8bc1b5957f86e | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with $AB=5$, $BC=9$, and $AC=10$, find the length of the tangent segment $BD$ and $CD$ from point $D$ to the circle $\omega$. Since $D$ is the intersection of the tangents at $B$ and $C$, $BD = CD$.
</path>
<path>
In triangle $ABC$ w... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 6 | false | true | false | aime24 | aime24_00001 | original | 0 | 4 | original/aime24/binary-mean-center/decompose/aime24_00001/4 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 6 | <action>
AGGREGATE
The Power of a Point theorem states that for a point $D$ outside the circle, with tangents $BD$ and $CD$ to the circle, and a secant $AD$ intersecting the circle at $P$, the relation $AD \cdot DP = BD^2$ holds. From the first path, $BD = CD = x$. From the second path, $AD \cdot DP = 25$. Since $AD \c... | [
{
"tree_index": 4,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be wr... | 2adf89798ded1558ca799f156aa148c30ff7f414588c3a72ffa3296c248506f4 | 4 | 856 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 5 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | d72b0f711e8f33f5109fa23f1ef14a7f7bcd3ab725a31aebbad8991d226f74fe | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$, the sides are $AB=5$, $BC=9$, and $AC=10$. Using the Power of a Point theorem, if $AD$ is a line intersecting the circle at $P$, and $D$ is the intersection of the tangents at $B$ and $C$, find the length of $AP \cdot AD$.
</path>
... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 7 | false | false | false | aime24 | aime24_00001 | original | 0 | 5 | original/aime24/binary-mean-center/decompose/aime24_00001/5 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 7 | <action>
AGGREGATE
Using the results from the paths:
1. $AP \cdot AD = 2$
2. $AD = 5\sqrt{2}$
Substituting $AD$ into the equation:
$AP \cdot (5\sqrt{2}) = 2$
$AP = \frac{2}{5\sqrt{2}} = \frac{2}{5\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{10} = \frac{\sqrt{2}}{5}$
Since $AP$ is expressed as $\frac{\... | [
{
"tree_index": 5,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be wr... | d0af7d5f87a8f5ed731b601ae0b6351b2a844198dfceade0476cde3af261a159 | 4 | 3,122 | true | parse_error | ["call_2: missing <action>...</action> block"] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 6 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | d6f32f3038cf2bc91fa184e6ef3974c48fbf01e810243813896dc18f3203f7e5 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with sides $AB=5$, $BC=9$, and $AC=10$, find the length of the tangent segment $BD$ from $B$ to the point of tangency $D$ on the circle $\omega$, given that $D$ is the intersection of the tangents at $B$ and $C$.
</path>
<path>
Give... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 38 | false | true | false | aime24 | aime24_00001 | original | 0 | 6 | original/aime24/binary-mean-center/decompose/aime24_00001/6 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 38 | <action>
AGGREGATE
The first path determined the length of $BD$ (which is equal to $CD$) to be 3. The second path determined the length of $AP$ to be $\frac{75\sqrt{14} + 5\sqrt{910}}{56}$. However, the second path's result is not in the form $\frac{m}{n}$, which suggests a miscalculation or misinterpretation of the pr... | [
{
"tree_index": 6,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be wr... | e1f70b659fed327468ed18e55d84299f06326102cee850a0d72eb690ff2a4df1 | 4 | 1,723 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 7 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 62f9f5b5b74f2d4c8eee5fd31e5608179af72fa0d8478b6aa639e735c20f6603 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with sides $AB=5$, $BC=9$, and $AC=10$, use the Law of Cosines to find $\cos(\angle BAC)$.
</path>
<path>
Given a triangle $ABC$ with sides $AB=5$, $BC=9$, and $AC=10$, and a point $P$ on the circumcircle $\omega$ such that $AD$ is ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 46 | false | true | false | aime24 | aime24_00001 | original | 0 | 7 | original/aime24/binary-mean-center/decompose/aime24_00001/7 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 46 | <action>
AGGREGATE
The Law of Cosines result $\cos(\angle BAC) = \frac{1}{15}$ was used to calculate the Power of a Point expression $AP = 45$. The problem asks for $AP$ in the form $\frac{m}{n}$ where $m$ and $n$ are relatively prime. Here, $AP = 45 = \frac{45}{1}$, so $m=45$ and $n=1$. Since 45 and 1 are relatively p... | [
{
"tree_index": 7,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be wr... | 17d3782d64fe6f99bf06707e20b8e7cbe521aa81c3165bba43ab725e1c3d24a6 | 4 | 612 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 8 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 70e5c90c91de2574dd50e8171126a84165d994752ba4edded5d299e1e3332bdd | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with sides $AB=5$, $BC=9$, and $AC=10$, use the Law of Cosines to find $\cos(\angle BAC)$. Then, use the formula for the power of a point $D$ (the intersection of tangents at $B$ and $C$) with respect to the circle $\omega$ to find $... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 17 | false | true | false | aime24 | aime24_00001 | original | 0 | 8 | original/aime24/binary-mean-center/decompose/aime24_00001/8 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 17 | <action>
AGGREGATE
The first path correctly applied the Law of Cosines and the power of a point theorem to find $AP = \frac{50}{35} = \frac{10}{7}$. The second path provided an alternative verification method, but the first path directly solved the problem using the given constraints.
The fraction $\frac{10}{7}$ is in ... | [
{
"tree_index": 8,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be wr... | 3fdde827108a2c90af1ae2b8ee98db667aad6c7092276c860b9de812fb1268da | 4 | 802 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 9 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 47988eabe301e30d47765906015d17ce147fcb85edba818256d9bf2b1a17df1c | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with sides $AB=5$, $BC=9$, and $AC=10$, use the Law of Cosines to find $\cos(\angle BAC)$. Then, use the formula for the power of a point $D$ with respect to circle $\omega$ (which is $DB^2 = DA \cdot DP$) to find the relationship be... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 177 | false | true | false | aime24 | aime24_00001 | original | 0 | 9 | original/aime24/binary-mean-center/decompose/aime24_00001/9 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 177 | <action>
AGGREGATE
The first path identified a dependency on the circumradius $R$, which was not provided. The second path successfully calculated $R$ using the side lengths and Heron's formula, and derived $AP$ as $\frac{m}{n}$.
The area $K = 24$ and $R = \frac{15}{4}$.
Using the formula $AP = \frac{R}{2 - \cos A}$:
$... | [
{
"tree_index": 9,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be wr... | 5b2d8c96b0f8669595d6df420568c966801c472f15b9c4079066af8b8515c37f | 4 | 1,332 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 10 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 331a1964aa6fa8a535d908193ab44d235d9e3f390d45b85e666ca0637d930cec | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $ABC$ be a triangle with $AB=5$, $BC=9$, and $AC=10$. Let $\omega$ be the circumcircle of $ABC$. Let $D$ be the intersection of the tangents to $\omega$ at $B$ and $C$. Prove that $D$ lies on the line $AD$.
</path>
<path>
Let $ABC$ be a triangle ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 302 | false | true | false | aime24 | aime24_00001 | original | 0 | 10 | original/aime24/binary-mean-center/decompose/aime24_00001/10 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 302 | <action>
AGGREGATE
The problem asks for the length $AP$ in the form $\frac{m}{n}$. The second path established that $P$ is the midpoint of the arc $BC$ not containing $A$. By the Power of a Point theorem, $AP \cdot AD = AB \cdot AC$.
Calculating $AD$:
$AD^2 = AB \cdot AC + BD^2$
Since $BD = CD$ (tangents from $D$ to $\... | [
{
"tree_index": 10,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be w... | d1152ba24f29374a2de3de06b146f5d496b57fa6df27bf363c98dc5501c10cd6 | 4 | 1,339 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 11 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | cc65e9c1fe8209b79fd81a17123547b150047c1f8042e53b6fec5b46000e93e1 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with $AB=5$, $BC=9$, and $AC=10$, let $D$ be the intersection of the tangents to the circumcircle $\omega$ at $B$ and $C$. Using the properties of tangents and the fact that $D$ is the exsimilicenter of the incircle and the circumcir... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 23 | false | true | false | aime24 | aime24_00001 | original | 0 | 11 | original/aime24/binary-mean-center/decompose/aime24_00001/11 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 23 | <action>
AGGREGATE
The first path calculated $AD = \frac{8\sqrt{30}}{9}$, which is the length of the symmedian. The second path calculated $AP = \frac{20}{3}$, which is the length of the segment from $A$ to the intersection $P$ of the symmedian with the circumcircle.
The problem asks for $AP$, which is $\frac{20}{3}$. ... | [
{
"tree_index": 11,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be w... | db2548123d878d876e1b523e886d048769b384804a000cc9b8183635296aae40 | 4 | 976 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 12 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 1a4be2c49bbdfead27e03d5cb730b4753cadbdb3337aa768d3eaf2e92b050565 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with $AB=5$, $BC=9$, and $AC=10$, find the length of the tangent segment $AD$ from vertex $A$ to the point $D$ where the tangents at $B$ and $C$ intersect. Use the fact that $AD$ is the symmedian of triangle $ABC$.
</path>
<path>
Giv... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 17 | false | true | false | aime24 | aime24_00001 | original | 0 | 12 | original/aime24/binary-mean-center/decompose/aime24_00001/12 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 17 | <action>
AGGREGATE
From the first path, we found that $AD = \frac{12\sqrt{7}}{7}$. From the second path, we found that the ratio $AP:PD = 5:1$. Since $AD = AP + PD$ and $AP = 5 \cdot PD$, we have $AP + PD = 6 \cdot PD = \frac{12\sqrt{7}}{7}$. Thus, $PD = \frac{2\sqrt{7}}{7}$ and $AP = 5 \cdot \frac{2\sqrt{7}}{7} = \fra... | [
{
"tree_index": 12,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be w... | 75b52652eb1005331259977615f01cb035d0ecf248587c17d698671e02fcfa9c | 4 | 1,286 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 13 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 591aa6468427e9581e65df31ef57f237818d29d6bf72ee0c0e3331f4cf8fef04 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with $AB=5$, $BC=9$, and $AC=10$, let $D$ be the intersection of the tangents to the circumcircle $\omega$ at $B$ and $C$. Prove that $D$ lies on the line $AD$ and that $AD$ is the symmedian from $A$.
</path>
<path>
In triangle $ABC... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 6 | false | true | false | aime24 | aime24_00001 | original | 0 | 13 | original/aime24/binary-mean-center/decompose/aime24_00001/13 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 6 | <action>
AGGREGATE
The first path established that $AD$ is the symmedian of triangle $ABC$. The second path calculated the ratio $AP:PD = 1:4$. This implies that if the length of $AD$ is $x$, then $AP = \frac{1}{5}x$ and $PD = \frac{4}{5}x$. Since $P$ lies on the circumcircle, the length $AP$ is a segment of the symmed... | [
{
"tree_index": 13,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be w... | 2168ff84a07753a2e4c73e9b03e0e05716a61021e1ec77029bb5f25d1496e2d8 | 4 | 907 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 14 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 4e85fd2a5eed288df6a2b109eab7d8cf37b41a2ddba9a77573e734383f4d81c8 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Let $ABC$ be a triangle with $AB=5$, $BC=9$, and $AC=10$. Using the Law of Cosines, find the value of $\cos B$.
</path>
<path>
Let $ABC$ be a triangle with $AB=5$, $BC=9$, and $AC=10$. Let $D$ be the intersection of the tangents to the circumcircle a... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 4859 | false | false | false | aime24 | aime24_00001 | original | 0 | 14 | original/aime24/binary-mean-center/decompose/aime24_00001/14 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 4859 | <action>
AGGREGATE
The first path provided the value of $\cos B = \frac{1}{15}$. This is a necessary trigonometric value for the symmedian length formula used in the second path.
The second path derived the length of $AP$ as $AP = \frac{3375\sqrt{14}}{1484}$.
The problem asks for $AP$ in the form $\frac{m}{n}$ where $m... | [
{
"tree_index": 14,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be w... | d40075707e6a1ae61050dcaf992d0cea2c5344c221ec94c1bb1bb3c5ccd634f7 | 4 | 2,679 | true | parse_error | ["call_3: missing <action>...</action> block"] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 15 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | f690410c4ee6c32fb5411aa1dca3fae91d0628cc4c5f189978791d4d09e2f1aa | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Let $ABC$ be a triangle inscribed in circle $\omega$. Let the tangents to $\omega$ at $B$ and $C$ intersect at point $D$, and let $\overline{AD}$ intersect $\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be written as the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m + n$. | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In triangle $ABC$ with $AB=5$, $BC=9$, and $AC=10$, find the length of the altitude $h$ from $A$ to $BC$. Use the formula for the area of a triangle: $Area = \frac{1}{2} \times base \times height$. First, calculate the area using Heron's formula. The ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 113 | 161 | false | true | false | aime24 | aime24_00001 | original | 0 | 15 | original/aime24/binary-mean-center/decompose/aime24_00001/15 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 161 | <action>
AGGREGATE
The altitude $h = \frac{4\sqrt{14}}{3}$ from $A$ to $BC$ is used to find the area of triangle $ABC$, which is $6\sqrt{14}$. Since $AD$ is the angle bisector, by the Angle Bisector Theorem, $AP$ is the harmonic mean of $AB$ and $AC$ with respect to the altitude. The formula for $AP$ in terms of the al... | [
{
"tree_index": 15,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Let $ABC$ be a triangle inscribed in circle $\\omega$. Let the tangents to $\\omega$ at $B$ and $C$ intersect at point $D$, and let $\\overline{AD}$ intersect $\\omega$ at $P$. If $AB=5$, $BC=9$, and $AC=10$, $AP$ can be w... | bea96a782550e3fda4857a286905fe280c2e19a5c9701e82f0900db7d4fdab18 | 4 | 1,151 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 16 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 54f98d55661e0b9d6555528b411d1ab5445f93852dbc0d5e5e09f513f7696e21 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 61 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the total number of ways Jen can pick 4 distinct numbers from the set $S=\{1,2,3,\cdots,9,10\}$. The total number of ways is given by the combination formula $C(10, 4) = \frac{10!}{4!6!}$.
</path>
<path>
Calculate the total number of ways Je... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 7552 | false | true | false | aime24 | aime24_00009 | original | 0 | 0 | original/aime24/binary-mean-center/decompose/aime24_00009/0 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 7552 | <action>
AGGREGATE
The probability of winning the grand prize given that she won a prize is the ratio of the number of ways to win the grand prize to the number of ways to win any prize.
- Number of ways to win the grand prize: 1.
- Number of ways to win any prize: 7551.
The probability is $\frac{1}{7551}$.
The problem... | [
{
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"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | 91ebbdc7d71bc70d13c1b0869292be2e1d5da396dfd6e26a3f5354995171c009 | 4 | 564 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 17 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 96ac222694173f5c31f0631953757d0e3b4fa274264638d9a4f0a5455545508d | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the total number of ways to choose 4 distinct numbers from the set $S=\{1,2,3,\cdots,9,10\}$. This is the total number of possible outcomes for the lottery.
</path>
<path>
Calculate the number of favorable outcomes for winning a prize (at le... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 6396 | false | true | false | aime24 | aime24_00009 | original | 0 | 1 | original/aime24/binary-mean-center/decompose/aime24_00009/1 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 6396 | <action>
AGGREGATE
The probability of winning the grand prize given that she won a prize is the number of ways to win the grand prize divided by the number of ways to win any prize.
Number of ways to win the grand prize = 1.
Number of ways to win any prize = 6395.
Probability = 1 / 6395.
The fraction 1/6395 is already ... | [
{
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"depth": 0,
"role": "root",
"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | e26f80089a090a0455ea6bbd8140b303d9d88d38c2cd0905fca4b53635b15ce3 | 5 | 696 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 18 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | aab56e5988587996c52a656a007bec1c1d7c3ffd7bf7eb6542414cb713027818 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a lottery where Jen picks 4 distinct numbers from the set $S=\{1,2,3,\cdots,9,10\}$ and 4 numbers are randomly chosen from $S$, calculate the probability that Jen's 4 numbers exactly match the 4 randomly chosen numbers (grand prize). The total num... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 116 | true | true | true | aime24 | aime24_00009 | original | 0 | 2 | original/aime24/binary-mean-center/decompose/aime24_00009/2 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 116 | <action>
AGGREGATE
We are given:
- $P(G) = \frac{1}{210}$ (probability of winning the grand prize)
- $P(W) = \frac{23}{42}$ (probability of winning a prize)
We need to find the conditional probability $P(G|W) = \frac{P(G \cap W)}{P(W)}$. Since winning the grand prize implies winning a prize ($G \subseteq W$), $P(G \ca... | [
{
"tree_index": 18,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | 3b09fa8735cea419bfb31b5437142cf546629043e2264a8ec8a551f791507e6b | 4 | 908 | false | completed | [] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 19 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 7953f3054b6d46abe94e24212ea3ad89a2e931bca77c06563726d6c8529dea46 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the total number of ways Jen can pick 4 distinct numbers from the set $S=\{1,2,3,\cdots,9,10\}$. This is the total number of possible combinations of 4 numbers from 10.
</path>
<path>
Calculate the number of ways Jen can win the grand prize.... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 116 | true | true | true | aime24 | aime24_00009 | original | 0 | 3 | original/aime24/binary-mean-center/decompose/aime24_00009/3 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 116 | <action>
AGGREGATE
The probability of winning the grand prize given that she won a prize is the ratio of the number of ways to win the grand prize to the number of ways to win any prize.
- Total ways to win a prize: The total number of ways to win a prize is the sum of the ways to win the grand prize and the ways to wi... | [
{
"tree_index": 19,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | bedc618219f006984f6326b8a93f3807695de6cc41b107296f38970720e0d31f | 4 | 553 | false | completed | [] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 20 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 0ebd54ed782f62538470abe20dda223d48453842221ba3b19ec62ba27ff3bed6 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the total number of ways to choose 4 distinct numbers from the set $S=\{1,2,3,\cdots,9,10\}$.
</path>
<path>
Calculate the number of favorable outcomes for winning the prize (at least 2 of her 4 chosen numbers match the 4 randomly chosen nu... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 827 | false | true | false | aime24 | aime24_00009 | original | 0 | 4 | original/aime24/binary-mean-center/decompose/aime24_00009/4 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 827 | <action>
AGGREGATE
The probability of winning the grand prize given that she won a prize is the number of ways to win the grand prize divided by the number of ways to win a prize.
Number of ways to win the grand prize (all 4 match) = 210.
Number of ways to win a prize (at least 2 match) = 3925.
The probability is 210 /... | [
{
"tree_index": 20,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | 8ef98d3271679a2f409b9ca6776ccfc8a64eef7b7ba45420f5f18da83e97d61c | 4 | 641 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 21 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | a8c8192512ea6cac612d3e95226f663dd479bc4c439a690a18c9f19a236af99d | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the total number of ways to pick 4 distinct numbers from the set $S=\{1,2,3,\cdots,9,10\}$.
</path>
<path>
Calculate the number of ways to pick 4 distinct numbers from $S$ such that at least two of the numbers match the 4 randomly chosen num... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 54 | false | true | false | aime24 | aime24_00009 | original | 0 | 5 | original/aime24/binary-mean-center/decompose/aime24_00009/5 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 54 | <action>
AGGREGATE
The probability of winning the grand prize given that she won a prize is the ratio of the number of ways to win the grand prize to the number of ways to win a prize. The number of ways to win a prize is 53, and the number of ways to win the grand prize is 1. Thus, the probability is $\frac{1}{53}$. S... | [
{
"tree_index": 21,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | 34e2c9ed2c11fb4dd6dea321fbde15828dceb33d5e754c0bf565841ac81454d4 | 4 | 509 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 22 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 8140f2949022100c01efda70661211c12b8572617de8d30588ec74e6857e3ca7 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the total number of ways Jen can pick 4 distinct numbers from the set $S = \{1,2,3,\cdots,9,10\}$. Then, calculate the number of ways to win the grand prize, which is the number of ways to choose exactly the 4 numbers that were randomly sel... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 210 | false | true | false | aime24 | aime24_00009 | original | 0 | 6 | original/aime24/binary-mean-center/decompose/aime24_00009/6 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 210 | <action>
AGGREGATE
The probability of winning the grand prize given that she won a prize is calculated as:
Prize-winning outcomes = (Ways to win the grand prize) + (Ways to win a prize but not the grand prize)
Total prize-winning outcomes = 1 + 208 = 209.
Probability = (Ways to win the grand prize) / (Total prize-winni... | [
{
"tree_index": 22,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | 7455511bb4607f764b9cab50eb6ad0fb7bc26e02fe44cc4faf66f79bb1457e9e | 4 | 563 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 23 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 3588649530ab851379c2df08cf1f5757bb668c3d80bb58508d7008b970579187 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Jen picks 4 distinct numbers from the set $S=\{1,2,3,\cdots,9,10\}$. Calculate the total number of ways to choose 4 numbers from 10.
</path>
<path>
Jen picks 4 distinct numbers from the set $S=\{1,2,3,\cdots,9,10\}$. Calculate the number of favorable ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 196 | false | true | false | aime24 | aime24_00009 | original | 0 | 7 | original/aime24/binary-mean-center/decompose/aime24_00009/7 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 196 | <action>
AGGREGATE
The probability of winning the grand prize given that she won a prize is the number of grand prize outcomes divided by the number of prize-winning outcomes: $\frac{1}{195}$.
Since 1 and 195 are relatively prime, $m=1$ and $n=195$.
$m+n = 1+195 = 196$.
Final answer: 196
</action> | [
{
"tree_index": 23,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | 97772795596b53ae4534aaba0597cead5f70e4a82f80eb1bd15e28a47bfa0af3 | 5 | 583 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 24 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 92b28b58a5b7da0dc68d7c0767763dbf20bd93d5007d4d908cd4cb667e2f51a8 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the total number of ways to choose 4 distinct numbers from the set $S=\{1,2,3,\cdots,9,10\}$, and the total number of ways to choose 4 distinct numbers from $S$ for the lottery draw. Then, calculate the probability of winning a prize (at le... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 233 | false | true | false | aime24 | aime24_00009 | original | 0 | 8 | original/aime24/binary-mean-center/decompose/aime24_00009/8 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 233 | <action>
AGGREGATE
The probability of winning the grand prize given a prize is $\frac{1}{232}$. The problem asks for the fraction $m/n$ where $m=1$ and $n=232$. Since 1 and 232 are relatively prime, $m+n=1+232=233$.
Final answer: 233
</action> | [
{
"tree_index": 24,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | d6cf26840c41650d5f0b6bab3cdcebe903d88521df725424689fe3040de4da3e | 4 | 547 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 25 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 16c9460b9008fc7a76d4ab9aafcce0126089f6eb55bce234a41d55d0aa2c409e | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the probability of Jen winning the grand prize given that she has won a prize. Let $W$ be the event of winning a prize and $G$ be the event of winning the grand prize. We need $P(G|W) = \frac{P(G \cap W)}{P(W)}$. Since $G$ implies $W$, $P(G... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 383 | false | true | false | aime24 | aime24_00009 | original | 0 | 9 | original/aime24/binary-mean-center/decompose/aime24_00009/9 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 383 | <action>
AGGREGATE
We have $P(G|W) = \frac{P(G)}{P(W)} = \frac{1/115}{23/42} = \frac{42}{115 \times 23} = \frac{42}{2645}$. Simplifying the fraction $\frac{42}{2645}$ by dividing numerator and denominator by 7 gives $\frac{6}{377}$. Since 6 and 377 are relatively prime, $m=6$ and $n=377$. Thus, $m+n=6+377=383$.
Final a... | [
{
"tree_index": 25,
"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | bb9952edfd813049bbb43593b2a8a1ae0250e1a4e12416d9004857c20907e909 | 4 | 1,108 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 26 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | bec7076af59d7150a5e8ec076b71a1c03102306765f73e6feb58520bffde1fe6 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Jen picks 4 distinct numbers from the set $S = \{1, 2, 3, \dots, 10\}$. The lottery draws 4 numbers from the same set. Calculate the total number of ways Jen can pick her 4 numbers and the total number of ways the lottery can draw 4 numbers. Express t... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 116 | true | false | false | aime24 | aime24_00009 | original | 0 | 10 | original/aime24/binary-mean-center/decompose/aime24_00009/10 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 116 | <action>
AGGREGATE
The probability of winning the grand prize given that she won a prize is the number of grand prize-winning outcomes divided by the number of prize-winning outcomes.
The number of grand prize-winning outcomes is $\binom{4}{4} = 1$.
The number of prize-winning outcomes is 115.
The probability is $\frac... | [
{
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"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | 66d7ced276bee8a12bd322241798fdb52ac4f451a6d467b4ef2a48ad1de3a90d | 4 | 2,618 | true | parse_error | ["call_2: missing <action>...</action> block"] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 27 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | e9c60f9af52f4c9a94dab1a414b9506b85d4924155a46249b3e000ad2eb9c3e1 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Jen picks 4 distinct numbers from the set $S = \{1, 2, 3, \dots, 10\}$. Calculate the total number of ways she can win a prize, defined as having at least 2 of her 4 chosen numbers match the 4 randomly selected numbers from $S$.
</path>
<path>
Jen pic... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 116 | true | true | true | aime24 | aime24_00009 | original | 0 | 11 | original/aime24/binary-mean-center/decompose/aime24_00009/11 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 116 | <action>
AGGREGATE
The probability of winning the grand prize given that she won a prize is the ratio of the number of grand prize-winning combinations to the total number of prize-winning combinations.
Probability = (Number of grand prize-winning combinations) / (Total number of prize-winning combinations) = 1 / 115.
... | [
{
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"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | 96df526497e4fb0eeed4ecfcbee77b2acd3454da5821631b0b1c086aa393bbde | 4 | 830 | false | completed | [] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 28 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 60a4d6ff434690c291c92ee7f861277a6f7acad1c6187ed3ceee0a7a638ec66c | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the total number of ways to choose 4 distinct numbers from the set $S=\{1,2,3,\cdots,9,10\}$. This is the denominator for the probability of winning a prize.
</path>
<path>
Calculate the number of favorable outcomes for winning the grand pr... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 116 | true | true | true | aime24 | aime24_00009 | original | 0 | 12 | original/aime24/binary-mean-center/decompose/aime24_00009/12 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 116 | <action>
AGGREGATE
The probability of winning the grand prize given that she won a prize is the number of grand prize outcomes divided by the number of prize-winning outcomes.
The number of prize-winning outcomes is the total number of ways to choose 4 numbers minus the number of ways to choose exactly 1 number from {2... | [
{
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"depth": 0,
"role": "root",
"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | 2632a442b23784770d2d77d069adb1943848f346a97308c70584346f2af1273c | 4 | 639 | false | completed | [] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 29 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 9b39f0a6bc20c169eb620fedd737b5f95f6a70bfd48876ac98f66bf28e793605 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the probability that Jen wins a prize (at least 2 of her 4 chosen numbers match the 4 randomly chosen numbers from the set $S = \{1,2,3,\cdots,9,10\}$).
Total ways to choose 4 numbers from 10: $\binom{10}{4} = 210$.
Total ways for Jen to c... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 211 | false | true | false | aime24 | aime24_00009 | original | 0 | 13 | original/aime24/binary-mean-center/decompose/aime24_00009/13 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 211 | <action>
AGGREGATE
The problem asks for the probability of winning the grand prize given that she won a prize.
From the first path, the probability of winning a prize is 1.
From the second path, the probability of winning the grand prize is 1/210.
Since the probability of winning a prize is 1, the conditional probabil... | [
{
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"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | 0ed4b015534ee73f24b63aeddbcd13373008b99b924dae35c5ffd6a4714a0494 | 4 | 527 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 30 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | d1240e9627ba7f2da4d27addbe8c23670505c66c29df3b34207dfdcb10e52b9d | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Calculate the total number of ways to pick 4 distinct numbers from the set $S=\{1,2,3,\cdots,9,10\}$. This is the denominator for the probability of winning a prize.
</path>
<path>
Calculate the number of favorable outcomes for winning the grand prize... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 116 | true | true | true | aime24 | aime24_00009 | original | 0 | 14 | original/aime24/binary-mean-center/decompose/aime24_00009/14 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 116 | <action>
AGGREGATE
The probability of winning the grand prize given that she won a prize is the number of grand prize outcomes divided by the number of winning outcomes.
From Path 1, the total number of ways to pick 4 distinct numbers is 210.
From Path 2, the number of grand prize outcomes is 1.
The total number of win... | [
{
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"depth": 0,
"role": "root",
"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | c2b1e411a9ea09ca274f639ca1ab30804f1d60f355e1eb5668a7fcb1ee392d43 | 4 | 584 | false | completed | [] | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | {"correct":true,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"exact","resolved":true,"transport_retries":0} | null | exact | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 31 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | fff7986fdce4d699d8c6560b378cbaea6b48d8e563f90ca7065154897d749474 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Jen enters a lottery by picking $4$ distinct numbers from $S=\{1,2,3,\cdots,9,10\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and wins the grand prize if all four of her numbers were the randomly chosen numbers. The probability of... | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
In a lottery where Jen picks 4 distinct numbers from the set $S=\{1,2,3,\cdots,9,10\}$ and 4 numbers are randomly chosen from $S$, calculate the probability that Jen wins a prize (at least 2 of her numbers match the 4 chosen numbers). Express this pr... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 116 | 2447 | false | true | false | aime24 | aime24_00009 | original | 0 | 15 | original/aime24/binary-mean-center/decompose/aime24_00009/15 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 2447 | <action>
AGGREGATE
The probability of winning the grand prize given that she won a prize is $P(\text{Grand Prize} | \text{Prize}) = \frac{P(\text{Grand Prize})}{P(\text{Prize})} = \frac{0.06997}{\frac{23}{42}} = \frac{0.06997 \times 42}{23} = \frac{2.93874}{23} = \frac{147}{2300}$.
The fraction $\frac{147}{2300}$ is in... | [
{
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"problem": "Jen enters a lottery by picking $4$ distinct numbers from $S=\\{1,2,3,\\cdots,9,10\\}.$ $4$ numbers are randomly chosen from $S.$ She wins a prize if at least two of her numbers were $2$ of the randomly chosen numbers, and... | 04405b5c55f86f7f73ddb7c5ac8295ca4d6b0485a69e735616ce71e4290d58ce | 4 | 1,168 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 32 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 3012c1b7cabe79812f7beba65c90891317b9e23a3d726b8722fc0549bd07ed54 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 69 |
Find the number of triples of nonnegative integers \((a,b,c)\) satisfying \(a + b + c = 300\) and
\begin{equation*}
a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.
\end{equation*} | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Find the number of triples of nonnegative integers \((a,b,c)\) satisfying \(a + b + c = 300\) and
\begin{equation*}
a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.
\end{equation*}
Specifically, find the number of solutions where \(a = 100\).
</p... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 601 | 203 | false | true | false | aime24 | aime24_00017 | original | 0 | 0 | original/aime24/binary-mean-center/decompose/aime24_00017/0 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 203 | <action>
AGGREGATE
The original problem asks for the total number of triples \((a,b,c)\) satisfying the given conditions. The path results indicate:
1. When \(a = 100\), there are 201 valid triples.
2. When \(a = 200\), there are 2 valid triples.
These are the only two non-trivial cases (since \(a = 0, 1, 2, 3, 4, 5, 6... | [
{
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"depth": 0,
"role": "root",
"problem": "Find the number of triples of nonnegative integers \\((a,b,c)\\) satisfying \\(a + b + c = 300\\) and\n\\begin{equation*}\na^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.\n\\end{equation*}",
"input_text": "You are... | a2ba70f1f559235e4ac84135e147eabd64327d0e4acf9a136f59d173fc755b04 | 4 | 3,395 | true | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 33 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | cf8a32e725aeaad6feba1f283fa08c693dad06dc597186c0f1f0d321044f7402 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 77 |
Find the number of triples of nonnegative integers \((a,b,c)\) satisfying \(a + b + c = 300\) and
\begin{equation*}
a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.
\end{equation*} | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Find the number of triples of nonnegative integers \((a,b,c)\) satisfying \(a + b + c = 300\) and the equation \(a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000\) where at least one of \(a, b, c\) is zero. Specifically, count the cases where \(a ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 601 | 7 | false | true | false | aime24 | aime24_00017 | original | 0 | 1 | original/aime24/binary-mean-center/decompose/aime24_00017/1 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 7 | <action>
AGGREGATE
The total number of triples is the sum of the results from the two paths: 6 (from the zero-one cases) + 1 (from the positive case) = 7.
Final answer: 7
</action> | [
{
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"call_id": 1,
"depth": 0,
"role": "root",
"problem": "Find the number of triples of nonnegative integers \\((a,b,c)\\) satisfying \\(a + b + c = 300\\) and\n\\begin{equation*}\na^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.\n\\end{equation*}",
"input_text": "You are... | 19637c64f4dad97f56c3e13b6a78dd8b9796b46c55e9a9c160d88629013441aa | 4 | 1,411 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 34 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 431be96d3c83113d3d4d4ed2a7bcb016217983ebfcd40ae421a23c9e4867e913 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 77 |
Find the number of triples of nonnegative integers \((a,b,c)\) satisfying \(a + b + c = 300\) and
\begin{equation*}
a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.
\end{equation*} | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Find the number of triples of nonnegative integers \((a,b,c)\) satisfying \(a + b + c = 300\). This is a stars and bars problem where we need to find the number of solutions to the equation \(a + b + c = 300\) in nonnegative integers. The formula for ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 601 | 6 | false | true | false | aime24 | aime24_00017 | original | 0 | 2 | original/aime24/binary-mean-center/decompose/aime24_00017/2 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 6 | <action>
AGGREGATE
The total number of valid triples is the product of the number of ways to choose the values of \(a, b, c\) and the number of valid permutations. From the first path, there are 45451 ways to choose the values (though none satisfy the second condition). From the second path, there are 6 valid triples. ... | [
{
"tree_index": 34,
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"depth": 0,
"role": "root",
"problem": "Find the number of triples of nonnegative integers \\((a,b,c)\\) satisfying \\(a + b + c = 300\\) and\n\\begin{equation*}\na^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.\n\\end{equation*}",
"input_text": "You are... | ca676b64e768b09a98b71d795ef8bd9ee651dd48e19b3e766a0336858efe1cc1 | 4 | 775 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 35 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 085f63b10fbc87fd022fd3b2b15458ae8603df8786b7eba2d4a7e76795192f9a | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 77 |
Find the number of triples of nonnegative integers \((a,b,c)\) satisfying \(a + b + c = 300\) and
\begin{equation*}
a^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.
\end{equation*} | === Call 1 | root ===
<action>
DECOMPOSE
</action>
<parallel>
<path>
Find the number of triples of nonnegative integers \((a,b,c)\) satisfying \(a + b + c = 300\) and \(a \leq b \leq c\). This is equivalent to finding the number of partitions of 300 into at most three parts.
</path>
<path>
Given the equation \(a + b + ... | GRPO-BINARY | forced_decomposition | DECOMPOSE | 601 | 15100 | false | true | false | aime24 | aime24_00017 | original | 0 | 3 | original/aime24/binary-mean-center/decompose/aime24_00017/3 | null | null | binary-mean-center | 100 | 7d5ffe3cf730f06f89cca32182896ef1570977bae7795fac5df4680d349985ab | 0.5 | 1 | 0 | 16 | 20,261,015 | 20,281,115 | 1 | 15100 | <action>
AGGREGATE
The first path determined that there are 45300 unordered triples \((a,b,c)\) where \(a + b + c = 300\). The second path determined that the only valid set of values for \(a, b, c\) that satisfy both the sum constraint \(a + b + c = 300\) and the product condition \(abc = 1,000,000\) is \((10, 10, 280... | [
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"problem": "Find the number of triples of nonnegative integers \\((a,b,c)\\) satisfying \\(a + b + c = 300\\) and\n\\begin{equation*}\na^2b + a^2c + b^2a + b^2c + c^2a + c^2b = 6,000,000.\n\\end{equation*}",
"input_text": "You are... | 9c2b09767176f8e0ffd846dd190b471d15b2ebd7d5d867164c9f5f00359a14c1 | 4 | 1,304 | false | completed | [] | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | {"correct":false,"grader_revision":"explicit-answer-timeout-safe-v3-parse-review","method":"math_verify","resolved":true,"transport_retries":0} | null | math_verify | null | outputs/tunix_root_ablation/20261014-audited-v3/collected/official-binary-mean-center-decompose-shard1/aime24/decompose/rollouts.jsonl | 36 | 7db824273b302d319c052162ca1a1d711aba34bdbc8a6cea72e5c5eb635fd221 | 7f43693225f52076aa03d53c2dcb32e38a42612145723467c899392851eac2a5 | math-ai/aime24 | 83a7f387baaa524a8bda0022eac0541582297103 | 77 |
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