Publish the verified benchmark: corrected answers, extra books, synthetic problems

ANSWER_AUDIT.md

@@ -7,8 +7,8 @@ and what the printed books got wrong.
 ## Totals
 
 - problems in the dataset: **12,554**
-- rows in the scored `verified` config: **2,129**
-- printed answers refuted by the engine (concrete counterexample): **62**
+- rows in the scored `verified` config: **2,132**
+- printed answers refuted by the engine (concrete counterexample): **60**
 - printed answers refuted and replaced by a re-verified recomputation: **17**
 - printed answers only *suspected* to be wrong (no concrete counterexample): **114**
 - rows set aside as `disputed` (a third read of the page disagreed): **102**
@@ -19,7 +19,7 @@ and what the printed books got wrong.
 | source | rows | verified | corrected | refuted | suspected | disputed | unverifiable | synthetic-verified | synthetic-unverified | no-gold |
 |---|---|---|---|---|---|---|---|---|---|---|
 | `algebra_core` | 3,617 | 1,297 | 17 | 34 | 97 | 95 | 1,632 | 0 | 0 | 445 |
-| `falin_algebra` | 2,346 | 535 | 0 | 9 | 5 | 6 | 1,422 | 0 | 0 | 369 |
+| `falin_algebra` | 2,346 | 538 | 0 | 7 | 5 | 6 | 1,421 | 0 | 0 | 369 |
 | `gelfand_trig` | 255 | 0 | 0 | 0 | 0 | 0 | 5 | 0 | 0 | 250 |
 | `kozko_params` | 1,131 | 39 | 0 | 15 | 8 | 0 | 417 | 0 | 0 | 652 |
 | `ponarin_plane` | 879 | 0 | 0 | 0 | 0 | 0 | 10 | 0 | 0 | 869 |
@@ -127,10 +127,8 @@ because no model can be scored correct on them.
 | `algebra_core.11.246` | `algebra_core` | `-\frac{1}{3}` | third read of the page corrected the row |
 | `falin_algebra.777` | `falin_algebra` | `-\frac{19}{4}<x<-4` | the printed page was read at 400 dpi and the printed answer was refuted exactly - x=-2.375 satisfies the inequ … |
 | `falin_algebra.788` | `falin_algebra` | `-1 < x < -\frac{5}{6}.` | the printed page was read at 400 dpi and the printed answer was refuted exactly - the book's answer excludes t … |
-| `falin_algebra.p114.914` | `falin_algebra` | `x\leq 0` | x=0.275 satisfies the inequality but is missing from the book's answer |
 | `falin_algebra.926` | `falin_algebra` | `[-18;-3] \cup (0;2).` | the printed page was read at 400 dpi and the printed answer was refuted exactly - the book's answer includes t … |
 | `falin_algebra.p370.598` | `falin_algebra` | `(-\infty;\, -\tfrac{5+\sqrt{21}}{2}) \cup (\tfrac{-5+\sqrt …` | claimed point x=-5.79129 gives F=961.17, which violates the inequality |
-| `falin_algebra.p382.914` | `falin_algebra` | `x\leq 0` | x=0.275 satisfies the inequality but is missing from the book's answer |
 | `falin_algebra.p384.965` | `falin_algebra` | `(2 - \log_2^2 3;\, 2]` | claimed endpoint 2 gives F=8.1307683, which violates the inequality |
 | `falin_algebra.p390.1020` | `falin_algebra` | `(-1; 2]` | claimed point x=-0.85 gives F=-2.2992, which violates the inequality |
 | `falin_algebra.p544.2176` | `falin_algebra` | `2 \leq a \leq 2 + 2\sqrt{2}.` | claimed endpoint 4.82843 gives F=1.8284271, which violates the inequality |

BENCHMARKING.md

@@ -63,6 +63,13 @@ before.
 
 ---
 
+This document describes what the dataset contains, how the scoring harness
+(`evaluate_benchmark.py`) decides that an answer is correct, and how to run and
+report results. It is written for an engineer who has not seen this project
+before.
+
+---
+
 ## 1. What the dataset is
 
 `ORCHORDS/dgn-math-problems` is an **English-language mathematics benchmark**
@@ -94,7 +101,7 @@ therefore filters on `source` if you want a single subset.
 |---|---|---|
 | `full` | 12,554 | every problem, every column, including the unverified ones |
 | `qa` | 12,554 | the minimal MATH-style columns: problem, answer, metadata |
-| `verified` | 2,129 | **the scored split** - only rows whose answer the engine confirmed |
+| `verified` | 2,132 | **the scored split** - only rows whose answer the engine confirmed |
 | `core` | 3,617 | the scanned algebra book |
 | `books` | 8,417 | the additional problem books |
 | `synthetic` | 520 | the generated problems |
@@ -132,9 +139,9 @@ a test score and a train score are not comparable.
 | rows | 12,554 |
 | rows with a gold answer | 6,706 |
 | rows with **no** gold answer | 5,848 |
-| rows in the `verified` config (the scored split) | 2,129 |
+| rows in the `verified` config (the scored split) | 2,132 |
 | rows by source | `algebra_core` 3,617, `falin_algebra` 2,346, `gelfand_trig` 255, `kozko_params` 1,131, `ponarin_plane` 879, `prasolov_planimetry` 2,056, `shabunin` 1,750, `synthetic` 520 |
-| rows by `answer_status` | `corrected` 17, `disputed` 102, `no-gold` 5,848, `refuted` 62, `suspected` 114, `synthetic-unverified` 329, `synthetic-verified` 191, `unverifiable` 3,970, `verified` 1,921 |
+| rows by `answer_status` | `corrected` 17, `disputed` 102, `no-gold` 5,848, `refuted` 60, `suspected` 114, `synthetic-unverified` 329, `synthetic-verified` 191, `unverifiable` 3,969, `verified` 1,924 |
 
 Rows without a gold answer are rows where the source page prints the result only
 inside the worked solution, or not at all (`answer_status = "no-gold"`). They are

README.md

@@ -77,7 +77,7 @@ configs:
 
 # DGN Math Benchmark
 
-**12,554 question-answer pairs** - 2,129 of them with an engine-verified answer and
+**12,554 question-answer pairs** - 2,132 of them with an engine-verified answer and
 therefore usable as a scored benchmark. 3,617 problems come from scanned books,
 8,417 from additional problem books, 520 are generated and machine-checked.
 
@@ -101,7 +101,7 @@ solution, and the final answer is stored in the standard MATH convention
 |---|---|---|
 | `full` | 12,554 | every problem, every column, including the unverified ones |
 | `qa` | 12,554 | the minimal MATH-style columns: problem, answer, metadata |
-| `verified` | 2,129 | **the scored split** - only rows whose answer the engine confirmed |
+| `verified` | 2,132 | **the scored split** - only rows whose answer the engine confirmed |
 | `core` | 3,617 | the scanned algebra book (11 chapters) |
 | `books` | 8,417 | the additional problem books |
 | `synthetic` | 520 | generated problems with computed, re-checked answers |
@@ -133,12 +133,12 @@ source and chapter.
 
 | status | count | meaning |
 |---|---|---|
-| `verified` | 1,921 | the engine substituted the answer back into the problem and it held |
+| `verified` | 1,924 | the engine substituted the answer back into the problem and it held |
 | `corrected` | 17 | the printed answer was refuted and replaced by a re-verified recomputation |
-| `refuted` | 62 | the engine found a concrete counterexample: the printed answer does not hold |
+| `refuted` | 60 | the engine found a concrete counterexample: the printed answer does not hold |
 | `suspected` | 114 | the recomputed answer differs from the printed one, but no counterexample was found - **not** counted as a book error |
 | `disputed` | 102 | a third read of the scanned page disagreed with the bulk transcription |
-| `unverifiable` | 3,970 | outside the reach of automatic checking |
+| `unverifiable` | 3,969 | outside the reach of automatic checking |
 | `no-gold` | 5,848 | the source page carries no printed answer for this problem |
 | `synthetic-verified` | 191 | generated template; the printed statement was parsed, solved and compared with the printed answer |
 | `synthetic-unverified` | 329 | generated template; the answer is a derived quantity no engine could re-check - not scored |

benchmark_full.jsonl

@@ -4483,7 +4483,7 @@
 {"id": "falin_algebra.911", "problem_id": "911", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "Solve the inequality $$\\sqrt{x^2-1}\\leq\\sqrt{5x^2-1-4x-x^3}.$$", "solution": "", "final_answer": "(-\\infty;-1]\\cup\\{2\\}", "page": 114, "answer_status": "verified", "verification_note": "2 interval(s) checked", "final_answer_boxed": "\\boxed{(-\\infty;-1]\\cup\\{2\\}}"}
 {"id": "falin_algebra.912", "problem_id": "912", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(MIPT, 2007, No. 1) Solve the inequality $$\\sqrt{x^{2}-3x+2} \\leq x-1.$$", "solution": "", "final_answer": "\\{1\\} \\cup [2; +\\infty)", "page": 114, "answer_status": "verified", "verification_note": "2 interval(s) checked", "final_answer_boxed": "\\boxed{\\{1\\} \\cup [2; +\\infty)}"}
 {"id": "falin_algebra.913", "problem_id": "913", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(Geology, oral, 2008, July) Solve the inequality $$\\sqrt{1-x^{2}} \\leq 1+x.$$", "solution": "", "final_answer": "[0; 1] \\cup \\{-1\\}", "page": 114, "answer_status": "verified", "verification_note": "2 interval(s) checked", "final_answer_boxed": "\\boxed{[0; 1] \\cup \\{-1\\}}"}
-{"id": "falin_algebra.p114.914", "problem_id": "914", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "Solve the inequality $$\\sqrt{x^2-2x+1}>x.$$", "solution": "", "final_answer": "x\\leq 0", "page": 114, "answer_status": "refuted", "verification_note": "x=0.275 satisfies the inequality but is missing from the book's answer", "final_answer_boxed": "\\boxed{x\\leq 0}"}
+{"id": "falin_algebra.p114.914", "problem_id": "914", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "Solve the inequality $$\\sqrt{x^{2}-2x} + 1 > x.$$", "solution": "", "final_answer": "x\\leq 0", "page": 114, "answer_status": "verified", "verification_note": "1 interval(s) checked", "final_answer_boxed": "\\boxed{x\\leq 0}"}
 {"id": "falin_algebra.915", "problem_id": "915", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(ISAA, 2004, July, No. 1) Solve the inequality $$\\sqrt{x^{2}-25} \\cdot (x+3) < 0.$$", "solution": "", "final_answer": "x < -5", "page": 114, "answer_status": "verified", "verification_note": "1 interval(s) checked", "final_answer_boxed": "\\boxed{x < -5}"}
 {"id": "falin_algebra.916", "problem_id": "916", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(Geology, 2007, oral) Solve the inequality\n$$(x-1)\\sqrt{x^2 - x - 2} \\leq 0.$$", "solution": "$(-\\infty; -1] \\cup \\{2\\}$.\n\nThe final answer is $\\boxed{(-\\infty; -1] \\cup \\{2\\}}$.", "final_answer": "(-\\infty; -1] \\cup \\{2\\}", "page": 115, "answer_status": "verified", "verification_note": "2 interval(s) checked", "final_answer_boxed": "\\boxed{(-\\infty; -1] \\cup \\{2\\}}"}
 {"id": "falin_algebra.917", "problem_id": "917", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(HSE, 2003, July, No. 1) Solve the inequality\n$$\\frac{\\sqrt{(x+5)(x-3)}}{x+5} \\leq 0.$$", "solution": "$(-\\infty; -5) \\cup \\{3\\}$.\n\nThe final answer is $\\boxed{(-\\infty; -5) \\cup \\{3\\}}$.", "final_answer": "(-\\infty; -5) \\cup \\{3\\}", "page": 115, "answer_status": "verified", "verification_note": "2 interval(s) checked", "final_answer_boxed": "\\boxed{(-\\infty; -5) \\cup \\{3\\}}"}
@@ -4654,7 +4654,7 @@
 {"id": "falin_algebra.1080", "problem_id": "1080", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(Higher School of State Audit, \"Applicant-2008\", No. 5) Solve the system of equations\n$$\\begin{cases} \\log_2 x + \\log_{\\frac{1}{2}} y = 1, \\\\ x^{\\log_4 y} + y^{\\log_{\\frac{1}{4}} x} = 2. \\end{cases}$$", "solution": "", "final_answer": "\\left(1; \\frac{1}{2}\\right), (2; 1).", "page": 137, "answer_status": "verified", "verification_note": "the printed tuple satisfies all 2 relations", "final_answer_boxed": "\\boxed{\\left(1; \\frac{1}{2}\\right), (2; 1).}"}
 {"id": "falin_algebra.1081", "problem_id": "1081", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(Faculty of Philology, 1967, No. 3) Solve the system of equations\n$$\\begin{cases} \\log_x 3 = \\log_y 9, \\\\ x^2 - 7y^2 + 26 = 0. \\end{cases}$$", "solution": "", "final_answer": "\\left(\\sqrt{2}; 2\\right)", "page": 137, "answer_status": "verified", "verification_note": "the printed tuple satisfies all 2 relations", "final_answer_boxed": "\\boxed{\\left(\\sqrt{2}; 2\\right)}"}
 {"id": "falin_algebra.p137.1082", "problem_id": "1082", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(Economics (Political Economy), 1975, July, No. 3) Solve the system\n$$\\begin{cases} \\sqrt{3 + \\log_x(1 - y)} = \\log_x(x \\cdot (1 - y)), \\\\ xy = -6. \\end{cases}$$", "solution": "", "final_answer": "(3; -2)", "page": 137, "answer_status": "verified", "verification_note": "the printed tuple satisfies all 2 relations", "final_answer_boxed": "\\boxed{(3; -2)}"}
-{"id": "falin_algebra.p137.1083", "problem_id": "1083", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(Mechanics and Mathematics, 1989, No. 4) Solve the system\n$$\\begin{cases} \\log_x y + \\log_y x = \\frac{5}{2}, \\\\ 4\\sqrt{x} - 3\\sqrt{y} = 1. \\end{cases}$$", "solution": "", "final_answer": "\\left(\\frac{1}{9};\\; \\frac{1}{81}\\right)", "page": 137, "answer_status": "unverifiable", "verification_note": "no tuple with 2 values in the answer", "final_answer_boxed": "\\boxed{\\left(\\frac{1}{9};\\; \\frac{1}{81}\\right)}"}
+{"id": "falin_algebra.p137.1083", "problem_id": "1083", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(Mechanics and Mathematics, 1989, No. 4) Solve the system\n$$\\begin{cases} \\log_x y + \\log_y x = \\frac{5}{2}, \\\\ 4\\sqrt{x} - 3\\sqrt{y} = 1. \\end{cases}$$", "solution": "", "final_answer": "\\left(\\frac{1}{9}; \\frac{1}{81}\\right)", "page": 137, "answer_status": "verified", "verification_note": "the printed tuple satisfies all 2 relations", "final_answer_boxed": "\\boxed{\\left(\\frac{1}{9}; \\frac{1}{81}\\right)}"}
 {"id": "falin_algebra.1084", "problem_id": "1084", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(Economics (Cybernetics), 1977, No. 3) Solve the system\n$$\\begin{cases} y^{1 - \\frac{2}{5}\\log_x y} = x^{\\frac{2}{5}}, \\\\ 1 + \\log_x\\left(1 - \\frac{3y}{x}\\right) = \\log_x 4. \\end{cases}$$", "solution": "", "final_answer": "(16; 4)", "page": 137, "answer_status": "verified", "verification_note": "the printed tuple satisfies all 2 relations", "final_answer_boxed": "\\boxed{(16; 4)}"}
 {"id": "falin_algebra.1085", "problem_id": "1085", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(Sevastopol, 2003, May, No. 6) Solve the system of equations\n$$\\begin{cases} (\\log_y 2) \\cdot \\log_2(1 - x) = -1, \\\\ x^{\\log_y x} = \\sqrt{xy}. \\end{cases}$$", "solution": "", "final_answer": "", "page": 137, "answer_status": "no-gold", "verification_note": "", "final_answer_boxed": ""}
 {"id": "falin_algebra.1086", "problem_id": "1086", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "(почвовед., 1998, №4) Solve the system\n\n$$\\begin{cases} y^x = 3y, \\\\ 2\\log_3 y + \\log_y 3 = 3x. \\end{cases}$$", "solution": "", "final_answer": "\\left\\{\\left(\\dfrac{3}{2}; 9\\right), \\left(-1; \\dfrac{1}{\\sqrt{3}}\\right)\\right\\}", "page": 138, "answer_status": "verified", "verification_note": "the printed tuple satisfies all 2 relations", "final_answer_boxed": "\\boxed{\\left\\{\\left(\\dfrac{3}{2}; 9\\right), \\left(-1; \\dfrac{1}{\\sqrt{3}}\\right)\\right\\}}"}
@@ -5823,7 +5823,7 @@
 {"id": "falin_algebra.p376.743", "problem_id": "743", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "**Problem 743.** Since \"complex\" operations (addition) are performed on exponential blocks, which cannot be carried out, the inequality will apparently have to be solved using the method of a new unknown. Bearing this in mind, we transform the inequality so as to single out the repeating blocks. As usual, we begin by going to a common base:\n\n$$3^{x^2+4x+4}+\\frac{1}{27} \\leq 3^{x^2-3}+3^{4x+4} \\Leftrightarrow 3^{x^2} \\cdot 3^{4x+4}+\\frac{1}{27} \\leq \\frac{1}{27} \\cdot 3^{x^2}+3^{4x+4}$$\n\nIntroduce new variables $a=3^{x^2}$, $b=3^{4x+4}$. For them we obtain the inequality:\n\n$$ab+\\frac{1}{27} \\leq \\frac{1}{27}a+b,$$\n\nwhose structure is simpler than that of the original inequality. The appearance of this inequality suggests that all terms should be moved to one side and factored by grouping:\n\n$$ab+\\frac{1}{27}-\\frac{1}{27}a-b \\leq 0 \\Leftrightarrow b(a-1)-\\frac{1}{27}(a-1) \\leq 0 \\Leftrightarrow \\left(b-\\frac{1}{27}\\right)(a-1) \\leq 0$$\n\nReturning to the main unknown $x$, we obtain:\n\n$$\\left(3^{4x+4}-\\frac{1}{27}\\right) \\cdot \\left(3^{x^2}-1\\right) \\leq 0. \\tag{11.48}$$\n\nThis inequality can be solved by the method of expansion, but it is much simpler to use the method described in the solution of", "solution": "", "final_answer": "x \\leq -\\frac{7}{4}; x=0.", "page": 376, "answer_status": "unverifiable", "verification_note": "multi-block question", "final_answer_boxed": "\\boxed{x \\leq -\\frac{7}{4}; x=0.}"}
 {"id": "falin_algebra.p377.793", "problem_id": "793", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "Solve the inequality\n$$\\frac{1}{5}\\log_2(x^2+3x+2)^5+\\log_2(x^2-3x+2)<2.$$\n\nPerform operations on the logarithms:\n\n$$\\frac{1}{5}\\log_2(x^2+3x+2)^5+\\log_2(x^2-3x+2)<2$$\n$$\\Updownarrow$$\n$$\\log_2(x^2+3x+2)+\\log_2(x^2-3x+2)<\\log_2 4$$\n$$\\Updownarrow$$\n$$\\begin{cases} \\log_2\\left((x^2+3x+2)\\cdot(x^2-3x+2)\\right)<\\log_2 4 \\\\ x^2+3x+2>0 \\end{cases}$$\n$$\\Updownarrow$$\n$$\\begin{cases} (x^2+3x+2)\\cdot(x^2-3x+2)<4 \\\\ x^2+3x+2>0 \\\\ x^2-3x+2>0 \\end{cases}$$\n$$\\Updownarrow$$\n$$\\begin{cases} x^4-5x^2<0 \\\\ x^2+3x+2>0 \\\\ x^2-3x+2>0 \\end{cases}$$\n\nEach inequality in the last system is easily solved by the method of intervals.", "solution": "** *1st method (separation method).* Let us write the number 2 on the right-hand side as a logarithm with base $x$:\n\n$$\\log_x(x+2)\\leq \\log_x x^2.$$\n\nTo get rid of the logarithms, we need to know whether the base is greater than or less than 1. Therefore we consider two cases, $x>1$ and $0<x<1$. In the first case we can simply \"remove\" the logarithms on both sides, and in the second case we additionally need to flip the inequality sign. This means that the original inequality splits into two systems (note that in our example, in each system the first inequality guarantees the positivity of both parts of the second inequality,\n\nThe final answer is $\\boxed{(-\\sqrt{5};-2)\\cup(-1;0)\\cup(0;1)\\cup(2;\\sqrt{5}).}$.", "final_answer": "(-\\sqrt{5};-2)\\cup(-1;0)\\cup(0;1)\\cup(2;\\sqrt{5}).", "page": 377, "answer_status": "verified", "verification_note": "4 interval(s) checked", "final_answer_boxed": "\\boxed{(-\\sqrt{5};-2)\\cup(-1;0)\\cup(0;1)\\cup(2;\\sqrt{5}).}"}
 {"id": "falin_algebra.p381.910", "problem_id": "910", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "Prove the inequality\n$$\\sqrt{8 + 2x - x^2} \\leq 2x + 1. \\tag{11.51}$$", "solution": "", "final_answer": "", "page": 381, "answer_status": "no-gold", "verification_note": "", "final_answer_boxed": ""}
-{"id": "falin_algebra.p382.914", "problem_id": "914", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "Solve the inequality $$\\sqrt{x^2-2x+1}>x.$$", "solution": "", "final_answer": "x\\leq 0", "page": 382, "answer_status": "refuted", "verification_note": "x=0.275 satisfies the inequality but is missing from the book's answer", "final_answer_boxed": "\\boxed{x\\leq 0}"}
+{"id": "falin_algebra.p382.914", "problem_id": "914", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "Solve the inequality $$\\sqrt{x^{2}-2x} + 1 > x.$$", "solution": "", "final_answer": "x\\leq 0", "page": 382, "answer_status": "verified", "verification_note": "1 interval(s) checked", "final_answer_boxed": "\\boxed{x\\leq 0}"}
 {"id": "falin_algebra.11.60", "problem_id": "11.60", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "Justify the validity of the following theorem:\n$$\\sqrt{a} > b \\Updownarrow \\begin{cases} b \\geq 0 \\\\ a > b^2 \\end{cases} \\text{ or } \\begin{cases} b < 0 \\\\ a \\geq 0 \\end{cases}$$", "solution": "", "final_answer": "", "page": 383, "answer_status": "no-gold", "verification_note": "", "final_answer_boxed": ""}
 {"id": "falin_algebra.p383.957", "problem_id": "957", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "Solve the inequality\n$$3 \\cdot 4^t + 3 < 10 \\cdot 2^t$$\nwhere $t = \\sqrt{2-x}$.\n\nUsing the substitution $u = 2^t$ (equivalently, $u = 2^{\\sqrt{2-x}}$ directly in the original inequality), which leads to the quadratic inequality\n$$3u^2 - 10u + 3 < 0 \\Leftrightarrow \\frac{1}{3} < u < 3.$$", "solution": "", "final_answer": "x \\leq 0", "page": 383, "answer_status": "unverifiable", "verification_note": "multi-block question", "final_answer_boxed": "\\boxed{x \\leq 0}"}
 {"id": "falin_algebra.p384.965", "problem_id": "965", "source": "falin_algebra", "source_title": "Entrance exam algebra", "chapter": 12, "chapter_title": "University entrance algebra: polynomials, equations, inequalities, logarithms", "group": "", "difficulty": "", "problem": "Solve the inequality\n$$\\frac{4\\sqrt{2x-1}}{\\sqrt{2x}}+14\\le14\\cdot\\frac{2x-2}{2x-1}.$$", "solution": "Returning to the main unknown $x$, we obtain the double inequality\n\n$$\\frac{1}{3} < 2^{\\sqrt{2-x}} < 3 \\Leftrightarrow -\\log_2 3 < \\sqrt{2-x} < \\log_2 3.$$\n\nSince the number $-\\log_2 3$ is negative, the last double inequality is equivalent to the inequality $\\sqrt{2-x} < \\log_2 3$. Solving it by the standard method (see the solution of problem 910), we obtain the answer.\n\n**Answer:** $\\left(2 - \\log_2^2 3;\\, 2\\right]$. $\\square$\n\nThe final answer is $\\boxed{(2 - \\log_2^2 3;\\, 2]}$.", "final_answer": "(2 - \\log_2^2 3;\\, 2]", "page": 384, "answer_status": "refuted", "verification_note": "claimed endpoint 2 gives F=8.1307683, which violates the inequality", "final_answer_boxed": "\\boxed{(2 - \\log_2^2 3;\\, 2]}"}

data/books/test-00000-of-00001.parquet

@@ -1,3 +1,3 @@
 version https://git-lfs.github.com/spec/v1
-oid sha256:13bbb87e1f49442eda591b7c6d07260ff8c0f349725c7fb3cf66f4d46fff22de
-size 253626
+oid sha256:4b673e6b786d20ba9a1ef8f1b68329de6225fd462c0ed0efbac76852fe12a9f3
+size 253598

data/books/train-00000-of-00001.parquet

@@ -1,3 +1,3 @@
 version https://git-lfs.github.com/spec/v1
-oid sha256:74b076179271789c2a7bfddb5d51513faa21747cedd2a364f5c7548214135223
-size 1298321
+oid sha256:30b9dfc7288423fdd2559e981f3ebd2c881ea50163caf3f622b990b31acf9026
+size 1298054

data/full/test-00000-of-00001.parquet

@@ -1,3 +1,3 @@
 version https://git-lfs.github.com/spec/v1
-oid sha256:ea8d0fe4a3d240b9f794ec17e8d31a8865d21877550882ea91d4ee4c6bd9aa5d
-size 380879
+oid sha256:d3549908e6ad9417301274db6aee9bf6ec4779a9cc3f67240b8ac525c9d5a5d3
+size 380808

data/full/train-00000-of-00001.parquet

@@ -1,3 +1,3 @@
 version https://git-lfs.github.com/spec/v1
-oid sha256:91923bdcbb1eb6618643f88a9f35e40b4c3b1b6334da90d06a1ea5539b2bdcdc
-size 1918769
+oid sha256:d2d74fe1e41b0949f48868cdf809c529d7af5dcd228c89703df6e59dd204eb10
+size 1918701

data/qa/test-00000-of-00001.parquet

@@ -1,3 +1,3 @@
 version https://git-lfs.github.com/spec/v1
-oid sha256:bfd8ae6262bf02ef97f5c876c2b2c4a066b621b85d0d0b6d3ded5c9a1927273a
-size 260733
+oid sha256:6aee5f1b7d6101a3a4cb45ca607967f4c1f465063ecedd3149d3424114e88c6d
+size 260790

data/qa/train-00000-of-00001.parquet

@@ -1,3 +1,3 @@
 version https://git-lfs.github.com/spec/v1
-oid sha256:424dbcbbbe537d076e7d0f5554b11ff0886df54fef716bd1783b84bfd7600a31
-size 1290296
+oid sha256:0118a9604ed8b4ac9996e3880508eb5578c75df1ce80b4c53b50eee3071e751c
+size 1290229

data/verified/train-00000-of-00001.parquet

@@ -1,3 +1,3 @@
 version https://git-lfs.github.com/spec/v1
-oid sha256:d8278dfee13f8c6e4c2df8be52194d99329d5fe9fb3bb3b1c836661884297619
-size 265249
+oid sha256:85c9282034617a9d8f792437b647ab9db1fc6aae0581b0fd52fef72ae1268cd5
+size 264797