Datasets:
Answer audit
Every printed answer in this dataset was checked against the problem it belongs to by a symbolic engine (SymPy). This file reports what was checked, what held, and what the printed books got wrong.
Totals
- problems in the dataset: 12,554
- rows in the scored
verifiedconfig: 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 - rows with no printed answer at all (
no-gold): 5,848
By source
| 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 | 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 |
prasolov_planimetry |
2,056 | 2 | 0 | 0 | 3 | 0 | 191 | 0 | 0 | 1,860 |
shabunin |
1,750 | 48 | 0 | 4 | 1 | 1 | 293 | 0 | 0 | 1,403 |
synthetic |
520 | 0 | 0 | 0 | 0 | 0 | 0 | 191 | 329 | 0 |
What each status means
| status | meaning |
|---|---|
verified |
the answer was substituted back into the problem and the equation held |
corrected |
the printed answer was refuted; it was replaced by a recomputed answer that the engine then re-verified |
refuted |
the engine produced a concrete counterexample: the printed root was substituted back and failed, the printed tuple satisfies no equation of the system, or the printed interval misses a point that satisfies the inequality. The printed answer is demonstrably wrong. |
suspected |
the recomputed answer differs from the printed one, but no concrete counterexample was found. A difference in a radical's scope or in a notational convention can produce this. These are not counted as book errors. |
disputed |
a third independent read of the scanned page disagreed with the bulk transcription, so the row is not trusted |
unverifiable |
the printed answer exists but is outside the reach of automatic checking - word problems, open-ended proofs, answers that depend on a figure |
no-gold |
the source page carries no printed answer for this problem |
synthetic-verified |
generated template; the printed statement was parsed, solved and compared with the printed answer |
synthetic-unverified |
generated template; the answer is a derived quantity (a remainder, a sum of coefficients, ...) that no engine could re-check from the printed statement, so the row is published but not scored |
Neither unverifiable, suspected, disputed nor no-gold means the answer is
wrong. They mean nobody checked it, or that the check was inconclusive.
The verified config is the honest scored split: every item in it has been
checked by a symbolic engine.
How the checking works
| problem type | check |
|---|---|
| equation | every printed root is substituted back into the equation; a solution family is checked at several integer values of its parameter |
| system of equations | every ordering of the printed tuple is substituted into every equation |
| inequality | the endpoints of the printed interval are evaluated exactly, and a grid of sample points is masked against the inequality |
identity (prove that A = B) |
both sides are compared numerically at a grid of sample points |
| simplify | the expression is evaluated numerically and compared with the printed result |
| limit / integral | the expression is evaluated symbolically with SymPy |
extremum (the greatest value is ...) |
a grid over the stated domain is searched for a counterexample to the claimed bound |
Answers printed in degrees are converted to radians before checking, and
degrees-minutes-seconds (3^\circ45') are parsed as 3.75 degrees.
Answers that were refuted and replaced
| id | printed answer | recomputed answer |
|---|---|---|
algebra_core.2.343 |
see the book | x = - 2 \sqrt{3} |
algebra_core.2.356 |
see the book | x = - \sqrt{2}; x = \sqrt{2} |
algebra_core.5.037 |
see the book | x = - \frac{5 \pi}{6}; x = - \frac{4 \pi}{5}; x = - \frac{3 \pi}{5}; x = - \frac{\pi}{2}; x = - \frac{2 \pi}{5}; x = - \frac{\pi}{5}; x = - \frac{\pi}{6}; x = 0; x = \frac{\pi}{6}; x = \frac{\pi}{5}; x = \frac{2 \pi}{5}; x = \frac{\pi}{2}; x = \frac{3 \pi}{5}; x = \frac{4 \pi}{5}; x = \frac{5 \pi}{6}; x = \pi |
algebra_core.5.107 |
see the book | x = - \frac{\pi}{4}; x = \frac{3 \pi}{4} |
algebra_core.5.123 |
see the book | x = \operatorname{atan}{\left(\frac{1}{5} \right)}; x = \frac{\pi}{4} |
algebra_core.5.162 |
see the book | x = 0; x = \frac{2 \pi}{3} |
algebra_core.5.217 |
see the book | x = - \operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{5}}{2} \right)}; x = - \frac{\pi}{8}; x = - \operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{5}}{2} \right)}; x = \frac{3 \pi}{8} |
algebra_core.5.232 |
see the book | x = - \operatorname{atan}{\left(\frac{1}{5} + \frac{\sqrt{26}}{5} \right)}; x = - \frac{\pi}{8}; x = - \operatorname{atan}{\left(\frac{1}{5} - \frac{\sqrt{26}}{5} \right)}; x = \frac{3 \pi}{8} |
algebra_core.5.413 |
see the book | x = 0; x = \pi |
algebra_core.5.421 |
see the book | x = - \frac{\pi}{4} |
algebra_core.7.038 |
see the book | x = \frac{1}{100}; x = 100 |
algebra_core.7.039 |
see the book | x = \frac{1}{10}; x = 1000 |
algebra_core.7.096 |
see the book | x = \left(\frac{3^{\frac{2}{3}}}{3}\right)^{\frac{\sqrt[3]{3} \left(- \left(9 + \sqrt{105}\right)^{\frac{2}{3}} + 2 \sqrt[3]{3}\right)}{\sqrt[3]{9 + \sqrt{105}}}} |
algebra_core.7.242 |
see the book | x = \frac{1}{10}; x = \sqrt{10}; x = 100 |
algebra_core.7.243 |
see the book | x = \frac{1}{1000}; x = 1; x = 10 |
algebra_core.7.250 |
see the book | x = \frac{2^{\frac{2}{5}}}{8}; x = 1; x = 4 |
algebra_core.10.122 |
see the book | x = \frac{9}{2}; x = 6 |
Printed answers the engine refuted
These are printed answers that do not hold: the engine produced a
concrete counterexample. They are excluded from the verified config
because no model can be scored correct on them.
| id | source | printed answer | what the engine found |
|---|---|---|---|
algebra_core.1.202 |
algebra_core |
\sqrt{mn} |
the printed page was read at 800 dpi and the printed answer was refuted exactly - third read of the page disag … |
algebra_core.1.206 |
algebra_core |
\frac{a(a+1)}{2a+1} |
the printed page was read at 800 dpi and the printed answer was refuted exactly - third read of the page disag … |
algebra_core.2.068 |
algebra_core |
(-4;24);\ (-6;-2) |
three independent reads agree on the statement and the printed answer |
algebra_core.2.169 |
algebra_core |
x_1=0;\ x_2=4. |
claimed solution 'x=4' fails the equation |
algebra_core.2.205 |
algebra_core |
(2;1),\ (6;-3),\ \left(6+2\sqrt{3};-2-2\sqrt{3}\right),\ … |
three independent reads agree on the statement and the printed answer - error of the book: the printed answer … |
algebra_core.2.240 |
algebra_core |
(10;1),\ \left(-21/2;53/12\right) |
three independent reads agree on the statement and the printed answer |
algebra_core.2.243 |
algebra_core |
(1;4),(4;1),(-1;-4),(-4;-1) |
three independent reads agree on the statement and the printed answer |
algebra_core.5.002 |
algebra_core |
x_1=35^\circ+120^\circ\cdot k,\quad x_2=55^\circ+120^\circ … |
three independent reads agree on the statement and the printed answer |
algebra_core.5.039 |
algebra_core |
x_1=\frac{\pi k}{5},\ x_2=\frac{\pi}{8}(4k+3). |
three independent reads agree on the statement and the printed answer |
algebra_core.5.121 |
algebra_core |
z_1=2\arctan3+2\pi k,\quad z_2=-2\arctan7+2\pi k |
three independent reads agree on the statement and the printed answer |
algebra_core.5.147 |
algebra_core |
x=\frac{\pi}{12}(6k+1) |
three independent reads agree on the statement and the printed answer |
algebra_core.5.152 |
algebra_core |
x_1=\pi k,\quad x_2=(-1)^k\frac{k\pi}{6}+\frac{\pi k}{2} |
claimed solution 'x=(-1)^k\frac{k\pi}{6}+\frac{\pi k}{2}' fails the equation |
algebra_core.5.174 |
algebra_core |
t=\frac{\pi}{2}(4k+1) |
three independent reads agree on the statement and the printed answer |
algebra_core.5.176 |
algebra_core |
x_1=\frac{\pi}{6}+\pi k;\quad x_2=(-1)^k\frac{\pi}{8}+\frac … |
three independent reads agree on the statement and the printed answer |
algebra_core.5.177 |
algebra_core |
x=(-1)^k\frac{\pi}{3}+\pi k |
claimed solution 'x=(-1)^k\frac{\pi}{3}+\pi k' fails the equation |
algebra_core.5.221 |
algebra_core |
x_1=-\frac{\pi}{16}(4k+1),\quad x_2=-\frac{1}{4}\arctan … |
three independent reads agree on the statement and the printed answer |
algebra_core.5.227 |
algebra_core |
t=\arctan\frac{4}{5}+2\pi k |
three independent reads agree on the statement and the printed answer |
algebra_core.5.259 |
algebra_core |
x=-\frac{\pi}{8}(4k+1) |
three independent reads agree on the statement and the printed answer |
algebra_core.5.269 |
algebra_core |
x=-\frac{\pi}{4}(4k+1) |
three independent reads agree on the statement and the printed answer |
algebra_core.5.334 |
algebra_core |
t=-\frac{\pi}{4}(4k+1) |
three independent reads agree on the statement and the printed answer |
algebra_core.5.344 |
algebra_core |
t=-\frac{\pi}{4}(4k-1) |
claimed solution 't=-\frac{\pi}{4}(4k-1)' fails the equation |
algebra_core.5.348 |
algebra_core |
t=(-1)^k\frac{k\pi}{6}+\pi k |
three independent reads agree on the statement and the printed answer |
algebra_core.5.445 |
algebra_core |
t=-\frac{\pi}{4}(4k+1) |
three independent reads agree on the statement and the printed answer |
algebra_core.5.472 |
algebra_core |
t_1=\frac{\pi}{4}(8k+1),\quad t_2=-\arctan3+\pi(2k+1) |
three independent reads agree on the statement and the printed answer |
algebra_core.7.151 |
algebra_core |
(9; 16) |
no assignment of the printed tuple (9; 16) satisfies the system |
algebra_core.7.160 |
algebra_core |
(3; 27) |
three independent reads agree on the statement and the printed answer |
algebra_core.7.263 |
algebra_core |
x_1=1/8,\ x_2=8. |
claimed solution 'x=1/8' fails the equation |
algebra_core.8.026 |
algebra_core |
(-1,\infty) |
three independent reads agree on the statement and the printed answer |
algebra_core.8.238 |
algebra_core |
(-\infty,-2]\cup[-1,(\sqrt{13}-1)/6) |
three independent reads agree on the statement and the printed answer |
algebra_core.8.276 |
algebra_core |
(\sqrt[5]{5},5) |
the printed page was read at 800 dpi and the printed answer was refuted exactly - third read of the page disag … |
algebra_core.8.283 |
algebra_core |
x=3 |
claimed integer 3 gives F=-1.2910247, which violates the inequality |
algebra_core.10.012 |
algebra_core |
(-3,2)\cup(2,3) |
three independent reads agree on the statement and the printed answer |
algebra_core.10.166 |
algebra_core |
(-4,0)\cup(0,4) |
three independent reads agree on the statement and the printed answer |
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.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.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 |
shabunin.p183.Example 3 |
shabunin |
(1;2), (-1;0) |
no assignment of the printed tuple (-1; 0) satisfies the system |
shabunin.p190.Example 2 |
shabunin |
(1;2;3),\ (-1;-2;-3) |
no assignment of the printed tuple (-1; -2; -3) satisfies the system |
shabunin.p194.Пример 8 |
shabunin |
(2;\sqrt[3]{3}; \sqrt[3]{9}),\ \left(\frac{1}{\sqrt[3]{2}}; … |
no assignment of the printed tuple (\frac{1}{\sqrt[3]{2}}; -\sqrt[3]{\frac{9}{2}}; \sqrt[3]{\frac{3}{2}} … |
shabunin.p222.Пример 2 |
shabunin |
(1; 3); \left(\frac{9}{4}; 4\right) |
no assignment of the printed tuple (\frac{9}{4}; 4\right) satisfies the system |
kozko_params.p163.3 |
kozko_params |
0 < x \leqslant 1/2;\ x \geqslant \log_2 3 |
claimed point x=2.58496 gives F=-24.8292, which violates the inequality |
kozko_params.2003-2-3 |
kozko_params |
[3/7; 1) \cup (3; +\infty) |
claimed endpoint 0.428571 gives F=-1.1818182, which violates the inequality |
kozko_params.p176.Задача 3 |
kozko_params |
[(-1-2\sqrt{29})/5;\,2] |
claimed point x=-2.81962 gives F=0.68419, which violates the inequality |
kozko_params.p195.3 |
kozko_params |
\left\{\left(-\frac{1}{2}; \log_9 \frac{9} … |
no assignment of the printed tuple (\log_4 \frac{4}{3}; -\frac{1}{2}\right) satisfies the system |
kozko_params.p237.7 |
kozko_params |
\left(\dfrac{\pi}{4}+2\pi m;\ \dfrac{3\pi}{4}+2\pi k … |
no assignment of the printed tuple (-\dfrac{\pi}{4}+2\pi m; \ \dfrac{\pi}{4}+2\pi k … |
kozko_params.ВАРИАНТ 2004 (июль), социологический факультет, 1 |
kozko_params |
[-5; -3]. |
claimed endpoint -5 gives F=0.66666667, which violates the inequality |
kozko_params.p271.2 |
kozko_params |
(-\infty; 1/4] |
claimed point x=-0.75 gives F=0.275412, which violates the inequality |
kozko_params.p274.1 |
kozko_params |
2 |
claimed integer 2 gives F=9, which violates the inequality |
kozko_params.ВАРИАНТ 2005 (июль), географический факультет, 1 |
kozko_params |
1. (x;y) = (-\pi/2 \pm \pi/3 + 2\pi k;\ 2\pi m),\ k,m \in … |
no assignment of the printed tuple (3\pi+4) satisfies the system |
kozko_params.p280.4 |
kozko_params |
(-\infty;\, 1-\sqrt{14}] \cup [-\sqrt{17}/3;\, -2) \cup (1; … |
x=-2.19 satisfies the inequality but is missing from the book's answer |
kozko_params.p281.1 |
kozko_params |
x=-2 |
claimed solution 'x=-2' fails the equation |
kozko_params.p289.2 |
kozko_params |
(-\infty;2] |
claimed endpoint 2 gives F=-0.17157288, which violates the inequality |
kozko_params.p290.2 |
kozko_params |
(-\infty; 0] |
claimed point x=-10 gives F=0.00537474, which violates the inequality |
kozko_params.Задача 1 (Вариант 2006, апрель) |
kozko_params |
[-2; (7+\sqrt{21})/21) |
claimed point x=6.14955 gives F=-0.294805, which violates the inequality |
kozko_params.p293.Задача 2 |
kozko_params |
[-3; (5+\sqrt{21})/2) |
claimed endpoint -3 gives F=-5, which violates the inequality |
Printed answers only suspected to be wrong
For these rows the recomputed answer differs from the printed one, but the engine could not produce a concrete counterexample. The difference can come from how the radical scope or a notational convention was read in the scan, so these are not counted as book errors and are kept out of the scored split as well.
| id | source | printed answer | what the engine found |
|---|---|---|---|
algebra_core.1.004 |
algebra_core |
0.04 |
numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from … |
algebra_core.1.023 |
algebra_core |
0.5a^3(a-1) |
the printed answer is contradicted at a=3: the expression is exactly 27/4, the printed answer gives 27 - repor … |
algebra_core.1.027 |
algebra_core |
-4 |
the printed answer is contradicted at a=1, b=4: the expression is exactly (-27649 + 242(1/3) + 5762**(2/3) … |
algebra_core.1.028 |
algebra_core |
\frac{2}{\sqrt[3]{r^2}}\left(r^2-\sqrt{r^4-16}\right) |
the printed answer is contradicted at r=2: the expression is exactly 2**(1/3), the printed answer gives 42*( … |
algebra_core.1.039 |
algebra_core |
y^2 |
the printed answer is contradicted at y=1: the expression is exactly (30**(1/10)(sqrt(2) + 3) + 6sqrt(2) + 8 … |
algebra_core.1.041 |
algebra_core |
-4 |
the printed answer is contradicted at t=1: the expression is exactly 152/25, the printed answer gives -4 - rep … |
algebra_core.1.050 |
algebra_core |
2\sqrt[6]{a^5/a} |
the printed answer is contradicted at a=1: the expression is exactly 2**(11/12)/2, the printed answer gives 2 … |
algebra_core.1.053 |
algebra_core |
\sqrt{6x} |
the printed answer is contradicted at x=3: the expression is exactly 0, the printed answer gives 3*sqrt(2) - r … |
algebra_core.1.054 |
algebra_core |
\sqrt[3]{20x} |
the printed answer is contradicted at x=1: the expression is exactly -2sqrt(3) + 2sqrt(2)(4sqrt(6) + 11)** … |
algebra_core.1.056 |
algebra_core |
1/\sqrt[12]{a^2b} |
the printed answer is contradicted at a=1, b=2: the expression is exactly 1/2, the printed answer gives 2**(11 … |
algebra_core.1.059 |
algebra_core |
-1 |
the printed answer is contradicted at a=2: the expression is exactly -2**(1/4) - 1 - 2**(3/4)/2 + sqrt(sqrt(2) … |
algebra_core.1.061 |
algebra_core |
-(\sqrt[4]{x}-\sqrt[4]{y}) |
the printed answer is contradicted at x=1, y=2: the expression is exactly 3*(1 - sqrt(2))/(-3 + 2**(1/4) + 2** … |
algebra_core.1.063 |
algebra_core |
a^{1/3}+b^{1/3} |
the printed answer is contradicted at a=1, b=2: the expression is exactly (-22(2/3) + 1 + 22**(1/3))/(1 - … |
algebra_core.1.065 |
algebra_core |
(\sqrt{m}-\sqrt{n})/m |
the printed answer is contradicted at m=1, n=2: the expression is exactly -sqrt(sqrt(5)/4 + 3/4), the printed … |
algebra_core.1.067 |
algebra_core |
1 |
the printed answer is contradicted at : the expression is exactly (-sqrt(5) + 2*sqrt(6))/(-sqrt(3) + sqrt(2)), … |
algebra_core.1.068 |
algebra_core |
\frac{1}{a\left(\sqrt[m]{a}-\sqrt[n]{a}\right)} |
the printed answer is contradicted at a=2, m=1, n=2: the expression is exactly 5*sqrt(2)/14 + 4/7, the printed … |
algebra_core.1.069 |
algebra_core |
6 |
the printed answer is contradicted at : the expression is exactly (-18sqrt(5) - 12(1/3) + 30sqrt(3))/(-3*s … |
algebra_core.1.073 |
algebra_core |
-\sqrt{ac} |
the printed answer is contradicted at a=1, b=1, c=2: the expression is exactly 2**(190/213)3(308/639)5(4 … |
algebra_core.1.077 |
algebra_core |
\frac{x^{1/3}+y^{1/3}}{\sqrt[6]{x^3y^2}} |
the printed answer is contradicted at x=2, y=1: the expression is exactly (-2**(1/6) - 1 + 2**(2/3) + 2**(5/6) … |
algebra_core.1.080 |
algebra_core |
0 |
the printed answer is contradicted at m=1, n=1: the expression is exactly -2, the printed answer gives 0 - rep … |
algebra_core.1.083 |
algebra_core |
5 |
the printed answer is contradicted at b=1, c=1, a=1/25: the expression is exactly 5sqrt(51)(sqrt(101) + 20)/ … |
algebra_core.1.086 |
algebra_core |
\frac{1}{\sqrt{a}+\sqrt{2}} |
the printed answer is contradicted at a=1: the expression is exactly -1/2 + 2*sqrt(2)/3, the printed answer gi … |
algebra_core.1.087 |
algebra_core |
-3n(m+p) |
the printed answer is contradicted at m=1, n=1, p=1: the expression is exactly (sqrt(6) + 23(1/3))(-2**(1/ … |
algebra_core.1.089 |
algebra_core |
(1-a)/\sqrt{a} |
the printed answer is contradicted at a=2: the expression is exactly sqrt(3)*(-4 + sqrt(2))/24, the printed an … |
algebra_core.1.098 |
algebra_core |
\sqrt[3]{x+y}-\sqrt[3]{x-y} |
the printed answer is contradicted at x=2, y=1: the expression is exactly 2**(1/3), the printed answer gives - … |
algebra_core.1.101 |
algebra_core |
2 |
the printed answer is contradicted at x=1: the expression is exactly 2/3 - 2*sqrt(2)/3, the printed answer giv … |
algebra_core.1.107 |
algebra_core |
a/2 |
the printed answer is contradicted at a=1, b=2, p=3: the expression is exactly sqrt(2)/(-22(5/6) - 22**(1/ … |
algebra_core.1.116 |
algebra_core |
1/m^2 |
the printed answer is contradicted at m=1, n=2: the expression is exactly sqrt(2)/(-62(3/4) - 82**(1/4) + … |
algebra_core.1.117 |
algebra_core |
-a^2 |
the printed answer is contradicted at a=1, b=2: the expression is exactly -2**(3/4)/2, the printed answer give … |
algebra_core.1.120 |
algebra_core |
3/5 |
the printed answer is contradicted at : the expression is exactly (32(1/3)7(1/4) + 602(1/3))/(22**(1 … |
algebra_core.1.121 |
algebra_core |
31/3 |
the printed answer is contradicted at : the expression is exactly 2**(1/3)(206**(1/3) + 63)/(2*(6**(2/3) + 1 … |
algebra_core.1.122 |
algebra_core |
\sqrt[12]{32} |
the printed answer is contradicted at : the expression is exactly -32(7/12) + 22**(1/6) + 22*(11/12), th … |
algebra_core.1.123 |
algebra_core |
0.1 |
the printed answer is contradicted at : the expression is exactly -36sqrt(5) - 23**(1/4)sqrt(5) + 8sqrt(30 … |
algebra_core.1.124 |
algebra_core |
0 |
the printed answer is contradicted at : the expression is exactly 3**(1/3)(-27sqrt(2)3(1/3) - 112**(1/6) … |
algebra_core.1.142 |
algebra_core |
-\frac{\sqrt{6}}{2} |
the printed answer is contradicted at x=sqrt(6): the expression is exactly 0, the printed answer gives -sqrt(6 … |
algebra_core.1.160 |
algebra_core |
\frac{1}{z+2} |
the printed answer is contradicted at x=1, z=2: the expression is exactly 1/3, the printed answer gives 1/4 - … |
algebra_core.1.173 |
algebra_core |
\frac{x+y}{2} |
the printed answer is contradicted at x=1, y=2: the expression is exactly 38/17 + 61*sqrt(2)/34, the printed a … |
algebra_core.1.176 |
algebra_core |
\frac{m-8}{2} |
the printed answer is contradicted at m=9: the expression is exactly 144*sqrt(5) + 322, the printed answer giv … |
algebra_core.1.180 |
algebra_core |
\frac{1}{a(3a+b)} |
the printed answer is contradicted at a=1, b=2: the expression is exactly 2/5 - 22*(1/6)/5, the printed answ … |
algebra_core.1.184 |
algebra_core |
\frac{1}{x-\sqrt{2x+1}} |
the printed answer is contradicted at x=2: the expression is exactly sqrt(6)/5 + 6/5, the printed answer gives … |
algebra_core.1.186 |
algebra_core |
\sqrt{x^3}-\sqrt{y^3} |
the printed answer is contradicted at x=1, y=1: the expression is exactly 1/2, the printed answer gives 0 - re … |
algebra_core.1.194 |
algebra_core |
x^2\sqrt[3]{y^1} |
the printed answer is contradicted at x=1, y=1: the expression is exactly -sqrt(3)(4sqrt(3) + 7)**(1/6) + 2* … |
algebra_core.1.197 |
algebra_core |
\frac{{}^{4}\!\sqrt a}{6} |
the printed answer is contradicted at a=2: the expression is exactly 32(1/12)(-(2 - sqrt(2))(1/6) + 2( … |
algebra_core.1.201 |
algebra_core |
1/(ab) |
the printed answer is contradicted at a=1, b=2: the expression is exactly (2**(1/3)/16 + 1/2)/(1 + 2**(1/4)), … |
algebra_core.1.222 |
algebra_core |
-\frac{\sqrt{x-2}}{x+2} |
the printed answer is contradicted at x=3: the expression is exactly -sqrt(5)/5, the printed answer gives -1/5 … |
algebra_core.1.224 |
algebra_core |
\sqrt{a}\left(\sqrt{2a+1}-\sqrt{a}\right) |
the printed answer is contradicted at a=1: the expression is exactly 1/2 + sqrt(3)/2, the printed answer gives … |
algebra_core.1.226 |
algebra_core |
2.4 |
the printed answer is contradicted at c=1, a=24/5, b=6/5: the expression is exactly 12*sqrt(163112586)/298741 … |
algebra_core.1.231 |
algebra_core |
\sqrt{p^2-q^2}\sqrt{p} |
the printed answer is contradicted at p=2, q=1: the expression is exactly 15 - 2*sqrt(6), the printed answer g … |
algebra_core.1.233 |
algebra_core |
\frac{4}{9} |
the printed answer is contradicted at b=1, a=9/4: the expression is exactly 2sqrt(-3960sqrt(2) + 3718*sqrt(3 … |
algebra_core.1.236 |
algebra_core |
\sqrt[3]{p-q} |
the printed answer is contradicted at p=3, q=1: the expression is exactly 2**(7/8)/2, the printed answer gives … |
algebra_core.1.237 |
algebra_core |
2 |
the printed answer is contradicted at m=3: the expression is exactly (-2*(4 - 3**(2/3))(15(2/3) + 10) + 55 … |
algebra_core.1.238 |
algebra_core |
1 |
the printed answer is contradicted at a=1, b=2: the expression is exactly (1 + sqrt(2))(4/3)/(2*2(5/6) + 3 … |
algebra_core.1.240 |
algebra_core |
\sqrt{a}(a+4b) |
the printed answer is contradicted at a=1, b=1: the expression is exactly sqrt(5)/5, the printed answer gives … |
algebra_core.1.241 |
algebra_core |
1 |
the printed answer is contradicted at p=1, q=2: the expression is exactly 4, the printed answer gives 1 - repo … |
algebra_core.1.324 |
algebra_core |
b-4a if a\in[0,\sqrt{2}); 2(a-1)^2 if a\in[\sqrt{2},+\infty) |
the printed value '2(a-1)^2' does not hold at a=3, where it is the only case that applies - reported as suspec … |
algebra_core.1.329 |
algebra_core |
-x if x\in(0,1); x if x\in(1,\infty) |
the printed value '-x' does not hold at x=1/2, where it is the only case that applies - reported as suspected, … |
algebra_core.1.330 |
algebra_core |
\frac{x-1}{x} if x\in(-\infty,0)\cup[1,\infty); \frac{1-x} … |
the printed value '\frac{x-1}{x}' does not hold at x=2, where it is the only case that applies - reported as … |
algebra_core.2.368 |
algebra_core |
x=2 |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.4.080 |
algebra_core |
\tan\frac{m+n}{2}\alpha |
the printed answer is contradicted at alpha=1, m=1, n=2: the expression is exactly (-cos(1) + cos(2))/(-sin(2) … |
algebra_core.4.117 |
algebra_core |
\tan^6\alpha |
numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from … |
algebra_core.4.145 |
algebra_core |
2\sqrt2\sin^2\alpha\cos\left(\frac\pi4+2\alpha\right) |
the printed answer is contradicted at alpha=1: the expression is exactly -sqrt(-sin(1) + tan(1)) + sqrt(sin(1) … |
algebra_core.4.241 |
algebra_core |
\displaystyle -\frac12\sin2\alpha |
the printed answer is contradicted at alpha=1: the expression is exactly sqrt(-cos(2))/sqrt(-cot(1)**2 + tan(1 … |
algebra_core.4.248 |
algebra_core |
\displaystyle \sin4\alpha\cos^{-2}4\alpha |
the printed answer is contradicted at alpha=1, x=2: the expression is exactly (sin(24) + sin(8))/(cos(16) + 1) … |
algebra_core.4.264 |
algebra_core |
\tan5\alpha |
the printed answer is contradicted at alpha=1, x=2: the expression is exactly tan(10), the printed answer give … |
algebra_core.4.265 |
algebra_core |
\sin3\alpha |
the printed answer is contradicted at alpha=1, x=2: the expression is exactly sin(6), the printed answer gives … |
algebra_core.4.266 |
algebra_core |
\cos3\alpha |
the printed answer is contradicted at alpha=1, x=2: the expression is exactly cos(6), the printed answer gives … |
algebra_core.4.270 |
algebra_core |
2\sin2\alpha |
the printed answer is contradicted at alpha=1, x=1: the expression is exactly (-sin(pi/4 + 2) + sin(pi/4 + 6)) … |
algebra_core.4.272 |
algebra_core |
\cot\left(\alpha-\frac{\pi}{4}\right) |
numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from … |
algebra_core.4.291 |
algebra_core |
-2\sin\frac{\alpha}{4} |
the printed answer is contradicted at alpha=1: the expression is exactly -sqrt(1 - sin(1/2)) + sqrt(sin(1/2) + … |
algebra_core.4.300 |
algebra_core |
\frac{4\sin^2\left(\frac{\pi}{4}-4\alpha\right)} … |
the printed answer is contradicted at alpha=1: the expression is exactly -tan(4) - cot(4) + 2, the printed ans … |
algebra_core.4.381 |
algebra_core |
a) (\cos 2\alpha)_1=-3/5; b) (\cos 2\alpha)_2=4/5 |
sides differ numerically - reported as suspected, not as a refutation: the difference can come from the readin … |
algebra_core.4.382 |
algebra_core |
a) 4/5; b) 3/5 |
sides differ numerically - reported as suspected, not as a refutation: the difference can come from the readin … |
algebra_core.4.412 |
algebra_core |
-\cos^32x |
the printed answer is contradicted at x=1: the expression is exactly -4*sin(1)*6 - 1 - 4cos(1)*6 + 4cos(1) … |
algebra_core.4.449 |
algebra_core |
(1 - a^2)/(2a) |
numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from … |
algebra_core.4.488 |
algebra_core |
A=B=|\cos\alpha\cos\beta\cos\gamma| |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.154 |
algebra_core |
x = \frac{2\pi}{3}(3k \pm 1) |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.155 |
algebra_core |
x_1=\frac{\pi}{2}(2k+1),\quad x_2=(-1)^k\frac{\pi}{6}+\pi k |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.156 |
algebra_core |
t_1=(-1)^k\frac{\pi}{6}+\pi k,\quad t_2=\frac{\pi}{2}(4k+1) |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.158 |
algebra_core |
x=\frac{\pi}{6}(6k\pm1) |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.186 |
algebra_core |
x_1=2\pi k,\ x_2=\frac{\pi}{2}(2k+1),\ x_3=\frac{\pi}{4}(4k- … |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.197 |
algebra_core |
x_1=\frac{\pi k}{3}; x_2=\frac{\pi}{12}(2k+1). |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.263 |
algebra_core |
t=\frac{\pi}{3}(3k\pm1) |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.301 |
algebra_core |
t_1 = 360^\circ \cdot k; \quad t_2 = 90^\circ (4k+1) |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.302 |
algebra_core |
t_1=90^\circ\cdot k;\quad t_2=\pm15^\circ+90^\circ\cdot k |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.311 |
algebra_core |
x=2\pi k. |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.337 |
algebra_core |
t=\pi k |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.467 |
algebra_core |
x=\frac{\pi}{2}(4k+1) |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.5.490 |
algebra_core |
x_1=\arccos\left(\sqrt5-2\right)+2\pi k,\quad x_2=\pi(2k+1). |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.6.088 |
algebra_core |
\approx 0.95 |
the printed answer is contradicted at approx=1, x=-1/20: the expression is exactly 160000/168421, the printed … |
algebra_core.7.045 |
algebra_core |
890 |
the printed answer is contradicted at : the expression is exactly sqrt(7) + 625 + 531441**(log(3)/log(6)), the … |
algebra_core.8.055 |
algebra_core |
[0, 4). |
the book's answer excludes the upper endpoint 4, but it satisfies the inequality - reported as suspected, not … |
algebra_core.8.148 |
algebra_core |
[-3,-\sqrt{6})\cup(-\sqrt{6},-2]\cup[2,\sqrt{6})\cup(\sqrt … |
the book's answer includes the lower endpoint -3, where equality holds - reported as suspected, not as a refut … |
algebra_core.8.185 |
algebra_core |
(a^4,a^{-1}) if 0<a<1; (a^{-1},a^4) if a>1 |
the book's answer excludes the upper endpoint 2, but it satisfies the inequality (for a=1/2) - reported as sus … |
algebra_core.10.093 |
algebra_core |
(\sqrt{x}-\sqrt{y})(\sqrt{x}-2\sqrt{y}) |
the printed answer is contradicted at x=1, y=1: the expression is exactly 2, the printed answer gives 0 - repo … |
algebra_core.10.145 |
algebra_core |
x is an integer. |
the two sides of the identity differ at every tested point - reported as suspected, not as a refutation: the d … |
algebra_core.11.078 |
algebra_core |
a>4 |
the book's answer excludes the lower endpoint 4, but it satisfies the inequality - reported as suspected, not … |
algebra_core.11.252 |
algebra_core |
0.5\ln 2-1.5 |
numeric difference over 2 point(s) - reported as suspected, not as a refutation: the difference can come from … |
falin_algebra.269 |
falin_algebra |
x = \dfrac{-3 + \sqrt{17}}{2}. |
sides differ numerically |
falin_algebra.540 |
falin_algebra |
\{\tfrac{2}{5};\, 10\}. |
the two sides of the identity differ at every tested point |
falin_algebra.907 |
falin_algebra |
x>\sqrt{3} |
the book's answer excludes the lower endpoint 1.73205, but it satisfies the inequality |
falin_algebra.1023 |
falin_algebra |
4017 |
integer 4014 satisfies the inequality but is not listed |
falin_algebra.p446.1397 |
falin_algebra |
**Answer:** \emptyset. |
the two sides of the identity differ at every tested point |
shabunin.p288.23 |
shabunin |
x > -\frac{1}{2} |
the book's answer excludes the lower endpoint -0.5, but it satisfies the inequality |
kozko_params.p18.30 |
kozko_params |
a \in (0; 1/54) |
the book's answer excludes the upper endpoint 0.0185185, but it satisfies the inequality |
kozko_params.p172.6 |
kozko_params |
(1; 18 - 12\sqrt{2}] |
a=500.545 satisfies the inequality but is missing from the book's answer - reported as suspected: the printed … |
kozko_params.p173.6 |
kozko_params |
(1; 8 - 4\sqrt{3}] |
a=500.565 satisfies the inequality but is missing from the book's answer - reported as suspected: the printed … |
kozko_params.p188.1 |
kozko_params |
[-7; -2) \cup (-2; -1]. |
the book's answer excludes the upper endpoint -2, but it satisfies the inequality |
kozko_params.198_2 |
kozko_params |
207 |
the printed answer is contradicted at x=1, y=1: the expression is exactly 2, the printed answer gives 207 |
kozko_params.p263.3 |
kozko_params |
x \in [1; 2) \cup (2; \sqrt{5}) |
the book's answer excludes the upper endpoint 2.23607, but it satisfies the inequality |
kozko_params.p266.Задача 1 |
kozko_params |
(0;\, (5 - \sqrt{17})/2) |
the book's answer excludes the upper endpoint 0.438447, but it satisfies the inequality |
kozko_params.p266.Задача 5 |
kozko_params |
a \in [\pi/2;\, 3\pi/2 - 3] \cup (5\pi/2 - 6;\, 7\pi/2 - 9] |
the book's answer excludes the lower endpoint 20.5708, but it satisfies the inequality |
prasolov_planimetry.6.37 |
prasolov_planimetry |
AB \cdot CD + AD \cdot BC = AC \cdot BD |
the two sides of the identity differ at every tested point |
prasolov_planimetry.13.31 |
prasolov_planimetry |
\boldsymbol{e}_1 \sin\alpha + \boldsymbol{e}_2 \sin\beta + … |
the two sides of the identity differ at every tested point |
prasolov_planimetry.31.45 |
prasolov_planimetry |
a + c = 0 |
the two sides of the identity differ at every tested point |
Reproducing the audit
python verify_all3.py 12 dataset/core_math_dataset_v2.jsonl v15 # the core book
python reverify_books2.py bookv15 # the additional books
python build_benchmark.py # assemble + split
python audit_pairing.py hf_dataset/benchmark_full.jsonl # question/answer integrity
The verification engine is verify3.py; verify.py holds the individual
checkers and verify2.py the newer ones (inequalities, limits, integrals,
factors).
Errors of the books that the engine cannot see
These are errors of the source books, found by reading the printed page. Two kinds occur: a wrong printed answer, which the symbolic engine does not reach because the exercise has several parts and a list of answers cannot be checked as a whole, and a misprinted statement, where the book prints a statement that does not match its own printed answer.
| id | kind | part | what the book prints | what is correct | why |
|---|---|---|---|---|---|
gelfand_trig.p158.26.6 |
wrong printed answer | (d) | \left(\dfrac{\sqrt{3}}{3}; 1\right] |
\left(\dfrac{\sqrt{3}}{2}; 1\right] |
the exercise asks to solve arccos x < pi/6, whose solution is x in (cos(pi/6), 1] = (sqrt(3)/2, 1]; the book prints (sqrt(3)/3, 1] |
algebra_core.2.205 |
misprinted statement | second equation of the system | \begin{cases}x+y+\frac{x^2}{y^2}=7,\\ \frac{(x-y)x^2}{y^2}=12.\end{cases} |
\begin{cases}x+y+\frac{x^2}{y^2}=7,\\ \frac{(x+y)x^2}{y^2}=12.\end{cases} |
the printed answer does not satisfy the printed second equation: (2; 1) gives (x-y)x^2/y^2 = 4, not 12, and (6; -3) gives 36, not 12. All four printed answers satisfy x+y+x^2/y^2 = 7 and (x+y)x^2/y^2 = 12 exactly, so the book misprinted x+y as x-y. The printed system as written has exactly one real solution, (1.3863205; -0.5580341), whose coordinates are roots of a quintic and are therefore certainly not the intended answer. |
algebra_core.1.206 |
wrong printed answer | \frac{a}{a+1} |
\frac{a(a+1)}{2a+1} |
the printed expression (a+1)/(2*cuberoot(sqrt(3)-sqrt(2))6throot(5+2sqrt(6)) + 1/a) equals a(a+1)/(2a+1), because 2cuberoot(sqrt(3)-sqrt(2))*6throot(5+2sqrt(6)) = 2 exactly; the difference between the printed answer and the true value is a^3/((a+1)(2a+1)), which is not an identity | |
algebra_core.1.202 |
wrong printed answer | \sqrt{mn} |
the expression has no closed form in m and n; its value at m = 8, n = 1 is exactly 40/3, so the printed answer cannot be right. The cube root of mn is not the value either (it would be 2 at that point) |
the printed expression (cube root of m^2 + ncube root of m + n^2) * (cube root of m^4 - n^3 + n^2cube root of (m - mn)) / (mn^-1 + n - n^4m^-1 - n^2) does not have the square root of mn as its value. At m = 8 and n = 1 the expression is exactly 40/3, a rational number, while the printed answer is sqrt(8) = 2sqrt(2), an irrational number, so the two can never be equal. The denominator is not zero there (it is 63/8), so the point lies inside the domain of the expression. The first factor and the denominator do factor neatly - the first one is (m - n^3)/(cube root of m - cube root of n) and the denominator is (m + n^2)(m - n^3)/(m*n) - which is why the statement was read carefully before the answer was called wrong | |
algebra_core.8.276 |
wrong printed answer | (\sqrt[3]{5},5) |
(\sqrt[5]{5},5) |
the printed answer is (cuberoot(5), 5) = (5^(1/3), 5), but the answer of the printed inequality log_x^2 sqrt(5) - log_x 5 sqrt(5) + 1,25 < 0 is (5^(1/5), 5). With t = log_x 5 the printed inequality becomes t^2/4 - 3t/2 + 5/4 < 0, i.e. (t-1)(t-5) < 0, so 1 < t < 5 and x = 5^(1/t) runs from 5^(1/5) to 5; the cube root 5^(1/3) is only an interior point of that interval, not its lower endpoint. The root index in the printed answer is a 3, not a 5: it was read from the page image at 800 dpi and the index glyph was compared pixel by pixel with the index of the cube root in 3sqrt35 printed on the same answer page; it matches the 3 far better than the 5 (IoU 0.61 against 0.51, and a 5 would have a horizontal top bar reaching the left edge and a narrow middle stem, neither of which the printed glyph has). | |
algebra_core.1.111 |
wrong printed answer | 100 |
the expression is negative at the stated values a = 25/7 and x = 7/25; under the transcribed structure its value is exactly -31250*sqrt(7)/343 - 84/25 |
the exercise asks for the value of the printed expression at a = 3 4/7 = 25/7 and x = 0,28 = 7/25. There ax = 1, so the dominant term -10a sqrt(x) sqrt((a x^-1)^2) is exactly -31250sqrt(7)/343 (about -241.1), and the remaining terms are bounded by 229 and 3, so the value is negative in every faithful reading of the printed expression (eight readings were evaluated exactly: the transcribed structure gives -31250sqrt(7)/343 - 84/25, the structure in which the first group is printed as ( ... )^3 gives about -1.39*10^7). A positive printed answer of 100 is therefore impossible. | |
falin_algebra.777 |
wrong printed answer | $-\frac{19}{4} < x < -4$ |
$\left(-\frac{11}{4};\, -2\right) \cup \left(-2;\, +\infty\right)$ |
the printed answer of log_{x+3}(1/2) > 1/2 lies entirely outside the domain of the logarithm, which needs x > -3 and x != -2. Solving the printed inequality gives x in (-11/4, -2) union (-2, +infinity): for 0 < x+3 < 1 the logarithm decreases, so 1/2 < (x+3)^(1/2) and x > -11/4, and for x+3 > 1 the inequality holds everywhere. The point x = -19/8 satisfies the inequality and is missing from the printed answer. The printed page 99 was read at 400 dpi and shows exactly -19/4 < x < -4. | |
falin_algebra.788 |
wrong printed answer | $-1 < x < -\frac{5}{6}$ |
$-1 < x < \frac{1}{\sqrt[3]{36}} - 1$ |
the printed answer of log_6 log_{1/6th-root-of-6}(x+1) > 1 is too narrow. Writing u = log_{6^(-1/9)}(x+1), the condition is u > 6, and because the base is below 1 this gives 0 < x+1 < 6^(-2/3), so the solution is -1 < x < 6^(-2/3) - 1 = -0.6972.... The printed upper end -5/6 = -0.8333 excludes the whole stretch from -0.8333 to -0.6972; the point x = -0.8 satisfies the inequality (log_6 log_{6^(-1/9)}(0.2) = 1.166 > 1) and is missing from the printed answer. The printed page 100 was read at 400 dpi. | |
falin_algebra.926 |
wrong printed answer | $[-18;-3] \cup (0;2)$ |
$[-18;-3) \cup (0;2)$ |
in sqrt((x^2+9x-162)/(x-2)) > x+9 equality holds at x = -3, because the radicand is 36 there and x+9 is 6 as well, so the strict inequality fails and -3 must not belong to the solution. The book closes the bracket at -3. The difference of the two sides is -x(x+3)(x+12)/(x-2), which changes sign exactly at -3. The printed page 116 was read at 400 dpi and shows the closed bracket. |