Datasets:
problem string | expected_answer string | original_expected_answer string | changed_answer_to_majority bool | data_source string | metadata dict | license string | url string | pass_bucket int8 | answer_tier int8 | question_md5 string | scan_index int64 | nemotron_n_samples int64 | nemotron_n_correct int64 | nemotron_pass_estimate float64 |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Given two vectors $\textbf{r} = (r_x, r_y)$ and $\textbf{s} = (1, s_y)$ that point in the same direction, derive the magnitude of $\textbf{s}$ in terms of $r_x$ and $r_y$. | \(\sqrt{1 + \frac{r_y^2}{r_x^2}}\) | \(\sqrt{1 + \frac{r_y^2}{r_x^2}}\) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/3034611/basic-ray-casting-vector-equality-problem | 5 | 2 | a72915d3e371411641be485787c724ea | 196,486 | 5 | 1 | 0.2135 |
Are there functions that satisfy the following properties?
1. Domain: \( x > 0 \)
2. \( f'(x) > 0 \)
3. \( f''(x) < \frac{-2f'(x)}{x} \)
Could anyone give an example of such a function? | \( f(x) = -\frac{1}{2x^2} \) | \( f(x) = -\frac{1}{2x^2} \) | false | stackflow | {
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"accuracy": 0.25
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"count": 8,
"pass": 3,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2817818/are-there-functions-that-satisfy-the-following-properties | 5 | 2 | 9902c225848944612cac343a75588f13 | 187,748 | 5 | 1 | 0.2135 |
Generate a primitive Pythagorean triple $(a, b, c)$ such that the radius of the inscribed circle is $n$. | \( (2n+1, 2n(n+1), 2n^2+2n+1) \) | \( (2n+1, 2n(n+1), 2n^2+2n+1) \) | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
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"count": 7,
"pass": 1,
"accuracy": 0.1429
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"count": 8,
"pass": 1,
"accuracy": 0.125
},
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"count": 8,
"pass": 1... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1625657/circle-inscribed-in-pythagorean-triangle | 4 | 2 | c8ce5226ac109b5746f15fb1be9d5495 | 139,206 | 5 | 2 | 0.3934 |
Six real numbers \( x_1 < x_2 < x_3 < x_4 < x_5 < x_6 \) are given. For each triplet of distinct numbers of those six, Vitya calculated their sum. It turned out that the 20 sums are pairwise distinct and are denoted by \( s_1 < s_2 < s_3 < \cdots < s_{19} < s_{20} \). It is known that \( x_2 + x_3 + x_4 = s_{11} \), \(... | 7 | 7 | false | aops | {
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"accuracy": 1
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"count": 8,
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"count": 8,
"pass": 6,
"accuracy": 0.75
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"reason_medium_no_tool": {
"count": 8,
"pass": 2,
"acc... | cc-by-4.0 | null | 6 | 1 | c6947f3099ba8b2ca769cc96315e6000 | 38,422 | 3 | 1 | 0.3447 |
Estimate the number of ways to concatenate strings of the form $01^k2^k$ (where $k \in \mathbb{N} \cup \{0\}$) to create a string of length $n$. | \( F_n \) | \( F_n \) | false | stackflow | {
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"pass": 4,
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"count": 8,
"pass": 5,
... | cc-by-sa-4.0 | https://mathoverflow.net/questions/139547/estimation-of-number-of-ways-to-concatenate-strings-of-the-form-01-k2-k-to-creat | 5 | 2 | a4171170c4c1752eaa7ef251668cf45d | 284,960 | 3 | 1 | 0.3362 |
Solve the second-order ordinary differential equation \(x^2y'' + 2y' - 2y = 0\). | \( y = C_1(x^2 + 2x + 2) + C_2x^2e^{2/x} \) | \( y = C_1(x^2 + 2x + 2) + C_2x^2e^{2/x} \) | false | stackflow | {
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"acc... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/4668770/second-order-ode-and-integral-of-exponential-divided-by-a-polynomial | 4 | 2 | d74afe4a0b110550ddb1bc2029652594 | 262,007 | 4 | 2 | 0.4811 |
Given that ABCD is a straight line, ABE is a sector of a circle with center B, and CED is a sector of a circle with center C. Angle ABE is a right angle, the length of AB is \( r \), and angle ACE is \( \frac{\pi}{4} \) radians. If \( r = 10 \) cm, find the area of the shape. | \( 100\pi + 50 \) | \( 100\pi + 50 \) | false | stackflow | {
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"count": 8,
"pass": 2,
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"count": 8,
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"acc... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/3679142/find-the-area-of-the-entire-shape | 4 | 2 | c535a97f5c7a14106a043fb752064299 | 225,415 | 5 | 1 | 0.2113 |
How many elements of order 6 are in the symmetric group S4? When can Euler's totient function be used to determine the number of elements of a given order in a group? | 0 | 0 | false | stackflow | {
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"pass": 6,
"accuracy": 0.75
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"count": 8,
"pass": 3,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2593521/general-formula-for-number-of-elements-of-a-given-order-in-finite-groups | 6 | 1 | b7dc0446ea56ec52b9043864ac85914a | 180,677 | 3 | 1 | 0.3447 |
Solve in $\mathbb{R}$ the equation: $\sum_{k=1}^{n}k^{[ x]}+\sum_{k=1}^{n}k^{ [\frac{1}{x}]}=n(n+1)$, where $[ . ]$ denotes the integer part. | \( x = 1 \) | \( x = 1 \) | false | aops | {
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"count": 8,
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"pass": 2,
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"count": 8,
"pass": 6,
... | cc-by-4.0 | null | 4 | 1 | 97b6517511c3846d79d385db8bc2f063 | 37,382 | 5 | 1 | 0.2113 |
Find an integer \( n \) such that the product \((n-7)(n+7)\) is a perfect square, where \( n \) represents a person's age and the perfect square is between 1 and 100. | 25 | 25 | false | stackflow | {
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"count": 8,
"pass": 5,
"accuracy": 0.625
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"count": 8,
"pass": 3,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1008106/n-7-n-7-some-perfect-square | 5 | 1 | 2d2d48477ff896ee5d3357526a8b50db | 113,765 | 5 | 2 | 0.396 |
Find the sum \(\sum_{n=2}^{\infty}\left(\sum_{j=1}^{n}\frac{1}{(2j-1)^2} \binom{2n}{n}\frac{1}{2^{2n}(2n+1)}\right)\). | \(\frac{\pi^3}{48} - \frac{1}{6}\) | \(\frac{\pi^3}{48} - \frac{1}{6}\) | false | stackflow | {
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"pass": 1,
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"ac... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/833680/how-find-the-sum-sum-n-2-infty-left-sum-j-1-n-frac-1-2j-1-2-binom-2n-n-frac-1-2- | 4 | 2 | b7f32fc506f4d5dfd7ae271368c97d22 | 107,319 | 5 | 3 | 0.5756 |
For \( x = \ln(2t + 7) \) and \( y = t^2 \) with \( t > 0 \), find \( \frac{d^2y}{dx^2} \). | \(\frac{(2t+7)(4t+7)}{2}\) | \(\frac{(2t+7)(4t+7)}{2}\) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1603311/parametric-differentiation | 5 | 2 | 7173fc1c79f7b0251b82894da5e20c48 | 254,459 | 3 | 1 | 0.3362 |
Is the integral in the textbook correct? Evaluate the integral \(\lambda^2 \int_0^s e^{-\lambda s} \, dt\) and determine if the result \(\lambda^2 s e^{-\lambda s}\) is accurate. | \(\lambda^2 s e^{-\lambda s}\) | \(\lambda^2 s e^{-\lambda s}\) | false | stackflow | {
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"pass": 2,... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1683716/incorrect-integral-in-textbook | 4 | 2 | 3777976bb790eded058f7c848089253a | 138,618 | 5 | 1 | 0.2113 |
From the positive integers $m, m+1, \dots, m+n$, only the sum of the digits of $m$ and the sum of the digits of $m+n$ are divisible by $8$. Find the maximum value of $n$. | 15 | 15 | false | aops | {
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... | cc-by-4.0 | null | 5 | 1 | a39341388b97fcc62f0d415a17b2a507 | 38,512 | 3 | 1 | 0.3362 |
A drunkard takes a step forward with a probability of 0.6 and a step backward with a probability of 0.4. If he takes 9 steps in total, what is the probability that he is exactly one step away from his initial position? | \(\binom{9}{5} \cdot (0.6)^4 \cdot (0.4)^4\) | \(\binom{9}{5} \cdot (0.6)^4 \cdot (0.4)^4\) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/718441/probability-of-1-step-fordward-and-1-step-backward-of-drunked-man | 6 | 2 | ee171d4c6cc2a0de455c29e8f343b9fd | 106,629 | 3 | 1 | 0.3447 |
You have an urn with a red ball and a blue ball. Every step, you draw a ball from the urn, return the ball, and add another ball of the same color as the one you drew. After N steps, what is the probability that the urn contains k red balls? | \(\frac{1}{N+1}\) | \(\frac{1}{N+1}\) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/3542425/non-inductive-solution-to-this-probability-riddle | 4 | 2 | d4ac1411b03646a584b045047720c38b | 218,751 | 3 | 1 | 0.3324 |
Determine the maximum value of the line integral
\[
\int_{C} \mathbf{F} \cdot d\mathbf{r}
\]
where
\[
\mathbf{F}(x,y,z) = xy^2 \mathbf{i} + (3z - xy^2) \mathbf{j} + (4y - x^2 y) \mathbf{k}
\]
and $C$ is a simple closed curve in the plane $x + y + z = 1$, oriented anticlockwise as seen from points far up on the $z$-axis... | \(\frac{\pi}{2}\) | \(\frac{\pi}{2}\) | false | aops | {
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"pass": 2,
... | cc-by-4.0 | null | 4 | 2 | 53447c4f56e25844adcd6e01d27253f3 | 23,457 | 5 | 1 | 0.2113 |
Determine whether the process $X_t = \sqrt{t}Z$, where $Z$ is a standard normal variate, is a Brownian motion. One of the reasons given is that the variance of $X_t - X_s$ is $(\sqrt{t} - \sqrt{s})^2$. However, the variance calculated is $t - s$. Identify the error in the calculation.
Calculation details:
- Given $X_t... | $(\sqrt{t} - \sqrt{s})^2$ | $(\sqrt{t} - \sqrt{s})^2$ | false | stackflow | {
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"count": 8,
"pass": 8,
"accuracy": 1
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"count": 8,
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1564119/what-is-wrong-with-my-calculation-for-variance | 5 | 2 | 564bab8233071227265a28dc4fbe9058 | 137,031 | 3 | 1 | 0.3362 |
Find the level curves of the equation \( f(x,y) = y(x^2 - y^2) \) for \( f(x,y) = -1 \). | \( x^2y - y^3 = -1 \) | \( x^2y - y^3 = -1 \) | false | stackflow | {
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"pass": 3,
"accuracy": 0.375
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"count": 8,
"pass": 7,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/3958575/the-level-surfaces-of-the-equation-f-x-y-y-x-2-y-2 | 6 | 1 | f4b4aad63057a1a88678e39e28329e4e | 235,350 | 5 | 2 | 0.4011 |
Solve the equation $(x^2-2x)^2+4(x^2-2x)+3=0$. | \(x = 1\) | \(x = 1\) | false | stackflow | {
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"count": 8,
"pass": 0,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/621379/how-to-solve-x-2-2x-2-4-x-2-2x-3-0 | 4 | 1 | aaf1dfccb82c5c25404993d609c5d0ac | 99,928 | 3 | 1 | 0.3324 |
Let \((X, \mathcal{M}, \mu)\) be a \(\sigma\)-finite measure space and let \((\mathbb{R}, \mathcal{L}, m)\) denote the set of real numbers with the Lebesgue measure. Let \(f: X \to \mathbb{R}\) be such that \(f \in L^1(X)\). Let \(\Gamma = \{((x, f(x))): x \in X\} \subset X \times \mathbb{R}\). Is \(\Gamma\) measurable... | 0 | 0 | false | stackflow | {
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"reason_medium_no_tool": {
"count": 8,
"pass": 2,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/3894723/measurable-set-in-product-space | 5 | 1 | de696aed5bbfc8bbf2ca9a50661417d2 | 235,701 | 3 | 1 | 0.3362 |
A couple has two children and at least one of them is female. What is the probability that the second child is also female? | \(\frac{1}{3}\) | \(\frac{1}{3}\) | false | stackflow | {
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"count": 8,
"pass": 1,
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"count": 8,
"pass": 7,
"accuracy": 0.875
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"reason_medium_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
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"reason_medium_no_tool": {
"count": 8,
"pass": 7,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/3339426/doubt-to-interpret-a-probability-question-and-its-variations | 6 | 2 | 262be89bfeda07b36daa86121ab17719 | 211,717 | 5 | 1 | 0.2196 |
If $(m-n)^2 = 48$ and $m^2 + n = 30$, then what is the value of $\frac{mn}{3}$? | -3 | -3 | false | stackflow | {
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"accu... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/3320518/if-m-n-2-48-m-2-n-30-then-what-is-the-value-of-mn-3 | 6 | 1 | c44862880ab2ac166be41f652ff85e4e | 206,293 | 3 | 1 | 0.3447 |
Solve and investigate the following systems of equations. The numbers $a$, $b$ and $c$ are such that the system $$\left\{\begin{array}{lll} ax-by=2a-b,\medskip \\ (c+1)x+cy=10-a+3b \end{array}\right.$$ has infinitely many solutions, and $x=1$, $y=3$ is one of the solutions. Find $a$, $b$ and $c$. | $\left(0,0,\frac{9}{4}\right),\ (2,-1,1)$ | $\left(0,0,\frac{9}{4}\right),\ (2,-1,1)$ | false | aops | {
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},
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"count": 7,
"pass": 0,... | cc-by-4.0 | null | 4 | 2 | dc0d4c3bc896475c89b5dd82c9528bb2 | 8,516 | 5 | 2 | 0.3934 |
Let \( A \) be a convex 90-gon. We place \( n \) points \( \{x_i\}_{i=1}^n \) in the interior of \( A \). What is the minimum number \( n \) such that the interior of any triangle, whose vertices are also vertices of \( A \), will contain at least one of the points \( x_i \)? | 88 | 88 | false | aops | {
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"reason_medium_no_tool": {
"count": 8,
"pass": 1,
"acc... | cc-by-4.0 | null | 4 | 1 | c54bb395e32d9d5e7254545e097e7cd5 | 38,035 | 5 | 3 | 0.5756 |
Determine \( \mu[0,c] \) for the exponential distribution \( X \sim \text{Exp}(\lambda) \). | \( 1 - e^{-\lambda c} \) | \( 1 - e^{-\lambda c} \) | false | stackflow | {
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Find solutions to the equation $\sqrt{-ix} = -i$. | $x = -i$ | $x = -i$ | false | stackflow | {
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Given the binary number \( x = (0.\underbrace{100100100100100100100100}_{24 \text{ bits}}10\overline{010})_{2} \times 2^{-1} \), compute \( x_+ - x \) where \( x_+ \) is obtained by increasing the value of the last bit by one unit and discarding all the bits after the last bit. | \(\frac{3}{7} \times 2^{-25}\) | \(\frac{3}{7} \times 2^{-25}\) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2418928/computing-x-x-where-x-is-the-rounded-binary-number-of-x | 6 | 2 | 5a159472f31d7f88a83aadb6588464a4 | 171,169 | 5 | 1 | 0.2196 |
Let \( z = f(x, y) = x^2 + y^2 \). Is it valid to compute \( \frac{\partial x^2}{\partial z} + \frac{\partial y^2}{\partial z} \)? If so, how would you compute the result? | 1 | 1 | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/4621525/fundamental-doubt-about-partial-differentiation-sum-of-derivates-equals-derivate | 5 | 1 | f39ecd73a39ca6f0bde0878513da9ba0 | 265,210 | 3 | 1 | 0.3362 |
From the 3-digit numbers formed using the set $\{0, 2, 3, 5, 6, 8\}$, how many of those numbers are odd? | 60 | 60 | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/4309842/finding-how-many-odd-numbers-we-can-get-from-a-given-set-of-numbers | 4 | 1 | c90446d8d2acde82b71913211cf9cdb5 | 250,734 | 3 | 1 | 0.3324 |
Find the derivative $\displaystyle \frac{d}{dx}\int^a_x \tan(\tan(t))\,dt$. | \(-\tan(\tan(x))\) | \(-\tan(\tan(x))\) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1176077/find-the-derivative | 5 | 2 | 11b927dc6769293107e09308da6f1ab4 | 100,734 | 3 | 1 | 0.3362 |
What is the largest power of 10 that divides 70!? | 16 | 16 | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2111294/largest-power-of-10-dividing-the-number-70 | 6 | 1 | 3f436d62d2be07a4955f30c9bacdfd5b | 156,474 | 5 | 1 | 0.2196 |
For each of six normal male infants, the amount of the amino acid alanine (mg/100mL) was determined while the infants were on an isoleucine-free diet, resulting in the following data: 2.84, 3.54, 2.80, 1.44, 2.94, 2.70. Compute a 97% confidence interval for the true median amount of alanine for infants on such a diet. | (1.44, 3.54) | (1.44, 3.54) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/750670/statistical-intervals-confidence-intervals | 6 | 2 | fc51bb1cccdf961cf084f46fb1a918df | 108,311 | 3 | 1 | 0.3447 |
Use $\mathcal{F}(e^{xb}) = 2\pi \delta_{ib}$ to calculate the Fourier-Transformation of $\sin x$. | \( \pi (\delta_{-1} - \delta_{1}) \) | \( \pi (\delta_{-1} - \delta_{1}) \) | false | stackflow | {
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... | cc-by-sa-4.0 | https://mathoverflow.net/questions/92998/fourier-transform | 5 | 2 | e3152d53d0ed7adbdc97ee5d23550ad2 | 80,766 | 3 | 1 | 0.3362 |
There are 1000 balls labeled $000, 001, \dots, 999$ and 100 boxes labeled $00, 01, \dots, 99$. A ball with label $\overline{ABC}$ can be placed into a box with label $\overline{XY}$ if and only if $\overline{XY} \in \{\overline{AB}, \overline{AC}, \overline{BC}\}$. Determine the smallest integer $n$ such that there exi... | 50 | 50 | false | aops | {
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... | cc-by-4.0 | null | 6 | 1 | aa04fd0a8da17ee7ff3e73935186e58c | 38,162 | 5 | 1 | 0.2196 |
Suppose that lightning strikes in any given ten-second window with probability $p$, where $p \in \{ \dfrac{1}{3}, \dfrac{2}{3} \}$ and all possible values of $p$ are equally likely. If lightning does not strike in a five-second window, then what is the probability that it will strike within the next five seconds? | \(\sqrt{3}-\sqrt{6}+1\) | \(\sqrt{3}-\sqrt{6}+1\) | false | aops | {
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... | cc-by-4.0 | null | 4 | 2 | f7da18f2cf8001c73eb957785dd107a0 | 22,470 | 3 | 1 | 0.3324 |
Given observations $z_j$ corrupted by Gaussian noise $w \sim \mathcal{N}(0, \sigma^2)$, where $z = x + w$, and the maximum likelihood function $\Lambda(x) = p(z|x) = \mathcal{N}(z; x, \sigma^2) = \frac{1}{\sqrt{2\pi} \sigma} \exp \left(\frac{(z-x)^2}{2\sigma^2}\right)$, how can one calculate $\Lambda(x)$ for an unknown... | 1.06672 | 1.06672 | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1071110/how-to-use-the-maximum-likelihood-function-to-acquire-the-maximum-likelihood-ml- | 4 | 1 | 1f5534495d3b301168bf231315d9cd69 | 118,823 | 5 | 1 | 0.2113 |
A rectangle and a triangle are inscribed in a circle of radius $1$. The base of the triangle is the same as the base of the rectangle. If the areas of the rectangle and triangle are equal, find the value of $h$. | \( \frac{2}{3} \) | \( \frac{2}{3} \) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1472747/if-the-areas-of-the-rectangle-and-triangle-are-equal-find-the-value-of-h | 6 | 2 | 0feb415bd52ec42e54898d52ecc6c2a4 | 133,042 | 5 | 1 | 0.2196 |
Given that \( b^x - x \) divides \( a^x - x \) for any natural number \( x \), find all pairs \( (a, b) \). | \( a = b \) | \( a = b \) | false | aops | {
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... | cc-by-4.0 | null | 5 | 2 | fc40bd7987fb4ef8c1ae14bf47c85c10 | 37,814 | 3 | 1 | 0.3362 |
Given a rectangle with vertices at $(5,0)$, $(20,0)$, $(5,5)$, and $(20,5)$, and a point inside the rectangle at $(6,1)$. After modifying the rectangle to have vertices at $(10,1)$, $(16,1)$, $(10,4)$, and $(16,4)$, determine the new location of the point corresponding to $(6,1)$ in the original rectangle. Also, genera... | (10.4, 1.6) | (10.4, 1.6) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/3730230/location-of-new-point-in-new-rectangle | 4 | 2 | 3f3618fad6d4f2636de2b8c5b7a949df | 228,406 | 5 | 2 | 0.3934 |
Using only the numbers 2, 4, 6, and 8, how many possible ways can you add up to 32 if the number 4 may only be used no more than once in any solution, order matters, any solution that uses the number 4 must begin with the number 4, there are two unique ways to include the value 2, four unique ways to include the value ... | 7424856 | 7424856 | false | stackflow | {
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"pass": 2,... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1338210/how-many-ways-to-add-to-32 | 4 | 1 | dfe3caa333959c1900ef1ea5df575ef4 | 126,555 | 5 | 3 | 0.5756 |
Find the order of the smallest non-commutative ring without unity. | 4 | 4 | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2542824/non-commutative-ring-without-unity | 5 | 1 | 948ef18c7a47d08d891cc77286262088 | 178,648 | 3 | 1 | 0.3362 |
Given a coin with arbitrary selectable probabilities for heads and tails, determine the probability of heads such that the expected length of uninterrupted heads is N. For example, if N=5, what is the probability of heads such that the average sequence of uninterrupted heads is 5? | \( p = \frac{5}{6} \) | \( p = \frac{5}{6} \) | false | stackflow | {
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"... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/4704918/compute-cumulative-consecutive-probability-with-a-given-size | 4 | 2 | 278170b8b4ddf301b7eb3729de0be175 | 265,836 | 3 | 1 | 0.3324 |
In a diagram where circles of the same color have the same radius, two chords of equal length meet at a point on a black circle, dividing it into two segments. In each segment, a largest possible green circle is inscribed, followed by a pair of largest possible blue circles, and then a pair of largest possible red circ... | 5 | 5 | false | stackflow | {
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"accura... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/4708430/self-made-sangaku-style-geometry-problem-involving-chords-and-inscribed-circles | 4 | 1 | 40fc60a32800a7f97aee27d6bdcfa1b7 | 265,228 | 3 | 1 | 0.3324 |
A 10-foot wall stands 5 feet from a building. A ladder, supported by the wall, touches the building. Express the length of the ladder as a function of \( x \), where \( x \) is the distance from the base of the ladder to the wall. | \( L=\sqrt{\left(\frac{10(x+5)}{x}\right)^2+(x+5)^2} \) | \( L=\sqrt{\left(\frac{10(x+5)}{x}\right)^2+(x+5)^2} \) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/845591/express-the-length-of-the-as-a-function-of-x | 5 | 2 | 1fcd636e143f42689dbb967cd63ae93b | 107,522 | 5 | 1 | 0.2135 |
What is the general formula for the number of permutations of $r$ elements taken from $n$ elements, where $q$ specific elements must always stay together without any specific order? | \(\frac{(n-q)!\,q!\,(r-q+1)!}{(r-q)!\,(n-r)!}\) | \(\frac{(n-q)!\,q!\,(r-q+1)!}{(r-q)!\,(n-r)!}\) | false | stackflow | {
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Evaluate the definite integral \(\int_{0}^{\pi} (1 + \tan x) \, \mathrm{d}x\) and explain why the substitution \(u = 1 + \tan x\) leads to a different result than directly evaluating the integral. | \(\pi\) | \(\pi\) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/4503630/bizarre-result-in-definite-integration | 5 | 2 | c2a1a61a9b8762387698964a5485ec7c | 256,612 | 3 | 1 | 0.3362 |
How many solutions are there to the equation $a + b + c + d = S$, where $a, b, c, d \in \{2, \ldots, S\}$? | \(\binom{S-5}{3}\) | \(\binom{S-5}{3}\) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/404777/number-of-solutions | 4 | 2 | 2a52e7dc13af39b269fc34a1ad6a58f6 | 154,864 | 3 | 1 | 0.3324 |
Given the probability distribution \( p(x) = \frac{a}{\pi (a^2 + x^2)} \) for \( -\infty < x < \infty \) with \( a < 0 \), calculate the probability distribution \( P(X) \) of the average \( X \) of \( M \) values of \( x \), each drawn from the distribution \( p(x) \). | \( P(X) = \frac{a}{\pi (a^2 + X^2)} \) | \( P(X) = \frac{a}{\pi (a^2 + X^2)} \) | false | stackflow | {
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"pass": 4,... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2238269/calculating-probability | 4 | 2 | be469bc1bff98d367b9c3b2d522d157a | 191,864 | 3 | 1 | 0.3324 |
Describe the left cosets of $SL_2(\mathbb{R})$ in $GL_2(\mathbb{R})$. What is the index of $SL_2(\mathbb{R})$ in $GL_2(\mathbb{R})$? | \(\infty\) | \(\infty\) | false | stackflow | {
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"pass": 4,... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1656250/index-of-sl-2-r-in-gl-2-r | 6 | 2 | bbd176909cdf3052104182b0ff93b77b | 138,883 | 5 | 1 | 0.2196 |
Find all Sylow 2-subgroups of \( S_6 \). | 45 | 45 | false | aops | {
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... | cc-by-4.0 | null | 5 | 1 | 62c41c5d885ed363f1e0b929b90fe544 | 17,696 | 5 | 1 | 0.2135 |
Solve the following linear programming problem:
\[
\max\ \ 6x_1 + 5x_2 + 4x_3 + 5x_4 + 6x_5
\]
subject to:
\[
x_1 + x_2 + x_3 + x_4 + x_5 \leq 3,
\]
\[
5x_1 + 4x_2 + 3x_3 + 2x_4 + x_5 \leq 14,
\]
\[
x_i \geq 0 \text{ for all } i.
\] | 18 | 18 | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2718804/multiple-optimal-solutions-for-a-linear-programming-problem | 5 | 1 | fe2a44f97f24cacfa8572bf94a19c4d2 | 185,736 | 3 | 1 | 0.3362 |
Given a cube with faces labeled $a_1, \ldots, a_6$, where $a_1$ and $a_6$ are opposite faces and $a_1$ is initially resting on the $x$-$y$ plane, and a transformation $r(p, q)$ defined as a $90$ degree rotation around a side between adjacent faces $a_p$ and $a_q$ such that at least one face is always touching the $x$-$... | 2 | 2 | false | stackflow | {
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"a... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2893219/cube-tumbleweeding-on-a-plane | 6 | 1 | 28ef465609ba4e0cda5c12848faa2d5e | 189,070 | 3 | 1 | 0.3447 |
Two identical regular quadrilateral pyramids are placed so that their bases coincide but their vertices do not. Let's look at the eight planes in which the side faces of these pyramids are located. How many parts do these planes divide space into? | 59 | 59 | false | aops | {
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"a... | cc-by-4.0 | null | 6 | 1 | e63b817c4600b728fec8d1dbfd0ed0ac | 6,734 | 5 | 2 | 0.4011 |
Estimate the order of magnitude of \( M = \max_{x \in [0, \infty)} x^{1000} e^{-0.001x} \) using only pencil-and-paper arithmetic. | \( 10^{5565} \) | \( 10^{5565} \) | false | aops | {
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"pass": 7... | cc-by-4.0 | null | 6 | 2 | 22ff70b4756957f38741b4129cbdff74 | 19,242 | 5 | 1 | 0.2196 |
Find the minimum of $\theta(x,y) = x^2 + y^2 - 2xy + 3y + 5$, subject to the constraint $g(x,y) = x^2 + y^2 + 3x - 2y - 3 \le 0$. Use the Kuhn-Tucker sufficient optimality theorem to solve the system of equations and inequalities:
$$
\left\{
\begin{array}{c}
\nabla \theta(x) + u\nabla g(x) = 0 \\
g(x) \le 0 \\
ug(x) ... | \( 0.5 \) | \( 0.5 \) | false | stackflow | {
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"accuracy": 1
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"count": 8,
"pass": 1,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1521834/about-a-nonlinear-problem-using-sufficient-optimality-kuhn-tucker-theorem | 4 | 1 | 29b35c62f6f49947e6f7b7d1c6aadf42 | 130,136 | 3 | 1 | 0.3324 |
In sector $OAB$ with center $O$, if $AQ = 5$, $BQ = 6$, and $OQ = PQ$, find the length of the radius of the sector. | \( 2\sqrt{10} \) | \( 2\sqrt{10} \) | false | aops | {
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"pass": 0,
"ac... | cc-by-4.0 | null | 4 | 2 | 2397b8a61a1b7b68c2eda8cce4223360 | 33,169 | 5 | 2 | 0.3934 |
Given the function \( f(x,y) = x^4 - 6x^2y^2 + y^4 - 2x^2 + 2y^2 \), if a particle is at the position \((x,y,z) = (2,1,f(2,1))\) at time \( t = 0 \), in which direction will it initially move if it always moves in the direction where \( f \) decreases the fastest? | (0, 40) | (0, 40) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1667254/a-particule-on-a-surface | 4 | 2 | 06ea1192ff238abf5993e89ffaa4cc79 | 141,050 | 3 | 1 | 0.3324 |
What is the maximum number of pairs of integers from 1 to 22 you can choose such that each pair has a distinct sum and the sum does not exceed 27? | 10 | 10 | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1809454/choosing-pairs-of-numbers-with-distinct-bounded-sum | 6 | 1 | 3f94de89758c5bc4ade9490ce732075d | 143,410 | 5 | 1 | 0.2196 |
Find the value of $\sqrt[3]{\dfrac{1-i}{1+i}}$ in binomial and polar form. | \(\frac{\sqrt{3}-i}{2}\) | \(\frac{\sqrt{3}-i}{2}\) | false | stackflow | {
"reason_low_with_tool": {
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"accuracy": 0.875
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1465838/how-to-solve-a-cube-root-of-unity-complex-numbers | 5 | 2 | 1a79b8563028eb968cd1aa6238eb0a20 | 131,051 | 5 | 1 | 0.2135 |
A speaks the truth 3 out of 4 times. If A reports that a 6 was the outcome of rolling a fair six-sided die, what is the probability that a 6 was actually rolled? | \(\frac{3}{8}\) | \(\frac{3}{8}\) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1691168/a-speaks-truth-3-out-of-4-times-conditional-probability | 6 | 2 | 26b7b45d3bb2ad1143ac32a51af1497b | 140,184 | 5 | 1 | 0.2196 |
Given the series \(1, 1, 2, 1, 2, 3, 1, 2, 3, 4, 1, 2, 3, 4, 5, \ldots\), find the value of the element at position \(K\). The method involves solving the quadratic equation \(x^2 - x - 2i = 0\) for \(i\) ranging from \(k-1\) to \(1\), and finding the positive natural root. The value of the element at position \(K\) is... | \( k - i \) | \( k - i \) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2153432/roots-of-square-equation-the-solution-for-the-value-of-element-k-in-a-series-of- | 4 | 2 | 5d6030c4b4e5552b290509df9dd8246b | 158,115 | 5 | 2 | 0.3934 |
Solve the recurrence relation \( T(n) = T\left(\left\lceil \frac{n}{2} \right\rceil\right) + T\left(\left\lfloor \frac{n}{2} \right\rfloor\right) + 1 \) with \( n = 2^m \). | \( T(2^m) = 2^m T(1) + 2^m - 1 \) | \( T(2^m) = 2^m T(1) + 2^m - 1 \) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/3505063/solving-the-recurrence-t-n-t-left-left-lceil-frac-n-2-right-rceil-right-t-left-l | 4 | 2 | 8545b914e1e41edf64640e6ff3ae536a | 218,970 | 5 | 1 | 0.2113 |
Given a monic polynomial \( P(x) \) of degree 2023 such that \( P(k) = k^{2023} P\left(1 - \frac{1}{k}\right) \) for all integers \( k \) from 1 to 2023, find \( P(-1) \). | \(\frac{4045!! - 2024!}{1 - 2023!}\) | \(\frac{4045!! - 2024!}{1 - 2023!}\) | false | aops | {
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"pass": 1,
"accura... | cc-by-4.0 | null | 4 | 2 | c8977ce47acc19d4c40bf0a4ab64e8ee | 8,205 | 3 | 1 | 0.3324 |
Let $X = X_1 \cup X_2$ be a scheme, and $F$ be a coherent sheaf on $X$. Consider the short exact sequence $0 \to I_{X_1} \to \mathcal{O}_{X_1 \cup X_2} \to \mathcal{O}_{X_1} \to 0$. Using the long exact sequence in cohomology, we have:
\[
\cdots \to H^i(FI_{X_1} \otimes L^n) \to H^i(F \otimes L^n) \to H^i(F \otimes \ma... | 0 | 0 | false | stackflow | {
"reason_low_with_tool": {
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"accuracy": 0.875
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"pass": 6,... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/890693/hartshorne-ex-iii5-7-c | 5 | 1 | 93f17cc41172f43c1a6b0e87a04bf069 | 110,006 | 3 | 1 | 0.3362 |
Find the differential equation of \( y = e^{-x} + ax + b \) where \( a \) and \( b \) are arbitrary constants. | \( y'' = e^{-x} \) | \( y'' = e^{-x} \) | false | stackflow | {
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"count": 8,
"pass": 4,
"accuracy": 0.5
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"count": 8,
"pass": 3,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1476510/find-the-differential-equation | 5 | 2 | 75930ea5682800974dbd4346ee4bce56 | 175,580 | 4 | 1 | 0.2612 |
Given the sequence defined by \( x_1 = 1 \) and \( x_{n+1} = \frac{x_n + nx_{n-1}}{n+1} \) for \( n > 1 \), find \( \lim_{n \to \infty} x_n \). | \(\ln 2\) | \(\ln 2\) | false | aops | {
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"count": 8,
"pass": 1,
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"count": 8,
"pass": 8,
"accuracy": 1
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"reason_medium_no_tool": {
"count": 8,
"pass": 6,
... | cc-by-4.0 | null | 6 | 2 | 34ac65bcfc35bc3af9496749d104d3e9 | 33,366 | 3 | 1 | 0.3447 |
Evaluate the integral $\int\frac{x^4+1}{x^2\sqrt{x^4-1}}dx$. | \(\sqrt{x^2 - \frac{1}{x^2}} + C\) | \(\sqrt{x^2 - \frac{1}{x^2}} + C\) | false | aops | {
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"count": 8,
"pass": 2,
... | cc-by-4.0 | null | 4 | 2 | b06ad1de4d70c3b2e2591fbf1557908d | 5,171 | 5 | 1 | 0.2113 |
In how many ways can 6 children each choose a single scoop of ice cream from 3 flavors such that exactly 3 children choose the same flavor? | 420 | 420 | false | stackflow | {
"reason_low_with_tool": {
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},
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/4875489/six-children-and-3-flavors-of-ice-cream | 5 | 1 | 6006d32297b9efe433d97756bec0a423 | 269,318 | 3 | 1 | 0.3362 |
In a survey of 100 school students on a snowy and chilly day, the following information was gathered:
- No kid was wearing all three of a scarf, gloves, and a hat.
- 19 out of 100 kids did not wear exactly one of the three accessories.
- At least 67 out of every 100 kids did not wear a scarf.
- At least 83 out of every... | 42 | 42 | false | stackflow | {
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"accuracy": 0.25
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"count": 7,
"pass": 1,
"accuracy": 0.1429
},
"reason_medium_with_tool": {
"count": 8,
"pass": 6,
"accuracy": 0.75
},
"reason_medium_no_tool": {
"count": 8,
"pass": 6,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2588548/percentages-and-inequalities | 5 | 1 | 764fc1f41abca65742307eca3aff64fa | 178,216 | 5 | 2 | 0.396 |
Given that it rained on Monday and Tuesday, what is the probability that it will rain on Thursday? Use the transition probability matrix:
$$P=
\begin{bmatrix}
0.7 & 0 & 0.3 & 0 \\
0.5 & 0 & 0.5 & 0 \\
0 & 0.4 & 0 & 0.6 \\
0 & 0.2 & 0 & 0.8 \\
\end{bmatrix}.
$$ | 0.61 | 0.61 | false | stackflow | {
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"ac... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1261597/transition-probability-matrix | 4 | 1 | d509bf2a690a6d7904f27bb1900187e0 | 133,384 | 5 | 1 | 0.2113 |
Write the permutation $(1,2,4,3)$ as a product of adjacent transpositions, i.e., transpositions of the form $(k \;\; k +1)$. | \((1,2)(3,4)(2,3)\) | \((1,2)(3,4)(2,3)\) | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/3420570/writing-permutations-as-products-of-adjacent-transposition | 6 | 2 | f8db51965689a92e6b66bc06a3c9a024 | 210,357 | 3 | 1 | 0.3447 |
Find all points that have the same distance from the line \( y = -1 \) and the point \( (1,0) \). | \( x^2 = 2(x + y) \) | \( x^2 = 2(x + y) \) | false | stackflow | {
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"... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2117754/finding-all-the-points-the-have-the-same-distance-between-a-line-and-a-point | 5 | 2 | 75d398eeee31b773d299632aab66eefe | 157,303 | 3 | 1 | 0.3362 |
$ABCD$ is a regular tetrahedron with side length 1. Points $X,$ $Y,$ and $Z,$ distinct from $A,$ $B,$ and $C,$ respectively, are drawn such that $BCDX,$ $ACDY,$ and $ABDZ$ are also regular tetrahedra. If the volume of the polyhedron with faces $ABC,$ $XYZ,$ $BXC,$ $XCY,$ $CYA,$ $YAZ,$ $AZB,$ and $ZBX$ can be written as... | 115 | 115 | false | aops | {
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"count": 8,
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"pass": 6,
"accuracy": 0.75
},
"reason_medium_no_tool": {
"count": 8,
"pass": 3,
... | cc-by-4.0 | null | 4 | 1 | b1a56843fc9c9c995cb039c7db5374f2 | 22,309 | 5 | 3 | 0.5756 |
Consider the sequence of numbers $a, a+1, a+2, \cdots, a+n$. Given that only $a$ and $a+n$ have a digit sum divisible by 8, what is the maximum possible value of $n$? | 15 | 15 | false | aops | {
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"count": 8,
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"count": 8,
"pass": 3,
"accuracy": 0.375
},
"reason_medium_no_tool": {
"count": 8,
"pass": 1,
... | cc-by-4.0 | null | 4 | 1 | 99f51d531062f2fd9e36343fa17807b0 | 37,160 | 3 | 1 | 0.3324 |
Suppose that \( f \in \mathbb{F}_p[x] \) is a degree \( d \) random univariate polynomial with coefficients from a finite field \( \mathbb{F}_p \). What is the probability that \( f \) can be written as:
\[ f(x) = \prod_{j=1}^d (x - \alpha_j), \]
for \( \alpha_j \in \mathbb{F}_p \)? | \(\frac{\binom{d+p-1}{d}}{p^d}\) | \(\frac{\binom{d+p-1}{d}}{p^d}\) | false | stackflow | {
"reason_low_with_tool": {
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"count": 8,
"pass": 5,
"accuracy": 0.625
},
"reason_medium_no_tool": {
"count": 8,
"pass": 5,... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1800709/probability-that-a-random-polynomial-over-a-finite-field-can-be-factorized-to-li | 5 | 2 | 5ee2da6c6c1a9a25c45da2812d31fbec | 143,187 | 3 | 1 | 0.3362 |
If a coin rolls without slipping around another coin of the same or different size, how many times will it rotate while making one revolution? | 2 | 2 | false | stackflow | {
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... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/149234/puzzle-a-coin-rolls-without-slipping-around-another-coin | 5 | 1 | c551c4e1760962cce4613bde9c2fe07e | 81,629 | 5 | 2 | 0.396 |
Given a channel \( Y = X + Z \) where \( Z \) is a random variable defined as
\[ Z = \begin{cases}
0, & \text{with probability } \frac{1}{10} \\
Z^*, & \text{with probability } \frac{9}{10}
\end{cases} \]
and \( Z^* \sim \mathcal{N}(0, \sigma^2) \), what is the capacity of the channel under the power constraint \( E[... | \(\infty\) | \(\infty\) | false | stackflow | {
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"pass": 1,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/4141494/calculating-capacity-of-a-channel-with-mixture-distribution | 4 | 2 | fcf0ad0824c8b7d8c585241dbf3ac52c | 242,608 | 3 | 1 | 0.3324 |
Given the recurrence relation \( c_{l+1,t} = c_{l,t+1} - c_{l-1,t+1} \) with initial conditions \( c_{k,t} = 0 \) for all \( k < 0 \) and \( c_{0,t} = \frac{{2t \choose t}}{t+1} \), find the closed form expression for \( c_{l,t} \). | \( c_{l,t} = {2t+l \choose t}\frac{l+1}{t+l+1} \) | \( c_{l,t} = {2t+l \choose t}\frac{l+1}{t+l+1} \) | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_medium_with_tool": {
"count": 8,
"pass": 6,
"accuracy": 0.75
},
"reason_medium_no_tool": {
"count": 8,
"pass": 3,
"... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2104519/solution-to-recurrence | 6 | 2 | 10eb31f81483f4d07673d08e1dac12fa | 224,818 | 3 | 1 | 0.3447 |
A given sequence $r_1, r_2, \dots, r_n$ of distinct real numbers can be put in ascending order by means of one or more "bubble passes". A bubble pass through a given sequence consists of comparing the second term with the first term, and exchanging them if and only if the second term is smaller, then comparing the thir... | 931 | 931 | false | aops | {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_medium_no_tool": {
"count": 8,
"pass": 1,
... | cc-by-4.0 | null | 6 | 1 | 4af136ed08810d63d123352fe0b58e8c | 10,620 | 5 | 2 | 0.4011 |
Given a sequence of pillars with heights $H = [2, 4, 6]$, find the minimum number of operations required to make the sequence beautiful. An operation consists of either adding or removing one stone slab from a pillar. | 2 | 2 | false | aops | {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 6,
"accuracy": 0.75
},
"reason_medium_no_tool": {
"count": 8,
"pass": 2,
"acc... | cc-by-4.0 | null | 6 | 1 | 72ee5e57f15d05c897afda48e51dff83 | 32,809 | 3 | 1 | 0.3447 |
What is the length of the side of the smallest square that will enclose three non-overlapping discs, each with a radius of 1 unit? | \( 2 + \frac{\sqrt{6} + \sqrt{2}}{2} \) | \( 2 + \frac{\sqrt{6} + \sqrt{2}}{2} \) | false | aops | {
"reason_low_with_tool": {
"count": 8,
"pass": 3,
"accuracy": 0.375
},
"reason_low_no_tool": {
"count": 8,
"pass": 3,
"accuracy": 0.375
},
"reason_medium_with_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_medium_no_tool": {
"count": 8,
"pass": 2,
... | cc-by-4.0 | null | 5 | 2 | 7179bea1b053b958f06408150f11a501 | 24,969 | 5 | 2 | 0.396 |
Calculate the expected value of the lowest number when \( N = 2 \). Given a known value \( a \), the expected value is \( a - \frac{a^2}{2} \). Use this to derive the expected value for \( N = 2 \). | \(\frac{1}{3}\) | \(\frac{1}{3}\) | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_no_tool": {
"count": 8,
"pass": 0,
"ac... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/736534/expected-value-of-lowest-x-in-sample-for-0-1-distribution | 5 | 2 | 38d37952e2be8813d3991f0e54097711 | 108,560 | 3 | 1 | 0.3362 |
Suppose the value of \$1 in Japanese yen decreases at 2% per year. Starting from \$1 = Y240, when will $1$ dollar equal $1$ yen? | \( t = \frac{\ln(1/240)}{\ln(0.98)} \) | \( t = \frac{\ln(1/240)}{\ln(0.98)} \) | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_low_no_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_medium_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_medium_no_tool": {
"count": 8,
"pass": 6,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2068276/when-will-1-dollar-equal-1-yen | 6 | 2 | faf11b07b21bf6f4ab3adbefd3ab5f57 | 158,308 | 3 | 1 | 0.3447 |
Solve the partial differential equation \( u_{xx} + u = 0 \). | \( u(x,y) = f(y)\cos(x) + g(y)\sin(x) \) | \( u(x,y) = f(y)\cos(x) + g(y)\sin(x) \) | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 6,
"accuracy": 0.75
},
"reason_low_no_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_medium_with_tool": {
"count": 8,
"pass": 5,
"accuracy": 0.625
},
"reason_medium_no_tool": {
"count": 8,
"pass": 5,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2723706/solve-the-pde-u-xx-u-0 | 4 | 2 | e8d7ccfc3c898d280f23dbc1193e897b | 183,435 | 3 | 1 | 0.3324 |
For which values of $k$ and $q$ (where $q$ is a prime power and $k > 1$) is the number $(q^k-1)/(q-1)$ a perfect square? Is the number of such perfect squares finite? | \( (k, q) = (2, 3), (5, 3), (4, 7), (2, 8) \) | \( (k, q) = (2, 3), (5, 3), (4, 7), (2, 8) \) | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 5,
"accuracy": 0.625
},
"reason_low_no_tool": {
"count": 8,
"pass": 6,
"accuracy": 0.75
},
"reason_medium_with_tool": {
"count": 8,
"pass": 7,
"accuracy": 0.875
},
"reason_medium_no_tool": {
"count": 8,
"pass": 5,
... | cc-by-sa-4.0 | https://mathoverflow.net/questions/177952/when-is-q-k-1-q-1-a-perfect-square | 6 | 2 | ccc8c7145b9f137564fcd16f227b4a59 | 278,669 | 4 | 1 | 0.2683 |
Let $D$ and $E$ be points on $[BC]$ and $[AC]$ of acute $\triangle ABC$, respectively. $AD$ and $BE$ meet at $F$. If $|AF| = |CD| = 2|BF| = 2|CE|$, and $Area(\triangle ABF) = Area(\triangle DEC)$, then what is $Area(\triangle AFC) / Area(\triangle BFC)$? | 4 | 4 | false | aops | {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": 3,
"accuracy": 0.375
},
"reason_medium_no_tool": {
"count": 7,
"pass": 0,... | cc-by-4.0 | null | 4 | 1 | dd6c49c66f7adec5d206dc06c5f9992f | 11,294 | 3 | 1 | 0.3324 |
For \( n = 3 \), calculate the expected number of splits \( E[s_3] \) required to separate any two elements. | 1.33333333333 | 1.33333333333 | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_medium_with_tool": {
"count": 8,
"pass": 5,
"accuracy": 0.625
},
"reason_medium_no_tool": {
"count": 8,
"pass": 5,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/3388518/expected-number-of-paths-required-to-separate-elements-in-a-binary-tree | 6 | 1 | d93e1faa9ba30fd1700d9abee414a21f | 212,631 | 5 | 2 | 0.4011 |
Let \(a, b, c > 0\) such that \(a + b + c = \frac{3}{2}\). Find the minimum value of the function \(F = 27abc + \frac{8}{ab + bc + ca}\). When does the equality occur? | 14 | 14 | false | aops | {
"reason_low_with_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_low_no_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_with_tool": {
"count": 8,
"pass": 5,
"accuracy": 0.625
},
"reason_medium_no_tool": {
"count": 8,
"pass": 2,
... | cc-by-4.0 | null | 4 | 1 | dce0bba4557daefa52afee047ec4d08b | 32,409 | 5 | 3 | 0.5756 |
In the definite integral $\displaystyle\int_0^1\frac{\mathrm dx}{1+x^2}$, why can't the substitution $x=\tan\theta$ with $\begin{cases}x=1\\\theta=\frac{5\pi}4\end{cases}$ be used, and what would the result be if this substitution were used? | \(\frac{\pi}{4}\) | \(\frac{\pi}{4}\) | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_medium_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_medium_no_tool": {
"count": 8,
"pass": 1,
"... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1776202/change-of-limits-in-definite-integration | 4 | 2 | 0efc9704142362c5f4c3e33a8a83b150 | 146,342 | 3 | 1 | 0.3324 |
In how many possible combinations can you add 5 numbers? Given 5 numbers A, B, C, D, and E, determine the number of unique combinations where any of the numbers can be added with any one or more of the other numbers. Note that A+B and B+A are considered the same combination. Provide the formula and steps to determine t... | 26 | 26 | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_medium_with_tool": {
"count": 8,
"pass": 6,
"accuracy": 0.75
},
"reason_medium_no_tool": {
"count": 8,
"pass": 4,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/4331634/in-how-many-possible-combinations-can-you-add-5-numbers | 5 | 1 | 3152b2f1e44e5cac7b6bff3e92524167 | 248,741 | 5 | 1 | 0.2135 |
Consider three storage facilities $s_1, s_2, s_3$ and two materials $m_1, m_2$. The storage proportions are as follows: $p_{s_1, m_1} = 0.5$, $p_{s_2, m_1} = 0.9$, and $p_{s_3, m_2} = 0.6$. Determine the minimum number of warehouses required to store all materials while covering all storage facilities. | 3 | 3 | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_medium_with_tool": {
"count": 8,
"pass": 5,
"accuracy": 0.625
},
"reason_medium_no_tool": {
"count": 8,
"pass": 6,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/3969508/problem-on-a-bipartite-graph-of-materials-and-storage-facilities | 5 | 1 | d2c6ed4f4fab09df0cc2e0c59358f80d | 230,896 | 3 | 1 | 0.3362 |
Rice University writes 10 questions every hour with a 10% mistake rate in their solutions, while Stanford University writes 20 questions every hour with a 20% mistake rate in their solutions. Each school works for 10 hours. Smartie checks the problems and correctly identifies 75% of the actually incorrect solutions. Sm... | \(\frac{1}{25}\) | \(\frac{1}{25}\) | false | aops | {
"reason_low_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_low_no_tool": {
"count": 8,
"pass": 3,
"accuracy": 0.375
},
"reason_medium_with_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_medium_no_tool": {
"count": 8,
"pass": 3,
... | cc-by-4.0 | null | 4 | 2 | 3d55ac3988712d4713edcbe492e60a9b | 14,570 | 5 | 1 | 0.2113 |
Calculate the determinant of a general matrix of order \( n \times n \) that looks like this:
\[
\left(
\begin{matrix}
a+b & b & \cdots & \cdots & b \\
a & a+b & b & \cdots & \vdots \\
\vdots & a & \ddots & \ddots & \vdots \\
a & \ddots & \ddots & \ddots & b \\
a & a & \cdots & a & a+b \\
\end{matrix}
\right)
\]
There ... | \(\frac{a^{n+1} - b^{n+1}}{a - b}\) | \(\frac{a^{n+1} - b^{n+1}}{a - b}\) | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_low_no_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_medium_with_tool": {
"count": 8,
"pass": 3,
"accuracy": 0.375
},
"reason_medium_no_tool": {
"count": 8,
"pass": 6,
... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/4605123/finding-the-determinant-of-a-special-matrix | 6 | 2 | c4ab7b604243fae71e8991239d7a7e22 | 261,123 | 5 | 1 | 0.2196 |
How many ways can 5 red balls, 6 white balls, and 7 black balls be arranged in a line such that no two adjacent balls are of the same color? Generalize this problem for R red balls, W white balls, and B black balls. | 31632 | 31632 | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 4,
"accuracy": 0.5
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 6,
"accuracy": 0.75
},
"reason_medium_no_tool": {
"count": 8,
"pass": 0,
"a... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/2921641/permutation-of-multiple-sets-of-elements-where-adjacent-elements-are-different | 5 | 1 | 21e412172186fcb603c2af00ea887405 | 189,445 | 4 | 1 | 0.2612 |
Let $\mathbb{F}_p$ be the set of integers modulo $p$. A function $f : \mathbb{F}_p^2 \to \mathbb{F}_p$ is called [i]quasiperiodic[/i] if there exist $a,b \in \mathbb{F}_p$, not both zero, such that $f(x + a, y + b) = f(x, y)$ for all $x,y \in \mathbb{F}_p$. Find the number of functions $\mathbb{F}_p^2 \to \mathbb{F}_p$... | \( p^{\frac{p(p+1)}{2}} \) | \( p^{\frac{p(p+1)}{2}} \) | false | aops | {
"reason_low_with_tool": {
"count": 8,
"pass": 3,
"accuracy": 0.375
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 7,
"accuracy": 0.875
},
"reason_medium_no_tool": {
"count": 8,
"pass": 0,
... | cc-by-4.0 | null | 4 | 2 | 105f33a035b02037834714a879e78436 | 7,135 | 3 | 1 | 0.3324 |
Find the digits \( A, B, C \) such that \( ABC + BAC + CAB = ABBC \), where \( A, B, C \) are distinct digits. Find the value of \( A + B + C \). | 18 | 18 | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_medium_no_tool": {
"count": 8,
"pass": 1,
"acc... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/588228/find-the-digits-a-b-c-such-that-abc-bac-cab-abbc | 4 | 1 | 494df58b6e9afbeacb6877f2935fb013 | 100,579 | 5 | 1 | 0.2113 |
Find the nth term of the sequence \(\{x_n\}\) such that
\[ x_{n+1} = x_n(2 - x_n). \] | \( x_n = 1 - (x_0 - 1)^{2^n} \) | \( x_n = 1 - (x_0 - 1)^{2^n} \) | false | aops | {
"reason_low_with_tool": {
"count": 8,
"pass": 6,
"accuracy": 0.75
},
"reason_low_no_tool": {
"count": 8,
"pass": 5,
"accuracy": 0.625
},
"reason_medium_with_tool": {
"count": 8,
"pass": 6,
"accuracy": 0.75
},
"reason_medium_no_tool": {
"count": 8,
"pass": 5,
... | cc-by-4.0 | null | 4 | 2 | 1f796d603bb691a80cb0732bb9583c5c | 14,493 | 5 | 1 | 0.2113 |
Evaluate the improper integral \( I_{13} = \int_{0}^{\infty} \left( \frac{1}{x^{3} + 1} \right) \left( \frac{1}{x^{4} + 1} \right) \, dx \). | \(\frac{2\pi}{9} \sqrt{3} - \frac{\pi}{8}\) | \(\frac{2\pi}{9} \sqrt{3} - \frac{\pi}{8}\) | false | aops | {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 8,
"pass": 0,
"accuracy": 0
},
"reason_medium_with_tool": {
"count": 8,
"pass": 8,
"accuracy": 1
},
"reason_medium_no_tool": {
"count": 8,
"pass": 1,
"acc... | cc-by-4.0 | null | 4 | 2 | a0de273caf30e532a11c5fc3fa44f088 | 21,695 | 3 | 1 | 0.3324 |
What is the optimal solution for maximizing \(X_1 + 2X_2 + 3X_3\) subject to the constraints:
1. \(X_1 + X_2 + X_3 \leq 9\),
2. \(-X_1 + 2X_2 + 5X_3 \leq 15\),
3. \(X_1 \geq 0\),
4. \(X_2 \geq 0\)? | 17 | 17 | false | stackflow | {
"reason_low_with_tool": {
"count": 8,
"pass": 2,
"accuracy": 0.25
},
"reason_low_no_tool": {
"count": 7,
"pass": 4,
"accuracy": 0.5714
},
"reason_medium_with_tool": {
"count": 8,
"pass": 1,
"accuracy": 0.125
},
"reason_medium_no_tool": {
"count": 8,
"pass": 6,... | cc-by-sa-4.0 | https://math.stackexchange.com/questions/1735456/linear-programming | 5 | 1 | c3b4fe9f7c9aad186ce6c07e59f93581 | 111,218 | 5 | 1 | 0.2135 |
End of preview. Expand in Data Studio
Nemotron-Math-v2-NoTool-Hard1100
1,100 hard math problems for no-tool reasoning RL with
nvidia/NVIDIA-Nemotron-3.5-Lightning-30B-A3B
as the policy. Every problem was solved by that model at least once but at a low rate:
its estimated single-sample pass rate over the set is about 33%.
Problems come from the train split of
JWei05/Nemotron-Math-v2-NoTool-B1-6,
pass buckets 4–6 only. That dataset's filtering, deduplication and columns apply unchanged; no
problem from its validation split is included.
| split | problems | bucket 4 | bucket 5 | bucket 6 | estimated pass rate |
|---|---|---|---|---|---|
| train | 1,000 | 367 | 327 | 306 | 0.326 |
| validation | 100 | 41 | 28 | 31 | 0.326 |
The split is a uniform random shuffle (seed 0) of the 1,100 selected problems.
How the problems were selected
- Pool. 7,200 problems: 2,400 drawn uniformly at random (seed 0) from each of pass buckets
4, 5 and 6 of the source
trainsplit. - Mixed screen. Each problem was sampled 2 at a time, fewest-sampled first, and stopped as soon as it had both a correct and an incorrect answer (at most 8 samples). Screening ended once 2,001 such mixed problems were found.
- Fresh sample. Every one of the 2,001 got one more independent sample (overall accuracy 48.1% on those fresh samples).
- Ranking. For each bucket, a Beta prior over per-problem pass rate was fitted by marginal likelihood on all 7,200 screened problems (the adaptive stopping rule depends only on observed outcomes, so it does not bias the fit). Each problem's pass rate was estimated as its posterior mean given all of its samples, and the 1,100 problems with the lowest estimate were kept.
Held-out check of the ranking: ranking on the screen samples alone and scoring the selected problems on the fresh sample from step 3 matched the predicted pass rate within noise (hardest 1,100: predicted 0.393, measured 0.391 ± 0.030). The 0.326 above uses all samples for ranking, so it has no held-out measurement; the per-problem estimates are noisy because most problems have 3–5 samples.
Sampling setup
- Model:
nvidia/NVIDIA-Nemotron-3.5-Lightning-30B-A3B, NVFP4 checkpoint (-NVFP4) on vLLM 0.31 for about 83% of all screening samples and BF16 on vLLM 0.20.2 for the rest (accuracy matched). - Single agent, one turn, 64K-token context, temperature 1.0, top-p 1.0, default chat template.
- System prompt:
Solve the math problem. Reason step by step, then put your final answer in \boxed{}. - Grading: last complete
\boxed{}answer againstexpected_answerwith the NeMo-compatiblemath_equalgrader.
Columns
All columns of JWei05/Nemotron-Math-v2-NoTool-B1-6 (problem, expected_answer,
original_expected_answer, changed_answer_to_majority, data_source, metadata, license,
url, pass_bucket, answer_tier, question_md5, scan_index), plus:
nemotron_n_samples: number of Nemotron-3.5-Lightning samples drawn for the problem (3–9).nemotron_n_correct: how many of them were correct (at least 1).nemotron_pass_estimate: posterior-mean pass rate used for selection.
The selection used these same samples, so nemotron_n_correct / nemotron_n_samples is biased
low (0.317 over the set). Use nemotron_pass_estimate for the expected pass rate.
License
Problems inherit the licenses of nvidia/Nemotron-Math-v2: 809 rows are CC BY-SA 4.0
(stackflow) and 291 are CC BY 4.0 (aops), as recorded per row in license.
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