Datasets:

toloka
/

u-math

uuid stringlengths 36 36	subject stringclasses 6 values	has_image bool 2 classes	image stringclasses 160 values	problem_statement stringlengths 32 784	golden_answer stringlengths 7 1.13k
00143e2e-2d9f-4fc3-a9c5-9d2e6aba5cba	multivariable_calculus	false	null	Determine the coordinates, if any, for which $f(x,y) = 6 \cdot x^2 - 3 \cdot x^2 \cdot y + y^3 + 12$ has 1. a Relative Minimum(s) 2. a Relative Maximum(s) 3. a Saddle Point(s) If a Relative Minimum or Maximum, find the Minimum or Maximum value. If none, enter None.	1. The function $f(x,y)$ has Relative Minimum(s) at None with the value(s) None 2. The function $f(x,y)$ has Relative Maximum(s) at None with the value(s) None 3. The function $f(x,y)$ has a Saddle Point(s) at $P(-2,2)$, $P(2,2)$
00282b7b-1bf6-415d-9f17-d04e9c9700e2	precalculus_review	true	data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAdAAAAGoCAYAAAD2LLSsAAAYU2lDQ1BJQ0MgUHJvZmlsZQAAeJyVeQVUVF8X77mTzDAM3d0l3SAxdHeDwNAdQ4NKioSKIKCUCioIIliEiIUgooigAgYiYVAqqKAIyLuEfv/v/6313npn1rn3N/vss+PsU3sGAM5UcmRkKIIOgLDwGIqtkS6fs4srH/YdgAAO/ggBEbJPdCTJ2tocwOXP+7/L8jDMDZdnUpuy/rf9/1roff2ifQCArGHs7RvtEwbjawCgMn0iKTEAYFRhum...	Estimate the average rate of change from $x=2$ to $x=6$ for the following function $f(x)$ whose graph is given below: Note that the average rate of change of a function from $x=a$ to $x=b$ is $\frac{ f(b)-f(a) }{ b-a }$.	The final answer: $0.5$
0075433c-0afb-41a9-a18a-986d8a81a653	multivariable_calculus	false	null	Evaluate the integral by choosing the order of integration: $$ \int_{0}^1 \int_{1}^2 \left(\frac{ y }{ x+y^2 }\right) \, dy \, dx $$	$\int_{0}^1 \int_{1}^2 \left(\frac{ y }{ x+y^2 }\right) \, dy \, dx$ = $\ln\left(\frac{25\cdot\sqrt{5}}{32}\right)$
00f6affb-905a-4109-a78e-2dde7a0b83ac	integral_calc	false	null	Solve the integral: $$ \int \frac{ 1 }{ \sin(x)^7 \cdot \cos(x) } \, dx $$	$\int \frac{ 1 }{ \sin(x)^7 \cdot \cos(x) } \, dx$ = $C+\ln\left(\left\|\tan(x)\right\|\right)-\frac{3}{2\cdot\left(\tan(x)\right)^2}-\frac{3}{4\cdot\left(\tan(x)\right)^4}-\frac{1}{6\cdot\left(\tan(x)\right)^6}$
0118dbbc-db0e-4ab9-a2ed-ddb2edd0eefc	sequences_series	false	null	Find the Fourier series of the periodic function $f(x) = x^2$ in the interval $-2 \cdot \pi \leq x < 2 \cdot \pi$ if $f(x) = f(x + 4 \cdot \pi)$.	The Fourier series is: $\frac{4\cdot\pi^2}{3}+\sum_{n=1}^\infty\left(\frac{16\cdot(-1)^n}{n^2}\cdot\cos\left(\frac{n\cdot x}{2}\right)\right)$
0146d63a-d9e9-4910-9267-10f87812aff4	multivariable_calculus	false	null	If $z = x \cdot y \cdot e^{\frac{ x }{ y }}$, $x = r \cdot \cos\left(\theta\right)$, $y = r \cdot \sin\left(\theta\right)$, find $\frac{ d z }{d r}$ and $\frac{ d z }{d \theta}$ when $r = 2$ and $\theta = \frac{ \pi }{ 6 }$.	The final answer: $\frac{ d z }{d r}$: $\sqrt{3}\cdot e^{\left(\sqrt{3}\right)}$ $\frac{ d z }{d \theta}$: $2\cdot e^{\left(\sqrt{3}\right)}-4\cdot\sqrt{3}\cdot e^{\left(\sqrt{3}\right)}$
014e9af5-759a-4711-aa22-0f9c76acb502	sequences_series	false	null	Consider the function $y = \left\| \cos\left( \frac{ x }{ 8 } \right) \right\|$. 1. Find the Fourier series of the function. 2. Using this decomposition, calculate the sum of the series $\sum_{n=1}^\infty \frac{ (-1)^n }{ 4 \cdot n^2 - 1 }$. 3. Using this decomposition, calculate the sum of the series $\sum_{n=1}^\infty...	1. The Fourier series is $\frac{2}{\pi}-\frac{4}{\pi}\cdot\sum_{n=1}^\infty\left(\frac{(-1)^n}{\left(4\cdot n^2-1\right)}\cdot\cos\left(\frac{n\cdot x}{4}\right)\right)$ 2. The sum of the series $\sum_{n=1}^\infty \frac{ (-1)^n }{ 4 \cdot n^2 - 1 }$ is $\frac{(2-\pi)}{4}$ 3. The sum of the series $\sum_{n=1}^\infty \fr...
01683c2c-a5b7-4bff-8e4f-d8fdad3cac45	differential_calc	false	null	For the function $r = \arctan\left(\frac{ m }{ \varphi }\right) + \arccot\left(m \cdot \cot(\varphi)\right)$, find the derivative $r'(0)$ and $r'(2 \cdot \pi)$. Submit as your final answer: 1. $r'(0)$ 2. $r'(2 \cdot \pi)$	1. $0$ 2. $\frac{1}{m}-\frac{m}{m^2+4\cdot\pi^2}$
01961276-06fd-4869-8e58-eb15a4eb4034	algebra	false	null	Rewrite the quadratic expression $x^2 + \frac{ 2 }{ 3 } \cdot x - \frac{ 1 }{ 3 }$ by completing the square.	$x^2 + \frac{ 2 }{ 3 } \cdot x - \frac{ 1 }{ 3 }$ = $\left(x+\frac{1}{3}\right)^2-\frac{4}{9}$
01c011c6-2528-46de-aa90-9a86ace5747a	sequences_series	false	null	Using the Taylor formula, decompose the function $f(x) = \ln(1+4 \cdot x)$ in powers of the variable $x$ on the segment $[0,1]$. Use the first nine terms. Then estimate the accuracy obtained by dropping an additional term after the first nine terms.	1. $\ln(1+4 \cdot x)$ = $4\cdot x-\frac{(4\cdot x)^2}{2}+\frac{(4\cdot x)^3}{3}-\frac{(4\cdot x)^4}{4}+\frac{(4\cdot x)^5}{5}-\frac{(4\cdot x)^6}{6}+\frac{(4\cdot x)^7}{7}-\frac{(4\cdot x)^8}{8}+\frac{(4\cdot x)^9}{9}$ 2. Accuracy is not more than $\frac{4^{10}}{10}$
01e4ed04-1ed3-4cda-af81-35686c0a16d0	differential_calc	false	null	Compute the limit: $$ \lim_{x \to 0}\left(\frac{ x-x \cdot \cos(x) }{ x-\sin(x) }\right) $$	$\lim_{x \to 0}\left(\frac{ x-x \cdot \cos(x) }{ x-\sin(x) }\right)$ = $3$
023674a7-236e-4771-9662-77157e370160	differential_calc	true	data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAASQAAAEYCAYAAAD1QYs6AABm4UlEQVR4nO39Z5AcV3anjT83s3x1ufbewXuCBAGCQ9AOSZAgh0Mzw1caiRppTcTGKnZjN2L1YT+8Efqwqw3F7sZfG1LEjEbi6F2NHw2HHA4tSNABhCU8YRvdQHtvqrury2Xd/4eszDLoBhpAo13lw2iiuyoz61bmzV+ee+455woppcTCwsJiEaAsdAMsFh/5z6jbfWYZ+1nPPIvZYgmShYkhHEKInNeFELckUtnHkVJedzwLi5mwBMnCxB...	Determine where the local and absolute maxima and minima occur on the graph given. Assume domains are closed intervals unless otherwise specified.	Absolute minimum at: $x=3$ Absolute maximum at: $x=-2.4$ Local minimum at: $x=-2$, $x=1$ Local maximum at: $-1, 2$
0296afc7-857a-46e2-8497-63653d268ac8	differential_calc	false	null	Compute the derivative of the function $y = \sqrt{\frac{ x^5 \cdot \left(2 \cdot x^6+3\right) }{ \sqrt[3]{1-2 \cdot x} }}$ by taking the natural log of both sides of the equation.	Derivative: $y'=\frac{-128\cdot x^7+66\cdot x^6-84\cdot x+45}{-24\cdot x^8+12\cdot x^7-36\cdot x^2+18\cdot x}\cdot\sqrt{\frac{x^5\cdot\left(2\cdot x^6+3\right)}{\sqrt[3]{1-2\cdot x}}}$
029deb6d-6866-4e2d-9d92-3a555ec4de2c	algebra	false	null	1. On a map, the scale is 1/2 inch : 25 miles. What is the actual distance between two cities that are 3 inches apart? 2. What are the dimensions of a 15 foot by 10 foot room on a blueprint with a scale of 1.5 inches : 2 feet? 3. The distance between Newark, NJ and San Francisco, CA is 2,888 miles. How far would that...	1. $150$ miles 2. $11.25$ feet by $7.5$ feet 3. $8.664$ cm 4. $8.3$ inches
02a70481-c574-417a-bf03-50e5080f4d19	differential_calc	true	data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAT0AAAFlCAIAAAAWJKhQAABqgklEQVR4nO39Z5Acx5nnAWeWr+qu9nZmeryF956gAyk6iKKk5VFakZI2dlfa29uL996Ni/twEaeI/XBfLmIvbmMvNvb2vePuSjxJJ1KURNFDoAFJEADhzQCYwXjT3tvqqsr3Q3bX9BgYAgNO90z+JA56aqqrq6rzX8+TTz75JEQIAUJjghCCEN5i45I7EFYBkOiWQGg4mJU+AcJtQAiVy2VVVSmK4jiOoigAgKZppVIJAMBxHE3TtUZV13...	List all intervals where $f$ is increasing or decreasing and the minima and maxima are located.	The function is increasing at: $(-1,0)\cup(0,1)\cup(1,\infty)$ The function is decreasing at: $(-\infty,-1)$ The minimum is at: $x=-1$ The maximum is at: None
02c57487-7f56-412a-8aeb-b1abc78ac85b	sequences_series	false	null	Find the radius of convergence $R$ and interval of convergence for the power series $\sum_{n=0}^\infty a_{n} \cdot x^n$: $$ \sum_{n=1}^\infty (-1)^n \cdot \frac{ x^n }{ \ln(2 \cdot n) } $$	* $R$ = $1$ * $I$ = $(-1,1]$
02d22949-d7a2-46ca-8af8-4bda85bc8e0e	differential_calc	false	null	Sketch the curve: $$ y = 16 \cdot x^2 \cdot e^{\frac{ 1 }{ 4 \cdot x }} $$ Submit as your final answer: 1. The domain (in interval notation) 2. Vertical asymptotes 3. Horizontal asymptotes 4. Slant asymptotes 5. Intervals where the function is increasing 6. Intervals where the function is decreasing 7. Intervals whe...	1. The domain (in interval notation): $(-\infty,0)\cup(0,\infty)$ 2. Vertical asymptotes: $x=0$ 3. Horizontal asymptotes: None 4. Slant asymptotes: None 5. Intervals where the function is increasing: $\left(\frac{1}{8},\infty\right)$ 6. Intervals where the function is decreasing: $\left(0,\frac{1}{8}\right)$, $(-\infty...
02d80ab4-c222-493c-a91d-448fba115380	multivariable_calculus	false	null	Evaluate $L=\lim_{P(x,y) \to P\left(1,\frac{ 1 }{ 2 }\right)}\left(f(x,y)\right)$ given $f(x,y) = \frac{ x^2 - 2 \cdot x^3 \cdot y - 2 \cdot x \cdot y^3 + y^2 }{ 1 + x + y - 2 \cdot x \cdot y - 2 \cdot x^2 \cdot y - 2 \cdot x \cdot y^2 }$.	The final answer: $L=\frac{1}{2}$
032cf885-c1e2-49f9-8584-c99b0161c08c	precalculus_review	false	null	Form the compositions $f\left(g(x)\right)$ and $g\left(f(x)\right)$ if $f(x) = \sin(3 \cdot x)$ and $g(x) = \frac{ 1 }{ \sqrt{2-x^2} }$. 1. Find $f\left(g(x)\right)$ and $g\left(f(x)\right)$. 2. Find the domain and range of $f\left(g(x)\right)$ and $g\left(f(x)\right)$.	1. $f\left(g(x)\right)$ = $\sin\left(\frac{3}{\sqrt{2-x^2}}\right)$ $g\left(f(x)\right)$ = $\frac{1}{\sqrt{2-\sin(3\cdot x)^2}}$ 2. Domain of $f\left(g(x)\right)$ is $\left(-\sqrt{2},\sqrt{2}\right)$ Range of $f\left(g(x)\right)$ is $[-1,1]$ Domain of $g\left(f(x)\right)$ is $(-\infty,\infty)$ Range of $g\l...
03b87419-fe7c-481a-9396-1d0723fc2b15	differential_calc	false	null	Sketch the curve: $y = 5 \cdot x \cdot \sqrt{4-x^2}$. Submit as your final answer: 1. The domain (in interval notation) 2. Vertical asymptotes 3. Horizontal asymptotes 4. Slant asymptotes 5. Intervals where the function is increasing 6. Intervals where the function is decreasing 7. Intervals where the function is co...	1. The domain (in interval notation): $[-2,2]$ 2. Vertical asymptotes: None 3. Horizontal asymptotes: None 4. Slant asymptotes: None 5. Intervals where the function is increasing: $\left(-\sqrt{2},\sqrt{2}\right)$ 6. Intervals where the function is decreasing: $\left(-2,-\sqrt{2}\right)$, $\left(\sqrt{2},2\right)$ 7. I...
03c28ad4-cfe4-430f-ab38-cb3e56091616	precalculus_review	false	null	Solve the following equation: $$ x^2 - x + 1 = \frac{ 1 }{ 2 } + \sqrt{x - \frac{ 3 }{ 4 }} $$	The final answer: $x=1$
040dbb94-2747-4799-89f8-dd544c248a9c	integral_calc	false	null	Consider the function $f(x) = x^2$ on $[-1,1]$ and the partition $\left\{-1, -\frac{ 1 }{ 2 }, \frac{ 1 }{ 4 }, 1\right\}$. Find the upper and lower sums.	The upper sum is: $\frac{23}{16}$ The lower sum is: $\frac{11}{64}$
0429a27b-d694-4e42-a60c-d446ae515ed2	differential_calc	true	data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAyAAAAMgCAYAAADbcAZoAABQIUlEQVR4nO3dfXDd1Zkf8EfBJhDDVYDFWbNXIQnZNZEEnkY7IRIQ0m5YIDNJs0PG7kxCPBCc7dIkDfVOt2leIKbTNjQE6CTsECekeduModl0mUljN25jCEhmi7drVsIlG0KEtLCGBqKLzZth3T88UpB/R7Ze7r2/l/v5zDADx7L0HF1J/L46zzmn6+DBgwejhL7yla9ERMRHPvKRnCtprV27dkVExMDAQM6VtFanzvO0007LvM1jjz3W1p...	The graph of $g'$ is given. Let $g$ be a differentiable function with $g(1)=-4$. The graph of $g'(x)$, the derivative of $g$, is shown. Write an equation for the line tangent to the graph of $g$ at $x=1$.	The equation for the tangent line is $y+4=-3\cdot(x-1)$
04960c86-a731-4641-a3a8-fd0529de5a51	multivariable_calculus	false	null	Evaluate $\int\int\int_{E}{(x+2 \cdot y \cdot z) \, dV}$, where $E = \left\{(x,y,z) \| 0 \le x \le 1, 0 \le y \le x, 0 \le z \le 5-x-y \right\}$.	$I$ = $\frac{439}{120}$

End of preview. Expand in Data Studio

U-MATH is a comprehensive benchmark of 1,100 unpublished university-level problems sourced from real teaching materials.

It is designed to evaluate the mathematical reasoning capabilities of Large Language Models (LLMs).
The dataset is balanced across six core mathematical topics and includes 20% of multimodal problems (involving visual elements such as graphs and diagrams).

For fine-grained performance evaluation results and detailed discussion, check out our paper.

Key Features

Topics Covered: Precalculus, Algebra, Differential Calculus, Integral Calculus, Multivariable Calculus, Sequences & Series.
Problem Format: Free-form answer with LLM judgement
Evaluation Metrics: Accuracy; splits by subject and text-only vs multimodal problem type.
Curation: Original problems composed by math professors and used in university curricula, samples validated by math experts at Toloka AI, Gradarius

Use it

from datasets import load_dataset
ds = load_dataset('toloka/u-math', split='test')

Dataset Fields

uuid: problem id
has_image: a boolean flag on whether the problem is multimodal or not
image: binary data encoding the accompanying image, empty for text-only problems
subject: subject tag marking the topic that the problem belongs to
problem_statement: problem formulation, written in natural language
golden_answer: a correct solution for the problem, written in natural language \

For meta-evaluation (evaluating the quality of LLM judges), refer to the µ-MATH dataset.

Evaluation Results

The prompt used for inference:

{problem_statement}
Please reason step by step, and put your final answer within \boxed{}

Licensing Information

All the dataset contents are available under the MIT license.

Citation

If you use U-MATH or μ-MATH in your research, please cite the paper:

@inproceedings{umath2024,
title={U-MATH: A University-Level Benchmark for Evaluating Mathematical Skills in LLMs},
author={Konstantin Chernyshev, Vitaliy Polshkov, Ekaterina Artemova, Alex Myasnikov, Vlad Stepanov, Alexei Miasnikov and Sergei Tilga},
year={2024}
}