text large_stringlengths 10 2k | source large_stringclasses 20
values | category large_stringclasses 9
values | subcategory large_stringclasses 16
values | split large_stringclasses 38
values | has_reasoning bool 2
classes | num_turns int64 0 377 | has_tools bool 2
classes | num_tools int64 0 4 | license large_stringclasses 13
values | data_source large_stringclasses 4
values |
|---|---|---|---|---|---|---|---|---|---|---|
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
In how many ways can we put 10 indistinguishable balls into 5 boxes?
a. The boxes are labeled and in order.
b. The boxes are unlabeled and their order doesn't matter. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Compute the area of a 12-gon with alternating side lengths of $1$ and $2$, given that it is equiangular. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let $\omega_A, \omega_B, \omega_C$ be three concentric circles with radii $2, 3,$ and $7,$ respectively. A point $P$ in the plane is called "nice" if there exist points $A, B,$ and $C$ on $\omega_A, \omega_B,$ and $\omega_... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
We want to place 2012 pockets, each containing variously colored balls, into \( k \) boxes such that for any box, either all pockets in the box must include a ball with the same color, or all pockets in the box must includ... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the equation $3^x = 2^y \cdot 5^z + 1$ in natural numbers. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
In how many ways can twelve loaves of bread be given away to 5 food banks? What is the number of ways if each food bank should get at least one loaf of bread? | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
How many non-overlapping $1 \times \frac{\sqrt{6}-\sqrt{2}}{4} \times \frac{\sqrt{6}+\sqrt{2}}{4}$ triangles can fit in a circle of radius $9$? | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find the value of \(\frac{1}{2012-\alpha} + \frac{1}{2012-\alpha^2} + \cdots + \frac{1}{2012-\alpha^{n-1}}\), where \(\alpha = e^{\frac{2\pi i}{n}}\). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find all positive integers $n$ such that there exist positive primes $p$ and $q$ with $p + 2 = q$, for which both $2^n + p$ and $2^n + q$ are primes. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Determine the maximum and minimum value of \( B = x(x + 2y - 2) \) where \( x^2 + 4y^2 - 8y = 0 \) and \( x, y \in \mathbb{R} \). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given \(a, b > 0\) such that \(\sin (2 - 2ab) - \sin (a + b) = 2ab + a + b - 2\), find the minimum value of \(S = a + 2b\). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
An ant on a hexagon has a $\frac{1}{2}$ chance of moving to the vertex to its left and a $\frac{1}{2}$ chance of moving to the vertex to its right. What is the probability that the ant will move from a vertex to the opposi... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the equation in $\mathbb{R}$:
\[ 5^{\log _2 (x^2 )} - 3^{2\log _4 \left(\frac{{x^2 }}{2}\right)} = \sqrt{3^{\log _{\sqrt 2 } \left((2x)^2 \right)}} - 5^{\log _2 \left(x^2 - 1\right)} \] | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
The lateral edge of a right prism is equal to $a$. Its base is a right triangle whose smaller angle is equal to $\alpha $. A section is drawn through the smaller leg of the base and the midpoint of the opposite lateral edg... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let \( a \) be a real number and define the sets \( A = \{(x, y) \in \mathbb{R} \times \mathbb{R} \mid x + y = a\} \) and \( B = \{(x, y) \in \mathbb{R} \times \mathbb{R} \mid x^3 + y^3 < a\} \). Find all values of \( a \)... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Two $5 \times 1$ rectangles have 2 vertices in common. Determine the area of overlap. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given a regular tetrahedron $SABC$ with edge $a$. Through the vertices of the base $ABC$ of the tetrahedron three planes are drawn each of which passes through the midpoints of two lateral edges. Find the volume of the por... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find the equation for the line normal to \( f(x) = -\frac{1}{x-3} \) and parallel to \( -9x - y - 1 = 0 \). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
On the Cartesian plane, consider the set \( V \) of all vectors with integer coordinates. Determine all functions \( f : V \rightarrow \mathbb{R} \) satisfying the conditions:
(i) \( f(v) = 1 \) for each of the four vector... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Two cows play a game on an octahedron, starting with their pieces on opposite vertices. They take turns moving their pieces to adjacent vertices. The first cow to move its piece to the vertex occupied by the opponent's pie... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Determine all pairs \((m, n)\) of positive integers that satisfy the equation \(m! + n! = m^n\). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
On each side of an equilateral triangle with side length \( n \) units, where \( n \) is an integer and \( 1 \leq n \leq 100 \), consider \( n-1 \) points that divide the side into \( n \) equal segments. Through these poi... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given positive real numbers \(a\), \(b\), and \(c\), under what conditions does the inequality \(a + b + c \geq abc\) hold? | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
A primary water tank is a sphere with a 4.5 metre radius. It needs to be filled so that the water is exactly 6.5 metres deep at the center. If one bucket of water holds exactly 10 litres, how many bucketfuls of water are r... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given the function \( f(x) = \frac{a}{x^2+1} + \frac{b}{x^2+2} + \frac{c}{x^2+3} - \frac{1}{x^2} \) with the conditions \( f(1) = f(2) = f(3) = 0 \), and if \( \gcd(m, n) = 1 \) and \( f(4) = \frac{m}{n} \), find \( m \). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Evaluate the definite integral $\int_{1}^{\sqrt{2}} \frac{1}{1+\sqrt{x^{2}-1}}\ dx$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve in reals:
\[ x^2 + 7x = (2x + 1)\sqrt{x^2 + x + 6} \] | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find the largest integer \( k \) such that
\[ f(x) \geq k \]
for all real numbers \( x \), where
\[ f(x) = \frac{2e^{100x} - 9900x e^x - 9900x - 2}{2e^x - 2}. \] | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find all real pairs $(x,y)$ that solve the system of equations \begin{align} x^5 &= 21x^3 + y^3 \\ y^5 &= x^3 + 21y^3. \end{align} | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the following equations. $7\left(x+\frac{1}{x}\right)-2\left(x^2+\frac{1}{x^2}\right)=9$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
A cone is inscribed in a sphere whose surface area is $S$. The angle between the generatrix of the cone and the plane of the base is $\alpha $. Find the area of the total surface of the cone. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Common tangents of three circles meet at point \( A \). The radius of the largest circle is 18. Points \( B \) and \( C \) are the points of contact of the tangent with the smallest circle. Given that \( \angle A = 60^\cir... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the following equations applying the methods presented above. $\tan x+2\cot 2x=\sin x\left(1+\tan x\tan \frac{x}{2}\right)$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
A balloon initially moving at 5 m/sec has a total mass of 210 kg. After a body mass of 70 kg falls from the balloon, find the distance the balloon travels in 5 seconds after the body falls. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the equation \(x^4 + y^2 = xy^2 + y\). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Denote by $A_1, B_1, C_1$ the midpoints of the corresponding sides of the acute triangle $ABC$, which has an area of $S$. The perpendicular $l$ on the side $AB$ is drawn from the point $B_1$, and the perpendicular $k$ on t... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Determine the vertex, x-intercept, and y-intercept of the quadratic function \( h(x) = -(x+2)(x+6) \). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the following equations applying the methods presented above. $\cos x-\cos 17x=1+2\sin 8x\sin x-\cos 16x$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given 10 straight lines on a plane, each parallel to one of the coordinate axes, what is the largest number of distinct squares that can be formed, where each square's vertices lie at the intersection of some two of those ... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the trigonometric equation:
\[
\frac{\sqrt{225\tan^{4}\frac{x}{2}+706\tan^{2}\frac{x}{2}+225}-34\tan\frac{x}{2}}{15(\tan^{2}\frac{x}{2}-1)}=\frac{17-2\sqrt{91}}{5\sqrt{3}}
\] | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
A trapezoid $ABCD$ is inscribed into a semi-circle of radius $l$ so that the base $AD$ of the trapezoid is a diameter and the vertices $B$ and $C$ lie on the circumference. Find the base angle $\varphi $ of the trapezoid $... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find all \( x \) in the interval \(\left[0, \frac{\pi}{2}\right]\) such that \(\sin x\), \(\cos x\), and \(\tan x\) form a geometric progression in any order. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
In a game, a participant chooses a nine-digit positive integer $\overline{ABCDEFGHI}$ with distinct non-zero digits. The score of the participant is $A^{B^{C^{D^{E^{F^{G^{H^{I}}}}}}}}$. Which nine-digit number should be ch... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find the equation for the graph passing through the points A(0,4), B(6,0), C(-4,0), and D(-6,0). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given a convex quadrilateral with $AB=AD=\sqrt{5}$, $CB=CD=2$, $\tan \angle DAB = -2$, and $\angle BCD = 2\alpha$ where $0 < 2\alpha < \pi$, find all integers $n$ such that $\cos n\alpha = \cos \alpha$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the following systems of equations: $\left\{\begin{array}{lll} \frac{x+y}{x-y}+\frac{x-y}{x+y}=5\frac{1}{5}\medskip \\ xy=6 \end{array}\right.$ | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Pete made a calendar using two cubes, each with a digital-looking number on each side. One cube has the numbers 3, 4, and 5 on it. What numbers did Pete put on the sides of the two cubes so that every date from 01 to 31 ca... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
A circle is called "interesting" if its center lies on the Y-axis and is tangent to the hyperbola \(x^2 - y^2 = 1\), whose vertex on the right branch is \(A\). When two "interesting" circles \(O_1\) and \(O_2\) are externa... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
What is the sum of all 4-digit numbers that can be formed using the digits $1, 2, 2, 3, 4, 5, 5, 6, 6, 7$ without repetition? | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find the coefficient of \( x^n \) in the expansion of \( \frac{\sqrt{1+x}}{\sqrt{1-x}} \). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Suppose \( f \in C^1 \), \( f(0) = 0 \), and \( f(1) = 0 \). What is the minimum value of
\[
\int_0^1 \left| f(x)^2 + (|f'(x)| - 1)^2 \right| \, dx?
\] | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Some squares of a $1999 \times 1999$ board are occupied with pawns. Find the smallest number of pawns such that for each empty square, the total number of pawns in the row or column of that square is at least $1999$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the equation $|x-1| + |x-2| + |x-3| = 3a$, where $a > 0$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Determine all positive integers \( n > 3 \) such that there exist \( n \) distinct non-zero integers \( a_1, a_2, \dots, a_n \) satisfying:
- \( a_1^2 + a_2^2 + \dots + a_n^2 \) is a perfect square.
- \( a_1a_2 \dots a_n \... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
One of the medians of a triangle is divided into three equal segments by the incircle. Determine the proportion of the sides of the triangle. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
A mirror is constructed in the shape of $y=\pm{\sqrt{x}}$ for $0\le{x}\le{1}$ and $y=\pm{1}$ for $1<x<9$. A ray of light enters at $(10,1)$ with slope $1$. How many times does it bounce before leaving? | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Determine the number of points \( P \) inside a cube \( W \) such that the perpendicular distances from \( P \) to the six faces of \( W \) are in the order \( a, 2a, 3a, 4a, 5a, \) and \( 6a \). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Simplify the following expressions: $\frac{\sqrt {a\sqrt {ab}}-(ab)^\frac{3}{4}:\sqrt a}{(a^2-b^2)a^{-1}} \left(\sqrt [4]{\frac{a}{b}}+\sqrt [4]{\frac{b}{a}}\right)+$ $+\frac{b}{a}\left(\frac{2a+2b}{a-4b}+\frac{a+3b}{2a+2... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let $A$ and $B$ be two points in the plane (for example, $A(1, 3)$ and $B(3, 7)$). Find the locus of points $C$ such that $r(A, C) = r(C, B)$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Triangle $\triangle ABC$ has side lengths $AB = 13$, $BC = 14$, and $CA = 15$. The incircle of $\triangle ABC$ has center $I$, and line $AI$ is extended to intersect the circumcircle $\omega$ at point $P$. What is the area... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the equation $$1+\frac{\log_a (p-x)}{\log_a (x+q)}=\frac{2-\log_{p-q}4}{\log_{p-q}(x+q)} (p>q>0).$$ | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
If the roots of the polynomial \( P(x) = 2012x^2 - 15853x + 2015 \) are \( n \) and \( w \), what is the value of \( (1 + n + n^2 + \cdots)(1 + w^{-1} + w^{-2} + \cdots) \)? | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
An isosceles trapezoid with base angle $60^\circ$ is such that it is possible to inscribe in it two circles tangent to each other, each one of which is tangent to the bases of the trapezoid and to one of the nonparallel si... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let \(ABC\) be a right triangle with \(\angle A = 90^\circ\) and \(D\) the foot of the altitude from \(A\). A variable point \(M\) traces the interior of the minor arc \(AB\) of the circumcircle of \(\triangle ABC\). The i... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find the minimum value of \( n \) such that each digit of the value of \( 2024n \) is either 0 or 1. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
A positive integer $k$ is said to be visionary if there are integers $a > 0$ and $b \geq 0$ such that $a \cdot k + b \cdot (k + 1) = 2020$. How many visionary integers are there? | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let \( n \) be a positive integer. A university accepts \( n^2 \) freshmen, where initially no two students know each other. The university organizes parties to ensure that there does not exist a group of \( n \) students ... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let \(a, b, c\) be non-negative real numbers such that \(a + b + c = 3\). Find the minimum value of the expression:
\[P = \frac{9}{{ab + bc + ca}} + abc\] | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
A circular pavement with diameter $2r$ has two vertical columns of height $h$ at its endpoints $A$ and $B$. These columns support a beam $A'B'$ of length $2r$. Numerous taut cables, which are perpendicular to the beam $A'B... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the equation: $[ x ]+[ 2x ]+...+[ 1995x ]=1995$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Simplify the expression \(\frac{\displaystyle\prod_{n = 2}^{10}\frac{2n - 2}{n + 2}}{\displaystyle\prod_{n = 0}^{10}\frac{2n - 1}{n + 1}}\). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
How many functions $f:\{1,2,3,4,5\} \to \{1,2,3,4,5\}$ are there such that $f(f(i)) \in \{1,2,3\}$ for all $1 \leq i \leq 5$? | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
In a circle with centre at $O$ and diameter $AB$, two chords $BD$ and $AC$ intersect at $E$. $F$ is a point on $AB$ such that $EF \perp AB$. $FC$ intersects $BD$ in $G$. If $DE = 5$ and $EG =3$, determine $BG$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Evaluate the integral $$\int_0^1\frac{\sin^2(rx)}{\sin^2\left(\frac{x}{2}\right)}\,dx$$ | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find the largest positive integer $k$ such that there exist $2k$ distinct positive integers $a_1, \ldots, a_k$, $b_1, \ldots, b_k$ for which the $k$ sums $a_i + b_i$ are distinct and less than $2014$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let \( W \) consist of all the points in the complex plane such that the taxicab distance between the point and the origin is at least 2. Let \( P \) be the set of all points of the form \( \frac{1}{z} \) where \( z \) is ... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find the sequence $\{ a_{n} \}$ given by $a_{n} = \frac{3}{2 + a_{n-1}}$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Three natural numbers greater than or equal to $2$ are written, not necessarily different. A sequence is constructed using the following procedure: in each step, if the penultimate number written is $a$, the second-to-last... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let \( n \) points lie on a circle. Exactly half of the triangles formed by these points are acute-angled. Find all possible values of \( n \). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let \( n \in \mathbb{N} \) with \( n \geq 4 \) and \( S = \{1, 2, 3, \dots, n\} \). Find the number of solutions to the system of equations
\[
\begin{cases}
x + z = 2y \\
y + t = 2z
\end{cases}
\]
where \( x, y, z, t \in S... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find the number of lines passing through exactly two centers of the $27$ unit cubes in a cube of edge length $3$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Evaluate the integral $$\int_0^1 \int_0^1\frac{\ln\left(x^{2}+y^{2}\right)}{\sqrt{x+y}}dx dy.$$ | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
In each side of a 9-sided polygon, place any one number from 1 to 9 such that each number is used exactly once and the sum of any three consecutive numbers is divisible by 3. Two arrangements are considered the same if one... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Compute the area of octagon $ABCDEFGH$ where $AB = BC = CD = DE = EF = FG = 1$, $\angle B = \angle D = \angle F = 120^\circ$, $\angle C = \angle E = 240^\circ$, and $\angle A = \angle G = 90^\circ$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given an isosceles triangle $\triangle ABC$ with $\angle A = 24^\circ$, and points $D$ such that $\angle DAC = x$, $\angle DBC = 2x$, and $\angle DCB = 5x$. Find the value of $x$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Factor the given expressions: $a^4+9$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
In a $5 \times 5$ grid with a border of empty squares surrounding a $3 \times 3$ subgrid, how many initial configurations will lead to a transformed grid consisting of a single filled square in the center after a single tr... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Consider a set \( A \) of positive integers such that the least element of \( A \) equals 1001 and the product of all elements of \( A \) is a perfect square. What is the least possible value of the greatest element of \( ... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Given that paint spots occur on batik cloths at a rate of 3 spots per 900 cm², find the probability that a 35 cm by 30 cm piece of cloth will contain exactly 6 paint spots. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Determine all $\alpha > 0$ such that the integral
\[
\int_0^{+\infty} \frac{(f(x))^{1+\alpha+\alpha^2}}{x^{2+2\alpha}} \, \mathrm{d}x
\]
converges, where $f : \mathbb{R} \to \mathbb{R}$ is a continuous bounded function sat... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the system of equations in $\mathbb{R}$:
\[ a^3 + b = c \]
\[ b^3 + c = d \]
\[ c^3 + d = a \]
\[ d^3 + a = b \] | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find all functions \( f: \mathbb{N} \to \mathbb{N} \) such that:
\[ (2^m + 1)f(n) \cdot f(2^m \cdot n) = 2^m f(n)^2 + f(2^m \cdot n)^2 + (2^m - 1)^2 \cdot n \]
for all \( m, n \in \mathbb{N} \). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
A $2019 \times 2019$ grid is made up of $2019^2$ unit cells. Each cell is colored either black or white. A coloring is called balanced if, within every square subgrid made up of $k^2$ cells for $1 \leq k \leq 2019$, the nu... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Find the maximum and minimum values of $\displaystyle \frac{x^4+y^4+z^4}{(x+y+z)^4}$ given that $x, y, z > 0$ and $(x+y+z)^3 = 32xyz$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Solve the differential equation \(x(x+y)dy - y^2dx = 0\) to find \(y = f(x)\). | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
For distinct prime numbers $p$ and $q$ with $p < q$, there exists a positive integer $n$ such that $n < p < q < 2n$ and $(2n)! - n! - 1$ is divisible by $p \cdot q$. Find the smallest possible value of $p \cdot q$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
When $(x+1)^{m}$ is multiplied out, the number of coefficients that are not divisible by $2$, $3$, and $5$ are $64$, $16$, and $8$ respectively. Find the value of $m$. | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let \( T = 2304 + 1152\sqrt{2} \) and \( k = \frac{T}{384} \). An isosceles right triangle with one vertex at the origin, another on the \( x \)-axis, and a third in the first quadrant has a perimeter \( k \) and leg lengt... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
Let \(ABCD\) be a rectangle with \(E\) a point on the diagonal \(BD\) such that \(\angle DAE = 15^\circ\). Let \(F\) be a point on \(AB\) such that \(EF\) is perpendicular to \(AB\) and \(EF = \frac{1}{2} AB\). Given \(AD ... | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
Solve the following math problem. Make sure to put the answer (and only answer) inside \boxed{}.
What is the smallest number of lines required to divide the plane into at least 2019 regions using three families of parallel lines? | Nemotron-Math-v2 | Math | training | high_part00 | true | 2 | false | 0 | cc-by-4.0 | aops |
End of preview. Expand in Data Studio
No dataset card yet
- Downloads last month
- 3