instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
Find the third term in the expansion of $(\\frac{a}{\\sqrt{x}}-\\frac{\\sqrt{x}}{a^2})^6$. Express your answer as a simplified fraction, including any necessary variables. | \\frac{15}{x} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\triangle ABC$ ~ $\triangle A_{1}B_{1}C_{1}$, and $\angle A=50^{\circ}$, $\angle B=95^{\circ}$, find the measure of $\angle C_{1}$. Express your answer in degrees as a numerical value. | 35 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ is a rational number and $|a-1|=4$, find all possible values of $a$. Express your answer as a list of exact values. | 5, -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Store A purchased several units of a certain model of mobile phone and set a selling price. Later, Store B purchased the same model at the same selling price but at a 10% lower purchase price, resulting in a 12 percentage point increase in profit margin compared to Store A. Calculate the profit margin of Store A for se... | 8\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Choose 3 types of vegetables from cucumber, cabbage, rape, and lentils to plant on three different types of soil, where cucumber must be planted. Calculate the total number of different planting methods. Express your answer as a single integer. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\cos \alpha =\dfrac{1}{4}$, calculate $\sin \left(\dfrac{\pi }{2}-2\alpha \right)$. Express your answer as a simplified fraction. | -\dfrac{7}{8} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system $xOy$, two acute angles $α$ and $β$ are constructed such that their terminal sides intersect the unit circle at points $A$ and $B$ respectively. Given that the x-coordinate of point $A$ is $\frac{\sqrt{5}}{5}$ and the y-coordinate of point $B$ is $\frac{\sqrt{2}}{10}$, calculate the v... | \frac{3\pi}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Two circles are given by the equations $x^2+y^2-2mx=0$ and $x^2+(y-2)^2=1$. The circles are externally tangent. Find the value of the real number $m$. Express your answer as a numerical value, including the ± symbol if necessary. | ± \frac {3}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(ωx + φ) (ω > 0, x ∈ R), which satisfies f(x) = f(x+1) - f(x+2), find the relationship between A and B, where A = sin(ωx + φ + 9ω) and B = sin(ωx + φ - 9ω). Express your answer as a specific relationship (e.g. A = B, A > B, etc.). | A = B | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
The function y=a^{2x-1}+1 (a > 0 and a!= 1) always passes through a fixed point. Find the coordinates of this fixed point. Express your answer as an ordered pair (x, y). | (\frac{1}{2},2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
On the number line, what is the integer point that is closest to the point representing $\sqrt {6}$? Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
If there exists an $a \in \mathbb{R}$ such that $|x + a| \leq \ln x + 1$ holds true on the interval $[1, m]$, find the maximum value of the integer $m$. Express your answer as an integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the arc length of a sector is $6$, and the central angle of the sector is $2\text{rad}$, calculate the area of the sector. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,k)$ and $\overrightarrow{b}=(2,2)$, and $\overrightarrow{a}+\overrightarrow{b}$ is colinear with $\overrightarrow{a}$, find the value of $\overrightarrow{a}\cdot\overrightarrow{b}$. Provide your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Write 0.0813 in the form of $a imes 10^n$ ($1 \leq a < 10$, $n$ is an integer). What is the value of $n$? | -2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $M=\{y|y=-x^2+2, x\in \mathbb{R}\}$, and the set $N=\{y|y=2^x, 0\leq x\leq 2\}$, calculate the intersection $(\complement_{\mathbb{R}}M) \cap N$. Express your answer as an interval, using square brackets or parentheses as necessary. | [2,4) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the vectors $\overrightarrow{a}=(3,1), \overrightarrow{b}=(1,3), \overrightarrow{c}=(k,7)$, find the value of $k$ such that $(\vec{a}-\vec{c})$ is parallel to $\vec{b}$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a^{m}=12$ and $a^{n}=3$, calculate the value of $a^{m-n}$. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(\lg 5)^2 + \lg 2 \cdot \lg 5 + \lg 20$. Provide your answer as a single number, the result of the simplification. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Daniel rode his bicycle at a constant speed. After 40 minutes, he cycled 24 km. How far did he cycle in 30 minutes? Express your answer in kilometers as a whole number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Point M is located in the second quadrant of the coordinate plane. The distance from point M to the x-axis is 1 unit, and the distance from point M to the y-axis is 2 units. Determine the exact coordinates of point M, expressing your answer as an ordered pair (x, y). | (-2,1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
If $a$ is a three-digit number, and now we place a 1 to its right to get a four-digit number, what is the algebraic expression for this four-digit number? Express your answer as an algebraic expression in terms of $a$. | 10a + 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) = cos(ωx + φ) (ω > 0, 0 ≤ φ ≤ π) is an odd function and is monotonically decreasing in the interval [-π/4, π/3], calculate the maximum value of ω. | 3/2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $$f(x)=ln( \sqrt {1+x^{2}}-x)+1$$, calculate the value of $$f(lg3)+f(lg \frac {1}{3})$$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
If $\frac{2}{3}$ of $10$ bananas are worth as much as $8$ oranges, how many oranges are worth as much as $\frac{1}{2}$ of $5$ bananas? Express your answer as a whole number of oranges. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that vectors $\overrightarrow {e_{1}}$ and $\overrightarrow {e_{2}}$ are non-collinear, $\overrightarrow {a} = 3\overrightarrow {e_{1}} - 4\overrightarrow {e_{2}}$, $\overrightarrow {b} = 6\overrightarrow {e_{1}} + k\overrightarrow {e_{2}}$, and $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$, find the... | -8 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence \(\{a_n\}\), where \(a_6 + a_8 = 16\) and \(a_4 = 1\), find the value of \(a_{10}\). | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is an even function on $\mathbb{R}$, and when $x \geq 0$, $f(x) = x^3 + \ln(x+1)$, find the expression for $f(x)$ when $x < 0$. Express your answer as a function of $x$. | -x^3 + \ln(1-x) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
A disk with a diameter of \( D \) is placed on a chessboard with an \( 8 \times 8 \) grid, where the width of each square is \( D \), and their centers coincide. Find the number of squares on the chessboard that are completely covered by the disk. Express your answer as a whole number. | 32 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the three vertices of a triangle A(2, -1, 4), B(3, 2, -6), and C(5, 0, 2), find the length of the median through point A. Express your answer as a simplified radical expression. | 2 \sqrt{11} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function $f(x)$ is an odd function, decreasing on the interval $(0,+\infty)$, and its range on the interval $[a,b]$ $(a < b < 0)$ is $[-3,4]$, find the minimum value of $f(x)$ on the interval $[-b,-a]$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Functions -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 0.000000125 in scientific notation in the form a × 10^b, where a is a number between 1 and 10 (not including 10) and b is an integer. | 1.25 \times 10^{-7} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
For any whole number $n$, find the numerical value that always divides the expression $n^2(n^2 - 1)$. | 12 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence ${a_n}$, $a_2=2008$, $a_{2008}=a_{2004}-16$. Find the value or range of values of $n$ for which the sum of the first $n$ terms, $S_n$, is maximized. Express your answer as a single value or a range of values, clearly describing what the answer represents. | 503 \text{ or } 504 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the line equation $\frac{x}{3} - \frac{y}{4} = 1$, calculate the sum of its intercepts on the two coordinate axes. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \\(\\frac{3-(-3)}{2-1}\\). Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ABC, the magnitudes of the sides are a, b, and c, as shown, with c ≤ b ≤ a. Through interior point P and the vertices A, B, C, lines are drawn meeting the opposite sides in A', B', C', respectively. Let s = AA' + BB' + CC'. Find the upper bound for s in terms of a, b, and c. | a+b+c | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $2x - 1 \geqslant 0$ to find the range of values for x that makes the square root $\sqrt{2x-1}$ meaningful within the real number range. Express your answer as an inequality. | x \geqslant \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $16.47 \times 24 = 395.28$, calculate the difference between the product of $16.47 \times 34$ and $395.28$. Express your answer as a numerical value. | 164.7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In right triangle $ABC$, where $\angle C=90^{\circ}$, the lengths of sides $AB$ and $AC$ are given as $6$ and $3$, respectively. Calculate the value of $\sin B$. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \( S_{n} = 1 - 2 + 3 - 4 + \cdots + (-1)^{n-1} n \) for \( n = 1, 2, \cdots \), compute the sum \( S_{17} + S_{33} + S_{50} \). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a function $y=f(x)$ defined on the interval $[-1,1]$ with a range of $[-2,0]$, find the range of the function $y=f(\cos 2x)$. Express your answer as an interval. | [-2,0] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
The function f(x) is an odd function defined on R, and for any x in R it satisfies f(x+6) = f(x) + f(3-x). Calculate the value of f(2010). Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. Find the complex conjugate of the complex number $\frac{17}{4-i}$. Express your answer in the form $a + bi$, where $a$ and $b$ are integers. | 4 - i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given points P(-1,-2) and Q(4,2) in the xy-plane, find the value of m at point R(1,m) such that the sum of the distances PR+RQ is minimized. Express your answer as a fraction. | -\frac{2}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that line segments a, b, c, and d are in proportion and a = 1, b = 2, c = 3, find the value of d. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Yasin donated one unit of blood on January 1st at noon. The effective period for one unit of blood is 10! seconds. Calculate the expiration date for his donated blood. Express your answer as a date (month and day). | February 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Find the x-axis intercept of the line given by the equation $x + 2y + 1 = 0$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of extreme points of the function $f(x)=3x^{2}-\ln x-x$. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Solve for \( y \) in the equation \( 5y - 100 = 125 \). Express your answer as a single integer. | 45 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{|x|-2}{x+2} = 0$, find the value of $x$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $S_n$ is the sum of the first $n$ terms of a geometric sequence $\{a_n\}$, and that $8a_2 - a_5 = 0$, calculate the value of $\frac{S_4}{S_2}$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of intersection points between the line $x + ay - 1 = 0$ ($a \in \mathbb{R}$) and the circle $x^2 + y^2 - 4x = 0$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
From 6 people, 4 are to be selected to visit cities A, B, C, and D respectively, with the requirement that each city is visited by one person, each person visits only one city, and among these 6 people, person A and person B will not visit city A. Calculate the total number of different selection schemes that meet thes... | 240 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a class, there are 26 male students and 24 female students. If one student is to be selected as the mathematics representative, how many different ways are there to make this selection? Express your answer as a numerical value. | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given $0 < \alpha < \pi$ and $-\sin \alpha = 2\cos \alpha$, calculate the value of $2\sin ^{2}\alpha - \sin \alpha \cos \alpha + \cos ^{2}\alpha$. Express your answer as a simplified fraction. | \dfrac{11}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Find the square root of 4. Express your answer using the ± symbol if there are two square roots. | \pm 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
In the space rectangular coordinate system $O-xyz$, calculate the distance from the point $(2,-1,2)$ to the coordinate origin $O$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence {a_n}, it is given that a_5 + a_10 = 12. Calculate the value of 3a_7 + a_9. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression 2x^2 + 3x^2 when x = 2. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( c \) is a real number and the negative of one root of \( x^{2} - 3x + c = 0 \) is a root of \( x^{2} + 3x - c = 0 \), find the roots of \( x^{2} - 3x + c = 0 \). Express your answer as two exact numerical values. | 0, 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Pat gives half of his marbles to his best friend and then a third of those remaining to his sister. If his sister receives 9 marbles, find the number of marbles Pat keeps. Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function \( f(x) \) is defined on the interval \([0,1]\), determine the range of values for \( a \) such that the function \( f(x-a) + f(x+a) \) is defined. Express your answer as an interval. | \left[-\frac{1}{2}, \frac{1}{2}\right] | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the functions $f(x)=\frac{1}{2}x^2+2ax$ and $g(x)=3a^2\ln x+b$, where the curves $y=f(x)$ and $y=g(x)$ have a common point with the same tangent line, find the maximum value of the real number $b$ when $a\in(0,+\infty)$. | \frac{3}{2}e^{\frac{2}{3}} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the function f(x) = sin(2x - π/6) is shifted to the right by π/12 units. Find the symmetry axis of the new graph. Express your answer as an equation for x in radians, using boxed notation for the final answer. | \frac{5\pi}{12} | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the vectors $\overrightarrow{a} = (1, 2)$ and $\overrightarrow{b} = (-2, x)$, if $\overrightarrow{a} + \overrightarrow{b}$ is parallel to $\overrightarrow{a} - \overrightarrow{b}$, find the value of the real number $x$. Express your answer as a single real number. | -4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle $C$: $x^{2}+y^{2}+2x-4y=0$, find the coordinates of the center of circle $C$. Express your answer as an ordered pair (x, y). | (-1,2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
In bag A, there are 3 white balls and 5 black balls. In bag B, there are 4 white balls and 6 black balls. Now, a ball is randomly taken from bag A and put into bag B. After thoroughly mixing, a ball is then randomly taken from bag B and put back into bag A. Find the probability that the number of white balls in bag A d... | \frac {35}{44} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( \triangle ABC \), \( AB=AC \) and \( \angle A=80^{\circ} \). Points \( D, E \), and \( F \) are on sides \( BC, AC \), and \( AB \) respectively, and \( CE=CD \) and \( BF=BD \). Calculate the measure of \( \angle EDF \) in degrees. | 50^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the complex number z satisfies the equation (z+i)(2+i)=5 (where i is the imaginary unit), find the value of \overset{-}{z} (which is the complex conjugate of z). Express your answer as a complex number in the form a+bi. | 2+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be a differentiable function with domain R and derivative f'(x), satisfying f'(x) - 2f(x) < 0, and f(0) = 1. Find the solution set of the inequality f(x) > e^(2x). Express your answer as an interval. | (-\infty, 0) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Let $A=\{x\in \mathbb{R} | -1 < x \leq 4\}$ and $B=\{x\in \mathbb{N} | x \geq 2\}$. Find the intersection $A \cap B$. Express your answer as a set of numbers using proper set notation. | \{2,3,4\} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be a non-decreasing function on the interval [0, 1], meaning that for any 0 ≤ x1 ≤ x2 ≤ 1, we have f(x1) ≤ f(x2). The function satisfies the following conditions:
1. f(0) = 0;
2. f(x/3) = (1/2)f(x);
3. f(1-x) = 1 - f(x).
Find the value of f(1/2016). Express your answer as a fraction. | 1/128 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The equation of the hyperbola is $4y^2-25x^2=100$. Find the coordinates of the foci. Express your answer as coordinates (x, y), including the ± symbol for the y-coordinate if applicable. | (0, \pm\sqrt{29}) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A school's participants in a math competition had an average score of 75. The number of male contestants was 80% more than the number of female contestants, and the average score of female contestants was 20% higher than that of male contestants. Calculate the average score of the female contestants. Express your answe... | 84 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A man is good at walking, increasing his distance equally every day. Over nine days, he walks a total of 1260 miles, and the sum of the distances he walks on the first, fourth, and seventh days is 390 miles. Find the distance he walks on the eighth day. Express your answer in miles as a whole number. | 170 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
If the letters in the English word "word" are written in the wrong order, what is the total number of possible mistakes? Express your answer as a whole number. | 23 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Permutations"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x) = x - 2013$ when $x > 0$, and $f(x)$ is an odd function in its domain, derive the expression for $f(x)$ when $x < 0$. Use the odd function property to arrive at your answer. Express your answer as a function of $x$. | x + 2013 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
A building has 4 units, and two people, A and B, live in this building. Find the probability that A and B live in the same unit. Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
If \( 2^n = 8^{20} \), what is the value of \( n \)? Express your answer as a whole number. | 60 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $2\cos ^{2}15^{\circ}-1$. Express your answer as a numerical value, including any necessary mathematical notation (e.g. square roots, fractions). | \dfrac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
The lateral surface of a cone is unfolded into a sector with a central angle of 120° and a radius of 1. Calculate the ratio of the surface area to the lateral surface area of this cone. Express your answer as a ratio of two integers (e.g., 4:3). | 4:3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of values for $x$ such that the square root $\sqrt{x-1}$ is meaningful. Express your answer as an inequality in the form of $x \geqslant a$. | x \geqslant 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality (x-1)/(2x+1) ≤ 0 and express the solution set in interval notation. | (-\frac{1}{2},1] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
The point (x, y) is mapped to (x+y, x-y) under the mapping f. Find the corresponding point of (2, 0) under the action of f. Express your answer as an ordered pair (x, y). | (2, 2) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Two identical equilateral triangles, A and B, each have a line segment connecting the midpoints of two sides. Starting from a symmetrical position about the left and right sides but with non-symmetrical internal line segments, triangle A is moved horizontally to the right until the two triangles overlap and then contin... | 11 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, assume the points corresponding to the complex numbers $z_1$ and $z_2$ are symmetrical about the imaginary axis, and $z_1=1+2i$ (where $i$ is the imaginary unit). Calculate the value of $z_1z_2$. Express your answer as a single number. | -5 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $6 \cdot 5(-1)^{2^{3^{5}}}+(-1)^{5^{3^{2}}}$. Express your answer as a single integer. | 29 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the equations $x^2-9x+2^a=0$ and $x^2-6x+2^b=0$ each have two distinct real roots, and these four roots form a geometric sequence with a common ratio of 2, find the sum of the exponents $a$ and $b$. Express your answer as a numerical value. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression $-\left(-2023\right)$. Provide your answer as a single numerical value. | 2023 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence defined recursively by a_0 = sin(x) and a_n = (-1)^floor(n/2) * sqrt(1 - a_(n-1)^2), find a_1989. Express your answer as a trigonometric function of x. | \cos x | big_math | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Calculus -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
One standard balloon can lift a basket with contents weighing not more than 80 kg. Two standard balloons can lift the same basket with contents weighing not more than 180 kg. Find the weight of the basket in kilograms. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the vertex coordinates of the parabola given by the quadratic function y = x^2 + 2x - 2. Express your answer as an ordered pair (x, y). | (-1, -3) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 small balls of the same size and texture in a bag, colored red, yellow, blue, green, and purple. Person A likes red and yellow, while person B likes yellow, green, and purple. If person A and person B each draw a ball from the bag, what is the probability that they both draw a ball of their preferred color?... | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the slope of the tangent line to the function $y = \sin x + e^x$ at the point $(0,1)$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Execute the following program and determine the final value of S:
\(S=0\)
\(i=0\)
\(WHILE i <=10\)
\(S= S+i\)
\(i=i^2+1\)
\(WEND\)
\(PRINT S\)
\(END\)
Provide your answer as a single integer value. | 8 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)=\ln x- \frac {2}{x}$. Find the value of $\cos\alpha+\sin\alpha$, where $\alpha$ is the slope angle of the tangent line to the curve $y=f(x)$ at $x=1$. Express your answer as a numerical value. | \frac{2 \sqrt{10}}{5} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
If the radius of circle $C$ is $1$, the center of the circle is in the first quadrant, and it is tangent to the line $4x-3y=0$ and the $x$-axis, then find the standard equation of the circle. Express your answer in the format $(x-h)^2 + (y-k)^2 = r^2$, where $(h,k)$ is the center of the circle and $r$ is the radius. | (x-2)^2 + (y-1)^2 = 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
What is the radius of a circle inscribed in a rhombus with diagonals of length $10$ and $24$? Express your answer as a fraction or decimal. | \frac{60}{13} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, 2)$ and $\overrightarrow{b} = (-3, 2)$, find the value of the real number $k$ such that $(k\overrightarrow{a} + \overrightarrow{b})$ is parallel to $(\overrightarrow{a} - 3\overrightarrow{b})$. Express your answer as a fraction or decimal. | -\frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
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