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 |
|---|---|---|---|---|---|
If $x^{2}+(x+1)^{7}=a_{0}+a_{1}(x+2)+a_{2}(x+2)^{2}+\ldots+a_{7}(x+2)^{7}$, find the value of $a_{2}$. Express your answer as an integer. | -20 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A sample with a capacity of 1000 is divided into several groups. It is known that the frequency rate of a certain group is 0.4. Calculate the frequency of that group. Express your answer as a whole number. | 400 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $15-2x \gt 7$ and find the number of positive integer solutions. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
For all \\(m\\) and \\(n\\) satisfying \\(1 \\leq n \\leq m \\leq 5\\), calculate the number of unique eccentricities \\(e > 1\\) represented by the polar equation \\(\\rho = \\frac{1}{1 - \\mathrm{C}_{m}^{n} \\cos \\theta}\\). | 6 | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Algebra -> Combinatorics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequalities $2\leqslant a+b\leqslant 5$ and $-2\leqslant a-b\leqslant 1$, find the range of $3a-b$. Express your answer as an interval, including the minimum and maximum values. | [-2,7] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{5}{x+1}=A-\frac{2x-3}{x+1}$, solve for the value of $A$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {a_n}, a_1 = 1 and a_{n+1} - a_n = 2. Find the value of a_{51}. Express your answer as a single numerical value. | 101 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is a monotonically increasing function on (0, +∞), f(n) ∈ N* for n ∈ N*, and f(f(n)) = 3n, find the value of f(1). Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The sequence {a_n} satisfies a_1 = 2, a_2 = 1, and (a_{n-1}-a_{n})/(a_{n} * a_{n-1}) = (a_{n}-a_{n+1})/(a_{n} * a_{n+1}) for n >= 2. Find the 100th term of the sequence {a_n}. Express your answer as a fraction. | \frac{1}{50} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A company hires two people from four college graduates A, B, C, and D, and the chances of these four being hired are equal. Calculate the probability of A being hired. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of equations: $\left\{\begin{array}{l}{x+3y=4-a}\\{x-y=-3a}\end{array}\right.$, and the solutions for x and y are opposite to each other, find the value of a. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x$ is the opposite number of $1$, find the absolute value of $x$. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
If $x$ and $\log_{10} x$ are real numbers and $\log_{10} x<0$, determine the range of $x$. Express your answer as an inequality. | 0<x<1 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given point P has coordinates (-2,4), find the coordinates of point P after reflecting it across the x-axis. Express your answer in the format (x, y). | (-2,-4) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ is a real number, and the complex number $z=\frac{{1+2i}}{{a-i}}$ is purely imaginary (where $i$ is the imaginary unit), find the value of $a$. Express your answer as a single value. | 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
A set of consecutive positive integers beginning with 1 is written on a blackboard. One number is erased. The average (arithmetic mean) of the remaining numbers is 35 7/17. What is the value of the erased number? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given $(3x-1)^{5}=a_{0}+a_{1}x+a_{2}x^{2}+\ldots+a_{5}x^{5}$, find the value of $a_{1}+2a_{2}x+3a_{3}x^{2}+4a_{4}x^{3}+5a_{5}x^{4}$ when x=1. Express your answer as a single number. | 240 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation |a+1| + (b-2023)^2 = 0, find the value of a^b. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
There are $1$ female teacher and $2$ male teachers participating in a speech contest. There are $2$ topics to choose from. If each contestant randomly selects one topic from the pool with replacement, find the probability that exactly one male and one female teacher choose the same topic. Express your answer as a simpl... | \dfrac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ whose product of the first $n$ terms is $T_n$, with a common ratio $q \neq 1$, if $T_7 = 128$, find the value of $a_4$. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( A B : B C = 1 : 2 \) and \( B C : C D = 8 : 5 \), calculate the ratio \( A B : B D \). Express your answer as a ratio, including both parts of the ratio. | 4 : 13 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the fixed point that the graph of y = 1 + log_a x always passes through. Express your answer as an ordered pair (x, y). | (1, 1) | big_math | [
"Mathematics -> Precalculus -> Functions -> Logarithmic Functions"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Freshmen go for a long walk in the suburbs after the start of school. They arrive at point \( A \) 6 minutes later than the originally planned time of 10:10, and they arrive at point \( C \) 6 minutes earlier than the originally planned time of 13:10. There is exactly one point \( B \) between \( A \) and \( C \) that ... | 11:40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A refrigerator's price was increased by 10% twice consecutively, and then by 5% once. Calculate the percentage by which the price should be reduced to restore the original price. Express your answer as an integer percentage. | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of $(1+x^{2})(1- \frac {1}{x})^{6}$. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Diane has five identical blue disks, two identical red disks, and one yellow disk. She wants to place them on an 8-cell grid so that each cell contains exactly one disk. The two red disks must not be placed in cells that share a common edge. Calculate the total number of different-looking completed grids that Diane can... | 108 | big_math | [
"Mathematics -> Applied Mathematics -> Combinatorics"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overline{z}$ is the complex conjugate of $z$, if $\overline{z} + z = 2$ and $(\overline{z} - z)i = 2$ (where $i$ is the imaginary unit), calculate the value of $z$ in the form $a + bi$. Express your answer in the form $\boxed{a + bi}$. | 1 + i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $2\overrightarrow{a} + \overrightarrow{b} = 0$ and $\overrightarrow{a} \cdot \overrightarrow{b} = -2$. Find the value of $(3\overrightarrow{a} + \overrightarrow{b}) \cdot (\overrightarrow{a} - \overrightarrow{b})$. Express your answer as a single numbe... | 3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence of real numbers -1, x, y, z, -2 forms a geometric sequence, calculate the value of $xyz$. Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Chantelle had two candles, one of which was 32 cm longer than the other. She lit the longer one at 3 p.m. and lit the shorter one at 7 p.m. At 9 p.m., they were both the same length. The longer one was completely burned out at 10 p.m. and the shorter one was completely burned at midnight. The two candles burned at diff... | 52 \text{ cm} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the solution set of the quadratic inequality $ax^{2}+bx+c > 0$ is $(-\frac{1}{3},2)$, find the solution set of $cx^{2}+bx+a < 0$. Express your answer in interval notation. | (-3, \frac{1}{2}) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let $z=(a+i)^{2}$ be a complex number whose corresponding point on the complex plane lies on the negative half of the imaginary axis. Find the value of the real number $a$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A certain school has a total of 2000 students, with the number of students in the first, second, and third year of senior high being 750, x, and y, respectively. It is known that when randomly picking one student from the whole school, the probability of it being a student from the second year is 0.3. Now, using strati... | 13 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola $C:\frac{x^2}{a^2}-\frac{y^2}{b^2}=1(a>0,b>0)$, there exist two points $M$ and $N$ symmetric about the origin $O$ on one of the asymptotes of the hyperbola. If the quadrilateral formed by the left and right foci $F_{1}$, $F_{2}$ of the hyperbola and points $M$ and $N$ is a rectangle, and the area of t... | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
An object with a mass of 5 kg is moving in a straight line according to the law $s = 2t + 3t^2$, where $s$ is in centimeters and $t$ is in seconds. Calculate the force acting on the object. Express your answer in Newtons (N). | 0.3 \text{ N} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A 3x3 grid is to be filled with the digits 1, 2, and 3, such that each number appears once in each row and each column. How many different arrangements of the grid are possible? Express your answer as a single integer. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = log3(1 - ax), find the range of 'a' for which f(x) is a decreasing function on the interval (-∞, 1]. Express your answer as an interval in standard notation. | (0,1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $xy\cdot \left(\ \right)=-x^{3}y^{2}$, find the algebraic expression that fits into the parentheses. Express your answer as an algebraic expression, including all necessary variables and their exponents. | -x^{2}y | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
In right triangle $ABC$, it is known that $AB=4$, $AC=2\sqrt{3}$, and $BC=2$. Calculate the dot product of vectors $\overrightarrow{BA}$ and $\overrightarrow{BC}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles",
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, it is given that a_4 = 2 and a_7 = 5. Find the sum of the first 10 terms of the sequence {log a_n}. Express your answer as a numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $\sin 4x \cos 5x = -\cos 4x \sin 5x$ for $x$. Express your answer in degrees. | 20 | big_math | [
"Mathematics -> Trigonometry -> Trigonometric Functions -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
What is the angle of inclination in degrees for the line $x + \sqrt {3}y + 5 = 0$? Express your answer as a numerical value. | 150 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(pi/7 - x) = -2/3, find the value of cos(9pi/14 - x). Express your answer as a single value. | \frac{2}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of lines that pass through the point P(-1,0) and have exactly one point in common with the parabola y^2 = x. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of the expression $-1-3$. Provide your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
An individual contracts influenza. After two rounds of contagion, a total of 100 people have caught influenza. Calculate the average number of people that one person will infect in each round of contagion. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $(\sqrt{3}+2)^{2023}(\sqrt{3}-2)^{2023}$. Express your answer as a single integer or a simplified expression. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
If the area of triangle $ABC$ is 64 square inches, and the geometric mean of sides $AB$ and $AC$ is 12 inches, then find the value of $\sin A$. Express your answer as a simplified fraction or exact decimal if necessary. | \frac{8}{9} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of three-digit numbers that can be formed by selecting any 3 numbers from the set {1, 2, 3, 4} without repetition. Express your answer as a whole number. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the slope of the line passing through points A(3,y) and B(2,-3) is 135 degrees, calculate the value of y. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
If each interior angle of a regular polygon is $150^{\circ}$, calculate the number of sides of the polygon. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of c such that the system of equations |x+y| = 2007 and |x-y| = c has exactly two solutions in real numbers. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given a set of data $x_{1}$, $x_{2}$, $x_{3}$, $x_{4}$, $x_{5}$ with a variance of $\frac{1}{3}$, what is the variance of the data set after applying the transformation $3x - 1$ to each data point? Express your answer as a numerical value. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
The distance from the Earth to the Moon is 384,401 kilometers. Express this distance in scientific notation, retaining three significant figures. Include units in your answer. | 3.84 \times 10^5 \text{ kilometers} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} 2^{|x-a|}, & x\leqslant 1 \\ x+1, & x > 1 \end{cases}$, and knowing that $f(1)$ is the minimum value of $f(x)$, find the range of the real number $a$. Express your answer as an interval. | [1,2] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
At the waterpark, Bonnie and Wendy decided to race each other down a waterslide. Wendy won by 0.25 seconds. If Bonnie's time was exactly 7.80 seconds, how long did it take for Wendy to go down the slide? Express your answer in seconds, rounded to two decimal places. | 7.55 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Let \( N = \left(7^{p+4}\right)\left(5^{q}\right)\left(2^{3}\right) \) be a perfect cube, where \( p \) and \( q \) are positive integers. Find the smallest possible value of \( p+q \) such that \( N \) is a perfect cube. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \\frac{i-5}{1+i}$ (where $i$ is the imaginary unit), calculate the imaginary part of $\\bar{z}$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $(x+3)(6-x) \geq 0$ and express the solution set in interval notation, including endpoints if they are part of the solution. | [-3,6] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\overrightarrow{a}$ and $\overrightarrow{b}$ are unit vectors with an angle of $60^{\circ}$ between them, and $\overrightarrow{c}=\lambda \overrightarrow{a}+\mu \overrightarrow{b}$. If $\overrightarrow{a} \perp \overrightarrow{c}$, find the equation that relates $\lambda$ and $\mu$. | 2\lambda+\mu=0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 2^{|2x-2|} - a cos(1-x) = 0, determine the value of a such that the equation has exactly one real solution. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Let \(\vec{a}, \vec{b}, \vec{c}\) be three unit vectors in the same plane, and \(\vec{a} \perp \vec{b}\). Calculate the maximum value of \((\vec{c} - \vec{a}) \cdot (\vec{c} - \vec{b})\). Express your answer as a single number. | 1 + \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the operation $\begin{vmatrix}a & b \\ c & d\end{vmatrix}=ad-bc$, find the complex number $\bar{z}$ that satisfies $\begin{vmatrix}z & -i \\ 1-i & -2i\end{vmatrix}=0$ (where $i$ is the imaginary unit). Express your answer in standard complex number format (a + bi) and specify the quadrant of the complex plane whe... | \frac{1 - i}{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
The three lines $ax+y+1=0$, $y=3x$, and $x+y=4$ intersect at one point. Find the value of $a$ by first determining the intersection point of the lines $y=3x$ and $x+y=4$, and then substituting this point into the line $ax+y+1=0$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x) = \lg(4 - x) + \frac{1}{\sqrt{x - 1}}$. Express your answer as an interval. | (1, 4) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of roots for the function $f(x) = 2x - \sin x$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of the interior angles of a polygon is three times the sum of its exterior angles, find the number of sides of the polygon. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of ways to form a team of 3 people from a group consisting of 5 boys and 5 girls, ensuring that at least one girl is selected. Express your answer as a whole number. | 110 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A fair coin is tossed twice. Calculate the probability of getting one heads and one tails. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
What is the minimum value of n (n ∈ ℕ) such that the expansion of (x^2 + 1/(2x^3))^n contains a constant term? Express your answer as a single integer value. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(2, -1, 3)$, $\overrightarrow{b}=(-1, 4, -2)$, $\overrightarrow{c}=(7, 5, \lambda)$, if vectors $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ are coplanar, find the real number $\lambda$. Express your answer as a real number. | \frac{65}{7} | big_math | [
"Mathematics -> Linear Algebra -> Vectors"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first n terms of the positive arithmetic geometric sequence {a<sub>n</sub>} is S<sub>n</sub>, and a<sub>2</sub>•a<sub>8</sub> = 16, a<sub>3</sub> = 1, find the value of S<sub>5</sub>. Express your answer as a fraction or decimal, including any necessary calculations or simplifications. | \frac{31}{4} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the side lengths of triangle \( \triangle ABC \) are all integers, and the cosine values of its angles are \( \cos A = \frac{11}{16} \), \( \cos B = \frac{7}{8} \), and \( \cos C = -\frac{1}{4} \), calculate the smallest possible value of the perimeter of \( \triangle ABC \). Express your answer as a single ... | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=e^{x}+a\cdot e^{-x}$, its derivative is $f′(x)$, which is an odd function. Determine the value of $a$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations $a^3 - b^3 = 19x^3$ and $a - b = x$, calculate the possible values of $a$ in terms of $x$. Express your answer as multiple possible values, using the format $a = \boxed{[value1]\text{ or }[value2]}$. | 3x\text{ or }-2x | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex fraction: $\frac{3-i}{1-i}$. Express your answer in the form a+bi, including both real and imaginary parts. | 2+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
How many times do the hour, minute, and second hands of a clock overlap completely in a 24-hour day? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $z$ satisfies $z(1+2i)=2$, calculate the imaginary part of $z$. Express your answer as a decimal fraction, including a negative sign if applicable. | -\dfrac{4}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of \(6102 - 2016\)? Express your answer as a single integer. | 4086 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the slope of the line passing through the points (2, 3) and (6, 5). Express your answer as a fraction or decimal. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} -6+e^{x-1},x < t \\ x^{2}-4x,x\geqslant t\\ \end{cases}$, determine the range of the real number t such that the equation f(x) = x - 6 has exactly three distinct real roots. Express your answer as an interval. | (1,2] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the line y = -x + 1, calculate the slope angle in radians. Express your answer as an exact value. | \dfrac{3\pi }{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
There are 15 girls in a class of 27 students. Calculate the simplified ratio of boys to girls in the class. Express your answer in the form a:b, where a and b are integers with no common factors other than 1. | 4:5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The digits 2, 3, 5, 7, and 8 can be used, each exactly once, to form many five-digit integers. Of these integers, find the one that is as close as possible to 30000. What is the tens digit of this integer? Express your answer as a single digit. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,1,0)$ and $\overrightarrow{b}=(-1,0,2)$, find the value of $k$ such that $k \overrightarrow{a}+ \overrightarrow{b}$ is perpendicular to $2 \overrightarrow{a}- \overrightarrow{b}$. Express your answer as a single number. | \frac{7}{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the projection of vector $\overrightarrow{a}$ onto vector $\overrightarrow{b}$ is $\frac{1}{2}\overrightarrow{b}$, and $|\overrightarrow{a}|=|\overrightarrow{b}|=1$, calculate the magnitude of $|\overrightarrow{a}-\frac{1}{2}\overrightarrow{b}|$. Express your answer as a numerical value. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
A high school senior is preparing to take the independent college entrance exams and needs to choose 3 out of 6 colleges to apply to. Among these colleges, two have exams scheduled at the same time. Calculate the number of different ways the student can register for the exams. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse C: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$), with F1 and F2 being its left and right foci, respectively, and P being any point on the ellipse C other than the endpoints of the major axis. Let G be a point inside $\triangle F_1PF_2$ such that $3\overrightarrow{PG} = \overrightarrow{P... | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
When goods purchased at a cost of 80 yuan each are sold for 90 yuan per unit, 400 units can be sold. If the price of the goods is increased by 1 yuan per unit, the number of units sold decreases by 20. Calculate the optimal selling price per unit to maximize profit. Express your answer in yuan, rounded to the nearest w... | 95 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance between the two points representing -9 and 9 on the number line. Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(x^{2} + x - 1 = 0\), calculate the value of \(2x^{3} + 3x^{2} - x\). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 people standing in a row. Calculate the number of arrangements where there is exactly 1 person between A and B. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A regular pentagon with center \( C \) is shown. Find the measure of angle \( x \) in degrees. | 144 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(alpha) = 3/5 and alpha is an angle in the second quadrant, calculate the value of tan(2*alpha + pi/4). Express your answer as a simplified fraction, including the negative sign if applicable. | -\frac{17}{31} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the directrix of the parabola y^2 = 2x. Express your answer in the form x = a. | x = -\frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
In the rectangular coordinate system xOy, one asymptote of the hyperbola C: y^2/a^2 - x^2/b^2 = 1 (a>0, b>0) is tangent to the circle (x-2)^2 + (y-1)^2 = 1. Find the value of b/a. Express your answer as a fraction. | \frac{3}{4} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In a bag, there are 6 balls, each with a number from 1 to 6 written on it. The ball with the number $n$ weighs $n^2 - 6n + 12$ grams. Assuming the balls are taken out of the bag with equal probability (unaffected by weight or number), what is the probability that the weight of a randomly withdrawn ball is greater than ... | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Observe the following equations: $a=c=\sqrt{6}$, $a^2+b^2=3$, $a^4+b^4=7$, $a^5+b^5=11$. Recognizing that each term is equal to the sum of the previous two terms, calculate the value of $a^{10}+b^{10}$. Express your answer as a single integer. | 123 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial equation \( z^{4} + a z^{3} + b z^{2} + c z + d = 0 \) with real coefficients \( a, b, c, d \), and all roots of the polynomial lying on the circle with center at the origin and radius 1 in the complex plane, find the sum of the reciprocals of these roots. Express your answer as a single value. | -a | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Rich and Ben ate an entire chocolate cake. The ratio of the amount eaten by Rich to the amount eaten by Ben is 3:1. Calculate the percentage of the cake that Ben ate. Express your answer as a percentage, including the percentage sign (%). | 25% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
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