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
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|---|---|---|---|---|---|
In the sequence $\{a_n\}$, it is known that $a_1=1$, $a_2=2$, and $a_{n+1}=a_n-a_{n-1}$ for $n \in \mathbb{N^*}$. Calculate the value of $a_5$. Express your answer as an integer. | -2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A die is rolled twice. What is the probability that the number of points on the second roll is three times the number of points on the first roll? Express your answer as a simplified fraction. | \frac{1}{18} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the first term of the sequence $\{ a_{n} \}$ is $-1$ and $a_{n+1} = 2a_{n} + 2$, derive the general term formula for the sequence $\{ a_{n} \}$. Express your answer as a formula in terms of $n$, using standard mathematical notation. | 2^{n-1}-2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A person has a probability of $0.6$ to hit a target with a single shot. After shooting 3 times, let $X$ denote the number of times the target is hit. Calculate the probability $P(X\geqslant 2)$. Express your answer as a simplified fraction. | \dfrac{81}{125} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Kartik wants to shade three of the squares in a grid blue, and Lucy wants to shade the remaining two squares red. How many possible finished grids are there where Lucy's red squares are next to each other? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let complex numbers $z_1=1+i$ and $z_2=x-i$ (where $x \in \mathbb{R}$). If $z_1 \cdot z_2$ is a real number, find the value of $x$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {a_n}, the initial term a_1 = -1 and each subsequent term is defined by the recursive formula a_{n+1} = a_n - 3. Calculate the value of a_8. Provide your answer as a single integer. | -22 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the unit digit of \(33 \cdot 83^{1001} \cdot 7^{1002} \cdot 13^{1003}\). Express your answer as a single digit. | 9 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that a real number $a$ satisfies $|a| = -a$, find the value of $|a - |a||$. Express your answer as a single algebraic expression in terms of $a$. | -2a | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given a set of three-digit numbers composed of "0" and "1", let $A$ represent the event that "the second digit is '0'", and let $B$ represent the event that "the first digit is '0'". Calculate the conditional probability $P(A|B)$. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = e^x + x, find its derivative f'(x). Express your answer as a function of x. | f'(x) = e^x + 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the asymptotes of the hyperbola $mx^{2}-y^{2}=1$ are $y=\pm3x$, find the value of $m$. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Hyperbolic Geometry"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the two sequences:
\[
\begin{array}{l}
1,3,5,7, \cdots, 1991 \\
1,6,11,16, \cdots, 1991
\end{array}
\]
Determine the number of numbers that appear in both sequences. Express your answer as a whole number. | 200 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that 1 mags = 8 migs and 1 mogs = 6 mags, find the value of 10 mogs + 6 mags in migs. Express your answer as a whole number. | 528 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n} where all terms are positive, given that a_3 * a_11 = 16, find the value of a_7. Express your answer as a positive number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-2,1)$, calculate the value of $\overrightarrow{a}+2\overrightarrow{b}$. Express your answer as an ordered pair (x, y). | (-3, 4) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.71875 | 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 $a$. | 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given $3\sin \alpha + \cos \alpha = 0$, calculate the value of $\dfrac{1}{{\cos }^{2}\alpha + \sin 2\alpha }$. Express your answer as a numerical value. | \dfrac{10}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integers n that satisfy the inequality 1/7 ≤ 6/n ≤ 1/4. Express your answer as a single integer. | 19 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
The equal-arm scale shown is balanced with 2 rectangles on one side and 6 circles on the other side. How many circles have the same mass as one rectangle? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of $\left( {{x}^{2}}+2 \right){{\left( \frac{1}{{{x}^{2}}}-1 \right)}^{5}}$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Polynomials"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The line $y=1$ is a line parallel to the x-axis. Find the slope angle of this line. Express your answer in degrees as a whole number. | 0 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Increasing the radius of a cylinder by 6 units increases the volume by y cubic units. Increasing the height of the cylinder by 6 units also increases the volume by y cubic units. If the original height is 2 units, find the original radius of the cylinder. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \\frac {1+ai}{2-i}$, where $i$ is the imaginary unit, find the value of the real number $a$ such that $z$ is a pure imaginary number. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the functions $f(x)=x^{3}-3x^{2}+1$ and $g(x)= \begin{cases} (x- \frac {1}{2})^{2}+1 &(x > 0) \\ -(x+3)^{2}+1 &(x\leqslant 0)\\ \end{cases}$, find the maximum number of real roots for the equation $g[f(x)]-a=0$, where $a$ is a positive real number. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Functions -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Observe the following equations: $a+b=1$, $a^2+b^2=3$, $a^3+b^3=4$, $a^4+b^4=7$, $a^5+b^5=11$, $a^6+b^6=18$. Using the pattern observed in these equations, calculate the value of $a^8+b^8$. Express your answer as a single integer. | 47 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 male and 3 female students in Group A, and 6 male and 2 female students in Group B. If 2 students are selected from each of Group A and Group B, find the number of different ways to select 4 students with exactly 1 female student. Express your answer as a whole number. | 345 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $C_{20}^{2x-1} = C_{20}^{x+3}$, find the value(s) of x. Express your answer as a numerical value(s) of x, including all possible solutions. | 4, 6 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence \(\left\{ a_n \right\}\), \(a_2=9\) and \(a_5=243\). Calculate the sum of the first 4 terms of \(\left\{ a_n \right\}\). Express your answer as a single integer. | 120 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the odd function f(x) defined on R, where for any x1, x2 in [0, +inf) (x1!= x2), (x2 - x1)(f(x2) - f(x1)) > 0, determine the order of f(-2), f(1), and f(3). Express your answer using less than (<) or greater than (>) symbols to indicate the order. | f(-2) < f(1) < f(3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given $\lg 2=0.3010$ and $\lg 3=0.4771$, calculate the position of the first non-zero digit after the decimal point in the number $\left(\frac{6}{25}\right)^{100}$. Express your answer as a whole number representing the position (e.g., 'the 62nd place'). | 62 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
If $a > 1$, find the minimum value of $a + \frac{1}{a-1}$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Use the digits 0, 1, and 2 to form natural numbers without repeating any digits. What is the total number of such natural numbers? | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence $\{a_n\}$ where all terms are positive, $a_1=3$, and $a_1+a_2+a_3=21$, find the value of $a_4+a_5+a_6$. Express your answer as a single integer. | 168 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A bag contains 6 balls, with 3 red balls and 3 white balls. Two balls are drawn successively without replacement. Given that the first ball drawn is red, calculate the probability of drawing a red ball again on the second draw. Express your answer as a simplified fraction. | \dfrac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A sphere with a radius of $\sqrt{6}$ has an inscribed right square prism with a height of $4$. What is the surface area of this right square prism? Express your answer as a whole number. | 40 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the constraints $\begin{cases} y \geqslant x, \\ x + 3y \leqslant 4, \\ x \geqslant -2, \end{cases}$ find the maximum value of $z = |x - 3y|$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
April, May, and June have 90 sweets between them. May has three-quarters of the number of sweets that June has. April has two-thirds of the number of sweets that May has. Calculate the number of sweets June has. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the first $n$ terms of a geometric sequence with positive terms is $3$, and the sum of the first $3n$ terms is $21$. Find the sum of the first $2n$ terms. Express your answer as a single numerical value. | 9 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $i$ is the imaginary unit, simplify the expression $\frac{i}{1 + \sqrt{3}i}$ in the form $a + bi$, where $a$ and $b$ are real numbers. | \frac{\sqrt{3}}{4} + \frac{1}{4}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given an equilateral triangle ABC with side length 2, vectors a and b satisfy the relationships: $\overrightarrow{AB} = 2\vec{a}$ and $\overrightarrow{AC} = 2\vec{a} + \vec{b}$. Find the angle between vectors a and b. Express your answer in degrees as a whole number. | 120 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the monomials $2x^{3}y^{4}$ and $-2x^{a}y^{2b}$ are like terms, find the values of $a$ and $b$ by comparing their exponents. Express your answer as a pair of values ($a$, $b$). | (3, 2) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Solve for x in the equation $\frac{{x}^{2}-4}{x} = 0$. Provide your answer as numerical value(s) for x, separating multiple values with commas or using 'or' if applicable. | 2, -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given a right triangle \( ABC \) with hypotenuse \( AB \). One leg \( AC = 15 \) and the altitude from \( C \) to \( AB \) divides \( AB \) into segments \( AH \) and \( HB \) with \( HB = 16 \). Calculate the area of triangle \( ABC \). Express your answer as a whole number. | 150 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function $f(x+1)$ is $\left[-1,0\right)$, find the domain of $f(2x)$. Express your answer in interval notation. | \left[0, \frac{1}{2}\right) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = a sin x - b cuberoot(x) + c ln(x + sqrt(x^2 + 1)) + 4, where a, b, c are real numbers, and f(lg log_3 10) = 5. Find the value of f(lg lg 3). Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=1+i$ (where $i$ is the imaginary unit), calculate the conjugate of $\frac{2}{z}-z^{2}$. Express your answer as a complex number in the form $a+bi$. | 1+3i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{2\sin^{2}\alpha}{\sin 2\alpha} \cdot \frac{2\cos^{2}\alpha}{\cos 2\alpha}$. Express your answer as a trigonometric function of alpha. | \tan 2\alpha | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Find the tens digit of $2015^{2016} - 2017$. Express your answer as a single digit. | 0 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
The "QQ Space" level is a symbol of user information and identity, divided into different levels according to space points. When a user is above level 10, there is a certain relationship between each level and the corresponding points. It is known that the points for level 10 are 90, for level 11 are 160, for level 12 ... | 17 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations \( a = \log_{8} 225 \) and \( b = \log_{2} 15 \), express \( a \) in terms of \( b \). Provide your answer as an equation or fraction using \( b \). | \frac{2b}{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Team A was tasked with installing 66 air conditioners in Community A, and Team B was tasked with installing 60 air conditioners in Community B. Both teams started and finished their tasks at the same time. If Team A installs 2 more air conditioners per day than Team B, calculate the daily installation numbers of Team A... | 22, 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given an even function $f(x)$ with the domain $(-1,0)\cup(0,1)$, and $f(\frac{1}{e})=0$. When $0 < x < 1$, $(1-x^{2})\ln x(1-x^{2})f′(x) > 2xf(x)$. Find the range of $x$ for which $f(x) < 0$. Express your answer as an interval or union of intervals. | (-1,-\frac{1}{e})\cup(\frac{1}{e},1) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of $(-3)^0$? Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {an}, a5 = 3 and a6 = -2. Find the value of a3 + a4 +... + a8. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of a circle (x+5)^2 + (y-12)^2 = 14^2, find the minimum value of x^2 + y^2. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, each of \( \triangle QPT, \triangle QTS \) and \( \triangle QSR \) is an isosceles, right-angled triangle, with \( \angle QPT = \angle QTS = \angle QSR = 90^{\circ} \). The combined area of the three triangles is 56. If \( QP = PT = k \), find the value of \( k \). Express your answer as a single numeri... | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of x^2 in the expansion of (1-x)^6(1+x)^4. Express your answer as an integer. | -3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
A counter is placed at position 0 on a circle with 10 points (indexed 0 through 9). On the nth move, the counter moves n^n steps clockwise. Find the position of the counter after 1234 moves. Express your answer as a single integer representing the final position. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
There are 2 doctors and 4 nurses assigned to perform physical examinations for students in 2 schools, with each school allocated 1 doctor and 2 nurses. Calculate the total number of different assignment methods for allocating the doctors and nurses to the two schools. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function f(x) = log2(x) in its domain. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Precalculus -> Functions -> Logarithmic Functions"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the curve y = -x^2 + 1 at the point (1, 0). Express your answer in the form y = mx + b, where m and b are numbers. | y = -2x + 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given $sinα=\frac{1}{3}$ and $α∈(\frac{π}{2},π)$, calculate the value of $ an \alpha$. Express your answer as a simplified fraction or radical if necessary. | -\frac{\sqrt{2}}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
In a wooden box, there are 3 red balls, 5 yellow balls, and 7 blue balls. If you blindly pick balls from the box, what is the minimum number of balls you need to pick to ensure that at least two of them are of the same color? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Let the set A = {x|x ≤ a} and B = (-∞, 2). Given that A ⊆ B, determine the range of the real number a. Express your answer as an inequality using standard notation (e.g., <, ≤, >, ≥). | a < 2 | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given that a < -1, evaluate the expression |1+a| - sqrt(a^2) and provide the simplified result as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
In the classroom, Teacher Yuan organized students to arrange triangles using small sticks. The lengths of the sticks are $8cm$, $7cm$, $13cm$, and $15cm$. Xiao Meng has already taken two sticks of $8cm$ and $7cm$. Determine the length of the third stick that Xiao Meng cannot take to form a triangle with the given stick... | 15 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A certain commodity has undergone two consecutive price increases, resulting in a current price that is 44% higher than its original price. If the percentage increase for both price adjustments is the same, what is this uniform percentage increase? Express your answer as a percentage (%). | 20\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
During the 2016 Spring Festival travel season, to check for drunk driving, three traffic police officers A, B, and C were arranged to conduct surprise inspections at two different intersections near a business center, with at least one officer at each intersection. Find the probability that officers A and B are not at ... | \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Three teachers and six students are to be arranged in three different places, A, B, and C, with each place being assigned 1 teacher and 2 students. Calculate the total number of different arrangements possible. | 540 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} f(x+5),x > 2 \\ e^{x},-2\leqslant x\leqslant 2 \\ f(-x),x < -2 \end{cases}$, find $f(-2016)$. Express your answer as a single value, possibly using e. | e | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Arrange 2016 coins into a triangle. The 1st row has 1 coin, the 2nd row has 2 coins, and the Nth row has N coins. Find the value of N such that the total number of coins is 2016, and then calculate the sum of the digits of N. Express your answer as a single integer. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
In a right triangle \(ABC\), the hypotenuse \(AB\) has median lengths \(AD = 7\) and \(BE = 4\). Find the length of \(AB\). Express your answer as a numerical value with a square root if necessary. | 2 \sqrt{13} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x / (x-1) = (y^2 + 2y - 1) / (y^2 + 2y - 2), solve for x in terms of y. | y^2 + 2y - 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Rational Equations"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
A person walks for 5 hours, first along a flat road, then up a mountain, and then back along the same route. The person walks at a speed of 4 kilometers per hour on the flat road, 3 kilometers per hour uphill, and 6 kilometers per hour downhill. Calculate the total distance walked in these 5 hours. Express your answer ... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function $y=x \cdot \ln x$. Express your answer in the format $y' =...$ and box the final answer. | \ln x + 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the two circles with equations x^2+y^2-2x-5=0 and x^2+y^2+2x-4y-4=0, find the equation of the perpendicular bisector of the line segment connecting their intersection points A and B. Express your answer as a linear equation in the form ax + by + c = 0. | x+y-1=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Convert the Cartesian coordinates of point M ($\sqrt{3}$, -1) into polar coordinates. Express your answer as an ordered pair of polar coordinates, with the radial distance and angle in radians. | (2, \frac{11\pi}{6}) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x)=ax^{3}+bx^{2}+cx+2$ have a derivative denoted by $f′(x)$. If $f′(x)$ is an even function, find the necessary condition on the coefficients $a$, $b$, and $c$. Express your answer as a mathematical condition (e.g. $b = 0$). | b = 0 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Applied to a bill for $10,000, calculate the difference in dollars between a single discount of 40% and two successive discounts of 36% and 4%. Express your answer in dollars. | \$144 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
A line l: x-2y-5=0 passes through one of the foci of the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) and is parallel to one of its asymptotes. Find the equation of this hyperbola. Express your answer in the standard form of a hyperbola equation. | \frac{x^2}{20} - \frac{y^2}{5} = 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
If an income of $100 is recorded as +100 in accounting, how should an expenditure of $100 be recorded? Express your answer as a dollar amount, including a negative sign if necessary. | -\$100 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=\\frac{{2i}}{{1-i}}$, find the imaginary part of $z$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of angle α passes through point P(-1,2), determine the value of tan(α + π/2). Express your answer as a numerical value. | 1/2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $y=\sqrt{x(3-x)}+\sqrt{x-1}$. Express your answer in interval notation. | [1,3] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given that line l intersects the ellipse x^{2}+2y^{2}=2 at points A and B, the midpoint of segment AB is P, the slope of line l is k_{1} (k_{1} ≠ 0), and the slope of line OP is k_{2}, calculate the value of k_{1}k_{2}. Express your answer as a single numerical value. | -\frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given an ellipse with its foci on the $x$-axis, described by the equation $\frac{x^2}{2} + \frac{y^2}{m} = 1$, and its eccentricity is $\frac{1}{2}$, calculate the length of the minor axis of this ellipse. Express your answer as a numerical value. | \sqrt{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of a circle is $(x-3)^2+y^2=9$, find the coordinates of the center of the circle. Express your answer as an ordered pair (x,y). | (3,0) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle ∆ABC, the sides opposite to angles A, B, and C are a, b, and c respectively. If {a}^{2}-{b}^{2}= \sqrt{3}bc and \sin C=2 \sqrt{3}\sin B, then find the measure of angle A in degrees. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{a} = (-2, 1)$, $\overrightarrow{b} = (k, -3)$, $\overrightarrow{c} = (1, 2)$, if $(\overrightarrow{a} - 2\overrightarrow{b}) \perp \overrightarrow{c}$, calculate the magnitude of vector $\overrightarrow{b}$. Express your answer as a numerical value, including the square root if necessary. | 3\sqrt{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
If x is 20% of y and x is 50% of z, then what percentage is z of y? Express your answer as a percentage, including the percentage sign (%). | 40% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
If the line $\left(3-m\right)x+\left(2m-1\right)y+7=0$ is perpendicular to the line $\left(1-2m\right)x+\left(m+5\right)y-6=0$, calculate the value of $m$. Express your answer as a numerical value or values. | -1, \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A TV factory plans to reduce the cost of a certain model of TV by 19% over the course of two years. If the percentage of reduction is the same each year, calculate the annual percentage reduction. Express your answer as a percentage. | 10\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is an odd function defined on R, and when x <= 0, f(x) = x(1 - x). The sequence {a_n} satisfies a_1 = 1/2, and a_{n+1} = 1/(1 - a_n). Calculate the value of f(-a_{2016}). Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Functions -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Let a random experiment have only two outcomes: $A$ and $\overline{A}$, with $P(A)=p$. Define the random variable $X$ as follows:
$$
X= \begin{cases}
1, & \text{if } A \text{ occurs} \\
0, & \text{if } A \text{ does not occur}
\end{cases}
$$
Calculate the variance of $X$ in terms of $p$. | p(1-p) | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the value of the fraction $\frac{x+3}{x-4}$ is $0$, find the value of $x$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of expanding the expression $\left(a-b\right)^{2}$. Express your answer as an algebraic expression. | a^{2} - 2ab + b^{2} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of $\left(x^3 - \frac{2}{x}\right)^4$. Express your answer as a single integer, including the negative sign if applicable. | -32 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given $a > 1$, $b > 1$, and $$\frac {1}{a-1} + \frac {1}{b-1} = 1$$, find the minimum value of $a + 4b$. Express your answer as a single number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, it is known that a_3 + a_5 = 10. Using the properties of the arithmetic sequence, find the value of a_4. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
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