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 |
|---|---|---|---|---|---|
The numbers 1 to 9 are randomly placed into a 3x3 grid such that each cell contains exactly one number and each number is used exactly once. Calculate the probability that the sum of the numbers in each row and each column is odd. Express your answer as a simplified fraction. | \frac{1}{14} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $\frac{x^{2}}{2}-y^{2}=1$ with left and right foci $F_1(-\sqrt{3}, 0)$ and $F_2(\sqrt{3}, 0)$, respectively, and a point $M(x_0, y_0)$ on the hyperbola that satisfies $\overrightarrow{MF_1} \cdot \overrightarrow{MF_2} \leq 0$, determine the range of values for $y_0$. Express your answer as an interv... | [-\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}] | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A woman is not good at weaving cloth, and the amount of cloth she weaves decreases by the same amount every day. She weaves 5 feet on the first day and 1 foot on the last day. Calculate the total amount of cloth she weaves in 30 days. Express your answer as a whole number in feet. | 90 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the eccentricity of the ellipse $\frac{x^{2}}{m} + \frac{y^{2}}{2} = 1$ with a focus on the $y$-axis is $\frac{2}{3}$, find the value of $m$. Express your answer as a single value. | \frac{10}{9} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the circle has center \( O \) and square \( OPQR \) has vertex \( Q \) on the circle. If the area of the circle is \( 72 \pi \), calculate the area of the square. Express your answer as a whole number. | 36 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $$\\overrightarrow {a}=(0，1，-1)$$ and $$\\overrightarrow {b}=(1，0，2)$$, find the value of k such that vector $$k \\overrightarrow {a}+ \\overrightarrow {b}$$ is perpendicular to vector $$\\overrightarrow {a}- \\overrightarrow {b}$$. Express your answer as a single number. | \\frac{7}{4} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)= \sqrt {2-x}+\log (x+1)$. Express your answer as an interval. | (-1,2] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the function y = x * e^x at the point (1, e). Express your answer in the form y = mx + b, where m and b are expressed in terms of e. | y = 2ex - e | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the derivative of the function f(x) = sin^2x. Express your answer as a mathematical expression using standard notation. | \sin 2x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, point $D$ is on side $AB$, and $\overrightarrow{BD}=\dfrac{1}{2}\overrightarrow{DA}$. Let $\overrightarrow{CB}=a$ and $\overrightarrow{CA}=b$. Calculate $\overrightarrow{CD}$ in terms of $a$ and $b$, expressing your answer as a linear combination of these vectors. | \dfrac{2}{3}a+ \dfrac{1}{3}b | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
If $x^{2}-2mx+16$ is a perfect square trinomial, calculate the value(s) of $m$. Express your answer in the format $\\boxed{[answer]}$, including all possible values of $m$. | 4 \text{ or } -4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = cos(ωx + π/4), find the minimum value of ω > 0 such that shifting the graph to the left by π/3 units results in an odd function. Express your answer as a single real number. | \frac{3}{4} | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Let \(a\) be a positive integer less than 100. If \(a^{3}+23\) is divisible by 24, how many such values of \(a\) are there? Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {x_n}, x_1 = 2, x_2 = 7, and x_{n+2} is equal to the units digit of x_n * x_{n+1}. Find the value of x_{1998}. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f\left(x\right)=\left\{\begin{array}{l}{{x}^{2}-2ax+2,x≤1}\\{x+\frac{16}{x}-3a,x＞1}\end{array}\right.$, find the range of values for $a$ such that the minimum value of $f\left(x\right)$ is $f\left(1\right)$. Express your answer as an interval or range, including any necessary mathematical notation o... | [1,5] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the polynomial $y^2+3y-a$ contains the factor $y-3$, find the value of the constant $a$. Express your answer as a single integer. | 18 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{1,2,3,4\}$ and $\complement_{U}A=\{2\}$, find the number of proper subsets of set $A$. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $\frac{S_3}{S_6} = \frac{1}{3}$, then find the value of $\frac{S_6}{S_12}$. Express your answer as a simplified fraction. | \frac{3}{10} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of linear equations in two variables x and y:
\[
\left\{
\begin{array}{l}
3x+2y=k+1 \\
2x+3y=k
\end{array}
\right.
\]
If the solution satisfies x + y = 3, calculate the value of k. | 7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sum of the first $n$ terms of the sequence $\{a_{n}\}$ as ${S}_{n}={n}^{2}+\frac{{a}_{1}}{2}$, calculate the value of $a_1 + a_5$. Express your answer as a single integer. | 11 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In an isosceles triangle \(ABC\), the base \(BC\) is \(\frac{1}{4}\) of the perimeter. From any point \(P\) on the base \(BC\), draw a line parallel to the two legs, intersecting the legs at points \(E\) and \(F\), respectively. Find the ratio of the perimeter of quadrilateral \(AEPF\) to the perimeter of \(\triangle A... | \frac{3}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The sum of two numbers is S. Suppose 3 is added to each number and then each of the resulting numbers is doubled. Express the sum of the final two numbers in terms of S. | 2S + 12 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A line and three non-collinear points not on the line are given. Determine the maximum number of planes that can be formed using these. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\{(x,y)|y=x^{2}\}$ and set $B=\{(x,y)|y=1-|x|\}$, find the number of proper subsets of the intersection of sets A and B, $A \cap B$. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\log_{10}{m}= b-\log_{10}{n}$, solve for $m$. Express your answer as a mathematical expression in terms of $b$ and $n$. | \frac{10^b}{n} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation $a(b-c)x^2+b(c-a)x+c(a-b)=0$ with one root known to be $1$, find the other root using Vieta's formulas. Express your answer as a fraction in terms of $a$, $b$, and $c$. | \frac{c(a-b)}{a(b-c)} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (2x-y)^2 + A = (2x+y)^2, solve for the algebraic expression A. Express your answer as a simplified algebraic expression. | A = 8xy | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given set A = {x | |x-1| ≤ 2} and set B = {x | x^2 - 4x > 0, x ∈ ℝ}, calculate A ∩ (ℝ \ B). Express your answer as an interval. | [0, 3] | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In triangle \( ABC \), \(\angle A\) is \(21^{\circ}\) more than \(\angle B\), and \(\angle C\) is \(36^{\circ}\) more than \(\angle B\). Find the size of \(\angle B\). Express your answer in degrees as a whole number. | 41 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^5 - 5x^4 + 10x^3 - 10x^2 + 5x - 1, where x is a real number, calculate f^{-1}(1). Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (m-2)x^|m| + 3mx + 1 = 0, find the value of m that makes the equation a quadratic equation in x. Express your answer as a single value. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Let O be the circumcenter of an acute triangle ABC with ∠C = 75°. If the areas of triangles AOB, BOC, and COA satisfy the relation S_{△AOB} + S_{△BOC} = √3S_{△COA}, then find the measure of ∠A in degrees. | 45 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given two finite decimal numbers where the integer parts are 7 and 10 respectively, determine the number of possible integer values for the integer part of their product. Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, given that point P(m,n) is in the second quadrant and the distance from point P to the x-axis is 3, what is the correct conclusion about the coordinates of point P? Express your answer in the format n =?, clearly stating the value of n based on the given conditions. | n = 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the planar vectors $\overrightarrow{a}=(1,x)$ and $\overrightarrow{b}=(-2,3)$, determine the value of the real number $x$ if $\overrightarrow{a}$ is parallel to $\overrightarrow{b}$. Express your answer as a real number. | -\frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
If $9-mx+x^2$ is a perfect square, find the value of $m$. Express your answer as a numerical value, including the ± symbol if applicable. | \pm6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the regression line equation $\hat y= \frac {1}{4}x+a$ and the data points $(x_i, y_i)$ ($i=1,2,\ldots,6$), where $x_1+x_2+\ldots+x_6=10$ and $y_1+y_2+\ldots+y_6=4$, calculate the value of the real number $a$. Express your answer as a single number. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given that point P(m,2) lies on the y-axis, what is the value of m? Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear function equation y = 3x + 2, identify the slope and intercept. Express your answer as a pair of numerical values (slope, intercept). | (3, 2) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Linear Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $$f(x)= \begin{cases} f(x+3) & (x<6) \\ \log_{2}x & (x\geq 6) \end{cases}$$, calculate the value of $f(-1)$. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Convert the decimal number 51 into its binary representation, denoted with a subscript 2 (e.g., XXXXXXX_{(2)}). | 110011_{(2)} | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the fractional equation $\frac{x}{x-2}-2=\frac{m}{x-2}$, find the value of $m$ when the equation has a root. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $x^{2} \leqslant 4$ and express the solution set in interval notation. | [-2, 2] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
If the value of the fraction (x+5)/(x-2) is 0, then what is the value of x? Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given a positive sequence $\{a_n\}$ that satisfies $a_{n+1}(a_{n+1}-2a_{n})=9-a_{n}^{2}$, and $a_{1}=1$, find the value of $a_{10}$. Express your answer as a single integer. | 28 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
If $m$ and $n$ are positive real numbers, and both equations $x^2 + mx + 2n = 0$ and $x^2 + 2nx + m = 0$ have real roots, then find the minimum value of $m + n$. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_2 and a_6 are the two roots of the equation x^2 - 34x + 64 = 0. Find the value of a_4. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $M=\{-1,1,2,3,4,5\}$, $N=\{1,2,4\}$, and $P=M\cap N$, calculate the number of proper subsets of set $P$. Express your answer as a single numerical value. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 1$ and $|\overrightarrow{b}| = 2$, and the projection of $\overrightarrow{a}$ onto $\overrightarrow{b}$ is equal to the projection of $\overrightarrow{b}$ onto $\overrightarrow{a}$, find the magnitude of the vector differen... | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Let $\overrightarrow{a} = (1, -2)$, $\overrightarrow{b} = (-3, 4)$, $\overrightarrow{c} = (3, 2)$. Calculate the value of $(\overrightarrow{a} + 2\overrightarrow{b}) \cdot \overrightarrow{c}$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=e^{x}+ae^{-x}$, which is an even function, if the slope of a tangent line to the curve $y=f(x)$ is $\frac{3}{2}$, then find the $x$-coordinate of the tangent point. Express your answer as a single value, including any necessary mathematical notation. | \ln 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function $f(x)$ is $\left[1,+\infty \right)$, determine the domain of the function $y=f(x-1)+f(4-x)$. Express your answer as an interval. | \left[2,3\right] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {0} and B = {0, 1, 2}, find the intersection of sets A and B, denoted as A ∩ B. Express your answer as a set using standard set notation with curly braces {}. | {0} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given that the value of $x^{2}-4x+1$ is $3$, calculate the value of the algebraic expression $3x^{2}-12x-1$. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of linear inequalities in one variable x: 2(x+1) < x+3 and x-a ≤ a+5, where the solution set for x is x < 1, and a is a non-positive integer, calculate the number of values of a that satisfy the conditions. | 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
The time on a digital clock is 10:25. What is the shortest length of time, in minutes, until all the digits on the clock will be equal to one another? Provide your answer as a whole number. | 46 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
In an election, Harold received 60% of the votes and Jacquie received all the rest. If Harold won by 24 votes, calculate the total number of votes cast in the election. Express your answer as a whole number. | 120 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A regular octahedron is formed by joining the centers of adjoining faces of a cube. Calculate the ratio of the volume of the octahedron to the volume of the cube. Express your answer as a simplified fraction. | \frac{1}{6} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given \(f(x)=3^{x}+3^{-x}\), if \(f(a)=4\), then calculate \(f(2a)\). Express your answer as a single integer. | 14 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
A rectangular solid has dimensions of length, width, and height as 2, 2, and 1, respectively. If its vertices lie on the same spherical surface, calculate the surface area of the sphere. Express your answer in terms of π. | 9\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence {a_n} where a_3 + a_4 + a_5 = 12, find the sum a_1 + a_2 +... + a_7. Express your answer as a single numerical value. | 28 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Find the complex conjugate of the complex number $z= \frac{1}{1+i}$. Express your answer as a complex number in the form $a+bi$, and include it in the boxed notation. | \frac{1}{2}+\frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of (a - 1/sqrt(a))^7, find the coefficient of a^(-1/2). Express your answer as a single integer. | -21 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
The volume of a rectangular box is 144 cm³. If its length is 12 cm and its width is 6 cm, what is its height? Express your answer in centimeters. | 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
What is the difference between 25% of £37 and 25% of £17? Express your answer in pounds (£) with two decimal places. | £5.00 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x ln x, find the value of x_0 such that f'(x_0) = 2. | e | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\frac{1000^2}{252^2-248^2}$. Provide your answer as a single number. | 500 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ with the first term $a_1 = 1$ and common ratio $q = 2$, calculate the sum $\log_2 a_1 + \log_2 a_2 + \ldots + \log_2 a_{11}$. Express your answer as a single integer. | 55 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence $\{a_n\}$, $a_1 + a_3 = 10$ and $a_2 + a_4 = 30$. Calculate the sum of the first five terms of the sequence $S_5$. | 121 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Let $M$ be a point inside $\triangle ABC$, and the area of $\triangle ABC$ is $2$. Define $f(M)=(m,n,p)$, where $m$, $n$, $p$ are the areas of $\triangle MBC$, $\triangle MCA$, $\triangle MAB$ respectively. If a moving point $P$ inside $\triangle ABC$ satisfies $f(P)=(1,x,y)$, find the minimum value of $\dfrac{1}{x}+ \... | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
What is the correct format of an assignment statement in programming? Use the variables S and a, and the expression a+1, to demonstrate the correct format. Express your answer as a specific assignment statement. | S=a+1 | big_math | [
"Mathematics -> Other -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Let $T = \{(x, y) | ax + y - 3 = 0\}$ and $S = \{(x, y) | x - y - b = 0\}$. If $S \cap T = \{(2, 1)\}$, then find the values of $a$ and $b$. Express your answer in the form $a = [value], b = [value]$. | a = 1, b = 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Three balls numbered 1, 2, and 3 are placed in a bag. A ball is drawn from the bag and the number is recorded. The ball is returned to the bag. After this has been done three times, find the probability that the sum of the three recorded numbers is less than 8. Express your answer as a fraction. | \frac{23}{27} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A square floor is tiled with a large number of regular hexagonal tiles, coloured blue or white. Each blue tile is surrounded by 6 white tiles, and each white tile is surrounded by 3 white and 3 blue tiles. Ignoring part tiles, calculate the exact ratio of the number of blue tiles to the number of white tiles. Express y... | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Two circles with radii 13 and 5 intersect, and the distance between their centers is 12. Find the length of their common chord. Express your answer as a single number. | 10 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $i(1-z)=-1$, find the imaginary part of $z$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
There are 4 students signing up for two extracurricular activity groups. Each student can only sign up for one of the groups. Calculate the total number of different sign-up methods. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $(a^{3})^{4}$. Express your answer as a single algebraic expression. | a^{12} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\left\{0,1,2,3\right\}$ and $B=\left\{x\left|\right.x^{2}-2x-3\geqslant 0\right\}$, describe the union of set $A$ and the complement of set $B$ in the real numbers, $A\cup \complement _{R}B$, using interval notation. | (-1,3] | big_math | [
"Mathematics -> Set Theory -> Set Operations"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, the logarithmic equation log_2(a_2a_{98}) = 4 holds. Calculate the product a_{40}a_{60}. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given ten piano keys, calculate the total number of unique chords that can be played by simultaneously pressing 3, 4, 5,..., 10 keys, where any difference in at least one key results in a different chord. Express your answer as a single integer. | 968 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
5 people stand in a row. Calculate the total number of ways to arrange them so that person A and person B are not adjacent to each other. Express your answer as a whole number. | 72 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of lattice points on the line segment with endpoints (3, 17) and (48, 281), including the endpoints. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.046875 | 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 -> Equations and Inequalities -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the following four functions:
1. $f(x) = x + 1$;
2. $f(x) = \frac{1}{x}$;
3. $f(x) = 2x^2$;
4. $f(x) = 2^x + \lg(x - 1)$.
Determine the number of functions that are increasing on the interval $(0, +\infty)$. Provide your answer as a whole number. | 2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given two points O(0,0) and Q(a,b), point P1 is the midpoint of the line segment OQ, point P2 is the midpoint of the line segment QP1, P3 is the midpoint of the line segment P1P2, and so on, Pn+2 is the midpoint of the line segment PnPn+1. Calculate the limit position of point Pn as n approaches infinity. Express your ... | \left( \frac{2a}{3}, \frac{2b}{3} \right) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{z}{{1-2i}}=i$, where $i$ is the imaginary unit, calculate the value of $z$. Express your answer as a complex number in the form $a+bi$. | 2+i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C: \frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$ with its left vertex at $A$ and point $B(0, \frac{\sqrt{15}}{3}b)$, if the perpendicular bisector of segment $AB$ passes through the right focus $F$, find the eccentricity of hyperbola $C$. Express your answer as a single numerical ... | 2 | big_math | [
"Mathematics -> Geometry -> Analytic Geometry -> Conics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ satisfying $z+2\bar{z}=-3-i$, calculate the magnitude of $z$, denoted as $|z|$. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given two circles C1: $x^{2}+y^{2}+2x+y-2=0$ and C2: $x^{2}+y^{2}-4x-2y+4=0$, determine the exact number of common tangent lines between them. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = ax^3 + x + 1, find the value of the real number 'a' such that the graph of f(x) intersects the x-axis at exactly two points. Express your answer as a single real number. | -\frac{4}{27} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In a right triangle \( \triangle ABC \), the lengths of the two legs are 5 and 12. Two arcs are drawn with centers at points \( A \) and \( B \) and radii 12 and 5, respectively. These arcs intersect the hypotenuse at points \( M \) and \( N \). Calculate the length of the segment \( MN \). Express your answer as a sin... | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=2^{-3}$, $b=3^{\frac{1}{2}}$, and $c=\log_{2}{5}$, list the variables in order from smallest to largest, separated by commas. | a, b, c | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the lengths of two sides of a triangle are 5 and 9, use the triangle inequality theorem to find the range for the length of the third side. Express your answer in the format a < x < b. | 4 < x < 14 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Ryan had finished cutting 1/2 of his lawn by 9 a.m. and 7/8 of his lawn by 10 a.m. If Ryan cut his lawn at a constant rate, at what time did he finish cutting his lawn? Express your answer in the format HH:MM a.m./p.m. | 10:20 a.m. | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function $y=\frac{x^2}{x+3}$. Express your answer as a mathematical expression in the form $\boxed{\text{derivative expression}}$. | \frac{x^2+6x}{(x+3)^2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Chris received a mark of 50% on a recent test. Chris answered 13 of the first 20 questions correctly. Chris also answered 25% of the remaining questions on the test correctly. If each question on the test was worth one mark, calculate the total number of questions on the test. | 32 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of zeroes for the function f(x) = 2x - 5 - ln x. Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_4 = 2 and a_7 = 5. Calculate the sum of the first 10 terms of the sequence {log a_n}. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
A town experiences four consecutive yearly changes in its population census: a 25% increase, a 25% increase, a 25% decrease, and a 25% decrease. Calculate the net percentage change over these four years. Express your answer as a percentage, rounded to the nearest percent. | -12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
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