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 |
|---|---|---|---|---|---|
Given $\cos\left(\alpha+ \frac{\pi}{4}\right) = \frac{7\sqrt{2}}{10}$ and $\cos 2\alpha = \frac{7}{25}$, calculate $\sin\alpha + \cos\alpha$. Express your answer as a fraction. | \frac{1}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
How many three-digit numbers have at least one 2 and at least one 3? Express your answer as a single integer. | 52 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Suppose that \(x\) and \(y\) are real numbers that satisfy the two equations:
\[
\begin{gathered}
x^{2} + 3xy + y^{2} = 909 \\
3x^{2} + xy + 3y^{2} = 1287
\end{gathered}
\]
Find a possible value for \(x + y\). Express your answer as a single numerical value. | 27 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence {a_n}, a_7 = 12. Find the value of a_3 + a_11. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given $3^{m}=5$ and $9^{n}=10$, calculate the value of $3^{m+2n}$. Express your answer as a single integer. | 50 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of positive integers k < 100 such that 2(3^(6n)) + k(2^(3n+1)) - 1 is divisible by 7 for any positive integer n. Express your answer as a single integer. | 14 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}=(-1,4)$ and $\overrightarrow {b}=(x,2)$, if $\overrightarrow {a}$ is parallel to $\overrightarrow {b}$, find the value of $x$. Express your answer as a single number. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equations of a hyperbola and an ellipse, $\frac{x^{2}}{1-m} + \frac{y^{2}}{m+2}=1$ and $\frac{x^{2}}{2m} + \frac{y^{2}}{2-m}=1$, respectively, find the range of values for $m$ that satisfy the compound proposition $p \land q$, where $p$ states that the first equation represents a hyperbola and $q$ states that... | 1 < m < 2 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let the set $U=\{x|x<5, x\in \mathbb{N}^*\}$, and $M=\{x|x^2-5x+6=0\}$. Calculate the complement of $M$ in $U$, denoted as $\complement_U M$. Express your answer as a set of integers, including all elements of the complement set. | \{1, 4\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x) = \begin{cases} x^{3}+\sin x, & -1\leq x\leq 1 \\ 2, & 1<x\leq 2 \end{cases}$, find the value of the definite integral $\int_{-1}^{2} f(x) \, dx$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
If none of the faces of a tetrahedron are isosceles triangles, what is the minimum number of edges with different lengths? Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Find the total number of non-negative integer solutions to the equation $x_1+2x_2+x_3+x_4+\ldots+x_{10}=3$. Express your answer as a single integer. | 174 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($0 < b < a$), a line passing through the center intersects the ellipse at points $A$ and $B$, and the right focus is $F_2(c,0)$. Calculate the maximum area of $\triangle ABF_2$ and express your answer as a product of two variables. | bc | big_math | [
"Mathematics -> Geometry -> Analytic Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A square has a perimeter of 24 units. A rectangle has the same area as the square. If the width of the rectangle is 4 units, find the perimeter of the rectangle. Express your answer in units as a whole number. | 26 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
A school clinic wants to distribute 23 identical thermometers among 10 senior classes, with each class receiving at least 2 thermometers. Calculate the number of different distribution methods that satisfy this condition. | 220 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the progression $10^{\dfrac{1}{11}}, 10^{\dfrac{2}{11}}, 10^{\dfrac{3}{11}}, 10^{\dfrac{4}{11}},\dots, 10^{\dfrac{n}{11}}$, find the least positive integer $n$ such that the product of the first $n$ terms of the progression exceeds $100,000$. Express your answer as a single integer. | 11 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given $x > 2$, find the minimum value of the expression $\frac{4}{x-2}+4x$. Express your answer as a single number. | 16 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given that f(x) is an odd function and g(x) is an even function, and f(x) + g(x) = 1/(x-1), calculate the value of f(3). Express your answer as a simplified fraction. | \dfrac {3}{8} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = 1 + 2i, find its conjugate \\overline{z}. Express your answer as a complex number in the form a + bi, and include it in the boxed notation. | 1 - 2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $i(1+i)$. Express your answer as a numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the quadratic equation $x^{2} = 9$ for its roots. Express your answer using the ± symbol if the roots have the same absolute value but opposite signs. | \pm 3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Line $l_2$ intersects line $l_1$ and line $l_3$ is parallel to $l_1$. The three lines are distinct and lie in a plane. Calculate the number of points equidistant from all three lines. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\{1,2,3\}$, find the total number of proper subsets of set $A$. Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} (\frac{1}{2})^x, & x \geq 4 \\ f(x+1), & x < 4 \end{cases}$, evaluate $f(\log_2 3)$. Express your answer as a fraction. | \frac{1}{24} | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\sin \left(\frac{\pi}{4}-\alpha\right)=\frac{3}{5}$, and $-\frac{\pi}{4} < \alpha < 0$, calculate the value of $\cos 2\alpha$. Express your answer as a simplified fraction. | \frac{24}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The distance from the top of the can on the floor to the top of the bottle on the table is $150 \textrm{ cm}$. The distance from the top of the bottle on the floor to the top of the can on the table is $110 \textrm{ cm}$. Find the height of the table. Express your answer in centimeters as a whole number. | 130 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that sin(alpha) = 3/4 and alpha is an angle in the second quadrant, find the value of sin(2alpha + pi/6). Express your answer as a simplified fraction or decimal, including any negative signs and square roots if applicable. | -\frac{3\sqrt{21}+1}{16} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given $(1-2x)^{11} = a_0 + a_1x + a_2x^2 + \ldots + a_{11}x^{11}$, calculate the sum $a_2 + a_3 + \ldots + a_{11}$. Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let complex numbers $z_1$ and $z_2$ correspond to points in the complex plane that are symmetric about the origin, and $z_1 = 2-i$. Calculate the product $z_1 \cdot z_2$. Express your answer as a complex number in the form $a+bi$. | -3+4i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Distribute 7 recommended places among 3 key high schools in such a way that each school receives at least one place. Calculate the number of different ways to achieve this distribution. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac{a+i}{2i}$, where $a \in \mathbb{R}$ and $i$ is the imaginary unit, and the real part of $z$ is equal to its imaginary part, calculate the modulus of $z$. Express your answer as a numerical value. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
A school decided to send 2 Chinese language teachers and 4 mathematics teachers to two schools, A and B. It is stipulated that each school should receive 3 teachers, including both Chinese language and mathematics teachers. Find the number of different arrangement plans. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Let vector $\overrightarrow {m} = (2x-1, 3)$, and vector $\overrightarrow {n} = (1, -1)$. If $\overrightarrow {m} \perp \overrightarrow {n}$, calculate the value of the real number $x$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $A=\left\{ y \mid y=e^{x}, x \in \mathbb{R} \right\}$, and $B=\left\{ x \in \mathbb{R} \mid x^{2}-x-6 \leqslant 0 \right\}$, find the intersection $A \cap B$. Express your answer as an interval. | (0,3] | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function f(x) = sin^2 x + sqrt(3) sin x cos x in the interval [pi/4, pi/2]. Express your answer as a single number in the boxed notation. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Starting from one vertex of a rectangular solid, the lengths of the three edges are $2$, $3$, and $4$. Calculate the length of its diagonal. Express your answer as a single value, possibly with a square root. | \sqrt{29} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that 1 pimi = 0.001 nanometers and 1 nanometer = 10^{-9} meters, calculate the value of 1 pimi in meters. Express your answer in scientific notation (e.g., 10^{-x}) and include the unit (meters). | 10^{-12} \text{ meters} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given that point \( A \) is the left vertex of the hyperbola \( x^{2} - y^{2} = 1 \), and points \( B \) and \( C \) are on the right branch of the hyperbola such that \( \triangle ABC \) is an equilateral triangle, calculate the area of \( \triangle ABC \). Express your answer as a single value with radicals if necess... | 3 \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
The surface area of the Earth is approximately 780,000,000 km^2. Represent this value using scientific notation in the form of a×10^n, where 1≤|a|＜10, and n is an integer. Include units in your answer. | 7.8×10^8 km^2 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Define the set operation: $A*B=\{z|z=xy, x \in A, y \in B\}$. Let $A=\{1, 2\}$, $B=\{0, 1, 2\}$. Calculate the sum of the elements of all subsets of the set $A*B$, including the empty set. Express your answer as a single integer. | 56 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality system for x: $\left\{\begin{array}{l}{\frac{x-a}{3}＜0}\\{2(x-5)＜3x-8}\end{array}\right.$, where the solution set contains only the integers -1 and 0, and the equation $10a = 2m + 5$, find the range of m. Express your answer as an interval or range. | (-2.5, 2.5] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient $a_2$ of the $x^2$ term in the expansion of $(3x+1)^5$. Express your answer as a numerical value. | 90 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
If the monomials $x^{a+1}y^{3}$ and $\dfrac{1}{2}y^{b}x^{2}$ are like terms, find the values of $a$ and $b$. Express your answer as a pair of values in the format $(a, b)$. | (1, 3) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the value of the algebraic expression $x-2y$ is 3, calculate the value of the algebraic expression $4y+1-2x$. Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Suppose for an arithmetic sequence $\{a_n\}$, the sum of the first $n$ terms is defined as $S_n$. Given that $S_{m-1}=-2$, $S_m=0$, and $S_{m+1}=3$, find the value of $m$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x|x^{2}-4x\leqslant 0,x\in Z\}$ and $B=\{x\left|\right.-1\leqslant x \lt 4\}$, calculate the intersection $A\cap B$. Express your answer as a set of integers, including all elements of the intersection. | \{0,1,2,3\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the expression $\sqrt{5^{2}-4^{2}}$. Provide your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $F_1$ and $F_2$ are the left and right foci of the ellipse $\frac{x^2}{9}+\frac{y^2}{4}=1$, and $P$ is a point on the ellipse. If $|PF_1|=m$ and $|PF_2|=n$, find the minimum value of $\frac{1}{m}+\frac{4}{n}$. Express your answer as a single number. | \frac{3}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $\triangle ABC$, the sides opposite to angles $\angle A$, $\angle B$, and $\angle C$ are $a$, $b$, and $c$ respectively. If $\angle A = \frac{\pi}{3}$, $a = \sqrt{6}$, and $b = 2$, calculate the measure of angle $\angle B$ in radians. Express your answer in the format of a single value, e.g., $\frac{\pi}{4}... | \frac{\pi}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)$ defined on $\mathbb{R}$ as $f(x) = \begin{cases} \log_{2}(1-x), & x \leqslant 0 \\ f(x-1) - f(x-2), & x > 0 \end{cases}$, find the value of $f(2017)$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Functions -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the set equality $\{a, \frac{b}{a}, 1\} = \{a^{2}, a+b, 0\}$, find the value of $a^{2012} + b^{2013}$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that -5 is a root of the quadratic equation $x^2 = c^2$, find the constant $c$. Express your answer in the form $\boxed{±\text{ a number}}$. | ±5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy $|\overrightarrow{a}| = 2$, $|\overrightarrow{b}| = 1$, and $\overrightarrow{a} \cdot \overrightarrow{b} = -1$, calculate the magnitude of the vector $2\overrightarrow{a} + \overrightarrow{b}$. Express your answer as a square root, including the s... | \sqrt{13} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \(5 x_{1} + 6 x_{2} - 7 x_{3} + 4 x_{4} = 1\), find the minimum value of the expression \(3 x_{1}^{2} + 2 x_{2}^{2} + 5 x_{3}^{2} + x_{4}^{2}\). Provide your answer as a single value. | \frac{15}{782} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
For a geometric sequence {a_n}, the sum of the first n terms is S_n. Given that a_1 + a_2 + a_3 = 3 and a_4 + a_5 + a_6 = 6, find S_{12}. Express your answer as a single integer. | 45 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given two points in the Cartesian coordinate system $xOy$, $A(\cos 110^{\circ}, \sin 110^{\circ})$ and $B(\cos 50^{\circ}, \sin 50^{\circ})$, calculate the dot product $\overrightarrow{OA} \cdot \overrightarrow{OB}$. Express your answer as a numerical value. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = \begin{cases} x - 2, & x \geqslant 10 \\ f(x + 6), & x < 10 \end{cases}$, find the value of $f(5)$. Express your answer as a single number. | 9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (sin(alpha), -3) and b = (1, cos(alpha)) that are perpendicular, calculate tan(alpha + pi/4). Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Trigonometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of inequalities about x: \left\{\begin{array}{l}1-2x>-3\\ x-a\geq 0\end{array}\right., and knowing that this system has a total of 5 integer solutions, describe the range of values for a. | -4 < a \leq -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $(-\frac{1}{2})×(-2)$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation $(x+1)^2 + (y-1)^2 = 4$, find the center coordinates $(h, k)$ and the radius $r$. Express your answer in the format: $\boxed{(h, k), r}$. | (-1, 1), 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, -2)$ and $\overrightarrow{b} = (-3, 2)$. If the directed line segments representing vectors $3\overrightarrow{a}$, $2\overrightarrow{b} - \overrightarrow{a}$, and $\overrightarrow{c}$ can be connected head-to-tail to form a triangle, then calculate the value of $\overrightarrow{a... | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
If the graph of the function f(x) = sin(2x) + cos(2x) is shifted to the left by φ units, and the resulting graph is symmetric about the y-axis, find the minimum positive value of φ. Express your answer in radians as a fraction of π. | \frac {\pi}{8} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
There are 18 teams participating in the opening ceremony of a competition. When entering the venue, the first team has 27 members, the second team has 26 members,..., and the 18th team has 10 members. If they all enter the venue in a single file and are assigned numbers from 1 to 333 in the order of their entrance, how... | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function y=f(x) is an even function on R, and when x_1, x_2∈(0,+∞), we always have (x_1-x_2)⋅[f(x_1)-f(x_2)] < 0. Let a=\\ln \\frac {1}{π}, b=(\\ln π)^{2}, c=\\ln \\sqrt {π}. Determine the relationship between f(a), f(b), and f(c). Express your answer as a relationship between f(a), f(b), and f(c), e.g.,... | f(c) > f(a) > f(b) | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Peyton puts 30 liters of oil and 15 liters of vinegar into a large empty can. He then adds 15 liters of oil to create a new mixture. What percentage of the new mixture is oil? Express your answer as a percentage. | 75 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\{a_n\}$ is a geometric sequence with $a_n > 0$, and $a_2a_4 + 2a_3a_5 + a_4a_6 = 25$, find the value of $a_3 + a_5$. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Joan and Grant are shopping at the deli for lunchmeat. If Joan buys 3 pounds of bologna for $2.80 per pound, and Grant buys 2 pounds of pastrami for $1.80 per pound, then what is the average (arithmetic mean) price per pound of all the lunchmeat they buy? Express your answer in dollars per pound, rounded to 2 decimal p... | 2.40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A moving point P lies on the circle $C: (x - 3)^2 + (y + 4)^2 = 4$. If point O is the origin of the coordinate system, then find the maximum value of $|OP|$. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the greatest common divisor (GCD) of 840 and 1764 using the Euclidean algorithm. Express your answer as a single integer. | 84 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of \\( \\left( \\sqrt{2}x- \\frac{1}{{x}^{2}}\\right)^{3} \\). Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is an even function defined on R, when x <= 0, f(x) = 3x^2 - x + 2a + 1. If f(2) = 13, find the value of a. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
What number should go in the $\square$ to make the equation $\frac{3}{4}+\frac{4}{\square}=1$ true? Provide your answer as a single number that solves the equation. | 16 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola equation y = 2x^2 + ax - 5 and the condition a < 0, determine the quadrant in which the vertex of the parabola lies. Express your answer as the quadrant number (I, II, III, or IV). | IV | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Akira and Jamie weigh 101 kg together, Akira and Rabia weigh 91 kg together, and Rabia and Jamie weigh 88 kg together. Find Akira's weight in kilograms. | 52 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The probability of a family telephone being answered at the first ring is 0.1, at the second ring is 0.3, at the third ring is 0.4, and at the fourth ring is 0.1. Calculate the probability that the telephone is answered within the first four rings. Express your answer as a decimal value between 0 and 1. | 0.9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Randomly select two different numbers from 1, 2, and 3, denoted as m and n, for the coordinates of point A, (m,n). Calculate the probability that point A lies on the hyperbola y = 6/x. Express your answer as a simplified fraction. | 1/3 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Two fair cubic dice, each with faces labeled with dot numbers (1), (2), (3), (4), (5), (6), are rolled sequentially. Let (x, y) represent the dot numbers on the upper faces of the two dice. Find the probability that (log_{2x} y = 1). Express your answer as a simplified fraction. | \frac{1}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Probability"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Count the number of positive integers less than 1000 that are not divisible by 5 and 7. Express your answer as a whole number. | 686 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
The parabola $C: x^{2}=4ay$ passes through the point $\left(-2,1
ight)$. Find the equation of the directrix of $C$. Express your answer in the form $y =...$. | y = -1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of real solutions to the equation 4x^2 - 40[x] + 51 = 0, where [x] represents the greatest integer less than or equal to x. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given a non-zero sequence {a_n} that satisfies a_{n+1}^{2}=a_{n}a_{n+2}, and 32a_{8}-a_{3}=0, let S_{n} be the sum of the first n terms of the sequence. Determine the value of S_{6}/(a_{1}-S_{3}). Express your answer as a fraction in simplest form. | -\frac{21}{8} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the area enclosed by the curve $y=x^2$ and the line $y=x$. Express your answer as a single number. | \frac{1}{6} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
If $\cos(\alpha + \beta) = \frac{1}{5}$ and $\cos(\alpha - \beta) = \frac{3}{5}$, find the value of $\tan \alpha \cdot \tan \beta$. Express your answer as a simplified fraction. | \frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given the real numbers \( a \) and \( b \) satisfying \(\frac{4}{a^{4}}-\frac{2}{a^{2}}-3=0\) and \(b^{4}+b^{2}-3=0\), respectively, calculate the value of the algebraic expression \(\frac{a^{4} b^{4}+4}{a^{4}}\). Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Operations"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = -x^2 + 4x - 3, if a real number x_0 is randomly chosen from the interval [2,6], calculate the probability that f(x_0) >= 0. Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the definite integral $\int_{0}^{1}\left(2x+e^{x}\right)dx$. Express your answer as a single value, potentially involving $e$. | e | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
In the binomial expansion of $( \sqrt{x} - \frac{1}{x})^{10}$, calculate the coefficient of the $x^{2}$ term. Express your answer as an integer. | 45 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric form of the ellipse $\begin{cases} x=3\cos \phi \\\\ y=4\sin \phi\\end{cases}$ where $\phi$ is the parameter, find the length of the major axis. Express your answer as a single number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
The minute hand on a clock points at the 12. The minute hand then rotates 120 degrees clockwise. What number on the clock face will the minute hand be pointing at after the rotation? Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Let the arithmetic sequence $\{a_n\}$ have a common difference $d \neq 0$, and $a_1 = 9d$. If $a_k$ is the geometric mean of $a_1$ and $a_{2k}$, find the value of $k$ that satisfies this condition. Express your answer as a single positive integer. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $|x-1|-|x-5| < 2$ and express the solution set in interval notation. | (-\infty,4) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x= \frac {1+yi}{1+i}$, where $i$ is the imaginary unit, $x$, $y$ are real numbers, calculate the value of $x+y$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\dfrac{11!-10!}{9!}$. Express your answer as a single integer. | 100 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function $f(x)=\begin{cases} x^{2}-ax+5,x < 1, \\ 1+ \frac{1}{x},x\geqslant 1 \end{cases}$ is monotonic on $\mathbb{R}$, determine the range of values for the real number $a$. Express your answer as an interval (e.g. [a, b]). | [2,4] | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the fraction: \(\frac{1+2+3+4+5}{2+4+6+8+10}\). Express your answer as a simplified fraction in the form \(\boxed{\frac{a}{b}}\). | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
The fiscal revenue \(x\) and expenditure \(y\) of a certain region satisfy the linear regression equation \(y = bx + a + e\) (unit: billion yuan), where \(b = 0.8\), \(a = 2\), and \(|e| \leqslant 0.5\). If the fiscal revenue of this region this year is 10 billion yuan, calculate the maximum expected annual expenditure... | 10.5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the foci of the ellipse $\frac{x^{2}}{2}+ \frac{y^{2}}{m}=1$ lie on the $x$-axis and its eccentricity is $\frac{1}{2}$, calculate the value of $m$. Express your answer as a single value. | \frac{3}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given $\tan(\pi+\alpha) = 2$, compute the value of
$$\frac {\sin(\alpha-\pi)+\cos(\pi-\alpha)}{\sin(\pi +\alpha )-\cos(\pi -\alpha )}.$$
Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear regression equation y = -0.7x + a, and the information that the mean of x is \overline{x} = 1 and the mean of y is \overline{y} = 0.3, predict the value of y when x = 2. Express your answer as a single number. | -0.4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.