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 equation of a parabola is given by $x=4y^2$. Calculate the exact coordinates of its focus. Express your answer as an ordered pair (x, y). | \left(\frac{1}{16},0\right) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Find the value of c that places the vertex of the parabola y = x^2 - 8x + c on the x-axis. Express your answer as a single integer. | 16 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
A circle is inscribed in an equilateral triangle, and a square is inscribed in this circle. Calculate the ratio of the area of the triangle to the area of the square. Express your answer as a simplified ratio, using a colon (:) to separate the parts of the ratio. | 3 \sqrt{3}: 2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Two meteorological stations, A and B, independently forecast the weather for a certain area. If the probability of station A making an accurate forecast is $p$, and the probability of station B making an inaccurate forecast is $q$, then find the probability that both stations make an accurate forecast in one prediction... | p(1-q) | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Given the solution $\left\{\begin{array}{l}x=-2\\ y=1\end{array}\right.$ to the equation $mx+3y=7$, find the value of $m$ through substitution. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression \((5 \times 5) + (5 \times 5) + (5 \times 5) + (5 \times 5) + (5 \times 5)\) to find its numerical value. | 125 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of a line in the form ax+by+c=0, what condition must be true for the coefficients a and b such that the line has an acute angle of inclination? Express your answer as a mathematical expression or inequality involving the coefficients a and b. | ab < 0 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the term x^6y^2 in the expansion of (x - sqrt(2)y)^8. Express your answer as a single integer. | 56 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Binomial Expansions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the general term of a sequence $a_n = 2n - 7$, find the value of $|a_1| + |a_2| + \ldots + |a_{15}|$. Express your answer as a single integer. | 153 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
A wooden cube has edges of length 3 meters. Square holes, of side one meter, centered in each face are cut through to the opposite face. The edges of the holes are parallel to the edges of the cube. Calculate the entire surface area of the cube, including the inside surfaces exposed by the holes. Express your answer in... | 72 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_{n}\}$, given that $a_{7}=19$ and $a_{2}+a_{8}=26$, calculate the sum of the first $5$ terms of the sequence. | 35 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x|x^{2}-2x\leqslant 0,x\in Z\}$ and $B=\{x|-2 < x < 2\}$, calculate the intersection $A\cap B$. Express your answer as a set of integers, including all elements of the intersection. | \{0,1\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $$\sqrt[3]{(2-\pi )^{3}} + \sqrt {(3-\pi)^{2}}$$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The product of the first three terms of a geometric sequence is 3, the product of the last three terms is 9, and the product of all terms is 729. Find the number of terms in the sequence. | 12 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A square floor is tiled with a large number of regular hexagonal tiles. The tiles are 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 ratio of the number of blue tiles to the number of white tiles. Ex... | \frac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of real solutions to the equation $x^{\ln3} + x^{\ln4} = x^{\ln5}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
In the complex plane, the points corresponding to the complex numbers $\frac{1}{1+i}$ and $\frac{1}{1-i}$ are $A$ and $B$, respectively. Find the complex number corresponding to point $C$, which is the midpoint of line segment $AB$. Express your answer as a simplified complex number in the form $\boxed{a+bi}$, where $a... | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
A pot can hold 2 cakes at a time, and it takes 5 minutes to cook both sides of a cake. What is the minimum time required to cook 3 cakes thoroughly? Express your answer in minutes as a whole number. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Joe had walked half way from home to school when he realized he was late. He ran the rest of the way to school. He ran 3 times as fast as he walked. Joe took 6 minutes to walk half way to school. Calculate the total time it took Joe to get from home to school. Express your answer in minutes as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
If \(2^n = 8^{20}\), what is the value of \(n\)? Express your answer as a whole number. | 60 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of inequalities $\begin{cases} x+y \leqslant 4 \\ y - 2x + 2 \leqslant 0 \\ y \geqslant 0 \end{cases}$, and $z = x + 2y$, find the maximum value of $z$. Express your answer as a single numerical value. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
In the positive sequence {a_n}, if a_1 = 1, and for all n in the set of positive integers, it satisfies na_{n+1} - (n+1)a_n = 0, calculate the value of a_{2014}. Express your answer as a single integer. | 2014 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The temperature at the foot of Mount Everest was $24^{\circ}\mathrm{C}$, and the temperature at the summit was $-50^{\circ}\mathrm{C}$. Calculate the temperature difference between the foot and the summit of the mountain. Express your answer in degrees Celsius ($^{\circ}\mathrm{C}$). | 74^{\circ}\mathrm{C} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given a positive sequence {a_n} with the sum of the first n terms S_n, where lg a_{n+1} = 1/2(lg a_n + lg a_{n+2}), and a_3 = 4, S_2 = 3, find the relationship between S_n and a_n. Express your answer as an equation relating S_n and a_n. | S_n = 2a_n - 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the proposition $p: \\forall x\\geqslant 0$, $x-2 \\gt 0$, what is the correct negation of $p$? Express your answer using logical notation for quantifiers and inequalities. | \\exists x\\geqslant 0, x-2\\leqslant 0 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Three people, A, B, and C, sit down randomly. Calculate the probability that B does not sit in the middle. Express your answer as a simplified fraction. | \dfrac {2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation iz = 1 - 2i, calculate the magnitude of the complex number z. Express your answer as a numerical value. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The domain of the function $y = 3^{|x|} - 1$ is $[-1, 2]$. Find the range of the function. Express your answer in interval notation, including the endpoints. | [0, 8] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence $\{a_n\}$, $a_1+a_4+a_7=39$ and $a_2+a_5+a_8=33$. Find the value of $a_5+a_8+a_{11}$. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Sofia entered an elevator. The elevator went up seven floors, then down six floors, and finally up five floors. If Sofia got out on the twentieth floor, calculate the floor number she entered on. Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
For the geometric sequence \\(\\{a_n\\}\\) where all terms are positive, and \\(a_5a_6 + a_4a_7 = 18\\), find the value of \\(\\log_3a_1 + \\log_3a_2 + \\cdots + \\log_3a_{10}\\). Express your answer as a single numerical value. | 10 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \frac{1}{e} \cdot e^{x}+ \frac{a}{2}x^{2}-(a+1)x+a (a > 0)$, where $e$ is the base of the natural logarithm. If the function $y=f(x)$ and $y=f[f(x)]$ have the same range, find the maximum value of the real number $a$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\sin \left(x- \frac {\pi}{3}\right)-1$, find the smallest positive period of $f(x)$. Express your answer as a single value. | 2\pi | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ is a root of the equation $x^{2}+3x+2=0$, find the value of the algebraic expression $a^{2}+3a$. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
What is the minimum number of acute angles that a triangle must have, given that the sum of the internal angles of a triangle is 180°? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Genevieve makes a small kite and a large kite for a bulletin board display. The kites look like the one in the diagram. For her small kite, Genevieve draws the kite on a one-inch grid. Calculate the area of the small kite in square inches.
[asy]
for (int a = 0; a < 7; ++a)
{
for (int b = 0; b < 8; ++b)
{
dot((a,b));
}... | 21 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $3x + (2k-1) = x - 6(3k+2)$, and the solution with respect to $x$ is $x=1$, calculate the value of $k$. Express your answer as a fraction. | -\dfrac{13}{20} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the function $y = 3x^2 + \frac{6}{x^2 + 1}$. Express your answer as a single value in the form $\boxed{[answer]}$. | 6\sqrt{2} - 3 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
If a diagonal drawn from a vertex of an n-sided polygon can divide it into at most 2023 triangles, what is the value of n? Express your answer as a whole number. | 2025 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
There are 6 people lining up for a photo, with the requirement that three specific individuals, named A, B, and C, must stand together. Find the number of different arrangements. Express your answer as a whole number. | 144 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Class 1 of Senior Year 3 has 52 students. Now, all students are randomly numbered, and a sample of size 4 is drawn using the systematic sampling method. Given that students numbered 5, 31, and 44 are in the sample, find the number of the remaining student in the sample. Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that |a|=5, |b|=1, and a-b < 0, find the possible values of a+b. Express your answer as numerical values, including both possible values if there are multiple solutions. | -4, -6 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set U = {0, 1, 2} and set A defined by the equation x^2 - x = 0, find the complement of A in U (∁_U A). Express your answer as a set using standard set notation. | \{2\} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In a sea area of 10,000 km^2, there are 40 km^2 of continental shelf with oil reserves. If drilling is carried out at any point in the sea area, what is the probability of reaching the oil layer? Express your answer as a simplified fraction. | \dfrac{1}{250} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
The measures of the six interior angles of a convex hexagon form an arithmetic sequence of natural numbers. Let \( m^{\circ} \) be the degree measure of the largest interior angle of this hexagon. Calculate the maximum possible value of \( m^{\circ} \). Express your answer in degrees. | 175^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given an even function f(x) and its derivative f'(x), and the condition x/2*f'(x) + f(-x) < 0 for x >= 0, find the solution set of the inequality g(x) < g(1-2x), where g(x) = x^2*f(x). Express your answer as an interval or union of intervals. | (\frac{1}{3}, 1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $abc^{2}$ in the expansion of $(a+2b-3c)^{4}$. Express your answer as a single integer. | 216 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the formula for the circumference of a circle, C = 2πr, where C is the circumference and r is the radius, identify the independent variable. Express your answer as the variable itself (e.g., 'r' or 'C'). | r | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Four drivers and four ticket collectors are to form four groups, each consisting of one driver and one ticket collector. Find the total number of different grouping arrangements. Express your answer as a numerical value. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that an object is in a state of equilibrium under the action of three forces F1=(1,2), F2=(-1,-3), and F3, find the value of F3. Express your answer as an ordered pair (x, y). | (0,1) | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z that satisfies the equation (1+i)z = 1-i, calculate the modulus of z. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a and b satisfying |a| = 1, |b| = 2, and (5/2)a + b is perpendicular to a - b, find the angle theta between vectors a and b. Express your answer in radians as a fraction of pi. | \frac{2\pi}{3} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence {a_n} is an arithmetic sequence and S_n is the sum of its first n terms, with the condition a_2 = 3a_4 - 6, find the value of S_9. Express your answer as a single numerical value. | 27 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Let $\{a_n\}$ be an arithmetic sequence with a non-zero common difference, satisfying $a_4^2 + a_5^2 = a_6^2 + a_7^2$. Find the sum of the first 10 terms of $\{a_n\}$, denoted as $S_{10}$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos \alpha = -\dfrac{3}{5}$ and $0 < \alpha < \pi$, find the value of $\tan \alpha$. Express your answer as a fraction. | -\dfrac{4}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
How many monomials of degree 7 are there that contain the letters a, b, and c, and have a coefficient of 1? Express your answer as a whole number. | 15 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A line passes through point A (1, 4), and the absolute values of its horizontal and vertical intercepts are equal. How many such lines exist? Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z represented by the point (-2,1), calculate the value of \\frac{\\overline{z}}{1+i}. Express your answer in the form a + bi. | -\\frac{3}{2} + \\frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Abstract Algebra -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Laura and Dina have a running race. Laura runs at a constant speed and Dina runs n times as fast, where n > 1. Laura starts s meters in front of Dina. Find the distance, in meters, that Dina runs before she overtakes Laura. Express your answer as a formula in terms of n and s. | \frac{ns}{n-1} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
In a class, 3 out of 6 student leaders (among which there are 4 boys and 2 girls) are chosen to participate in the school's voluntary labor. Event A is "boy A is selected", while event B is "girl B is selected". Calculate the conditional probability P(B|A), which is the probability that girl B is also selected given th... | \frac{2}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Let $m\in\mathbb{N}$, and consider the function $f(x)=2x-m\sqrt{10-x}-m+10$. Find the number of possible values of $m$ for which $f(x)$ has an integer root. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $y^{2}=4x$ with its focal chord endpoints as $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$, and $AB \perp x$-axis, find the length of the focal chord $|AB|$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given $P\left(A|B\right)=\frac{3}{7}$ and $P\left(B\right)=\frac{7}{9}$, calculate the exact value of $P\left(AB\right)$ using the formula for conditional probability. Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(ωx) + cos(ωx) with ω > 0, where the graph's adjacent axes of symmetry are separated by a distance of π/3, find the value of ω. Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function f(x) = x + e^x. Express your answer as a function of x. | f'(x) = 1 + e^x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$ the ratio $AC:CB$ is $3:4$. The bisector of the exterior angle at $C$ intersects $BA$ extended at $P$ ($A$ is between $P$ and $B$). Find the ratio $PA:AB$. Express your answer as a ratio of two integers. | 3:1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the proposition $p: \exists x\in \mathbb{R}$, $\sin x + \cos x > \sqrt{2}$, what is the negation of proposition $p$? Express your answer as a logical statement with proper mathematical notation. | \forall x \in \mathbb{R}, \sin x + \cos x \leq \sqrt{2} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression S = (x-1)^4 + 4(x-1)^3 + 6(x-1)^2 + 4(x-1) + 1. Express your answer as a simplified algebraic expression. | x^4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
In January 2000, the price of a kilogram of chicken was R$1.00, and it started to triple every 6 months. Calculate the time it will take for the price to reach R$81.00. Express your answer in years, rounding to 1 decimal place if necessary. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
In a dark box, there are $a$ balls with all the same characteristics except for color, and among these $a$ balls, only $3$ are red. After stirring the balls evenly, one ball is randomly picked, its color is noted, and then it is put back into the dark box. After conducting a large number of ball-picking experiments, it... | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given $\overrightarrow{OA}=(\cos 15^{\circ},\sin 15^{\circ})$ and $\overrightarrow{OB}=(\cos 75^{\circ},\sin 75^{\circ})$, calculate the magnitude of $\overrightarrow{AB}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
The side lengths of a cyclic quadrilateral are 25, 39, 52, and 60. Find the diameter of the circle. Express your answer as a whole number. | 65 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\left|x-\sqrt{(x-1)^2}\right|$ when $x < 0$. Express your answer as an algebraic expression. | 1-2x | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\left(-0.25\right)^{2022}\times 4^{2023}$. Express your answer as a single number. | 4 | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the line l: y=kx intersects the circle C: x^2 + (y-2)^2 = 4 at points A and B, if |AB| = 2√3, calculate the slope k of the line. Express your answer as a simplified radical or decimal, including a ± symbol if necessary. | ±√3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a > 0$, $b > 0$, the line $ax + by = 1$ passes through the point $(1, 3)$. Find the minimum value of $\frac{1}{a} + \frac{1}{3b}$. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
When \( k \) candies were distributed among seven people so that each person received the same number of candies and each person received as many candies as possible, there were 3 candies left over. If instead, \( 3k \) candies were distributed among seven people in this way, then how many candies would be left over? E... | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
The complex number $z$ satisfies $\bar{z} + |z| = 8 - 4i$. Find the value of $z$ in the form $a + bi$, where $a$ and $b$ are real numbers. | 3 + 4i | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence defined by a_1 = 1 / (1 - x) and a_n = 1 / (1 - a_(n-1)) for n >= 2, x!= 1 and x!= 0, calculate a_107. Express your answer as an expression involving x. | \frac{x - 1}{x} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Let f(x) be a monotonically decreasing odd function defined on R. If x1+x2>0, x2+x3>0, and x3+x1>0, determine the sign of f(x1)+f(x2)+f(x3). Express your answer as a single inequality in the format f(x1)+f(x2)+f(x3) [inequality] 0. | f(x1)+f(x2)+f(x3) < 0 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Orvin went to the store with just enough money to buy 30 balloons. When he arrived, he discovered that the store had a special sale on balloons: buy 1 balloon at the regular price and get a second at 1/3 off the regular price. What is the greatest number of balloons Orvin could buy? Express your answer as a whole numbe... | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{1}{1-\sqrt[4]{3}}+\frac{1}{1+\sqrt[4]{3}}+\frac{2}{1+\sqrt{3}}$. Express your answer as a single numerical value. | -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $1 \le n \le 455$ and $n^3 \equiv 1 \pmod{455}$, calculate the number of solutions for $n$. | 9 | big_math | [
"Mathematics -> Number Theory -> Congruences"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A circle has a diameter of 10cm. Find the distance d between the center of the circle and a line such that the line is tangent to the circle. Express your answer in centimeters (cm). | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the sequence: $1 - 2 - 3 + 4 + 5 - 6 - 7 + 8 + \ldots + 2005 - 2006 - 2007 + 2008$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that each diagonal of a square has a length of 2, calculate the area of the square. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=\sqrt{2}$, and $\overrightarrow{a}\perp(\overrightarrow{a}+2\overrightarrow{b})$, calculate the projection of $\overrightarrow{b}$ in the direction of $\overrightarrow{a}$. Express your answer as a single... | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = a^x where a > 0 and a!= 1, the sum of the maximum and minimum values of the function in the interval [1,2] is 12. Determine the value of the real number a. Express your answer as a single real number. | 3 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \( 2^{2} + 1^{2} + 0^{2} + (-1)^{2} + (-2)^{2} \). Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the arithmetic sequence -3, -7, -11, …, find the general term formula in terms of n. Express your answer as a formula using standard algebraic notation. | -4n+1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Among the numbers 1, 2, 3,..., 100, find the last digit of the product of all integers that cannot be divided by 2 or 5. Express your answer as a single digit. | 1 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $M=\{-1,0,1,2\}$ and $N=\{y|y=2x+1, x \in M\}$, calculate the intersection of sets $M$ and $N$, $M \cap N$. Express your answer as a set. | \{-1,1\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Four students from a university went to the school's practice base for an internship during the summer vacation. Each student can choose one from base A, base B, and base C. If other conditions are not considered, calculate the total number of different ways the students can choose their practice bases. Express your an... | 81 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
John lists the integers from 1 to 20 in increasing order. He then erases the first half of the integers in the list and rewrites them in order at the end of the second half of the list. What is the integer in the new list that has exactly 12 integers to its left? Provide your answer as a single integer. | 3 | big_math | [
"Mathematics -> Discrete Mathematics -> Logic"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Find the next year after 2002 that is a palindrome and calculate the product of its digits. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Viola has been practising the long jump. At one point, the average distance she had jumped was 3.80 m. Her next jump was 3.99 m, and that increased her average to 3.81 m. After the following jump, her average had become 3.82 m. Calculate the distance of her final jump. Express your answer in meters, rounded to two deci... | 4.01 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $M=\left\{{x|{x=\frac{k}{4}+\frac{1}{4},k∈Z}}\right\}$, and the set $N=\left\{{x|{x=\frac{k}{8}-\frac{1}{4},k∈Z}}\right\}$, determine the subset relationship between sets M and N. Express your answer using standard subset notation (e.g., M⊆N, N⊆M, etc.). | M⊆N | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\sin \frac{\pi }{12}-\sqrt{3}\cos \frac{\pi }{12}$ to obtain a single numerical value. | - \sqrt{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
A cylindrical container filled with water has a base radius of $5cm$. Two glass spheres with a diameter of $5cm$ each are submerged in the water. If these two spheres are removed, by how much will the water level drop? Express your answer in centimeters (cm) as a fraction or decimal value. | \dfrac {5}{3} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Subesha wrote down Davina's phone number in her math binder. Later that day, while correcting her homework, Subesha accidentally erased the last two digits of the phone number, leaving 893-44__. Subesha tries to call Davina by dialing phone numbers starting with 893-44. What is the least number of phone calls that she ... | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
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