instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
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Given a sequence $\{a_{n}\}$ where ${a_2}=\sqrt{3}$ and ${a_1}=1$, and $a_{n+1}^2-a_n^2=2{a_n}-2{a_{n-1}}+1$ for $n\geq 2$, calculate the value of $a_{2023}^2-2{a_{2022}}$. Express your answer as a single integer. | 2022 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the functions f(x) = [x]{x} and g(x) = x - 1, where [x] denotes the greatest integer that is not greater than x, and {x} = x - [x], find the length of the interval that represents the solution set for the inequality f(x) < g(x) within the given interval 0 ≤ x ≤ 3. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $\frac{2+i}{a-i}$ (where $a \in \mathbb{R}$, $i$ is the imaginary unit), if it is a pure imaginary number, calculate the modulus of $a+i$. Express your answer as a single value, including the square root symbol if necessary. | \frac{\sqrt{5}}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given set $A=\{x|x \gt 0\}$ and set $B=\{x|x^{2}-2x-3 \lt 0, x\in \mathbb{R}\}$, calculate the intersection $A \cap B$. Express your answer as an interval. | (0,3) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A rectangle has sides of integer length (when measured in centimeters) and an area of 36 square centimeters. Find the maximum possible perimeter of the rectangle. Express your answer in centimeters as a whole number. | 74 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $f(x)$ is a function defined on $\mathbb{R}$ and satisfies the property $f(-x) = -f(x)$, and for $x > 0$, $f(x) = 2x(x+1)$, evaluate $f(-1)$. Express your answer as a single integer. | -4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\dfrac {1}{i}+ \dfrac {1}{i^{3}}+ \dfrac {1}{i^{5}}+ \dfrac {1}{i^{7}}+ \dfrac {1}{i^{9}}$. Express your answer as a simplified complex number. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the range of the independent variable x in the function y = sqrt(x + 2) such that the function is defined. Express your answer as an inequality. | x >= -2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $| \overrightarrow {a}|=1$, $| \overrightarrow {b}|=2$, and $( \overrightarrow {a}+ \overrightarrow {b})\cdot (2 \overrightarrow {a}- \overrightarrow {b})=-1$, find the angle between vectors $\overrightarrow {a}$ and $\overrightarrow {b}$ (in terms of π). Express your answer as a fraction if necessary. | \frac{\pi}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the proposition "There exists an $x \in \mathbb{R}$, such that $x^2+1<0$", what is the direct statement of its negation? Express your answer as a complete sentence. | For all $x \in \mathbb{R}$, $x^2+1 \geq 0$. | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Last year, Century Middle School had 100 students participating in "AHSME," and their average score was 100. The number of non-graduating students was 50% more than the number of graduating students, and the average score of graduating students was 50% higher than that of non-graduating students. Calculate the average ... | 125 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {2}{1-i}+i$, find the conjugate of $z$. Express your answer as a complex number in the form $a + bi$. | 1-2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be an odd function defined on $\mathbb{R}$ and have a period of $4$. If $f(1)=1$, then calculate the value of $f(-1)+f(8)$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Let the complex number $Z$ satisfy $(1+i)Z=i$. Find the modulus of $Z$, denoted $|Z|$. Express your answer as a simplified fraction or decimal, including any necessary square roots or other mathematical operations. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the product of all real roots of the equation \( x^{\log x} = 10 \). Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of different values of the expression $C_{10}^{r+1} + C_{10}^{17-r}$, where $r$ is a positive integer satisfying the inequalities $0 \leq r+1 \leq 10$ and $0 \leq 17-r \leq 10$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Five people are to line up in a row. Person A can only be in the first or second position, and person B can only be in the second or third position. Find the total number of different possible arrangements. Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$, $b$, and $c$ form a geometric sequence, determine the number of zeros of the function $y=ax^{2}+bx+c$. Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the quadratic equation $2x^{2}-3x-1=0$ using the completing the square method. Express your answer in the form $(x-a)^2=b$, where $a$ and $b$ are the values calculated from the completing the square process. | (x-\frac{3}{4})^2=\frac{17}{16} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression $6 \cdot 5(-1)^{2^{3^{5}}}+(-1)^{5^{3^{2}}}$. Provide your answer as a single integer. | 29 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Find a unit vector perpendicular to the vector $\overrightarrow{a}=(1,- \sqrt{3})$. Express your answer as an ordered pair (x, y). | \left(\dfrac{\sqrt{3}}{2}, \dfrac{1}{2}\right) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
There are 6 boxes, and the task is to choose 3 of them to put things in, with the condition that at least one of the boxes A or B must be chosen. Calculate the number of ways to do this. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfying $z(2+i)=3+i$ (where $i$ is the imaginary unit), find the conjugate complex number $\overline{z}$. Express your answer in the form $a + bi$, where $a$ and $b$ are real numbers. | \overline{z} = \frac{7}{5} + \frac{1}{5}i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a real number $a<0$, and the function $$f(x)= egin{cases} x^{2}+2a, & x<1 \ -x, & x\geq 1 \end{cases}$$, find the range of the real number $a$ such that $f(1-a) \geq f(1+a)$. Express your answer as an interval using standard interval notation. | [-2, -1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $2x+m+\frac{2}{x-1} > 0$ holds for all $x \in (1, +\infty)$, find the range of the real number $m$ that satisfies this inequality. Express your answer as an inequality for $m$. | m > -6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the plane vectors $a=(1,2)$ and $b=(-3,2)$, if $ka+b$ is perpendicular to $a$, then find the value of the real number $k$. Express your answer as a single real number. | -\dfrac{1}{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=1$ and $b=9$, calculate the geometric mean $c$ of $a$ and $b$, where $c$ satisfies $c^2=ab$. Provide your answer in the form $\boxed{c}$, including both positive and negative solutions, if applicable. | \pm3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
The mean of five numbers is 25. If 5 is added to the first number, 10 to the second number, 15 to the third number, 20 to the fourth number, and 25 to the fifth number, what is the mean of the numbers in the new set? Express your answer as a single number. | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
An amusement park has a series of miniature buildings and landscape models from various places in the United States, scaled at $1: 20$. If the height of the United States Capitol Building is 289 feet, calculate the height of its model in feet, rounded to the nearest integer. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Find the interval on which the function y = xln(x) is monotonically decreasing. Express your answer as an interval, specifying the endpoints and the type of bracket (open or closed). | (0, e^{-1}) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,2)$, $\overrightarrow{b}=(n,3)$, if the vector $\overrightarrow{b}- \overrightarrow{a}$ is collinear with vector $\overrightarrow{c}=(4,-1)$, find the value of $n$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=\frac{{1+3i}}{{1-2i}}$, calculate the imaginary part of the conjugate of $z$. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
For the quadratic function f(x) = x^2 - 1 defined on the closed interval [k, 2] (where constant k < 2), find the value of k that makes the function an even function, if such a value exists. | k = -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: \(\frac{20-20}{20+20}\). Provide your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the curve equation (x - arcsin a)(x - arccos a) + (y - arcsin a)(y + arccos a) = 0 and the line equation x = pi/4, find the minimum value of the chord length d as the parameter a varies. Express your answer as a single value. | \frac{\pi}{2} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
There is a bamboo stick divided into 9 sections, and the volumes of these sections from top to bottom form an arithmetic sequence. The total volume of the top 4 sections is 5 liters, and the total volume of the bottom 3 sections is 4 liters. Find the volume of the 5th section. Express your answer as a fraction. | \frac {85}{66} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\alpha \in (0, 2\pi)$, $\sin{\alpha} > 0$, and $\cos{\alpha} < 0$, determine the range of values for angle $\alpha$. Express your answer as an interval. | (\dfrac{\pi}{2}, \pi) | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Let the function f(x) be an even, derivable function on R with a period of 5. Find the slope of the tangent line to the curve y=f(x) at x=5. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
In an arithmetic sequence where all terms are positive, the sum of the first 20 terms is 100. Find the maximum value of the product of the third and eighteenth terms. Express your answer as a single numerical value. | 25 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality system about x: $\left\{{\begin{array}{l}{x≤5}\\{x＞a}\end{array}}\right.$, determine the range of values for a that results in the system having no solution. Express your answer using mathematical notation (e.g., a ≥ 5 or a ≤ 5). | a \geq 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the direction vector of line $l_1$ is $\vec{a} = (1, 2, -2)$, and the direction vector of line $l_2$ is $\vec{b} = (-2, 3, m)$, if $l_1 \perp l_2$, find the value of the real number $m$. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} with a_1 = 1, and for any m, n in N*, a_{m+n} = a_m + a_n + mn, find the value of the sum from t=1 to 2019 of 1/a_i. Express your answer as a single fraction. | = 2019/1010 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Adriano, Bruno, César, and Daniel are four good friends. Daniel had no money, but the others had. Adriano gave Daniel one-fifth of his money, Bruno gave Daniel one-fourth of his money, and César gave Daniel one-third of his money. Each gave Daniel the same amount. What fraction of the total initial money that his three... | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence with the first term a_1 = 1 and common difference d = 3, find the position n when a_n = 2005. Express your answer as a whole number. | 669 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
A small ink cartridge has enough ink to print 600 pages. Three small cartridges can print as many pages as two medium cartridges. Three medium cartridges can print as many pages as two large cartridges. How many pages can be printed using a large cartridge? Express your answer as a whole number. | 1350 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
If the radius of a sphere is $r$, find the maximum lateral surface area of a cylinder inscribed in the sphere. Express your answer in terms of $r$. | 2\pi r^2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
An integer \( n \) is decreased by 2 and then multiplied by 5. If the result is 85, what is the value of \( n \)? Express your answer as a single integer. | 19 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic equation x^2 - px + (p^2 - 1)/4 = 0, find the difference between the roots. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, n)$ and $\overrightarrow{b} = (-1, n-2)$, if $\overrightarrow{a}$ and $\overrightarrow{b}$ are collinear, find the value of $n$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfying $|\overrightarrow{a} + \overrightarrow{b}| = \sqrt{9}$ and $|\overrightarrow{a} - \overrightarrow{b}| = \sqrt{5}$, find the value of $\overrightarrow{a} \cdot \overrightarrow{b}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of the first 7 terms of an arithmetic sequence is 77, find the value of the 4th term (a_4). Express your answer as a single integer. | 11 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The sum of the first $n$ terms of a geometric sequence with positive terms is $3$, and the sum of the first $3n$ terms is $21$. Find the sum of the first $2n$ terms. Express your answer as a single numerical value. | 9 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $F$ is a focus of the hyperbola $C$: $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ ($a > 0, b > 0$), and the distance from point $F$ to one of the asymptotes of $C$ is $2a$, calculate the eccentricity of the hyperbola $C$. Express your answer as a single numerical value. | \sqrt{5} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
The parametric equation of an ellipse is given by $\begin{cases} x=4\cos \phi \\ y=5\sin \phi \end{cases}$. Calculate the eccentricity of the ellipse. Express your answer as a fraction. | \frac{3}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
There are 8 classes in a certain high school's sophomore year, and there are 10 "Merit Student" spots to be distributed among these classes. Each class must receive at least one spot. Calculate the number of possible distribution methods. | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In a round-robin tournament, each team plays every other team exactly once. If a total of 21 games are played, how many teams are participating in the tournament? Express your answer as a whole number. | 7 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The set $M$ is defined as $M=\{x|x=i^n+i^{-n}, n\in \mathbb{N}\}$. Calculate the number of elements in set $M$. Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $\frac{1998- 998}{1000}$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) defined on R is a monotonic function, and for any x in R, it always satisfies f(f(x) - 2^x) = -1/2, find the zero point(s) of the function f(x). Express your answer as a single numerical value. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given four points A, B, C, and D arranged sequentially on a circle, find the total number of coloring schemes using four different colors, such that no two adjacent points have the same color. Express your answer as a single integer. | 72 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the circumference of a sector is $4cm$ and its area is $1cm^2$, calculate the radian measure of the central angle of the sector. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Three faces of a rectangular box meet at a corner of the box. The centers of these faces form the vertices of a triangle having side lengths of 4 cm, 5 cm, and 6 cm. Calculate the volume of the box in cubic centimeters (cm³). | 90 \sqrt{6} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x=1$ is a root of the equation $x^2 + bx - 2 = 0$, find the value of the other root using the relationship between the roots and coefficients. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,3)$ and $\overrightarrow{b}=(-2,m)$, if $\overrightarrow{a}$ is parallel to $\overrightarrow{a}+2\overrightarrow{b}$, find the value of $m$. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
What is the sum of three tenths and four thousandths? Express your answer as a decimal. | 0.304 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Decimals"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the greatest common divisor (GCD) of the products $360 \times 473$ and $172 \times 361$. Express your answer as a single integer. | 172 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Freda wants to write a number in each of the nine cells of this figure so that the sum of the three numbers on each diameter is 13 and the sum of the eight numbers on the circumference is 40. What number must be written in the central cell? Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat? Express your answer as a simplified fraction. | \frac{1}{8} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle equation (x-1)^2 + y^2 = 4 and the point M(2,1), find the equation of the line passing through M and intersecting the circle at points A and B, such that the angle ∠ACB is minimized. Express your answer in the standard form of a line equation (Ax + By + C = 0). | x + y - 3 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\dfrac {1}{2^{2}-1}+ \dfrac {1}{4^{2}-1}+ \dfrac {1}{6^{2}-1}+ \dfrac {1}{8^{2}-1}+ \dfrac {1}{10^{2}-1}$. Express your answer as a simplified fraction. | \dfrac {5}{11} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{-1,1,2,4\}$ and $B=\{x| -1 < x < 3\}$, find $A\cap B$. Express your answer as a set of elements, separating multiple elements with commas. | \{1,2\} | big_math | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
In a round-robin tournament format, a win earns 3 points, a draw earns 1 point, and a loss earns 0 points. There are six teams participating, and at the end of the tournament, the top three teams have the same score. What is the maximum possible score for these three teams? Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Five consecutive integers are represented by p, q, r, s, and t. The two integers in the leftmost circle add to 63, and the two integers in the rightmost circle add to 57. Find the value of r. Express your answer as an integer. | 30 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A parabola is given by the equation $y^{2}=2px (p > 0)$, with a focus at point $F$. A line with a slope angle of $\frac{\pi}{4}$ passes through the focus $F$, intersecting the parabola at points $A$ and $B$. If the perpendicular bisector of chord $AB$ passes through the point $(0, 2)$, calculate the value of $p$. Expre... | \frac{4}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $2-3i$, where $i$ is the imaginary unit, find the imaginary part, which is the coefficient of $i$. Express your answer as a real number. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number z satisfying i * z = 1 - 2i, 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"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
To obtain the graph of the function $y=\sin \left(x+1\right)$, describe the horizontal shift required relative to the graph of the function $y=\sin x$. Express your answer in the format 'X unit(s) to the [left/right]'. | 1 unit to the left | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given sin(π/2 + 2α) = -4/5, where α is in the interval (π/2, π), find the value of tan(α + π/4). Express your answer as a single fraction or decimal value. | -\frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(AB = AC\) and \(D\) is a point on \(AC\) such that \(BD = BC\). Angle \(BAC\) is \(40^\circ\). Find the measure of angle \(ABD\). Express your answer in degrees as a whole number. | 30^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Let $f$ be a real-valued function defined over ordered pairs of integers such that \[f(x+3m-2n, y-4m+5n) = f(x,y)\] for every integers $x,y,m,n$. Determine the maximum number of distinct values in the range set of $f$. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=2-i$, calculate the imaginary part of $z(\overline{z}+i)$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given that a line passes through the point (2, 1) and has an inclination angle of 135°, derive the equation of the line in the standard form Ax + By + C = 0. Express your answer in the form \boxed{Ax + By + C = 0}. | x + y - 3 = 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\frac{2\sin^{2}\alpha}{\sin 2\alpha} \cdot \frac{2\cos^{2}\alpha}{\cos 2\alpha}$. Express your answer as a trigonometric function of alpha. | \tan 2\alpha | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that α is an angle in the third quadrant and sin α + cos α = 2m, sin 2α = m^2, find the value of m. Express your answer as a simplified fraction if possible, including a negative sign if the value is negative. | -\dfrac{\sqrt{3}}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
The graph of the function y = 3 + log_a(2x + 3) necessarily passes through a fixed point P. Find the coordinates of point P. Express your answer as a coordinate pair (x, y). | (-1,3) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\frac{x+3}{x-4} = 0$, solve for the value of $x$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
If p(m, n) is a point on the terminal side of a 600° angle, calculate the value of n/m using the definition of trigonometric functions and reduction formulas. Express your answer as a single value. | \sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
A student is browsing a bookstore and finds 3 books that they like. If they decide to buy at least two of them, how many different purchasing options are available? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
In a school of 100 students, 39 participated in a Chinese competition, 49 participated in a Math competition, and 41 participated in an English competition. Fourteen students participated in both the Chinese and Math competitions, thirteen participated in both the Math and English competitions, nine participated in bot... | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Find the period of the function y = tan(2x + $\frac{\pi}{6}$). Express your answer in radians as a fraction of $\pi$ if possible. | \frac{\pi}{2} | big_math | [
"Mathematics -> Trigonometry -> Functions -> Periodic Functions"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Suppose that 7 boys and 13 girls line up in a row. Let $S$ be the number of places in the row where a boy and a girl are standing next to each other. Calculate the average value of $S$ if all possible orders of these 20 people are considered. Round your answer to the nearest whole number. | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear function y = -1/2x + 1, identify the quadrant of the coordinate plane that the graph of this function does not pass through. Express your answer as the name of the quadrant (e.g., First, Second, Third, Fourth). | Third | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Find the eccentricity of the hyperbola $C$: $\frac{x^{2}}{4} - \frac{y^{2}}{2} = 1$. Express your answer as a fraction with a square root in the form $\boxed{[answer]}$. | \frac{\sqrt{6}}{2} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle of inclination of the line y = $- \frac{4}{3}$x + 1 is α, simplify the expression $\frac{cos2α}{cos(\frac{5π}{4} + α)sin(π + α)}$ and express your answer as a numerical value. | \frac{\sqrt{2}}{4} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the intersection point of the line $x+y=5$ and the line $x-y=1$. Express your answer as an ordered pair (x, y). | (3,2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $\sin30^\circ\sin75^\circ+\sin60^\circ\sin15^\circ$. Express your answer as a single value, potentially involving square roots. | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the length of the major axis of an ellipse is twice its focal length, calculate the eccentricity of the ellipse. Express your answer as a fraction or decimal value. | \dfrac{1}{2} | big_math | [
"Mathematics -> Geometry -> Differential Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given $\sin\left( \frac{\pi}{2}-\alpha \right)=-\frac{4}{5},0 < \alpha < \pi$, find the value of $\sin 2\alpha$. Express your answer as a fraction. | -\frac{24}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
If the complex number $z= \frac {1+ai}{2-i}$ (where $i$ is the imaginary unit) is a pure imaginary number, find the value of the real number $a$. Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of initial configurations where Ayşe can guarantee a win in a turn-based game with Burak, given the initial configurations of the boxes: (1,2012,2014), (2011,2011,2012), (2011,2012,2013), (2011,2012,2014), (2011,2013,2013). Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
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