instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
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|---|---|---|---|---|---|
Given the binomial expansion $(1 - 2x)^7 = a_0 + a_1x + a_2x^2 + \dots + a_7x^7$, find the value of $|a_0| + |a_1| + |a_2| + \dots + |a_7|$. Express your answer as a single integer. | 2187 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=\frac{{2+i}}{{1+{i^2}+{i^5}}}$, calculate its conjugate $\overline{z}$. Express your answer as a complex number in the form $a+bi$, where $a$ and $b$ are real numbers. | 1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Express 4600 in scientific notation in the form $a \times 10^{b}$. | 4.6 \times 10^{3} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Jill was given a large jar of jam. She gave one sixth of the jam to Jan. Jill then gave one thirteenth of the remaining jam to Jas. Jill was left with 1 kg of jam. What was the original weight of the jam in Jill's jar? Express your answer as a decimal value in kg. | 1.3 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Let set $P = \{1, 2\}$ and set $Q = \{x \mid |x| < 2\}$. Calculate the intersection of set $P$ and set $Q$, expressing your answer as a set. | \{1\} | big_math | [
"Mathematics -> Discrete Mathematics -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of equations to find the equivalent set of ordered pairs:
$$
\begin{align*}
x + y &= 1 \\
2x - y &= 2
\end{align*}
$$
Express your answer as a set of ordered pairs (x, y). | \{(1, 0)\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Two rays starting from a point O form an angle of 30 degrees. Point A is on one ray and point B is on the other ray, with AB = 1. What is the maximum length of OB? Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \frac {(1-i)^{2}}{1+i}$, calculate the modulus of $z$, denoted as $|z|$. | \sqrt{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(\sin 15^{\circ},\cos 15^{\circ})$ and $\overrightarrow{b}=(\cos 15^{\circ},\sin 15^{\circ})$, find the angle between the vectors $\overrightarrow{a}+ \overrightarrow{b}$ and $\overrightarrow{a}- \overrightarrow{b}$. Express your answer in degrees. | 90 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = log2(3x + a/x - 2) is monotonically increasing in the interval [1, +∞), find the range of values for the real number a. Express your answer as an interval, including any endpoint that is part of the solution. | (-1, 3] | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos \alpha + \sin \alpha = \frac{2}{3}$, calculate the value of $\frac{\sqrt{2}\sin(2\alpha - \frac{\pi}{4})+1}{1+\tan \alpha}$. Express your answer as a simplified fraction. | -\frac{5}{9} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x \in \mathbb{R}$, vector $\vec{a} = (3, 2)$, vector $\vec{b} = (x, 4)$, and $\vec{a} \parallel \vec{b}$, find the value of $x$. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
In a certain sports competition, the scoring rules are: winning a match gives 3 points, drawing a match gives 1 point, and losing a match gives 0 points. If a particular athlete plays 15 matches and scores a total of 22 points, how many possible scenarios are there? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A certain company plans to build a container (neglecting thickness, with length unit in meters). The bottom of the container is cylindrical with a height of 1 and a bottom radius of r. The top is a hemisphere with a radius of r. According to the design requirements, the volume of the container is 28/3π cubic meters. As... | \sqrt[3]{4} | big_math | [
"Mathematics -> Calculus -> Applications of Derivatives"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence {a_n}, it is known that a_1 = 0 and a_{n+2} - a_n = 2. Find the value of a_7. Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given a list of positive integers \( p, q, r, s, t, u, v, w \) where the sum of the values of each group of four consecutive letters in the list is 35, and \( q + v = 14 \), find the largest possible value of \( p \). Express your answer as a single integer. | 20 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Let \( p \) be a prime number. Find the value of \( p \) such that the equation \( x^2 - px - 580p = 0 \) has two integer roots. | 29 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that angle $\alpha \in \left(0,\frac{\pi}{2}\right)$, and $\cos 2\alpha + \cos^2\alpha = 0$, calculate $\tan \left(\alpha +\frac{\pi}{4}\right)$. Express your answer as a single value. | -3-2\sqrt{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $U=\{x|-3 < x < 3,x∈Z\}$, $A=\{1,2\}$, $B=\{-2,-1,2\}$, find $A∪∁\_UB$. Express your answer as a set of integers, including all elements of the resulting set. | \{0,1,2\} | big_math | [
"Mathematics -> Discrete Mathematics -> Set Theory -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation x/(x-1) = (y^2 + 2y - 1)/(y^2 + 2y - 2), solve for x in terms of y. Express your answer as an algebraic expression in terms of y. | x = y^2 + 2y - 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Five identical squares form rectangle \(PQRS\), as shown. The perimeter of rectangle \(PQRS\) is 48. Calculate the area of \(PQRS\). Express your answer as a whole number. | 80 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Ben and David go to buy bagels. For every 1 yuan Ben spends, David spends 25 cents less. As a result, Ben spends 12.50 yuan more than David. Calculate the total amount they spend together. Express your answer in yuan, rounded to 2 decimal places. | 87.50 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the geometric sequence $\{a_n\}$ is an increasing sequence, and $S_n$ is the sum of its first $n$ terms. If $a_1+a_5= \frac {17}{2}$ and $a_2a_4=4$, then calculate the value of $S_6$, the sum of the first 6 terms of the sequence. Express your answer as a fraction or decimal. | \frac{63}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the positive arithmetic geometric sequence {a_{n}}, a_{1} and a_{4029} are the two roots of the equation x^{2}-10x+16=0. Calculate the value of log_{2}a_{2015}. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
An object starting from rest can roll without slipping down an incline. Using the principle of energy conservation, derive an expression for the speed of the object's center of mass after it has moved through a vertical distance h. Assume the object is a uniform solid sphere. | \sqrt{\frac{10gh}{7}} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\vec{m} = (x-1, 1)$ and $\vec{n} = (1, y)$, if $\vec{m} \perp \vec{n}$, find the minimum value of $2^{x} + 2^{y}$. Express your answer as a single value. | 2\sqrt{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sketch of triangle $\triangle ABC$ is the oblique drawing of $\triangle {A'}{B'}{C'}$, where ${A'}{B'}=4$, ${B'}{C'}=3$, and $\angle {A'}{B'}{C'}=60^{\circ}$, and the ratio of the area of $\triangle ABC$ to the area of $\triangle {A'}{B'}{C'}$ is $2\sqrt{2}$, calculate the area of triangle $\triangle ABC$. Ex... | 6\sqrt{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the product $10 \cdot 71 \left(1 - \frac{1}{2^2}\right)\left(1 - \frac{1}{3^2}\right) \cdots \left(1 - \frac{1}{9^2}\right) \left(1 - \frac{1}{10^2}\right)$. Express your answer as a single number. | 71 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = ((x+1)^2 + a*sin(x)) / (x^2 + 1) + 3, where a is a real number, and f(ln(log_2(5))) = 5, calculate the value of f(ln(log_5(2))). Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z satisfies the equation z + |z| = 2 + 8i, calculate the value of |z|^2. Express your answer as a single integer. | 289 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sets $A=\{x|\log _{2}x \lt 2\}$ and $B=\{x|x\geqslant 2\}$, find the intersection $A \cap B$. Express your answer in interval notation. | [2,4) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given an isosceles triangle with two sides of lengths $a$ and $b$, and $a$ and $b$ satisfy $\sqrt{2a-3b+5}+\left(2a+3b-13\right)^{2}=0$, calculate the possible values for the perimeter of this isosceles triangle. Express your answer as two possible values. | 7 \text{ or } 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = x^3 - 3x + a with three distinct zeros, find the range of the real number a. Express your answer in interval notation. | (-2,2) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence \( 10^{\frac{1}{11}}, 10^{\frac{2}{11}}, 10^{\frac{3}{11}}, \ldots, 10^{11} \), find the smallest value of \( n \) such that the product of the first \( n \) terms of this sequence exceeds 100,000. Express your answer as a single integer. | 11 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum positive period of the function f(x) = (√3 sin x + cos x)(√3 cos x - sin x). Express your answer in radians. | π | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If all terms of the geometric sequence $\{a_n\}$ are positive, and $a_1+2a_2=3$, $a_3^2=4a_2a_6$, find the value of $a_4$. Express your answer as a fraction. | \dfrac{3}{16} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
In $\triangle ABC$, $AB = AC = 3.6$, point $D$ lies on $AB$ with $AD = 1.2$, and point $E$ lies on the extension of $AC$. The area of $\triangle AED$ is equal to the area of $\triangle ABC$. Find the length of $AE$. Express your answer as a single number. | 10.8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given $a=\ln \frac{\pi}{3}, b=\ln \frac{e}{3}, c=e^{0.5}$, determine the ordering of the values from largest to smallest. Express your answer in the format "x > y > z", where x, y, and z are the variables a, b, and c. | c > a > b | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given that {a<sub>n</sub>} is an arithmetic sequence, if a<sub>2</sub> = 2a<sub>3</sub> + 1 and a<sub>4</sub> = 2a<sub>3</sub> + 7, find the value of a<sub>5</sub>. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $sinθ+sin({θ+\frac{π}{3}})=1$, calculate $cos({2θ+\frac{π}{3}})$. Express your answer as a single fraction or decimal. | \frac{1}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Let \( f(x) = x + \sqrt{x^{2} + 1} + \frac{1}{x - \sqrt{x^{2} + 1}} \). Calculate the value of \( f(2015) \). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Two numbers, x and y, are randomly selected from the interval [0,π]. Calculate the probability that the event "y ≤ sin(x)" occurs. Express your answer as a fraction or decimal, including the π symbol if necessary. | \dfrac{2}{\pi^2} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A coin travels along a path that starts in an unshaded square in the top row of the figure, uses only diagonal moves, and ends in an unshaded square in the bottom row. A diagonal move takes the coin either one square down and one square left, or one square down and one square right. Calculate the total number of differ... | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given an ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$, where the left focus $F$ draws a perpendicular line to the x-axis, intersecting the ellipse at points $P$ and $Q$. Point $A$ is the intersection point of the ellipse and the positive x-axis. If $|PQ|=|FA|$, calculate the eccentricity of the el... | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the coefficient of the $x$ term in the expansion of $(1+x)(2-x)^4$. Express your answer as a single integer. | -16 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
Given a parabola $y^2 = 2px$, where a straight line $l$ passes through its focus and is perpendicular to the x-axis, intersecting the parabola at points A and B. If $|AB| = 10$ and P is a point on the directrix of the parabola, calculate the area of the triangle $\triangle ABP$. Express your answer as a single numerica... | 25 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of the expression $-10 + (-12)$? Provide your answer as a single integer. | -22 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = a + 4i, and the equation z / (z + b) = 4i, determine the value of b. Express your answer as a single number. | 17 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given real numbers $a, b > 0$, and the arithmetic mean of $a$ and $b$ is $\frac{1}{2}$, let $m=a+ \frac{1}{a}, n=b+ \frac{1}{b}$. Calculate the minimum value of $m+n$. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Applied Mathematics -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given $$a= \frac {1}{log_{2}\pi }+ \frac {1}{log_{3}\pi }+ \frac {1}{log_{4}\pi }+ \frac {1}{log_{5}\pi }$$ and y=|x-a|, where x is a natural number, find the value of x that minimizes y. Express your answer as a single natural number. | 4 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
A certain DNA fragment consists of 500 base pairs, with A+T making up 34% of the total number of bases. If this DNA fragment is replicated twice, calculate the total number of free cytosine deoxyribonucleotide molecules required. Express your answer as a whole number. | 1320 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
When two dice are thrown consecutively, the numbers obtained are m and n. Calculate the probability that point P(m, n) lies on the line x+y=4. Express your answer as a simplified fraction. | \frac{1}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Express the length of the virus, 0.0000043 meters, in scientific notation. Use the format $a \\times 10^{b}$, where $a$ is a number between 1 and 10, and $b$ is an integer. | 4.3 \times 10^{-6} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
The sequence $\{a_n\}$ satisfies $a_{n+1}= \begin{cases} 2a_{n},0 \leqslant a_{n} < \dfrac {1}{2} \\ 2a_{n}-1, \dfrac {1}{2} \leqslant a_{n} < 1 \end{cases}$, and $a_{1}= \dfrac {6}{7}$. Find the exact value of $a_{2016}$. Express your answer as a fraction in simplest form. | \dfrac {3}{7} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Let $F$ be the focus of the parabola $y^2=4x$, and $A$, $B$, $C$ be three points on this parabola. If $\overrightarrow{FA}+\overrightarrow{FB}+\overrightarrow{FC}=\overrightarrow{0}$, calculate the sum of the magnitudes of the vectors $\overrightarrow{FA}$, $\overrightarrow{FB}$, and $\overrightarrow{FC}$. Express your... | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=-\frac{1}{2}+\frac{{\sqrt{3}}}{2}i$, calculate the value of $\sum_{i=1}^{2023}{{z^i}}$. Express your answer as a complex number in the form $a + bi$. | -\frac{1}{2}+\frac{\sqrt{3}}{2}i | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In a best-of-three badminton match, the probability of player A winning each game is p. Let random variable X represent the total number of games played. If the expected value of X is 22/9, calculate the value of p. Express your answer as a fraction. | \frac{1}{3} \text{ or } \frac{2}{3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ and $b$ are two distinct real numbers, and the sets $A=\{a^2-4a, -1\}$, $B=\{b^2-4b+1, -2\}$. If the mapping $f: x \rightarrow x$ represents mapping the element $x$ in set $A$ to set $B$ while still being $x$, find the value of $a+b$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
In a certain math test, the students' scores ξ follow a normal distribution N(100, σ^2) (σ > 0). If the probability of ξ being in the interval (80,120) is 0.8, calculate the probability of ξ being in the interval (0,80). Express your answer as a decimal value between 0 and 1, rounded to 2 decimal places if necessary. | 0.1 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x^3$ in the expansion of $(1-2x^{2})(1+x)^{4}$. Express your answer as an integer. | -4 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Determine the values of $a$ for which the polynomial $x^{2}+\left(a-1
ight)x+9$ is a perfect square trinomial. Express your answer as a list of values of $a$. | 7, -5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given two circles C1: (x+a)^2+(y-2)^2=1 and C2: (x-b)^2+(y-2)^2=4 that are externally tangent, where a, b are positive real numbers, find the maximum value of ab. Express your answer as a single numerical value. | \frac{9}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x+2016)=\frac{x^2+1}{2x} (x>0)$, find the minimum value of the function $f(x)$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Point P is the circumcenter of triangle ABC, and the vector equation $\overrightarrow{PA} + \overrightarrow{PB} + \lambda \overrightarrow{PC} = 0$ holds, where $\angle C = 120^{\circ}$. Calculate the value of the real number lambda. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the piecewise function $f(x) = \begin{cases} f(x-5), & x > 0 \\ 2^{x}+\int_{0}^{\frac{\pi}{6}}{\cos 3t \ dt}, &x \leqslant 0 \end{cases}$, calculate the value of $f(2014)$. Express your answer as a single fraction in simplest form. | \frac{5}{6} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
If $x^{2}+(x+1)^{7}=a_{0}+a_{1}(x+2)+a_{2}(x+2)^{2}+\ldots+a_{7}(x+2)^{7}$, find the value of $a_{2}$. Express your answer as an integer. | -20 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x) = x^3 - 12x$, find the value of the maximum value point $a$. Express your answer as a single number. | -2 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the direction vector of line $l$ is $\overrightarrow{a}=\left(-2,1,t\right)$, and the normal vector of plane $\alpha$ is $\overrightarrow{m}=\left(4,-2,-2\right)$, and knowing that $l\perp \alpha$, find the real number $t$. Express your answer as a single real number. | 1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
On a 6x6 grid, place Go pieces in all squares, with one piece in each square. The number of white pieces in each row must be different from one another, and the number of white pieces in each column must be the same. Calculate the total number of black Go pieces on the grid. | 18 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
If $2137^{753}$ is multiplied out, what is the units digit in the final product? Express your answer as a single digit. | 7 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given $0 < \alpha < \pi$, satisfying $3\sin 2\alpha = \sin \alpha$, find the value of $\cos (\pi - \alpha)$. Express your answer as a single number. | -\frac {1}{6} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Scott challenges Chris to a 100-meter race. Scott runs 4 meters for every 5 meters that Chris runs. How far will Scott have run when Chris crosses the finish line? Express your answer in meters as a whole number. | 80 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Two circles with equations $x^2+y^2-4x+6y=0$ and $x^2+y^2-6x=0$ intersect at points A and B. Find the equation of the line passing through points A and B. Express your answer in the form $Ax+By=C$. | x+3y=0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the point $P(a+3,a)$ is located in the first quadrant, and its distance to the $y$-axis is $5$. What is the value of $a$? Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given that x is an integer, find the value of x that minimizes the expression |x - √30|. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive integer $n$ such that the expansion of $(3x^{2}+ \frac {2}{x^{3}})^{n}$ contains a constant term. | 5 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Triangle OAB has O = (0,0), B = (5,0), and A in the first quadrant. In addition, angle ABO = 90 degrees and angle AOB = 30 degrees. Suppose that OA is rotated 90 degrees counterclockwise about O. What are the coordinates of the image of A? Express your answer as coordinates (x, y). | \left(-\frac{5\sqrt{3}}{3}, 5\right) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given positive numbers $a$ and $b$ satisfying $a+2b=6$, find the minimum value of $\frac{1}{a+2}+\frac{2}{b+1}$. Express your answer as a fraction in simplest form. | \frac{9}{10} | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point $P(m,n)$ satisfies ${(m+3)^2}+\sqrt{4-n}=0$. Calculate the exact coordinates of point $P$ as a pair $(m, n)$, including negative signs if applicable. | (-3, 4) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A line passing through point P(1,1) divides the circular region {(x, y) | x^2 + y^2 ≤ 4} into two parts. Find the equation of this line that makes the difference in area between these two parts the largest. Express your answer as a linear equation in the form ax + by + c = 0, in the simplest form. | x + y - 2 = 0 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
There is a fan-shaped field with an arc length of 30 steps and the diameter of its circle is 16 steps. Find the area of this field in square steps. Express your answer as a whole number. | 120 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) with its left focus at F and eccentricity sqrt(2), and the line passing through F and P(0, 4) is parallel to one of the asymptotes of the hyperbola, find the equation of the hyperbola. Express your answer in the standard form of a hyperbola equation. | \dfrac{x^2}{8} - \dfrac{y^2}{8} = 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (z+2i)(z-2i) = 2, solve for z. Express your answer in terms of i (imaginary unit), including the ± symbol if applicable. | ±\sqrt{2}i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation 1/s + 1/2s + 1/3s = 1/(d^2 - 2d), where s and d are positive integers, find the smallest possible value of s + d. Express your answer as a single integer. | 50 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Among the integers between 1000 and 9999, how many four-digit numbers have all even digits and are divisible by 5? Express your answer as a whole number. | 100 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the imaginary part of the complex number $\left(\frac{1+i}{1-i}\right)^{2023}$. Express your answer using the imaginary unit $i$. | -i | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = |x-1| - |x+1|, solve for the real number 'a' in the equation f(f(a)) = f(9) + 1. Express 'a' as a real number, including the negative sign if applicable. | -\frac{1}{4} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A flowchart is executed with the following initial conditions and loop structure:
- Initial values: S = 1, n = 1
- Loop condition: n > 3
- Loop body: S = S + 2n, n = n + 1
Calculate the final value of S after executing the loop. | 13 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given $|\vec{a}| = 1$, $|\vec{b}| = 2$, and the angle between vector $\vec{a}$ and vector $\vec{b}$ is $60^\circ$, calculate the magnitude of the vector $\vec{a} + \vec{b}$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{7} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Vectors"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Given an equilateral triangle with vertices R, S, and T, and midpoints of its sides X, Y, and Z, how many noncongruent triangles can be formed using any three of these six points as vertices? Provide your answer as a whole number. | 4 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, the circle is divided into equal regions. Calculate the fraction of the circle that is shaded. Express your answer as a simplified fraction. | \frac{1}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Divide the Cartesian coordinate system into a 6 x 5 grid. Let points A(1,3), B(5,1), and C(4,4). Calculate the fraction of the area of this grid that is occupied by the area S1 of triangle ABC. Express your answer as a simplified fraction in the form of \frac{numerator}{denominator}. | \frac{1}{6} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of real roots of the equation
$$
\sqrt{x^{2}+2x-63}+\sqrt{x+9}-\sqrt{7-x}+x+13=0.
$$
Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
The numbers 1 to 8 are to be placed, one per circle, in the circles shown. The number next to each arrow shows what the product of the numbers in the circles on that straight line should be. Calculate the sum of the numbers in the three circles at the bottom of the diagram. Use each number from 1 to 8 exactly once. | 17 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A sequence {a_n} is defined by a_1 = 1/2 and a_{n+1} = 1 - 1/a_n. Calculate the value of a_{2010}. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is ${60}^{\circ },\left|\overrightarrow{a}\right|=2,\left|\overrightarrow{b}\right|=5$, find the value of $\left|2\overrightarrow{a}-\overrightarrow{b}\right|$. Express your answer as a numerical value, including the square root symbol i... | \sqrt{21} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
In a certain region are five towns: Freiburg, Göttingen, Hamburg, Ingolstadt, and Jena. On a certain day, 40 trains each made a journey, leaving one of these towns and arriving at one of the other towns. Ten trains traveled either from or to Freiburg. Ten trains traveled either from or to Göttingen. Ten trains traveled... | 40 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integers n from the set {2000, 2001,..., 2010} such that 2^(2n) + 2^n + 5 is divisible by 7. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Three squares with the same center and corresponding parallel sides are drawn. The distance between the sides of each successive square is 3 units, and the side length of the largest square is 22 units. Find the perimeter of the smallest square. Express your answer in units as a whole number. | 40 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Let $\overrightarrow{a}$, $\overrightarrow{b}$, and $\overrightarrow{c}$ be unit vectors, and $\overrightarrow{a} \cdot \overrightarrow{b} = 0$. Find the minimum value of $(\overrightarrow{a} - \overrightarrow{c}) \cdot (\overrightarrow{b} - \overrightarrow{c})$. Express your answer as a single numerical value. | 1 - \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
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