instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
Given that $\sin A = \dfrac{1}{2}$, calculate the value of $\cos (\dfrac{3\pi}{2} - A)$. Express your answer as a fraction, including the negative sign if applicable. | -\dfrac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
To obtain the graph of the function y = 2 sin 2x, we need to shift the graph of the function y = √3 sin 2x - cos 2x by a certain amount. Calculate the horizontal shift required to achieve this. Express your answer in radians and specify the direction (left/right). | \dfrac {\pi }{12} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In the right triangle ABC, side AC is divided into 8 equal parts, and from each division point, a segment parallel to BC is drawn, resulting in a total of 7 segments. Given that BC = 10, calculate the sum of the lengths of these 7 segments. | 35 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Liam spent all his money buying 50 soda bottles at the corner shop for £1 each. He sold each bottle at the same higher price. After selling 40 bottles, he had £10 more than he started with. He then sold all the remaining bottles. Calculate the total amount of money Liam had once all the bottles were sold. Express your ... | £75 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $\left(-2\right)-\left(-4\right)$. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Claire buys chocolates such that for every 3 chocolates at the regular price, she gets a fourth chocolate for 25 cents. If Claire buys 12 chocolates in total for $6.15, what is the regular price of one chocolate in cents? Express your answer as a whole number. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Determine the domain of the function $f(x)=\sqrt{\log_{2}(1-x)}$. Express your answer in interval notation. | (-\infty,0] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \\frac{i-5}{1+i}$, where $i$ is the imaginary unit, calculate the imaginary part of $\\bar{z}$. Express your answer as a single numerical value. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the equation $2x-1=3x+2$ for x. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Suppose the success rate of a certain experiment is twice the failure rate. Let the random variable X describe the number of successful outcomes in a single trial. Find the value of P(X=0), expressing your answer as a simplified fraction. | 1/3 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a\in\left(0, \frac{\pi}{6}\right)$ and $\sin\left(a+\frac{\pi}{3}\right)=\frac{12}{13}$, find the value of $\cos\left(\frac{\pi}{6}-a\right)$. Express your answer as a fraction. | \frac{12}{13} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
A person starts from a point 378 'li' away from the pass. On the first day, they walk briskly. From the second day onwards, due to foot pain, they halve their walking distance each day. They reach the pass on the sixth day. Using the formula for the sum of the first n terms of a geometric sequence, calculate the distan... | 192 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear equation $mx + 2x - 12 = 0$ with positive integer solutions, how many possible values of the positive integer $m$ exist? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
A disease control station is conducting a health survey among students, using stratified sampling to draw a sample. A high school has a total of 2000 students, and a sample with a size of 200 is drawn. If there are 103 males in the sample, calculate the total number of female students in this high school. Express your ... | 970 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the statements $p$: $\frac{1}{x-2} \geqslant 1$, and $q$: $|x - a| < 1$, if $p$ is a sufficient but not necessary condition for $q$, determine the range of values for the real number $a$. Express your answer as an interval, specifying the left and right endpoints. | (2, 3] | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
If $A \subseteq \{1, 2, 3\}$, calculate the number of sets $A$ that satisfy this condition. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that an integer m is a perfect cube exactly when it is equal to n^3 for some integer n, find the smallest positive integer k for which the integer 2^4 * 3^2 * 5^5 * k is a perfect cube. Express your answer as a single positive integer. | 60 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A number x is randomly selected from the interval [-5, 4]. Find the probability that the inequality 3/(x+2) > 1 holds. Express your answer as a simplified fraction. | 1/3 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Solve for □ in the equation 320÷(□+26)=4. Express your answer as a single integer. | =54 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given sinα = $\frac{2}{3}$, calculate cos(-2α). Express your answer as a simplified fraction. | \frac{1}{9} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the operation |a+4|-10 and starting with a = -12, determine the result after applying this operation 2022 times. Express your answer as a single rational number. | -8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
There are 20 parts, among which 16 are first-class and 4 are second-class. If 3 parts are randomly selected from these 20 parts, find the number of different ways to select at least one first-class part. Express your answer as a single numerical value. | 1136 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
If ${(2x+1)^{100}}={a_0}+{a_1}x+{a_2}{x^2}+⋯+{a_{100}}{x^{100}}$, calculate the remainder when $2(a_{1}+a_{3}+\cdots +a_{99})-3$ is divided by $8$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
A 12th-grade student wishes to apply for independent admission tests at 3 of 6 universities. However, since two of these universities have overlapping test schedules, the student cannot apply to both of these at the same time. Calculate the number of different application methods available to the student. | 16 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $M=\{x|x^2 \leq 4\}$ and $N=\{x|1 < x \leq 3\}$, find the intersection $M \cap N$. Express your answer in set notation. | \{x|1 < x \leq 2\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Find the largest of four consecutive primes whose sum is also a prime number. | 7 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Two containers with vertices facing down are vertically placed and contain equal masses of liquid. The radii of the liquid surface circles are 3 and 6, respectively. A small ball with a radius of 1 is placed in each container, fully submerged in the liquid and resting at the bottom without causing any overflow. Calcula... | 4 : 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, \(\triangle PQR\) is isosceles with \(PQ = PR\), and \(QRST\) is a rectangle. If \(\angle QPR = 70^\circ\), \(\angle PQR = x^\circ\), and \(\angle RQT = y^\circ\), find the value of \(x + y\). Express your answer in degrees as a whole number. | 145 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given that p is a prime number and the roots of the equation x^2 + px - 444p = 0 are integers, find the exact value of p that satisfies these conditions. | 37 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the slope of line $l_1$ is 2 and $l_1 \parallel l_2$, and that line $l_2$ passes through the point (-1,1) and intersects the y-axis at point P, find the coordinates of point P. Express your answer as an ordered pair (x, y), where x = 0. | (0,3) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Express 2.6 million in scientific notation in the format $a×10^n$, where $1≤|a|＜10$, and $n$ is an integer. | 2.6×10^6 | big_math | [
"Mathematics -> Other -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the trapezoid with base 1 = 9, base 2 = 11, and height = 3. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set of numbers { -3, -2, -1, 4, 5 }, find the largest possible product when three different numbers are multiplied. Express your answer as a single numerical value. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation of the line $2x-y-3=0$, find the value of $\tan\theta$, where $\theta$ is the slope angle of the line. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that point $A(a,4)$ and point $B(-2,b)$ are symmetric about the $x$-axis, find the sum $a+b$. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $({m}^{2}-m) \cdot {4}^{x}-{2}^{x} < 0$ and the condition $x \in (-\infty, -1]$, determine the range of values for the real number $m$ that satisfies the inequality. Express your answer as an interval. | (-1, 2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ that satisfies $\left(3+4i\right)\cdot z=5\left(1-i\right)$, calculate the real part of $z$. Express your answer as a single fraction or decimal. | -\frac{1}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Solve the inequality $x^{2}-4x\geqslant 0$. Express your answer as an interval or set of intervals. | (-\infty, 0] \cup [4, \infty) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
The longest professional tennis match ever played lasted a total of 11 hours and 5 minutes. Calculate the total duration of the match in minutes. | 665 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
In triangle $ABC$, if $AB=4$, $AC=3$, and $\angle A=30^\circ$, calculate the area of $\triangle ABC$. Express your answer as a single number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex number expression $\dfrac{1}{(1+i)^{2}}$ (where $i=\sqrt{-1}$). Express your answer in standard form for complex numbers, and include the boxed notation around your final answer. | -\dfrac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
If the line $ax+by-1=0$ ($a>0$, $b>0$) passes through the center of symmetry of the curve $y=1+\sin(\pi x)$ ($0<x<2$), then calculate the smallest positive period for $y=\tan\left(\frac{(a+b)x}{2}\right)$. Express your answer in terms of π. | 2\pi | big_math | [
"Mathematics -> Calculus -> Other",
"Mathematics -> Geometry -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
In the diagram, $ABC$ represents a triangular jogging path. Jack jogs along the path from $A$ to $B$ to $F$. Jill jogs from $A$ to $C$ to $F$. Each jogs the same distance. Given that $AB = 120$ m and $AC = 160$ m, find the distance from $F$ to $B$. Express your answer in meters as a whole number. | 80 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{OA}=(1,-3)$, $\overrightarrow{OB}=(2,-1)$, $\overrightarrow{OC}=(k+1,k-2)$, find the value of k such that points A, B, and C are collinear. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
The function $f(x)$ is an odd function defined on $\mathbb{R}$ and $f(1+x)=f(1-x)$. If $f(1)=9$, then calculate $f(2019)$. Express your answer as an integer in a boxed format. | -9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
In the sequence $\{a_n\}$, $a_1=2$. When $n$ is an odd positive integer, $a_{n+1}=a_n+2$; when $n$ is an even positive integer, $a_{n+1}=2a_n$. Calculate the value of $a_6$. Express your answer as a single integer. | 22 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function f(x) = x^2 + ax + b, where a, b ∈ R, find the minimum value of m such that m is the maximum value of |f(x)| when x ∈ [-1,1]. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Algebra -> Quadratic Functions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain information transmission process, each piece of information can only be a permutation of a 6-digit number composed of 0 or 1 (digits can be repeated and the first digit can be 0). Different permutations represent different information. Calculate the probability that a randomly selected piece of information... | \frac{15}{64} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the reciprocals of the first 2002 triangular numbers, where the nth triangular number is given by the formula $t_n = \\frac{n(n+1)}{2}$. Express your answer as a fraction in simplest form. | \\frac{4004}{2003} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} which satisfies 5^{a_{n+1}} = 25 * 5^{a_n}, and a_2 + a_4 + a_6 = 9, find the value of log_{1/3}(a_5 + a_7 + a_9). | -3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions",
"Mathematics -> Algebra -> Sequences and Series"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors \(\overrightarrow{a} = (-2, 2, 0)\) and \(\overrightarrow{b} = (1, 0, -1)\), use the dot product formula to find the angle between them. Express your answer in degrees, rounded to the nearest whole number if necessary. | 120^\circ | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
A sample consists of 5 positive integers with an arithmetic mean of 10 and a median of 12. Find the minimum possible difference between the largest and smallest numbers in this sample. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 35,800,000,000 in scientific notation. Provide your answer in the form of a number between 1 and 10 multiplied by a power of 10. | 3.58 \times 10^{10} | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation $\log_x 8 = 3$, find the value of $x$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given two points P(1, 3) and Q(4, -1), calculate the distance between these two points using the distance formula. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function f(x) = e^x - x on the interval [0,1]. Express your answer as a single value involving the natural logarithm base e. | e - 1 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
If the line 2x - y + 2 = 0 is parallel to the line y = kx + 1, what is the value of the real number k? Express your answer as a single real number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of distinct solutions to the equation $|x-|2x+1||=3$. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $x= \dfrac {\pi}{12}$, calculate the exact value of $\sin ^{4}x-\cos ^{4}x$. Express your answer as a simplified fraction or exact value, including any necessary negative sign or radical. | - \dfrac {\sqrt {3}}{2} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{BA}=(3,-2)$ and $\overrightarrow{AC}=(0,6)$, calculate the magnitude of $\overrightarrow{BC}$. Express your answer as a single number. | 5 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In isosceles triangle ABC, angle A measures 92 degrees. Line segment CB is extended to a point D. Find the measure of angle ABD. Express your answer in degrees as a whole number. | 136 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function y = sin(2x + φ) + 1, if its graph is symmetric about the line x = -π/8, calculate the possible value(s) of φ. Express your answer in radians. | φ = 3π/4 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of (x - 1/x^2)^9. Express your answer as a single integer. | -84 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(2+x,1-y)$, and knowing that $\overrightarrow{a}⊥\overrightarrow{b}$, calculate the value of $2y-x$. Provide your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. If $(2a+i)(1-2i)$ is a pure imaginary number, then find the real number $a$. Express your answer as a single real number. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
In the right triangle ABC, side AC is divided into 8 equal parts, and from each division point, a segment parallel to BC is drawn, resulting in a total of 7 segments. Given that BC = 10, find the sum of the lengths of these 7 segments. Express your answer as a whole number. | 35 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the binomial expansion of $(\sqrt{x} + \frac{2}{x})^n$, if only the fifth term has the maximum binomial coefficient, find the value of $n$. Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
The bar graph below shows the numbers of boys and girls in Mrs. Kuwabara's class. Calculate the percentage of students in the class who are girls. Express your answer as a percentage, including the percentage sign (%). | 40\% | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Point A on the number line represents the number -3. If point A is moved 5 units to the right and then 7 units to the left to point A', what is the number represented by A'? Provide your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Divide 6 boys and 4 girls into two groups, each containing 5 individuals, with each group comprising 3 boys and 2 girls. How many different distribution methods are there? Express your answer as a single integer. | 60 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {a_n}, the sum of the first n terms is S_n, a_4+a_7+a_{10}=9, S_{14}-S_{3}=77. Find the value of n that makes S_n minimum. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, the sum of the first 10 terms (S_{10}) is 120. Calculate the value of a_{1} + a_{10}. Express your answer as a single integer. | 24 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Let S<sub>n</sub> be the sum of the first n terms of the sequence {a<sub>n</sub>}. Given that S<sub>n</sub> = $$\frac {1}{2}$$a<sub>n</sub> + 1, n ∈ N<sub>+</sub>, calculate the value of a<sub>5</sub>. Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given a regression equation y = 3 - 5x, when the variable x increases by one unit, what is the average change in y? Express your answer as a single numerical value and specify whether it is an increase or decrease. | -5 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number equation $(3-4i)z=|4+3i|$, calculate the imaginary part of $z$. Express your answer as a simplified fraction. | \frac {4}{5} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors in the plane, $\overrightarrow{a}=(m,4)$ and $\overrightarrow{b}=(1,-2)$, and $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of m. Express your answer as a single integer. | -2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
If w is a positive integer and w^3 = 9w, calculate the value of w^5. | 243 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Polynomial Operations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=\frac{1+i}{2-2i}$, calculate its magnitude, denoted as $|z|$. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Josanna's test scores to date are $90, 80, 70, 60,$ and $85$. Her goal is to raise her test average by at least $3$ points with her next test. What is the minimum test score she would need to accomplish this goal? Express your answer as a whole number. | 95 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality x^2 + x < a/b + b/a, which holds for any positive real numbers a and b, determine the range of real number x. Express your answer as an interval or range using interval notation (e.g., (-2, 1)). | (-2, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given two vectors in the plane, $\overrightarrow{a}$ and $\overrightarrow{b}$, with an angle of $\frac{\pi}{3}$ between them, and $|\overrightarrow{b}|=1$, $|\overrightarrow{a}+2\overrightarrow{b}|=2\sqrt{3}$, calculate the magnitude of $\overrightarrow{a}$. Express your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
An exhibition hall has two entrances, $A$ and $B$, and three exits, $D$, $E$, and $F$. What is the probability of entering from entrance $A$ and exiting from exit $F$? Express your answer as a simplified fraction. | \dfrac{1}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the closed figure formed by the curve y = 3x^2 and the lines x = 1, x = 2, and the x-axis. Express your answer as a single number. | 7 | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
When travelling from London to Edinburgh by train, you pass a sign saying 'Edinburgh 200 miles'. Then, \(3 \frac{1}{2}\) miles later, you pass another sign saying 'Half way between London and Edinburgh'. Calculate the total distance in miles from London to Edinburgh. | 393 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the complex number expression: \frac{1-\sqrt{3}i}{{{(\sqrt{3}+i)}^{2}}}. Express your answer in the form a+bi, where a and b are real numbers, and include the boxed notation around your answer. | -\frac{1}{4}-\frac{\sqrt{3}}{4}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
From among ten students, including students A and B, four are to be selected to participate in a certain charity activity. The requirement is that at least one among students A and B must participate. Calculate the total number of different selection methods. Express your answer as a single integer. | 140 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Using Horner's method to calculate the polynomial function f(x) = 6x^5 - 4x^4 + x^3 - 2x^2 - 9x, calculate the number of additions (or subtractions) and multiplications required. Express your answer as two integers separated by a comma, where the first integer represents the number of additions (or subtractions) and th... | 4, 5 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
A circular truncated cone has a top radius of 2, a bottom radius of 4, and a slant height of 2√10. Calculate the volume of the truncated cone. Express your answer in terms of π. | 56\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given points A(0,30), B(20,10), and O(0,0), use Pick's theorem to calculate the number of lattice points inside triangle ABO. Pick's theorem states that the area of a polygon with lattice points as vertices is S = N + 1/2L - 1, where N and L represent the number of lattice points inside and on the boundary of the polyg... | 271 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Convert 4.15 billion to scientific notation. Express your answer in the form $a \times 10^b$, where $a$ is a number between 1 and 10, and $b$ is an integer. | 4.15 \times 10^9 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_{n}$ be the sum of the first $n$ terms of the arithmetic sequence ${a_{n}}$. Given that $a_{2}+a_{3}+a_{4}=3$, find the value of $S_{5}$. Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
In a positive geometric sequence $\{a_n\}$, the sum of the first 2 terms is 7 and the sum of the first 6 terms is 91. Find the sum of the first 4 terms. Express your answer as a single numerical value. | 28 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$), its left vertex is $A$, and point $B\left( 0,\frac{\sqrt{15}}{3}b \right)$. If the perpendicular bisector of segment $AB$ passes through the right focus $F$, find the eccentricity of the hyperbola $C$. Express your answer as a single num... | 2 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
A point starts from the point representing $-3$ on the number line, first moves $5$ units to the left, and then moves $10$ units to the right. What is the number represented by the endpoint? Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the proposition: "If \( n \) is a positive odd number, and the sum of its digits is 4, and none of the digits are zero, then \( n \) is a prime number." How many counterexamples exist that meet all the given conditions but are not prime numbers? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
A sequence of monomials is given by $a$, $3a^{2}$, $5a^{3}$, $7a^{4}$, $9a^{5}$, $\ldots$. Based on the pattern observed in the sequence, determine the formula for the nth monomial in terms of n and a. Express your answer as a formula. | (2n - 1)a^{n} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$=(2,-3,1) and $\overrightarrow{b}$=(1,x,4) that satisfy the condition $\overrightarrow{a} \cdot \overrightarrow{b}=0$, calculate the value of x. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the polar coordinate system, derive the equation of the line that passes through the point (1, 0) and is perpendicular to the polar axis. Express your answer as a polar equation using standard polar coordinate notation (e.g., rho, theta). | \rho\cos\theta=1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of elements in the set $\{x|C_{10}^{x} \leq 20\}$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
For a certain type of corn seed, the probability of each seed germinating is 90%. If 5 seeds are sown, find the probability that exactly two seeds do not germinate. Express your answer as a decimal approximation rounded to two decimal places. | 0.07 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
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