instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
value |
|---|---|---|---|---|---|
The average of four test marks is 60. The first three marks were 30, 55, and 65. What is the fourth mark? Provide your answer as a whole number. | 90 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f$ where $f(1) = 1$, $f(2) = 3$, and for $n \geq 3$, $f(n) = f(n-2) + f(n-1)$, find the value of $f(10)$. Express your answer as a single integer. | 123 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
A tangent line is drawn to the circle $C: x^{2}+y^{2}-2x+4y-1=0$ at a point on the line $x+y=5$. Calculate the minimum length of the tangent line. Express your answer as a numerical value, including the square root symbol if necessary. | 2\sqrt{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $C$: $y^{2}=2x$ with a focus at $F$, and $A(x_{0},y_{0})$ is a point on $C$ with $|AF|= \frac {3}{2}x_{0}$, find the value of $x_{0}$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
To obtain the graph of $y=\cos 2x$, we need to shift the graph of the function $y=\sin (2x-\frac{π}{6})$ by some amount. Calculate the shift required to obtain the graph of $y=\cos 2x$. Express your answer in radians and specify the direction of the shift (left or right). | \frac{π}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that 1 meter = 10^9 nanometers, express 125 nanometers in scientific notation in meters. | 1.25 \times 10^{-7} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Three of the vertices of a parallelogram are \((0,1)\), \((1,2)\), and \((2,1)\). Calculate the area of the parallelogram. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Three circles, each with a radius of 10 cm, are drawn tangent to each other so that their centers are all in a straight line. These circles are inscribed in a rectangle which is inscribed in another circle. Calculate the area of the largest circle. Express your answer in terms of pi and include units (cm^2). | 1000 \pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A tetrahedron is formed such that each face is not an isosceles triangle. What is the minimum number of edges with different lengths in this tetrahedron? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
What is the result of the calculation $5-7$? Express your answer as an integer. | -2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Integers"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Find the constant term in the expansion of $(x^2 - x^{-4})^6$ (where $x \in \mathbb{R}$). Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
For a positive geometric sequence {a_n}, if a_1a_5 + 2a_3a_6 + a_1a_{11} = 16, then find the value of a_3 + a_6. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
In a production process, there are 4 stages and each stage requires one person to oversee it. Now, from among 6 workers named A, B, C, etc., 4 people are to be arranged to oversee each stage respectively. The first stage can only be overseen by one person from workers A and B, and the fourth stage can only be overseen ... | 36 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
According to the Central Committee's requirements for "precise" poverty alleviation, the agricultural economic department of a city dispatched four experts to investigate three counties. At least one expert should be assigned to each county. Calculate the probability that experts A and B are dispatched to the same coun... | \frac{1}{6} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of sets of two or more consecutive positive integers whose sum is 100. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given line segments a = 5, b = 10, c = 15, and d = 20, how many non-congruent trapezoids can be formed using these segments as sides? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
What is the reciprocal of $-\dfrac{5}{4}$? Express your answer as a fraction. | -\dfrac{4}{5} | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given that vectors a and b are both unit vectors, and the angle between them is 60 degrees, calculate the magnitude of the vector difference |a-3b|. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{7} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Find the modulus of the complex number $z= \frac{|1-i|}{3+i}$. Express your answer as a simplified fraction with a square root. | \frac{\sqrt{5}}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the terminal side of acute angle $\alpha$ passes through point P($\cos40°+1$, $\sin40°$), calculate the measure of angle $\alpha$ in degrees. | 20° | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given 5 consecutive movie tickets are to be distributed among 5 people, including A and B, with each person receiving one ticket and the tickets A and B receive being consecutive, calculate the total number of different distribution methods. Express your answer as a single integer. | 48 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z$ satisfying $|z|+z=2+4i$, calculate the value of $z$ in the form $a + bi$, where $a$ and $b$ are real numbers. | -3+4i | big_math | [
"Mathematics -> Algebra -> Complex Numbers -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Given a parabola $y^2=8x$ and a point $P$ on it, the distance from point $P$ to the y-axis is 4. Find the distance from point $P$ to the focus of this parabola. Express your answer as a whole number. | 6 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x|-1 \lt x \lt 2\}$ and $B=\{x|x^{2}+2x\leqslant 0\}$, calculate the intersection $A\cap B$. Express your answer in interval notation. | (-1, 0] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the value of \\( \dfrac {\sin (2\pi-\theta)\cos (\pi+\theta)\cos ( \dfrac {\pi}{2}+\theta)\cos ( \dfrac {11\pi}{2}-\theta)}{\cos (\pi-\theta)\sin (3\pi-\theta)\sin (-\pi-\theta)\sin ( \dfrac {9\pi}{2}+\theta)}\\). Express your answer as a simplified trigonometric expression. | -\tan \theta | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
There are 6 numbers forming an arithmetic sequence with the first term being 2 and the common difference also being 2. If two numbers are randomly selected from these 6 numbers, find the probability that the sum of these two numbers is less than 12. Express your answer as a simplified fraction. | \frac{4}{15} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)=\left\{{\begin{array}{l}{{x^2}-{2^x}-1, x≥0}\\{{{log}_2}|x|+1, x＜0}\end{array}}\right.$, calculate $f\left(f\left(1\right)\right)$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A program uses the following loop statement:
i = 1
Do
i = i + 2
S = 2i + 3
i = i - 1
Loop While i < 8
What is the final value of S after the loop exits? | 21 | big_math | [
"Mathematics -> Other -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Abraham's mathematics exam had 30 algebra questions and 50 geometry questions, each worth 1 mark. He got 70% of the algebra questions correct, and his overall exam mark was 80%. Calculate the number of geometry questions he answered correctly. Express your answer as a whole number. | 43 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The first four terms of an arithmetic sequence are $a$, $x$, $b$, $2x$. Find the value of $\frac{a}{b}$. Express your answer as a simplified fraction. | \frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Find the value range of the function y = 1 + 2^x in the interval x ∈ [0, 1]. Express your answer as a closed interval. | [2, 3] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
On the highway, starting from 3 kilometers, there is a speed limit sign every 4 kilometers, and starting from 10 kilometers, there is a speed monitoring device every 9 kilometers. The first time both types of facilities are encountered simultaneously is at 19 kilometers. Calculate the distance at which both types of fa... | 55 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Let the increasing sequence of positive terms \( a_{1}, a_{2}, \cdots, a_{n} \) be the simplest proper fractions with the denominator 60. Calculate the sum \( \sum_{i=1}^{n} \cos(a_{i} \pi) \). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence $\{a_n\}$ where all terms are positive, and $a_6 \cdot a_{10} + a_3 \cdot a_5 = 26$, $a_5 \cdot a_7 = 5$, find the value of $a_4 + a_8$. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy: $|\overrightarrow{a}+ \overrightarrow{b}|=2 \sqrt {3}$ and $|\overrightarrow{a}- \overrightarrow{b}|=2$, calculate the dot product $\overrightarrow{a}\cdot \overrightarrow{b}$. Provide your answer as a numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
From 3 type A and 4 type B televisions, select 3 in total, with at least one of each type. Find the number of different ways to do this. Express your answer as a whole number. | 30 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
A square is divided by 5 lines parallel to one pair of its sides and 3 lines parallel to the other pair of sides into 24 rectangles. If the sum of the perimeters of these 24 rectangles is 24, what is the area of the original square? Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $125^{2015}\\times \\left(-0.008\\right)^{2016}$. Express your answer as a fraction or decimal. | \frac{1}{125} | big_math | [
"Mathematics -> Algebra -> Exponential Functions"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $$z= \frac {(cos45 ° +isin45 ° )^{2}}{1-i}$$, calculate the coordinates of the point corresponding to $z$ on the complex plane. Express your answer as a coordinate pair (a, b). | \left(-\frac{1}{2}, \frac{1}{2}\right) | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $|\vec{a}|=1$, $|\vec{b}|=1$, and the angle between $\vec{a}$ and $\vec{b}$ is 60°, calculate the value of $|2\vec{a} - \vec{b}|$. Express your answer as a single value, including any necessary mathematical notation. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Other",
"Mathematics -> Algebra -> Other",
"Mathematics -> Applied Mathematics -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $tan(α-\frac{π}{8})=2$, calculate the exact value of $tan(2α-\frac{π}{4})$. Express your answer as a single numerical value. | -\frac{4}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence {a_n} where each term is positive, a_1 = 2 and a_2, a_4+2, a_5 form an arithmetic sequence. Let S_n be the sum of the first n terms of the sequence {a_n}. Calculate S_5. | 62 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions",
"Mathematics -> Algebra -> Sequences and Series"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U=\{x\in\mathbb{N}|x\leqslant 5\}$ and the set $A=\{x\in\mathbb{N}|2x-5 < 0\}$, find the complement of $A$ in $U$, denoted as $\complement_{U}A$. Express your answer as a set of numbers in the format $\{a, b, c,...\}$. | \{3,4,5\} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive period of the function $y = \tan(\frac{x}{3} + \frac{\pi}{4})$. Express your answer in terms of pi. | 3\pi | big_math | [
"Mathematics -> Precalculus -> Functions -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
The Shan City Agricultural Science Research Institute planted 5 different types of seeds in 5 adjacent experimental fields. Among them, the seeds of types A and B are required to be planted in two adjacent fields, and neither can be planted in the fields at the ends. Find the number of different planting methods. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given $f_1(x)=\cos x$, $f_2(x)=f_1'(x)$, $f_3(x)=f_2'(x)$,..., $f_n(x)=f_{n-1}'(x)$, find the function $f_{2010}(x)$. Express your answer as a function of $x$ using standard trigonometric function notation. | -\sin x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sequence {a_n} is a geometric sequence with the sum of its first n terms denoted as S_n, if S_4 = 1 and S_8 = 4, find the value of a_{13}+a_{14}+a_{15}+a_{16}. Express your answer as a single integer. | 27 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a_n}, a_1 = 3 and a_3 = 12. Find the value of a_5. Express your answer as a single numerical value. | 48 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Sequences and Series"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that the base radius of a cylinder is 3cm, and the generatrix (lateral edge) is 5cm, calculate the lateral surface area of the cylinder. Express your answer in square centimeters (cm^2), including π if necessary. | 30\pi | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Surface Area"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
One can holds 12 ounces of soda. What is the minimum number of cans needed to provide at least 128 ounces of soda? Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the area of the plane region represented by the system of inequalities $$\begin{cases} (x-y+5)(x+y)\geqslant 0, \\ 0\leqslant x\leqslant 3 \end{cases}$$. | 24 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
For how many positive integer values of n is the expression n^2 + 2n a prime number? Express your answer as a whole number. | 1 | big_math | [
"Mathematics -> Number Theory -> Prime Numbers"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given the trinomial $a^{2}-ka+25$, find the value of $k$ such that the trinomial is a perfect square. Express your answer as a numerical value, including both positive and negative possibilities if applicable. | 10 \text{ or } -10 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Xiaoming has 4 identical coins, each with a head and a tail. He wants to stack the 4 coins in such a way that the heads of adjacent coins do not face each other. Find the number of different stacking methods. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Among the following numbers: $\dfrac{22}{7}$, $-3.5$, $0$, $\sqrt {8}$, $\pi$, $0.1010010001\ldots$, how many are irrational? Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Four college students are to be divided into two units, with the condition that each unit must receive at least one student. Calculate the total number of possible distribution schemes, considering all different ways the students can be divided between the two units. | 14 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given the system of equations:
\[
\begin{align*}
x - y &= 2, \\
cx + y &= 3,
\end{align*}
\]
find the range of values for c such that the solution (x, y) is in Quadrant I (where both x > 0 and y > 0). Express your answer as an inequality or a range of values for c. | -1 < c < \frac{3}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\tan \theta = \sqrt{5}$, calculate $\cos 2\theta$. Express your answer as a simplified fraction. | -\frac{2}{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
In a regular octagon, three points are randomly chosen from the eight vertices to form a triangle. What is the probability that the triangle has at least one side that is also a side of the octagon? Express your answer as a simplified fraction. | \frac{5}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let the random variable X follow the normal distribution N(0,1). If P(X>1) = p, find the probability P(X>-1). Express your answer as a probability value between 0 and 1. | 1-p | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(2x) = 2/(2+x), calculate 2f(x) in terms of x. Express your answer as a fraction or algebraic expression. | \frac{8}{4+x} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
A farmer divides his herd of n cows among his four sons so that one son gets one-half the herd, a second son gets one-fourth, a third son gets one-fifth, and the fourth son gets 7 cows. What is the total number of cows, n? Express your answer as a whole number. | 140 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {x | x = a^2 + 1, a in N+} and B = {y | y = b^2 - 4b + 5, b in N+}, determine the correct set relation between A and B. Express your answer using standard set notation (e.g., A = B, A ⊃ B, A ⊂ B, A ∩ B = ∅). | A ⊂ B | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that i is the imaginary unit, calculate the exact value of i^2033. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given a hyperbola $C: x^2 - y^2 = 2$ with its foci $F_1$ and $F_2$, and let $O$ be the origin. A point $P$ lies on the hyperbola $C$ such that $|OP| = 2$. Calculate the area $S_{\triangle PF_{1}F_{2}}$ of triangle $PF_1F_2$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=\frac{2i}{1-i}$, calculate the value of $z•\overline{z}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Point M is located in the second quadrant, above the x-axis, with a distance of 2 to the x-axis and a distance of 1 to the y-axis. Find the coordinates (x, y) of point M. Express your answer as a coordinate pair (x, y), where x and y are integers. | (-1, 2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A die is rolled consecutively three times. Calculate the probability that the sequence of numbers facing upwards forms an arithmetic progression. Express your answer as a simplified fraction. | \frac{1}{12} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Given that \(a, b, c\) are distinct real roots of the polynomial equation \(x^3 - 3x + 1 = 0\), find the value of \(a^8 + b^8 + c^8\). Express your answer as a single integer. | 186 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
A polyhedron has 6 faces and 8 vertices. Using Euler's formula, F + V - E = 2, calculate the number of edges (E) of the polyhedron. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
On a line segment MN of length 16cm, a point P is chosen at random. A rectangle is formed with MP and NP as adjacent sides. Calculate the probability that the area of this rectangle is greater than 60cm². Express your answer as a simplified fraction in the form \boxed{\frac{a}{b}}. | \frac{1}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Other"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given \(x = 11\) and \(y = -8\), solve for \(z\) in the equation \(2x - 3z = 5y\). Express your answer as a fraction in the form \(\boxed{\frac{a}{b}}\). | \frac{62}{3} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Simple Equations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $f(x)=\frac{3x^2}{\sqrt{1-x}}+\lg (3x+1)$. Express your answer as an interval. | \left(-\frac{1}{3},1\right) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
In the geometric sequence {a<sub>n</sub>}, a<sub>1</sub>a<sub>3</sub> = a<sub>4</sub> = 4, find the value of a<sub>6</sub>. Express your answer as a single numerical value. | 8 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
If the sum of the first $3n$ positive integers is $150$ more than the sum of the first $n$ positive integers, then find the sum of the first $4n$ positive integers. Express your answer as a single integer. | 300 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
Using weights of 1 gram, 3 grams, 9 grams, 27 grams, and 81 grams, calculate the total number of different weights that can be measured on a balance scale (the weights can be placed on either side of the scale). Express your answer as a whole number. | 121 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola y^2 = 4x, if the distance from point M on the parabola to its focus is 10, find the distance from M to the y-axis. Express your answer as a whole number. | 9 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a \in \mathbb{R}$, if $\frac{1 + ai}{2 + i}$ is a real number, then find the value of $a$. Express your answer as a single value. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Among the real numbers $-\frac{22}{7}$, $0$, $\sqrt{2}$, $\sqrt{4}$, $\frac{\pi}{2}$, how many are irrational? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of the expression $50 + 50 \div 50 + 50$. Provide your answer as a single integer. | 101 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
The coordinates of point P are (2,-1). Find the coordinates of point P after reflecting it across the origin. Express your answer as an ordered pair (x, y). | (-2,1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
A school wants to select 6 students from 10 to participate in a study seminar. However, two students, A and B, cannot attend at the same time. Calculate the total number of different selection methods. Express your answer as a numerical value. | 140 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of pairs of positive integers \(a\) and \(b\) that satisfy the equation \(\frac{a}{7} + \frac{2}{b} = 1\). Express your answer as a single integer. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.40625 | SynthLabsAI/Big-Math-RL-Verified |
Find the last four digits of $5^{2017}$. Express your answer as a four-digit number. | 3125 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Let P be any point on the curve $y=x^2-\ln x$. Calculate the minimum distance from point P to the line $y=x-4$. Express your answer as a numerical value. | 2\sqrt{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Real numbers $\sqrt[3]{27}$, $0$, $-\pi$, $\sqrt{16}$, $\frac{1}{3}$, $0.1010010001\ldots$ (with one more zero between adjacent ones) are given. Count the number of irrational numbers among them. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Estimate the value of $\sqrt{7}$ by comparing it with the square roots of perfect squares close to $7$. Express your answer as an inequality or a range of values. | 2 < \sqrt{7} < 3 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Using the bisection method, find a root of the function f(x) = log10(x) + x - 3 accurate to 0.1. Reference data: log10(2.5) ≈ 0.398, log10(2.75) ≈ 0.439, log10(2.5625) ≈ 0.409. Express your answer as a single number accurate to 0.1. | 2.6 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
There is a 3-hour time difference between Vancouver and Toronto, with Vancouver being behind Toronto. For example, when it is 1:00 p.m. in Vancouver, it is 4:00 p.m. in Toronto. What time is it in Vancouver when it is 6:30 p.m. in Toronto? Express your answer in 12-hour clock format, including the 'p.m.' or 'a.m.' desi... | 3:30 p.m. | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain residential area, there are two independent security systems, $A$ and $B$. The probability of system $A$ and system $B$ malfunctioning at any given moment is $\dfrac{1}{8}$ and $p$, respectively. If the probability of exactly one system functioning properly at any given moment is $\dfrac{9}{40}$, then find... | \dfrac{2}{15} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $θ$, the "cross product" of $\overrightarrow{a}$ and $\overrightarrow{b}$, denoted as $\overrightarrow{a} × \overrightarrow{b}$, is defined as a vector with magnitude $|\overrightarrow{a} × \overrightarrow{b}| = |\overrightarrow{a}| \... | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Find the greatest common divisor (GCD) of 459 and 357. Express your answer as a single integer. | 51 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the point $M(m+3, m+1)$ lies on the $y$-axis. Find the coordinates of point $M$. Express your answer as an ordered pair $(x, y)$, where $x$ and $y$ are integers. | (0, -2) | big_math | [
"Mathematics -> Algebra -> Simple Equations"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
A boat's speed in still water is $15 \, \mathrm{mph}$, and a creek's current speed is $5 \, \mathrm{mph}$. If the boat travels back and forth over a certain distance, what is the ratio of the boat’s average speed for the round trip to its speed in still water? Express your answer as a fraction. | \frac{8}{9} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the regions bounded by the curves x^2 = 4y, x^2 = -4y, x = 4, and x = -4 for V1, and the conditions x^2 y <= 16, (y - 2)^2 >= 4x^2, and (y + 2)^2 >= 4 for V2, find the relationship between the volumes V1 and V2 obtained by rotating these regions about the y-axis. Express your answer using mathematical notation (e... | V1 = V2 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Volume"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$, $b$, and $c$ are the sides opposite to the internal angles $A$, $B$, and $C$ of $\triangle ABC$, and $\frac{a}{b}+\frac{b}{a}=4\cos C$, calculate the value of $\frac{\tan C}{\tan A}+\frac{\tan C}{\tan B}$. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other",
"Mathematics -> Trigonometry -> Trigonometric Functions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the greatest common divisor (GCD) of 459 and 357 using successive division. Express your answer as a single integer. | 51 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $\frac{1}{z} = -5i$, find the value of $\overset{.}{z}$. Express your answer as a complex number, including the imaginary unit i if necessary. | -\frac{i}{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation (1+i)(x+yi)=2, where x and y are real numbers, calculate the value of |2x+yi|. Express your answer in simplest radical form if necessary. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Find the minimum value of the expression |x-1| + |x-2| + |x-3|. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.