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 |
|---|---|---|---|---|---|
Find the minimum value of the function f(x) = sin(x)cos(x). Express your answer as a single numerical value, including the negative sign if the answer is negative. | -\dfrac{1}{2} | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation sinθ + cosθ = 1/3, calculate the value of sin2θ. Express your answer as a fraction. | -\frac{8}{9} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of trailing zeroes in the product of \(25^5\), \(150^4\), and \(2008^3\). Express your answer as a whole number. | 13 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {{a_n}}, let {S_n} denote the sum of its first n terms. If {a_4 + a_6 + a_8 = 15}, find the value of {S_{11}}. Express your answer as a numerical value. | 55 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
In a geometric sequence with positive terms, the product of the first and ninth terms is 10 (a_1a_9 = 10). Calculate the value of the fifth term (a_5). | \sqrt{10} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
From the five numbers $1$, $2$, $3$, $4$, $5$, if any two numbers are selected, calculate the probability that both numbers are even. Express your answer as a simplified fraction. | \frac{1}{10} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the combinatorial expression $$C_n^0 - 2C_n^1 + 4C_n^2 - 8C_n^3 + \ldots + (-2)^n C_n^n.$$ Express your answer as a mathematical expression. | (-1)^n | big_math | [
"Mathematics -> Applied Mathematics -> Other",
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a = (-1, 1)$ and $b = (2, x)$, if vector $a$ is perpendicular to $(a + b)$, then find the value of the real number $x$. Express your answer as a single real number. | 0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\left\{x_{n}\right\}$, with $x_{1}=1$ and $x_{n+1}=\frac{\sqrt{3} x_{n}+1}{\sqrt{3}-x_{n}}$, determine the value of $\sum_{n=1}^{2008} x_{n}$. Express your answer as a single integer or a mathematical expression. | 0 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given the Fibonacci sequence $\{a_n\}$, satisfying $a_1=1$, $a_2=1$, $a_{n+2}=a_{n+1}+a_n$ ($n\in\mathbb{N}^*$), find the remainder of $a_{2012}$ divided by 4. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Given that P(m, 2) is a point on the terminal side of angle α, and tan(α + π/4) = 3, find the value of cosα. Express your answer as a fraction or decimal value. | \frac{2\sqrt{5}}{5} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = a*sin(x) - b*tan(x) + 4*cos(pi/3), and f(-1) = 1, calculate the value of f(1). Express your answer as a single numerical value. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Find the y-intercept of the line $x+2y-1=0$. Express your answer as a single number. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Linear Equations"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given plane vectors $\overrightarrow{a}=(1,-\sqrt{3})$ and $\overrightarrow{b}=(2,m)$. If $(\overrightarrow{a}+\overrightarrow{b})∥(\overrightarrow{a}-\overrightarrow{b})$, calculate the value of $m$. Express your answer as a single value, including the negative sign if applicable. | -2\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$ and $b$ are opposite numbers, and the reciprocal of $c$ is $4$, find the value of $3a + 3b - 4c$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
If the mutually prime integers \(A\), \(B\), and \(C\) satisfy \( A \log_{200} 5 + B \log_{200} 2 = C \), then calculate \( A + B + C \). Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
The ratio of the land area to the ocean area on the Earth's surface is 29:71. If three-quarters of the land is in the northern hemisphere, then what is the ratio of the ocean area in the southern to the northern hemisphere? Express your answer as a simplified ratio, using a colon (:) to separate the parts of the ratio. | 171:113 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a random variable $X$ with the probability density function (PDF) $P(x) = \\begin{cases} 0, & x\\leq 0 \\\\ e^{-x}, & x > 0 \\end{cases}$, find the probability that $X$ falls within the interval (1, 3). Express your answer as a fraction or decimal value. | \\frac{e^2 - 1}{e^3} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point P is located in the second quadrant. The distance from point P to the x-axis is 2, and its distance to the y-axis is 3. Find the coordinates of point P. Express your answer as an ordered pair (x, y), including negative signs if applicable. | (-3, 2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given a function $f(x) = g(x) + x^2$, for any $x \in \mathbb{R}$ we always have $f(-x) + f(x) = 0$. Also given that $g(-1) = 1$, find the value of $g(1)$. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arbitrary point in space, three non-coplanar lines are drawn through it. Determine the number of planes that these lines can form. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
A direction vector of line l is (4, -4), and its intersection with the y-axis is at point (0, -4). If M and N are two moving points on line l such that |MN| = 4, find the minimum value of OM · ON. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A physical education teacher has 9 identical footballs to be placed into three boxes labeled 1, 2, and 3. The number of footballs in each box must not be less than its own label number. Calculate the total number of different ways to place the footballs into the boxes while adhering to this requirement. | 10 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
What is the resulting function expression after translating the parabola y = x^2 3 units to the right and then 2 units up? Express your answer as a function of x, using the format y = f(x). | y = (x - 3)^2 + 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.46875 | SynthLabsAI/Big-Math-RL-Verified |
The number in an unshaded square is obtained by adding the numbers connected to it from the row above. Using the values given in the second row (11, 6+x, x+7) and knowing that the single entry in the final row equals 60, calculate the exact value of x. Express your answer as a single numerical value. | 10 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A forestry farm has 30,000 saplings, of which there are 4,000 pine saplings. To investigate the growth condition of the saplings, a stratified sampling method is used to take a sample with a size of 150. Calculate the expected number of pine saplings in the sample. Express your answer as a whole number. | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $y^2=2px$ with the latus rectum equal to 4, calculate the value of $p$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given that vectors $a$ and $b$ are both unit vectors, and the angle between them is $60^{\circ}$, calculate the magnitude of the vector $a-3b$, i.e., $|a-3b|$. | \sqrt{7} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the radius of a sector is 16 and the central angle is 2 radians, find the length of the arc of the sector. Express your answer as a whole number. | 32 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that July 1, 1997, was a Tuesday, determine the day of the week for June 9, 1898. Express your answer as the full day of the week (e.g., Monday, Tuesday, etc.). | Thursday | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
How many three-digit numbers greater than 200 can be written using only the digits 1, 3, and 5? Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
If the line $l$ passes through the origin and $(3,-3)$, calculate its angle of inclination in degrees. Express your answer as an angle in the range 0° to 360°, including the degree symbol (°). | 135° | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Ann and Sue bought identical boxes of stationery. Ann used hers to write 1-sheet letters and Sue used hers to write 3-sheet letters. Ann used all the envelopes and had 50 sheets of paper left, while Sue used all of the sheets of paper and had 50 envelopes left. Calculate the number of sheets of paper in each box. | 150 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
What is the exact value of $\sin (-600^{\circ})$? Express your answer as a fraction with a square root if necessary. | -\frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=1+(1-i)^2$, find the value of its magnitude $|z|$. Express your answer as a single value, including the square root symbol if necessary. | \sqrt{5} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the ellipse C defined by the equation $\frac {x^{2}}{8}+ \frac {y^{2}}{4}=1$, find the number of points P on the ellipse that satisfy the condition that the line segments PF<sub>1</sub> and PF<sub>2</sub>, connecting P to the two foci of the ellipse, are perpendicular to each other. Express your answer as a whole... | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of the function $y = f(x)$ is $\left[ 1,2 \right]$, determine the domain of $y=f(x+1)$. Express your answer in interval notation. | \left[ 0,1 \right] | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
In the figure shown, $PQRS$ has three sides of equal length and $SR = 16 \text{ cm}$. If the perimeter of $PQRS$ is $40 \text{ cm}$, then what is the length of $PQ$? Express your answer in centimeters as a whole number. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Polygons"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given a real constant R and a number N (N ≠ 0), if N diminished by four times its reciprocal equals R, then find the sum of all possible values of N for this given R. Express your answer in terms of R. | R | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given the trinomial $16x^{2}+mxy+25y^{2}$, find the value of $m$ that makes it a perfect square trinomial. Express your answer as a numerical value, including the ± symbol if applicable. | ±40 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the function f(x) = x * e^x at the point (1, e). Express your answer in the form y = mx + b, where m and b should be expressed in terms of e. | y = 2ex - e | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f\left(x\right)=\left\{\begin{array}{c}{x}^{2}+1\begin{array}{c},x＜1\end{array}\right.\\ 2x\begin{array}{c},x≥1\end{array}\end{array}$, find all real numbers $a$ such that $f\left(a\right)=10$. Express your answer as a list of real numbers. | -3, 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( S_n \) is the sum of the first \( n \) terms of an arithmetic sequence \( \{a_n\} \). It is known that \( S_9 = 18 \) and \( a_{n-4} = 30 \) for \( n > 9 \), and \( S_n = 336 \). Calculate the value of \( n \). Express your answer as a single integer. | 21 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence ${a_n}$, where $a_3=2$ and $a_7=1$, and the sequence ${\frac{1}{1+{a_n}}}$ is an arithmetic sequence, find the value of $a_{11}$. Express your answer as a numerical value in the format $\boxed{[answer]}$. | \frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Using systematic sampling, 160 students are divided into 20 groups of 8 students each. The students are numbered from 1 to 160. If the number to be drawn from the 16th group is 126, calculate the number that would be drawn from the first group. | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
Find the interval where the function $f(x) = \frac{1}{2}x^2 - \ln x$ is strictly decreasing. Express your answer as an interval using interval notation (e.g., (a, b)). | \left(0, 1\right) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 6,390,000 in scientific notation. Use the format $a \\times 10^{b}$, where $a$ is a number between 1 and 10, and $b$ is an integer. | 6.39 \times 10^{6} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given two points in space $P(-1, 2, -3)$ and $Q(3, -2, -1)$, calculate the distance between points $P$ and $Q$. Express your answer as a single number. | 6 | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\[x\}$ represents the largest integer not exceeding the real number $x$ $(x∈R)$, for example: $\[-1.3]=-2$, $\[0.8]=0$, $\[3.4]=3$. Define $\{x}=x-\[x\}$. Calculate the sum of the fractional parts of the sequence: $\{ \frac{1}{2015} \}+\{ \frac{2}{2015} \}+\{ \frac{3}{2015} \}+\cdots+\{ \frac{2015}{2015} \}... | 1007 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the universal set $U={x∈Z∣|x|<5}$, sets $A={-2,1,3,4}$, and $B={0,2,4}$, find the set $A∩{∁}_UB$. Express your answer in set notation, including all elements of the resulting set. | {-2,1,3} | big_math | [
"Mathematics -> Set Theory -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of \(3 + 3^{3}\). Express your answer as a single integer. | 30 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Convert the decimal number 3 into a binary number using the method of repeated division by 2 and recording remainders. Express your answer as a binary number. | 11_2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Given the function $f(x)= \begin{cases} x^{2}+1,x\leqslant 1 \\ \dfrac {2}{x},x > 1\\ \end{cases}$, find the value of $f(f(2))$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
In isosceles triangle \( \triangle ABC \), \( AC = BC \) and \( \angle ACB = 40^\circ \). A circle with diameter \( BC \) intersects \( AC \) and \( AB \) at points \( D \) and \( E \) respectively. The diagonals of quadrilateral \( BCDE \) intersect at point \( F \). Find the measure of \( \angle BFC \) in degrees. | 110^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( E \) is the midpoint of side \( BC \) of the square \( ABCD \), and perpendicular lines are drawn from points \( B \) and \( D \) to \( AE \) with feet of the perpendiculars at points \( F \) and \( G \) respectively, calculate the measure of \( \angle FBG \) in degrees. | 45^\circ | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x|x^{2}+5x-6 \lt 0\}$ and $B=\{x|x^{2}-5x-6 \lt 0\}$, find the union of sets A and B, $A\cup B$, in interval notation. | (-6, 6) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
In the pigeon house of Ma'an Pool Park, a number of pigeons are being raised. The breeder mentioned that if 75 pigeons are sold, then the pigeon feed can last for 20 days. If another 100 pigeons are bought, then the pigeon feed can only last for 15 days. Calculate the initial number of pigeons in the pigeon house. Expr... | 600 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given the parabola $x^{2}=2py$ with its focus at $F(0,2)$, find the value of $p$. Express your answer as a single numerical value. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $x^{2}+ax+1\geqslant 0$ that holds for all $x\in(0, \frac {1}{2}]$, find the minimum value of the real number $a$. Express your answer as a single real number. | -\frac {5}{2} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) defined on R is a monotonic function, and for any x in R, it always satisfies f(f(x) - 2^x) = -1/2, find the zero point(s) of the function f(x). Express your answer as a single number in a boxed format. | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
What is the smallest possible value of x that makes the square root expression $\sqrt{x-2}$ meaningful? Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given a complex number $z=\frac{1-i}{{i}^{3}}$, calculate the imaginary part of its conjugate, $\overline{z}$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Express the number 350,000,000 in scientific notation. Include the base and exponent in your answer. | 3.5 \times 10^{8} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given that line $l_1: y = k(x - 4)$ is symmetric to line $l_2$ about the point $(2, 1)$, find the fixed point that line $l_2$ consistently passes through. Express your answer as an ordered pair (x, y). | (0, 2) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a grid with 150 rows and 150 columns, numbered from 1 to 150, the shading pattern is as follows:
- In row 1, every box is shaded.
- In row 2, every second box is shaded.
- In row 3, every third box is shaded.
- This pattern continues such that every $n$th box in row $n$ is shaded.
Find the column number that has th... | 120 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given the inequality $(x-1)^2 < 1$ holds under the sufficient but not necessary condition that $x \in (1-m, 1+m)$, find the range of the real number $m$ in interval notation. | (0, 1) | big_math | [
"Mathematics -> Algebra -> Inequalities"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Find the arithmetic square root of $\sqrt{16}$. Provide your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the following pseudo-code:
```
a=3;
b=-5;
c=8;
a=b;
b=c;
c=a;
```
What is the output of the print statement `Print(“%d %d %d”, a, b, c)`? Express your answer as a sequence of three integers separated by commas. | -5, 8, -5 | big_math | [
"Mathematics -> Discrete Mathematics -> Algorithms"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
Given $(1-x)^{10} = a_0 + a_1(1+x) + a_2(1+x)^2 + \ldots + a_{10}(1+x)^{10}$, find the value of $a_9$. Express your answer as an integer. | -20 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of $x$ in the expansion of $(x^2 + 3x + 2)^5$. Express your answer as a single numerical value. | 240 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In quadrilateral ABCD, ∠BAC = ∠BDC = 90°, AB = AC = √5, and CD = 1. The intersection point of the diagonals is M. Find DM. Express your answer as a fraction or decimal. | \frac{1}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the greatest common divisor (GCD) of 840 and 1764. Express your answer as a whole number. | 84 | big_math | [
"Mathematics -> Number Theory -> Greatest Common Divisors (GCD)"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a$, $b$, $c$ form a geometric sequence, determine the number of intersection points between the graph of the function $y = ax^2 + bx + c$ and the x-axis. | 0 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.53125 | SynthLabsAI/Big-Math-RL-Verified |
Given the quadratic function y = -3(x+2)^2 + 1, find the vertex coordinates. Express your answer as an ordered pair (x, y). | (-2, 1) | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Quadratic Functions"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
A tangent is drawn from a point on the line $y=x+1$ to the circle $(x-3)^{2}+y^{2}=1$. Calculate the minimum distance from that point to the point of tangency. | \sqrt{7} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Among the real numbers $3.14159$, $\sqrt[3]{64}$, $1.010010001$, $\sqrt{7}$, $\pi$, $\frac{2}{7}$, how many are irrational? Provide your answer as a whole number. | 2 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
There are six people standing in a row. Person A can only be at the head or the tail of the row, and persons B and C must be adjacent. Find the number of ways to arrange them that meets the requirements. Express your answer as a numerical value. | 96 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
A certain middle school randomly surveyed 15 students to understand the time they spend on physical exercise at school per week. The data is listed below. Find the median of the time spent on physical exercise per week by these 15 students.
| Exercise Time (hours) | $5$ | $6$ | $7$ | $8$ |
|-----------------------|---... | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Starting at square 1 and finishing at square 7, how many possible routes are there if you move to a higher numbered adjacent square at each step? Express your answer as a single integer. | 13 | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
If in the fraction $\frac{x}{y}$ we decrease the numerator by $40 \%$ and the denominator $y$ by $60 \%$, what is the resulting percentage change in the fraction? Express your answer as a percentage increase or decrease. | 50\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Using Horner's method to calculate the value of the polynomial f(x) = 5x^5 + 4x^4 + 3x^3 + 2x^2 + x + 1 when x = 2, find the number of multiplications and additions required respectively. Express your answer as two integers separated by a comma, where the first integer represents the number of multiplications and the s... | 5, 5 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Let a random experiment have only two outcomes: A and not A, with P(A) = p. Define the random variable X as follows:
$$
X= \begin{cases}
1, & \text{if } A \text{ occurs} \\
0, & \text{if } A \text{ does not occur}
\end{cases}
$$
Calculate the variance of X in terms of p. Simplify your expression. | p(1-p) | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
In a certain population, the ratio of the number of women to the number of men is 11 to 10. If the average (arithmetic mean) age of the women is 34 and the average age of the men is 32, calculate the average age of the population. Express your answer as a mixed number or decimal. | 33\frac{1}{21} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Given $1 < a < 2$, calculate the exact value of the algebraic expression $|a-2|+|1-a|$. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.5 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(x, \sqrt {3})$ and $\overrightarrow{b}=(x,- \sqrt {3})$, if $(2 \overrightarrow{a}+ \overrightarrow{b}) \perp \overrightarrow{b}$, find the magnitude of vector $\overrightarrow{a}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
A person passes through three intersections with traffic lights on their way to work. Assuming that the events of encountering a red light at each intersection are independent, and the probability of encountering a red light is 0.3 at each intersection, calculate the expected number of times the person encounters a red... | 0.9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence $\{n(n+4)\left(\frac{2}{3}\right)^{n}\}$, find the value of $k$ for which the $k$-th term is the largest. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
The right focus of the hyperbola $\dfrac {x^{2}}{a^{2}}- \dfrac {y^{2}}{b^{2}}=1 (a > 0, b > 0)$ is the focus $F$ of the parabola $y^{2}=8x$. The two curves have a common point $P$ such that $|PF|=5$. Calculate the eccentricity of this hyperbola. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the eccentricity of the ellipse $\dfrac{x^2}{16} + \dfrac{y^2}{9} = 1$. Express your answer as a fraction with a square root in the numerator. | \dfrac{\sqrt{7}}{4} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A jar is filled with soybeans. In order to estimate the number of soybeans in the jar, 200 beans are taken out, dyed red, then put back into the jar. The soybeans in the jar are then thoroughly mixed. After that, 60 beans are randomly taken out, and it is found that 5 of them are red. Estimate the total number of soybe... | 2400 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos x = 2m - 1$, and $x \in \mathbb{R}$, determine the range of values for $m$. Express your answer as an interval in standard interval notation. | [0, 1] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.5625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = e^x, find its derivative f'(x). Express your answer as a function of x. | e^x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the functions f(x) and g(x) are both odd functions, and F(x) = af(x) + bg(x) + 2 has a maximum value of 5 on the interval (0, +∞), find the minimum value of F(x) on the interval (-∞, 0). Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Find the degree of the slope angle of the line $x - y + 8 = 0$. Express your answer in degrees, including the degree symbol (°). | 45° | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Find the largest number in the set {0.109, 0.2, 0.111, 0.114, 0.19}. Express your answer as a decimal number, including all necessary decimal places. | 0.2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, point A(1,-1) is moved 3 units to the left. Find the coordinates of the resulting point A'. Express your answer as an ordered pair (x, y). | (-2, -1) | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
For the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ $(a > 0, b > 0)$ with an eccentricity of $2$, find the minimum value of $\frac{b^2 + 1}{3a}$. Express your answer as a single value. | \frac{2\sqrt{3}}{3} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.546875 | SynthLabsAI/Big-Math-RL-Verified |
A sector of a circle has a central angle of 2 radians and a perimeter of 8 cm. Find the area of the sector. Express your answer in square centimeters (cm²). | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
There are 18 teams participating in the opening ceremony of a competition. When entering, the 1st team has 27 members, the 2nd team has 26 members, and the 18th team has 10 members. If they enter in a single file, and all 18 teams' members are assigned numbers from 1 to 333 in the order they enter, how many teams have ... | 9 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the linear function y = (k+1)x + k-2, find the range of values for k such that the function passes through the first, third, and fourth quadrants. Express your answer in interval notation. | (-1, 2) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
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