instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
value | domain listlengths 1 3 | llama8b_solve_rate float64 0 0.98 | _source stringclasses 1
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|---|---|---|---|---|---|
Given vectors $\overrightarrow{a}=(\cos 25^{\circ},\sin 25^{\circ})$ and $\overrightarrow{b}=(\sin 20^{\circ},\cos 20^{\circ})$, let $t$ be a real number and $\overrightarrow{u}=\overrightarrow{a}+t\overrightarrow{b}$. Calculate the minimum value of $|\overrightarrow{u}|$. Express your answer in the form $\boxed{[answe... | \frac{\sqrt{2}}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given the proposition $p: \forall x \in \left[1,5\right]$, $ax^{2}-x-4 \gt 0$, find the range of real number $a$ for which the proposition is true. Express your answer as an inequality in terms of $a$. | a > 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Find the derivative of the function $y=\cos (1+x^{2})$. Express your answer as a function of $x$ using standard mathematical notation. | -2x\sin (1+x^{2}) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Derivatives"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Find the last two digits of $2005^{1989}$. Express your answer as a two-digit number. | 25 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Let $f(x)$ be an odd function defined on $\mathbb{R}$. When $x \geq 0$, $f(x) = \lg(x+1) - b$ ($b$ is a constant). Calculate the value of $f(-9)$. Express your answer as a single numerical value. | -1 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.25 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions to the equation \((x+1)^{2}+(y-2)^{2}=1\). Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In the ellipse $\dfrac {x^{2}}{8}+ \dfrac {y^{2}}{6}=1$, find the slope of the line containing the chord with midpoint $M(2,1)$. Express your answer as a numerical value. | -\dfrac{3}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the distance from point (1, -1) to the line y = x + 1. Express your answer as a simplified fraction or exact decimal. | \frac{3\sqrt{2}}{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
Solve for \(n\) in the equation \(\sqrt{5+n}=7\). Express your answer as a single integer. | 44 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that point $P$ is on the curve $ρ \sin θ = 2$ and point $Q$ is on the curve $\begin{cases} x = 1 + \cos θ \
y = \sin θ \end{cases}$ (where $θ$ is a parameter), find the minimum value of $|PQ|$. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
The sequence is given by 1, (1+2), (1+2+2^2),..., (1+2+2^2+...+2^(n-1)). Find the sum of the first n terms of this sequence, expressed in terms of n. | 2^{n+1} - n - 2 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let the sum of the first n terms of an arithmetic sequence {a_n} be S_n, and a_1 = 2. If the sequence {S_n} is also an arithmetic sequence, find the value of S_{2014}. Express your answer as a single numerical value. | 4028 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
In the expansion of $(3+x)^5+(2-x)^6$, the binomial coefficient of the term containing $x^4$ is the term of the sequence $\{a_n\}$ with the general term $a_n=2n+2$. Which term of the sequence is it? Express your answer as the term number (e.g., 6th, 7th, etc.). | 9th | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let the function $f(x) = \begin{cases} x-3, & \text{if } x\geq 100 \\ f[f(x+5)], & \text{if } x<100 \end{cases}$. Calculate the value of $f(97)$. Provide your answer as a single number. | 98 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that p and q are two natural numbers that do not end in zero and their product is always a power of 10, find the last digit that p - q cannot be when p > q. Express your answer as a single digit. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Given $a+b=0$, and $a \neq b$, simplify the expression $(1-a) + (1-b)$. Provide your answer as a single value. | 2 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
The first three terms of an arithmetic sequence are $x-1$, $x+1$, and $2x+3$, respectively. Derive the general term formula of this sequence in the form $a_n =...$. | $a_n=2n-3$ | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
A, B, C, D, and E are standing in a row. If A and B must stand next to each other with B to the left of A, then what is the total number of different arrangements? Express your answer as a whole number. | 24 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Find the imaginary part of the complex number $2-3i$, where $i$ is the imaginary unit. Express your answer as the coefficient of $i$. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the vectors $\overrightarrow{a}=\left(8,-2,1\right)$ and $\overrightarrow{b}=(-4,1,k)$ are direction vectors of the same line, find the value of the real number $k$. Express your answer as a single real number. | -\frac{1}{2} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
The Jiujiang Meteorological Observatory has collected data indicating that on May 1st, the probability of rain in Xunyang District is $\frac{4}{15}$, the probability of windy conditions is $\frac{2}{15}$, and the probability of both wind and rain is $\frac{1}{10}$. Let A be the event of rain and B the event of wind. Us... | \frac{3}{4} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
Two straight lines intersect as shown. One angle adjacent to the angle marked $\square$ measures $120^{\circ}$. Find the measure of the angle marked $\square$. Express your answer in degrees as a whole number. | 60^{\circ} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Jorge starts watching three movies at 1:00 p.m. The first movie is 2 hours and 20 minutes long, followed by a 20-minute break. The second movie is 1 hour and 45 minutes long, followed by another 20-minute break. The last movie is 2 hours and 10 minutes long. What is the end time of the last movie? Express your answer i... | 7:55 p.m. | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Find the area of the triangle formed by the lines $y=5$, $y=1+x$, and $y=1-x$. Express your answer as a whole number. | 16 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given the set $M=\left\{{x∈{N}|{\frac{1}{{x-2}}}≤0}\right\}$, determine the elements of set M. Express your answer as a set of natural numbers. | \{1\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
The integers from 1 to 9 are listed on a blackboard. If an additional m eights and k nines are added to the list, the average of all of the numbers in the list is 7.3. Find the value of k + m. Express your answer as a single integer. | 21 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(x,-1)$, if $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of the real number $x$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given that \( p \) and \( q \) are positive integers such that \( pq = 75 \), calculate the sum of all possible values of \( p \). Express your answer as a single integer. | 124 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $a$ and $b$ that satisfy $|a| = 1$, $|b| = 4$, and $a \cdot b = 2\sqrt{3}$, determine the angle between $a$ and $b$ in radians. If possible, express your answer as an exact value; otherwise, round to 3 decimal places. | \frac{\pi}{6} | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Linear Algebra -> Vectors"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given that ξ follows the normal distribution N(4, σ^2) with σ > 0, and the probability that ξ takes a value in the interval (0, 4) is 0.4, calculate the probability that ξ takes a value in the interval (0, +∞). Express your answer as a decimal value between 0 and 1, rounded to 1 decimal place. | 0.9 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of f(x) is \u211d, and for any x \u2208 \u211d, it holds that f(-x) = -f(x), and f(x+2) = f(x), find the value of f(2010). | 0 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = sin(2x + θ), it is translated to the left by π/8 units to obtain the function g(x). The function g(x) is an even function. Calculate the smallest positive value of θ that makes this transformation valid. Express your answer in radians, as a fraction of π. | θ = π/4 | big_math | [
"Mathematics -> Precalculus -> Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Gill has four identical pictures to hang in a horizontal row on a wall that is 4800 mm wide. Each picture is 420 mm wide. She hangs the first two pictures so that one is on the extreme left of the wall and one is on the extreme right of the wall. Gill wants to hang the remaining two pictures such that all four pictures... | 730 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Find the conjugate of the complex number $\frac{2+i}{1-2i}$. Express your answer as a complex number in simplest form. | -i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the graph of the function f(x) defined on ℝ is symmetric about the line x = 2, and f(x) satisfies: for any x1, x2 ∈ (-∞, 2] (x1 ≠ x2), it holds that (f(x1) - f(x2))/(x1 - x2) < 0, and f(4) = 0. Find the solution set of the inequality f(x)/x < 0 with respect to x. Express your answer as an interval or union o... | (-∞, 0) ∪ (0, 4) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
If $a = \log_8 225$ and $b = \log_2 15$, express $a$ in terms of $b$. Provide your answer as an algebraic expression in terms of $b$. | \frac{2b}{3} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
In a polygon, each exterior angle measures $30^{\circ}$. Find the number of sides of this polygon. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The equation of a parabola is given by $x^{2}=4y$. Find the coordinates of the intersection point of the directrix of this parabola with the $y$-axis. Express your answer as an ordered pair $(x, y)$. | (0,-1) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Mary had ten pieces of paper. Some of them were squares, and the rest were triangles. She cut three squares diagonally from corner to corner. She then found that the total number of vertices of the 13 pieces of paper was 42. Calculate the number of triangles Mary had before making the cuts. Express your answer as a who... | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.265625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $$\frac {1+ai}{1-i}$$=-2-i (where i is the imaginary unit), find the value of the real number a. Express your answer as a single integer. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.125 | SynthLabsAI/Big-Math-RL-Verified |
Given the circle with equation x^{2}+y^{2}-4x+6y+3=0, find the equation of the circle that is concentric with it and passes through the point (-1,1). Express your answer in the standard form of the equation of a circle, (x-h)^{2}+(y-k)^{2}=r^{2}. | (x-2)^{2}+(y+3)^{2}=25 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
A particle moves in such a way that its velocity as a function of time is given by $v(t) = t^2 - t + 2$. If the particle moves in a straight line, calculate the displacement of this object in the time interval $[1, 2]$. Express your answer as a single number. | \frac{17}{6} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
A company has 1,000 employees, consisting of 450 young workers, 350 middle-aged workers, and 200 elderly workers. In order to understand the health status of the employees, stratified sampling is used to select a sample. If the sample contains 7 middle-aged workers, calculate the sample size. Express your answer as a w... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
The temperature in Vancouver is $22^{\circ} \mathrm{C}$. The temperature in Calgary is $19^{\circ} \mathrm{C}$ colder than the temperature in Vancouver. The temperature in Quebec City is $11^{\circ} \mathrm{C}$ colder than the temperature in Calgary. Calculate the temperature in Quebec City. Express your answer in degr... | -8^{\circ} \mathrm{C} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
A flag is in the shape of a right-angled triangle, with the horizontal and vertical sides being of lengths 72 cm and 24 cm respectively. The flag is divided into 6 vertical stripes of equal width. Calculate the difference between the areas of any two adjacent stripes. Express your answer in square centimeters (cm²). | 48 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Find the smallest positive integer N whose digits have a product of 2700, and then calculate the sum of the digits of N. Express your answer as a single integer. | 27 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Find the distance between line $l_1$: $x-y-1=0$ and line $l_2$: $x-y+1=0$. Express your answer as a numerical value, including the square root symbol if necessary. | \sqrt{2} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
Given the parametric equations of curve C are $\begin{cases}x=4\cos \theta \\ y=2\sin \theta\end{cases} (\theta$ is the parameter$)$, calculate the eccentricity of this curve. Express your answer as a simplified fraction. | \dfrac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
Solve the quadratic equation $x^{2}-8x+6=0$ using the completing the square method. Express your answer as a perfect square equation in the form $(x-a)^{2}=b$. | $(x-4)^{2}=10$ | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: $i^{3}+2i$. Express your answer as a simplified complex number. | i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Find the real part of the complex number $i^2(1+i)$. Express your answer as a single integer. | -1 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.90625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\frac{\overline{z}}{1+2i}=2+i$, calculate the sum of the real and imaginary parts of the complex number $z+5$. Express your answer as a single integer. | 0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation a^2 + b^2 = k * ab, where a and b are positive integers, find the number of positive integer values of k for which there exists at least one pair (a, b) satisfying the equation. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
Hagrid has 100 animals. Among these animals:
- Each is either striped or spotted but not both.
- Each has either wings or horns but not both.
- There are 28 striped animals with wings.
- There are 62 spotted animals.
- There are 36 animals with horns.
Calculate the number of Hagrid's spotted animals that have horns. ... | 26 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow {a}=(3,4)$ and $\overrightarrow {b}=(2,-1)$, if the vector $\overrightarrow {a}+x \overrightarrow {b}$ is perpendicular to $\overrightarrow {b}$, calculate the value of $x$. Express your answer as a single numerical value, including the negative sign if applicable. | - \frac {2}{5} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given the set of numbers $\{-7, -5, -1, 1, 3\}$, find the smallest product that can be obtained by multiplying two numbers from this set. Express your answer as an integer. | -21 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.444444 | SynthLabsAI/Big-Math-RL-Verified |
Among the real numbers $\sqrt{36}$, $\sqrt[3]{2}$, $\frac{{22}}{7}$, $0.1010010001\ldots$, $\frac{π}{3}$, $\sqrt{5}$, how many are irrational? Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
One hundred students at Century High School participated in the AHSME last year, and their mean score was 100. The number of non-seniors taking the AHSME was $50\%$ more than the number of seniors, and the mean score of the seniors was $50\%$ higher than that of the non-seniors. Calculate the mean score of the seniors.... | 125 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $\frac{2+i}{a-i}$ (where $a \in \mathbb{R}$, $i$ is the imaginary unit), if it is a pure imaginary number, calculate the modulus of $a+i$. Express your answer as a single value, including the square root symbol if necessary. | \frac{\sqrt{5}}{2} | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Given vector $\overrightarrow{a}$, calculate the result of the operation $\overrightarrow{a}+2\overrightarrow{a}$. Express your answer as a vector in terms of $\overrightarrow{a}$. | 3\overrightarrow{a} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
Given that point M is on the left branch of the hyperbola C: x^2 - y^2/8 = 1, and F1, F2 are the left and right foci of C respectively, find the value of |MF1| + |F1F2| - |MF2|. Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Hyperbolic Geometry"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the coefficient of the $x^3$ term in the expansion of the binomial $(2x- \frac {1}{x})^{5}$. Express your answer as an integer. | -80 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
A craft factory needs to produce 60 units of a certain type of craft per day. Each worker can produce x units of this craft per day. Express the number of workers required, y, as a function of x. | y = \frac{60}{x} | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.59375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the equation $\frac{x-3}{x-2}=\frac{m}{2-x}$ has no solution, calculate the exact value of $m$ that makes the equation undefined. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that is symmetric to the line 2x+y-1=0 about the line x=1. Express your answer in the standard form of a line equation (Ax + By + C = 0). | 2x-y-3=0 | big_math | [
"Mathematics -> Geometry -> Other"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Find the maximum value of the function f(x) = x^3 - 3x^2 on the interval [-2, 4]. | 16 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
The sum of seven consecutive integers is 77. Find the smallest of these integers. Express your answer as a whole number. | 8 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
If the expression $(x+y+z)^{10}$ is expanded into a polynomial and like terms are combined, calculate the total number of terms in the resulting polynomial. Express your answer as a single integer. | 66 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence of sums involving six integers, determine the minimum number of even integers among them. The sequence is as follows: the sum of the first two integers is 26, adding two more integers results in a sum of 41, and adding the final two integers to the previous sum gives 57. Express your answer as a single... | 1 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors a = (1, -2) and b = (3, 4), calculate the projection of a in the direction of b. Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
Three balls labeled 1, 2, and 3 are placed in a jar. A ball is drawn from the jar, its number is recorded, and it is then returned to the jar. This process is repeated three times, with each ball having an equal chance of being drawn in each trial. If the sum of the recorded numbers is 6, what is the probability that t... | \frac{1}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence of 10 terms, where each term is either 1 or -1, how many different possible values can the sum of the sequence take? Express your answer as a whole number. | 11 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.421875 | SynthLabsAI/Big-Math-RL-Verified |
In a triangle with sides of 30, 70, and 80 units, an altitude is dropped upon the side of length 80. Find the length of the larger segment cut off on this side. Express your answer in units as a whole number. | 65 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The manufacturing of a certain mechanical part consists of two operations. The scrap rate of the first operation is $a$, and the scrap rate of the second operation is $b$. Assuming that the production of scrap from these two operations is independent of each other, find the qualifying rate of the product. Express your ... | 1 - a - b + ab | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.640625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the sum of the first 3 terms of an arithmetic sequence is 6 and the sum of the first 9 terms is 27, calculate the sum of the first 6 terms. Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Algebra -> Sequences and Series -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
Given the cubic equation $x^3 - 3x^2 - a = 0$, find the range of $a$ for which the equation has three distinct real roots. Express your answer as an interval, using parentheses or brackets to denote the range. | (-4, 0) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
In the plane, there is a line segment AB with a length of 2 and a moving point P such that $|PA|+|PB|=6$. Calculate the range of values for $|PA|$ and express your answer as a closed interval. | [2, 4] | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
A certain school has 21 elementary teachers, 14 intermediate teachers, and 7 senior teachers. A random sample of 6 teachers is selected using stratified random sampling, with 3 elementary, 2 intermediate, and 1 senior teacher. If 2 teachers are randomly selected from this sample for further data analysis, what is the p... | \frac{1}{5} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Combinations"
] | 0.796875 | SynthLabsAI/Big-Math-RL-Verified |
Given vectors $\overrightarrow{a} = (1, 2)$ and $\overrightarrow{b} = (-3, 2)$, find the value of the real number $k$ such that $(k\overrightarrow{a} + \overrightarrow{b})$ is parallel to $(\overrightarrow{a} - 3\overrightarrow{b})$. Express your answer as a fraction or decimal. | -\frac{1}{3} | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.296875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets M and N defined as follows:
\[ M = \left\{ x \left\lvert\, -\frac{1}{2} < x < \frac{1}{2} \right. \right\} \]
\[ N = \left\{ x \mid x^2 \leq x \right\} \]
Calculate the intersection \( M \cap N \) and express your answer in interval notation, including any relevant brackets or parentheses. | \left[0, \frac{1}{2}\right) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
In a certain school, there are three grades: 1st grade (senior one), 2nd grade (senior two), and 3rd grade (senior three), with student populations of 1500, 1200, and 1000 respectively. A stratified sampling method is used to investigate the students' eyesight conditions. It is known that 60 students were sampled from ... | 185 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Mathematical Statistics"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence {a_n}, where a_1=1, and 4a_2, 2a_3, a_4 form an arithmetic sequence, find the value of a_2+a_3+a_4. Express your answer as a single integer. | 14 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Determine the number of common tangents to circle O1: x^2+y^2-2x=0 and circle O2: x^2+y^2-4y=0. Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z$ that satisfies $(2+i)z=2-i$ ($i$ is the imaginary unit), find the value of $z$ in standard form $a+bi$. Express your answer with both real and imaginary parts. | \frac{3}{5} - \frac{4}{5}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z represented by the point (-2,1) in the complex plane, calculate the value of \\frac{\\overline{z}}{1+i}. Express your answer in the form a+bi, where a and b are real numbers. | -\\frac{3}{2}+\\frac{1}{2}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Given the class average score of 85 and the differences between the scores of 14 students in a group and this average as 2, 3, -3, -5, 12, 12, 8, 2, -1, 4, -10, -2, 5, 5, calculate the average score of this group. Express your answer as a numerical value rounded to two decimal places. | 87.29 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $A=\{x|x>2\}$ and $B=\{x|(x-1)(x-3)<0\}$, find the intersection $A\cap B$. Express your answer in interval notation. | (2,3) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given sets A = {0, 1, 2} and B = {2, 3}, if x is not in set A and x is in set B, then what is the value of x? Express your answer as a single integer. | 3 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Given the equation \( 9210 - 9124 = 210 - \square \), solve for the value of \( \square \). Provide your answer as a whole number. | 124 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Simple Equations"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of $2\cos 30^{\circ}$. Express your answer as a single value, potentially including a square root. | \sqrt{3} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
In the adjoining figure, $AB$ is a diameter of the circle, $CD$ is a chord parallel to $AB$, and $AC$ intersects $BD$ at $E$, with $\angle AED = \alpha$. Calculate the ratio of the area of $\triangle CDE$ to that of $\triangle ABE$ in terms of $\alpha$. Express your answer using trigonometric functions if necessary. | \cos^2\alpha | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Solve the system of equations $\left\{\begin{array}{c}4x+3y=11\\ 4x-3y=5\end{array}\right.$ for the values of x and y. Express your answer as a pair of values (x, y). | (2, 1) | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Systems of Equations"
] | 0.65625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the value of the algebraic expression $2y^2+3y+7$ is $8$, calculate the value of the algebraic expression $4y^2+6y-9$. Express your answer as a single integer. | -7 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.21875 | SynthLabsAI/Big-Math-RL-Verified |
An unequal-sided triangle has sides that are integers, and its perimeter is less than 13. Calculate the number of such triangles. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the domain of $f(x)$ is $[-2,4]$, find the domain of $f(3x-2)$. Express your answer as an interval with endpoints. | [0,2] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.625 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence {an} such that a1 + a2 = 10, and a3 + a4 = 26, find the slope of the line passing through points P(n, an) and Q(n+1, an+1). Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Consider the following functions: f(x) = ln(x), f(x) = x^2 + 1, f(x) = sin(x), and f(x) = x^3. A function f(x) is said to have property M if the function e^x*f(x) is monotonically increasing over the domain of f(x). How many of these functions have property M? Express your answer as a single integer count. | 1 | big_math | [
"Mathematics -> Calculus -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
The equation of an ellipse is given by $\frac {x^{2}}{5}$+ $\frac {y^{2}}{4}$$=1$. Find the eccentricity of this ellipse. Express your answer as a fraction or decimal. | \frac{\sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.859375 | SynthLabsAI/Big-Math-RL-Verified |
Given a geometric sequence ${a_n}$ with positive terms not equal to 1, and a sequence ${b_n}$ defined as $b_n = \log{a_n}$, where $b_3 = 18$ and $b_6 = 12$, calculate the maximum value of the sum of the first n terms of ${b_n}$. Express your answer as a single integer. | 132 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
Consider a geometric sequence {a_n} with the first term 1 and a common ratio of 1/3. Let S_n denote the sum of the first n terms. Calculate S_n in terms of a_n and express your answer as a formula. Use the boxed notation around your final answer. | \frac{3 - a_n}{2} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
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