instruction_seed stringlengths 40 1.39k | answer stringlengths 1 95 | source stringclasses 1
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Let f be a real-valued function defined over ordered pairs of integers such that f(x+3m-2n, y-4m+5n) = f(x,y) for every integers x, y, m, n. What is the maximum number of distinct values that the function f can take? Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $sin(\frac{π}{6}+α)=\frac{3}{5}$, calculate $cos(2α-\frac{2π}{3})$. Express your answer as a fraction, including the negative sign if applicable. | -\frac{7}{25} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
There are 2 doctors and 4 nurses to be allocated to two community hospitals for a "health check-up" activity, with each hospital being allocated 1 doctor and 2 nurses. Calculate the total number of different allocation schemes possible for this scenario. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
In the given acute triangle ABC, the incenter I and circumcenter O intersect the line BC at point X. Let M be the midpoint of the arc BC (not containing point A) of the circumcircle of triangle ABC. If the points A, O, M, and X are concyclic, calculate the measure of angle BAC in degrees. | 60 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
At the grocery store last week, small boxes of facial tissue were priced at 4 boxes for $5. This week they are on sale at 5 boxes for $4. Calculate the percent decrease in the price per box during the sale. Express your answer as a percentage, rounded to the nearest whole number. | 36\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
A sedan and a truck are driving in the same direction on the highway. The lengths of the sedan and the truck are denoted as $a$ and $b$ respectively. During the process of overtaking, the sedan travels an additional distance compared to the truck. Express this additional distance as an algebraic expression involving $a... | a+b | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.1875 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the function $y=\frac{1}{{x}^{2}+3}$. Express your answer as an interval in standard notation. | (0,\frac{1}{3}] | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
There are 5 people standing in a row. If person A is not at the left end and not adjacent to person B, how many possible arrangements are there? Express your answer as a single integer. | 54 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Let \( i = \sqrt{-1} \). Given the equation \( z^{2} - z = 5 - 5i \), find the product of the real parts of the roots. Express your answer as a single integer. | -6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Given that the dihedral angle formed by a line $l$ with plane $\alpha$ is 45°, find the largest possible angle that can be formed by line $l$ and any line within plane $\alpha$ that does not pass through the foot of the perpendicular from $l$ to $\alpha$. Express your answer in degrees (°). | 90° | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the lengths of the three sides of a triangle are $3$, $1-2a$, and $8$, find the range of values for $a$ that satisfy the triangle inequality. Express your answer as an interval (lower bound, upper bound). | (-5, -2) | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Let \( a_1, a_2, \cdots, a_k \) be a finite arithmetic sequence. Given \( a_4 + a_7 + a_{10} = 17 \), \( a_4 + a_5 + a_6 + \cdots + a_{14} = 77 \), and \( a_k = 13 \), determine the value of \( k \). Express your answer as a whole number. | 18 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the value of the definite integral $\int_{0}^{1}\left(2x+e^{x}\right)dx$. Express your answer as a single value, using e if necessary. | e | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Applications of Integrals"
] | 0.875 | SynthLabsAI/Big-Math-RL-Verified |
Given a regular triangle ACD with side length 1, let $\overrightarrow {AB}= \overrightarrow {a}$, $\overrightarrow {BC}= \overrightarrow {b}$, $\overrightarrow {AC}= \overrightarrow {c}$. Calculate the magnitude of $\overrightarrow {a}- \overrightarrow {b}+ \overrightarrow {c}$. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the expressions $x = 1 + 2^p$ and $y = 1 + 2^{-p}$, express $y$ in terms of $x$ as a single algebraic expression. | \frac{x}{x-1} | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.8125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression $\sqrt{\frac{4}{3}} - \sqrt{\frac{3}{4}}$. Express your answer as a simplified fraction with square roots if necessary. | \frac{\sqrt{3}}{6} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the expression x + x(x^x) when x = 2. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Let $\overrightarrow {a} = (m, -1, 2)$, $\overrightarrow {b} = (3, -4, n)$. If $\overrightarrow {a} \parallel \overrightarrow {b}$, find the values of $m$ and $n$. Express your answer as an ordered pair $(m, n)$. | \left(\frac{3}{4}, 8\right) | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the number of positive integers not greater than 2009 that are multiples of 3 but not multiples of 5. Express your answer as a whole number. | 536 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
In a group consisting of male and female students totaling 8, two male students and one female student are to be selected. There are 30 different ways to do this. Calculate the number of female students in the group. Express your answer as a whole number. | 3 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
The product of six distinct integers is 36. Find the sum of these six integers. | 0 | big_math | [
"Mathematics -> Number Theory -> Factorization"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
The Five Goats Travel Group travels 50 kilometers by water and then 300 kilometers by land, with the land speed being three times the speed of water travel. Calculate the ratio of the time it takes for the travel group to travel by land to the time it takes to travel by water. Express your answer as a single numerical ... | 2 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z_1=2+i$, and that the points corresponding to the complex numbers $z_1$ and $z_2$ in the complex plane are symmetric with respect to the real axis, find the value of $\\frac{z_1}{z_2}$. Express your answer as a complex number in the form $a+bi$. | \\frac{3}{5} + \\frac{4}{5}i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given an arithmetic sequence with 10 terms, the sum of its even terms is 30, and the sum of its odd terms is 25. Calculate the common difference d of the sequence. Express your answer as a single numerical value. | 1 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.171875 | SynthLabsAI/Big-Math-RL-Verified |
The mean (average) of five consecutive even numbers is 12. What is the mean of the smallest and largest of these numbers? Express your answer as a single number. | 12 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 | SynthLabsAI/Big-Math-RL-Verified |
In the Cartesian coordinate system, the graph of the function y = 3x is translated 6 units upwards. Find the coordinates of the intersection point of the translated graph with the x-axis. Express your answer as an ordered pair (x, y), where y = 0. | (-2, 0) | big_math | [
"Mathematics -> Algebra -> Functions"
] | 0.671875 | SynthLabsAI/Big-Math-RL-Verified |
Given a sequence $\left\{a_{n}\right\}$ with $k$ terms $(k\in N^{*})$, where $a_{1}=2$, define vectors $\overrightarrow {c_{n}}=(a_{n}, a_{n+1})$ and $\overrightarrow {d_{n}}=(n, n+1)$ for $n=1,2,3,\ldots, k-1$. If $|\overrightarrow {c_{n}}|=|\overrightarrow {d_{n}}|$, calculate the number of sequences $\left\{a_{n}\ri... | 2^{k-1} | big_math | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The ancient Chinese mathematical text, "The Nine Chapters on the Mathematical Art," presents the following problem: "There is a gold bar that is 5 feet long. When 1 foot is cut from the thicker end, it weighs 4 pounds; when 1 foot is cut from the thinner end, it weighs 2 pounds. What is the weight of each successive fo... | 6 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given the coordinates of points P(2,6), Q(2,2), and R(8,5), calculate the area of the triangle formed by these points. Express your answer as a whole number. | 12 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
Given that $a=3^{0.4}$, $b=log_{3} \frac {1}{2}$, $c=( \frac {1}{3})^{0.2}$, determine the relationship among a, b, and c. Express your answer as a string describing the relationship using standard inequality notation (e.g., 'a > c > b'). | a > c > b | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Exponential Functions"
] | 0.703125 | SynthLabsAI/Big-Math-RL-Verified |
The ancient "Five Elements" theory posits that there are five types of properties in materials: metal, wood, water, fire, and earth. Metal overcomes wood, wood overcomes earth, earth overcomes water, water overcomes fire, and fire overcomes metal. If two different properties are randomly selected from these five, calcu... | \dfrac{1}{2} | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Other"
] | 0.453125 | SynthLabsAI/Big-Math-RL-Verified |
Let $S_n$ be the sum of the first $n$ terms of an arithmetic sequence $\{a_n\}$. If $a_3+a_5+a_7=27$, then find $S_9$. Express your answer as a single integer. | 81 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of ordered triplets (x, y, z) that satisfy the system of equations:
\[
\begin{cases}
x + 2y + 4z = 12, \\
xy + 4yz + 2xz = 22, \\
xyz = 6.
\end{cases}
\]
Express your answer as a single integer. | 6 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Given sets $S=\{x\in\mathbb{R}\ |\ x+1\geq 2\}$ and $T=\{-2,-1,0,1,2\}$, find the intersection $S \cap T$. Express your answer as a set of elements using standard set notation. | \{1,2\} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
Find the domain of the function $y = \sqrt{1-\lg(x+2)}$. Express your answer in interval notation. | (-2,8] | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the line equation $ax+by+1=0$ ($a>0$, $b>0$) and the circle equation $(x+4)^2+(y+1)^2=16$, find the minimum value of $\frac {1}{2a}+ \frac {2}{b}$, given that the line divides the circle into two parts of equal area. | 8 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence {a_n} satisfying 3a_{n+1} + a_n = 4 for n >= 1 and a_1 = 9, and the sum of the first n terms being S_n, find the smallest integer n that satisfies the inequality |S_n - n - 6| < 1/125. Express your answer as a single integer. | 7 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Distribute 4 identical balls into 3 different boxes. What is the total number of different ways to do this? Express your answer as a single integer. | 15 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Combinations"
] | 0.890625 | SynthLabsAI/Big-Math-RL-Verified |
If the radius of a circle is 1, calculate the arc length corresponding to a 60-degree central angle. Express your answer in terms of pi. | \frac{\pi}{3} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.9375 | SynthLabsAI/Big-Math-RL-Verified |
Among the four numbers -1.5, -3, -1, and -5, what is the largest number? Express your answer as a single number. | -1 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z=1+\sqrt{2}i$, calculate the value of $z^2-2z$. Express your answer as a single number, which may be complex. | -3 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.734375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $z= \\dfrac {2}{-1-i}$, let $\\overset{.}{z}$ be the conjugate of $z$. Calculate the coordinates of the point corresponding to $i \\overset{.}{z}$ in the complex plane. Express your answer as coordinates (a, b). | (1, -1) | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Each number from 1 to 6 replaces one of the letters \(P, Q, R, S, T\), and \(U\). The sum of \(P\) and \(Q\) is 5 and the difference between \(R\) and \(S\) is 5. If \(T\) is greater than \(U\), what number replaces the letter \(T\)? Express your answer as a single integer. | 5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the tangent line to the function y = e^x + cos(x) at the point (0, 2). Express your answer in the standard form of a linear equation (Ax + By + C = 0). | x - y + 2 = 0 | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.234375 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the result of $(\sqrt{2}-\sqrt{3})^{2020} \cdot (\sqrt{2}+\sqrt{3})^{2021}$. Express your answer as a simplified algebraic expression. | \sqrt{2}+\sqrt{3} | big_math | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Find the units digit of the product \(\left(5^{3}
ight)\left(7^{52}
ight)\). Express your answer as a single digit. | 5 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.984375 | SynthLabsAI/Big-Math-RL-Verified |
Find the sum of the coefficients in the expansion of $(1-2x)^{10}$. Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
The first term of an arithmetic sequence is 24. Starting from the 10th term, the sequence becomes negative. Find the range of the common difference d. Express your answer as an interval, including the endpoints if they exist. | [-3, -\frac{8}{3}) | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
If the function $f(x) = 1 + \frac{m}{e^{x} - 1}$ is an odd function, find the value of $m$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Find the number of integer solutions (x, y) for the equation 1/x + 1/y = 1/7. Express your answer as a whole number. | 5 | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.15625 | SynthLabsAI/Big-Math-RL-Verified |
A function f(x) defined on R satisfies f(x) * f(x+2) = 13. If f(1) = 2, then what is the value of f(2015)? Express your answer as a single value in the form of a fraction or integer. | \frac{13}{2} | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Rowena is able to mow 2/5 of a lawn in 18 minutes. If she began the job at 10:00 a.m., and mowed at this same constant rate, what time did she finish mowing the entire lawn? Express your answer in 12-hour time format (e.g., 10:45 a.m.). | 10:45 a.m. | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Derive a general term formula for the sequence $\left(1, \frac{2}{3}, \frac{3}{7}, \frac{4}{15}, \frac{5}{31}\right)$. Express your answer as a formula that relates the nth term to the value of n. | \frac{n}{2^n-1} | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.0625 | SynthLabsAI/Big-Math-RL-Verified |
Given the function f(x) = 1/2*x^2 - a*ln(x) + 1, find the range of 'a' such that the function has a minimum value in the interval (0,1). Express your answer as an interval. | (0, 1) | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Find the range of the function $f(x) = \cos(\dfrac{\pi}{3} - x) + \sin(\dfrac{\pi}{2} + x)$. Express your answer as an interval. | [-\sqrt{3}, \sqrt{3}] | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Calculate the sum of the digits of the integer equal to 10^2021 - 2021. Express your answer as a single integer. | 18185 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the polynomial P(x) satisfies the functional equation 2P(x) = P(x + 3) + P(x - 3) for every real number x, find the maximum degree of P(x). Express your answer as a single integer. | 1 | big_math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
A school is planning to assign 6 teachers to join New Year's Eve celebrations in two senior year classes, Class 1 and Class 2, with each class having no more than 4 teachers. How many different arrangements are there for assigning the teachers to the two classes? Provide your answer as a whole number. | 50 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.109375 | SynthLabsAI/Big-Math-RL-Verified |
Given the sequence of functions defined by f_0(x) = cos(x) and f_{n+1}(x) = f'_n(x) for n in the set of positive integers, compute f_2016(x) using the periodicity of the sequence. Express your answer as a function of x. | \cos x | big_math | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.765625 | SynthLabsAI/Big-Math-RL-Verified |
Three of the vertices of a parallelogram are \((0,1)\), \((1,2)\), and \((2,1)\). Find the area of the parallelogram. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
Equilateral triangle ABC has sides of length 4. The midpoint of BC is D, and the midpoint of AD is E. Calculate the value of EC^2. Express your answer as a single numerical value. | 7 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles"
] | 0.078125 | SynthLabsAI/Big-Math-RL-Verified |
Let \(a_{1}, a_{2}, \cdots, a_{n}\) be all the irreducible proper fractions with a denominator of 60. Calculate the value of \(\sum_{i=1}^{n}\left(\cos \frac{a_{i} \pi}{2}\right)^{2}\). Express your answer as a single integer. | 8 | big_math | [
"Mathematics -> Number Theory -> Other",
"Mathematics -> Trigonometry -> Other"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Two cones, A and B, have equal slant heights. The sum of the central angles of their unfolded lateral surfaces is $2\pi$. The lateral surface areas of the cones are $S_A$ and $S_B$, with $\frac{S_A}{S_B}=2$. Calculate the ratio of the heights of the cones, $\frac{h_A}{h_B}$. Express your answer as a simplified ratio. | \frac{\sqrt{10}}{4} | big_math | [
"Mathematics -> Geometry -> Solid Geometry -> 3D Shapes"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
In a certain city, there are a total of 270,000 junior high school students, among which there are 99,000 students in the first grade, 90,000 students in the second grade, and 81,000 students in the third grade. To understand the students' intentions to participate in "open scientific experiment activities", a stratifi... | 900 | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Other"
] | 0.78125 | SynthLabsAI/Big-Math-RL-Verified |
Let vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ satisfy $\vec{a}\parallel \vec{b}$ and $\vec{a}\perp \vec{c}$. Calculate the value of $\vec{c} \cdot (\vec{a} + 2\vec{b})$. Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
A household electric rice cooker with a rated power of 800W has a measured voltage U of 200V across the circuit when operating. At this time, the working resistance R is not less than 62.5Ω. Calculate the maximum current intensity I passing through the circuit. Express your answer in amperes (A). | 3.2 | big_math | [
"Mathematics -> Applied Mathematics -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Factorize the polynomial $a^2 - b^2 + 2a + 1$. Express your answer as a factored polynomial. | (a+1+b)(a+1-b) | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.328125 | SynthLabsAI/Big-Math-RL-Verified |
Under the constraints x >= 0, y >= 0, x + y <= s, and y + 2x <= 4, where 3 <= s <= 5, find the range of the maximum value of the objective function z = 3x + 2y. Express your answer as a closed interval. | [7, 8] | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that the angle between non-zero vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is $\dfrac{\pi}{3}$, and $|\overrightarrow{b}| = 2$, $|\overrightarrow{b} - 2\overrightarrow{a}| = 2$, find the value of $|\overrightarrow{a}|$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.484375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number z = (2+ai)/(1+2i), where a is an integer, and the point corresponding to z in the complex plane is in the fourth quadrant, find the maximum value of a. | 3 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
In a triangle with side lengths of 5, 7, and 8, find the sum of the largest and smallest angles. Express your answer in degrees as a whole number. | 120 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.203125 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression ${\sin ^{2}}{70^{\circ }} + \tan {225^{\circ }} + {\sin ^{2}}{20^{\circ }}$. Express your answer as a single numerical value. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.09375 | SynthLabsAI/Big-Math-RL-Verified |
Simplify the expression: i+i^{2}+i^{3}+...+i^{2017}. Express your answer in the simplest form using the imaginary unit i. | i | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
Find the arithmetic square root of 4. Express your answer as a single number. | 2 | big_math | [
"Mathematics -> Algebra -> Prealgebra -> Other"
] | 0.828125 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {a_n}, if a_2 = 2 and a_4 + a_5 = 12, then find the value of a_7. Express your answer as a single integer. | 10 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions -> Sequences and Series"
] | 0.75 | SynthLabsAI/Big-Math-RL-Verified |
The hyperbola x^2/a^2 - y^2/b^2 = 1 (a > 0, b > 0) has its right focus at F2. A line passing through F with an inclination angle of 30 degrees intersects the right branch of the curve at point M. If MF is perpendicular to the x-axis, find the eccentricity of the hyperbola. | \sqrt{3} | big_math | [
"Mathematics -> Geometry -> Non-Euclidean Geometry -> Other"
] | 0.046875 | SynthLabsAI/Big-Math-RL-Verified |
Points R, S, and T are vertices of an equilateral triangle, and points X, Y, and Z are midpoints of its sides. How many noncongruent triangles can be drawn using any three of these six points as vertices? Express your answer as a single integer. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Triangles",
"Mathematics -> Discrete Mathematics -> Combinatorics -> Counting Methods"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The sequence -1, 4, -7, 10,..., $(-1)^n(3n-2)$ has a sum of the first $n$ terms denoted as $S_n$. Calculate the value of $S_{11} + S_{20}$. Express your answer as a single integer. | 14 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
The sum of two numbers and their quotient both equal 6. Calculate the result when the product of these two numbers is reduced by their difference (larger minus smaller). Express your answer as a simplified fraction. | \frac{6}{49} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given $\log_{2}3=a$ and $\log_{2}5=b$, calculate the value of $\log_{2} \frac{9}{5}$ in terms of $a$ and $b$. Express your answer as an algebraic expression. | 2a-b | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Logarithmic Functions"
] | 0.921875 | SynthLabsAI/Big-Math-RL-Verified |
How many solutions does the equation sin(pi/2 * cos(x)) = cos(pi/2 * sin(x)) have in the closed interval [0, pi]? Express your answer as a whole number. | 2 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
In the arithmetic sequence {an}, the first term a1 = 0, and the common difference d ≠ 0. If ak = a1 + a2 + a3 +... + a7, find the value of k. Express your answer as a whole number. | 22 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.28125 | SynthLabsAI/Big-Math-RL-Verified |
Find the equation of the line that passes through point A(2,3) and is perpendicular to the line 2x+y-5=0. Express your answer in the standard form of a line equation (Ax + By + C = 0). | x-2y+4=0 | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Linear Equations"
] | 0.390625 | SynthLabsAI/Big-Math-RL-Verified |
A grid is composed of small squares of the same size. Given that the total number of small squares is 64 and the number of shaded squares is 48, calculate the percentage of the grid that is shaded. Express your answer as a percentage, including the percent sign (%). | 75% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 | SynthLabsAI/Big-Math-RL-Verified |
A person has a probability of 0.6 to hit the target in a single shot. Calculate the probability that this person hits the target at least twice in three shots. Express your answer as a fraction. | \frac{81}{125} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Other"
] | 0.515625 | SynthLabsAI/Big-Math-RL-Verified |
A sequence of numbers is given: 7, 7^2, 7^3, 7^4, …, 7^2011. How many terms in this sequence have a last digit of 3? Express your answer as a whole number. | 503 | big_math | [
"Mathematics -> Number Theory -> Other"
] | 0.359375 | SynthLabsAI/Big-Math-RL-Verified |
Given that the function f(x) is an odd function defined on (-∞, 0) ∪ (0, +∞), and f(-1) = 0, and the inequality (f(x1) - f(x2))/(x1 - x2) > 0 holds for any two real numbers x1, x2 ∈ (0, +∞) and x1 ≠ x2, find the solution set of the inequality xf(x) > 0. Express your answer as an interval or a union of intervals. | (-∞, -1) ∪ (1, +∞) | big_math | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other",
"Mathematics -> Functions -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Given that $\cos\theta - 3\sin\theta = 0$, calculate the value of $\tan\left(\theta - \frac{\pi}{4}\right)$. Express your answer as a single numerical value. | -\frac{1}{2} | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
The sales volume of the "Optimization Plan" series in its third year increased by 44% compared to its first year. If the average annual growth rate is the same (denoted as $x$), calculate the value of $x$ as a percentage. | 20\% | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.578125 | SynthLabsAI/Big-Math-RL-Verified |
In a positive geometric sequence {a_n}, it is known that a_1a_2a_3 = 4, a_4a_5a_6 = 12, and a_{n-1}a_na_{n+1} = 324. Find the value of n. Express your answer as a whole number. | 14 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.015625 | SynthLabsAI/Big-Math-RL-Verified |
Evaluate the definite integral $\int_{0}^{\frac{\pi}{3}} \left( 1-2\sin^{2}\frac{\theta }{2} \right)d\theta$. Express your answer as a single value, including any necessary radicals. | \frac{\sqrt{3}}{2} | big_math | [
"Mathematics -> Calculus -> Integral Calculus -> Other"
] | 0.609375 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit. If the complex number $z= \frac{i}{1+i}$, calculate the complex conjugate $\bar{z}$ in standard form $a+bi$. | \frac{1}{2} - \frac{i}{2} | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.71875 | SynthLabsAI/Big-Math-RL-Verified |
A class is scheduling 5 lessons for Monday morning, including Chinese, Mathematics, English, Music, and Physical Education, each for one lesson. Considering the effectiveness of students' learning, Mathematics cannot be arranged for the first lesson, Chinese and English must be adjacent, and Music and Physical Educatio... | 20 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.03125 | SynthLabsAI/Big-Math-RL-Verified |
Given that $P$ is a moving point on the circle $(x-3)^2 + (y+1)^2 = 4$, and $Q$ is a moving point on the line $x = -3$, find the minimum value of $|PQ|$. Express your answer as a whole number. | 4 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.3125 | SynthLabsAI/Big-Math-RL-Verified |
Let $i$ be the imaginary unit, and $\overline{z}$ be the conjugate of the complex number $z$. If $z + 2\overline{z} = 3 + 4i$, find the value of $z$. Express your answer as a complex number in the form $a + bi$. | 1 - 4i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.6875 | SynthLabsAI/Big-Math-RL-Verified |
Given the operation on $\mathbb{R}$ defined as $$\begin{vmatrix} a & c \\ b & d\end{vmatrix} =ad-bc$$, find the range of $x$ that satisfies the inequality $$\begin{vmatrix} x & 3 \\ -x & x\end{vmatrix} < \begin{vmatrix} 2 & 0 \\ 1 & 2\end{vmatrix}$$. Express your answer in interval notation. | (-4,1) | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.84375 | SynthLabsAI/Big-Math-RL-Verified |
What is the value of $(2(2(2(2(2(2+1)+1)+1)+1)+1)+1)$? Express your answer as a single integer. | 127 | big_math | [
"Mathematics -> Algebra -> Algebraic Expressions"
] | 0.375 | SynthLabsAI/Big-Math-RL-Verified |
Given the complex number $$\frac {3-bi}{2+i}$$ where $b\in\mathbb{R}$, and the condition that the real part is equal to the imaginary part, calculate the value of $b$. Express your answer as a single real number. | -9 | big_math | [
"Mathematics -> Algebra -> Complex Numbers"
] | 0 | SynthLabsAI/Big-Math-RL-Verified |
Given the line equation mx + (5-2m)y - 2 = 0 and the circle equation x^2 + y^2 = 4, calculate the maximum distance from the center of the circle to the line. Express your answer as a single numerical value, including any necessary mathematical notation. | \frac{2\sqrt{5}}{5} | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 | SynthLabsAI/Big-Math-RL-Verified |
Given the imaginary unit $i$ and a complex number $z$ that satisfies $\frac {2}{1+i}= \bar{z}+i$, determine the value of $z$. Express your answer as a complex number in the form $a+bi$. | 1+2i | big_math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.34375 | SynthLabsAI/Big-Math-RL-Verified |
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