problem stringlengths 10 3.15k | answer stringlengths 1 1.22k | source stringclasses 11
values | domain listlengths 1 4 | llama8b_solve_rate float64 0 1 ⌀ |
|---|---|---|---|---|
A circle is inscribed in a square, and then another square is inscribed in that circle. Find the ratio of the areas of these squares. | 2 | olympiads | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.234375 |
Determine the lateral sides of an isosceles trapezoid if its bases and area are equal to 8 cm, 14 cm, and 44 cm², respectively. | 5 \,\text{cm} | olympiads | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.4375 |
Given the line \( y = 2\sqrt{2} x + 2\sqrt{3} \) intersects with the ellipse \( \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) at points \( A \) and \( B \), with the origin \( O \), and given \( \angle AOB = 90^\circ \) and \( AB = \frac{12\sqrt{11}}{17} \), find the equation of the ellipse. | \frac{x^2}{4} + \frac{y^2}{2} = 1 | olympiads | [
"Mathematics -> Geometry -> Solid Geometry -> Other"
] | 0.03125 |
Find all functions $f: \mathbb R \to \mathbb R$ such that \[ f( xf(x) + f(y) ) = f^2(x) + y \] for all $x,y\in \mathbb R$. | f(x) = x \text{ or } f(x) = -x | omnimath | [
"Mathematics -> Other -> Other"
] | 0.078125 |
Sam was collecting cans for recycling. On Saturday he filled 4 bags up and on Sunday he filled 3 more bags. If each bag had 6 cans in it, how many cans did he pick up total? | 42 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 |
Let $X_1,X_2,..$ be independent random variables with the same distribution, and let $S_n=X_1+X_2+...+X_n, n=1,2,...$ . For what real numbers $c$ is the following statement true: $$ P\left(\left| \frac{S_{2n}}{2n}- c \right| \leqslant \left| \frac{S_n}{n}-c\right| \right)\geqslant \frac{1}{2} $$ | c \in \mathbb{R} | aops_forum | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.03125 |
Calculate the limit of the function:
$\lim_{x \rightarrow 0} \sqrt[x^{2}]{2-\cos x}$ | \sqrt{e} | olympiads | [
"Mathematics -> Calculus -> Limits -> Other"
] | 0.203125 |
A woodworker is crafting enough furniture legs for their projects, which include chairs, tables, and cabinets. They have made a total of 80 furniture legs so far, and this is the exact amount they needed for everything they're building. They will need 4 legs for each chair, 4 legs for each table, and 2 legs for each ca... | 12 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 |
Through the point $M(1,1)$, a line with slope $- \frac {1}{2}$ intersects an ellipse $C: \frac {x^{2}}{a^{2}}+ \frac {y^{2}}{b^{2}}=1 (a > b > 0)$ at points $A$ and $B$. If $M$ is the midpoint of line segment $AB$, then the eccentricity of the ellipse $C$ equals ____. | \frac { \sqrt {2}}{2} | cn_k12 | [
"Mathematics -> Geometry -> Analytic Geometry -> Conic Sections"
] | 0.078125 |
Find all natural numbers \( n \) and \( k \) (where \( k > 1 \)) such that \( k \) divides each of the binomial coefficients \( C_{n}^{1}, C_{n}^{2}, \cdots, C_{n}^{n-1} \). | t \prod_{i=1}^{m} p_i^{a_i + 1} | olympiads | [
"Mathematics -> Number Theory -> Other"
] | 0.015625 |
Vasily Petrov is completing an English language assignment. The assignment consists of 10 English phrases and their translations into Russian in random order. He needs to match the phrases with their correct translations. Each correct match earns 1 point. Hence, the possible scores range from 0 to 10 points. Vasya know... | 0 | olympiads | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Other"
] | 0.015625 |
Given the sequence $\{a\_n\}$ satisfies $a\_1=5$, $a\_2=13$, $a_{n+2}=5a_{n+1}-6a_{n}$, find the smallest natural number $n$ such that the sum of the first $n$ terms of the sequence, denoted as $S\_n$, is not less than $2016$. | 7 | cn_k12 | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.21875 |
Given that α is an angle in the second quadrant, let point P (x, $\sqrt{5}$) be a point on the terminal side of α, and cosα = $\frac{\sqrt{2}}{4}x$. Determine the value of $4\cos(\alpha + \frac{\pi}{2}) - 3\tan\alpha$. | \sqrt{15} - \sqrt{10} | cn_k12 | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.03125 |
Given that the function $f(x) = 4x^2 - kx - 8$ is monotonically decreasing on the interval $[5, 20]$, find the range of real numbers for $k$. | B: [160, +\infty) | cn_k12 | [
"Mathematics -> Algebra -> Intermediate Algebra -> Inequalities"
] | 0.203125 |
Tickets for a play at the community theater cost $12 for an adult and $4 for a child. A certain number of tickets were sold and the total receipts were $840, with 90 adult tickets and 40 child tickets sold. How many tickets were sold in total? | 130 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.3125 |
Given that $\tan \alpha = -3$, find the value of $\frac{\cos \alpha + 2 \sin \alpha}{2 \cos \alpha - 3 \sin \alpha}$. | -\frac{5}{11} | cn_k12 | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.5 |
In a graduation ceremony, each graduate can bring their 2 parents, and 30% of the graduates will bring 1 additional family member. There will be twenty-five teachers attending, and for every 5 teachers, there will be 2 administrators attending as well. If there are 75 graduates, how many chairs must be prepared for the... | 283 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.359375 |
In the school election, Eliot got some votes, Shaun got 5 times as many votes as Randy, and Randy got 16 votes. Eliot got 160 votes. What is the ratio of the number of votes Eliot got to the number of votes Shaun got? | 2:1 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.609375 |
Given the parametric equation of the line $l$ as
$$
\begin{cases}
x=t \\
y=2t+1
\end{cases}
$$
where $t$ is the parameter, and the parametric equation of the circle $C$ as
$$
\begin{cases}
x=a\cos\theta \\
y=a\sin\theta
\end{cases}
$$
with $a>0$ and $\theta$ as the parameter. Let point $P$ be any point on circle $C$. I... | 1 | cn_k12 | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.078125 |
Given the complex number z = $$\frac{a^2i}{2-i} + \frac{1-2ai}{5}$$ (where a ∈ R, i is the imaginary unit), find the value(s) of a if z is a purely imaginary number. | -1 | cn_k12 | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.046875 |
The negation of the proposition "There exists \\(x_0 \in \mathbb{R}\\), such that \\(x_0^2 + x_0 + 1 \geqslant 0\\)" is __________. | \forall x \in \mathbb{R}, x^2 + x + 1 < 0 | cn_k12 | [
"Mathematics -> Discrete Mathematics -> Logic -> Other"
] | 0.71875 |
The equation of the tangent line to the curve $y=\sin x + e^x$ at the point $(0,1)$ is \_\_\_\_\_\_. | y = 2x + 1 | cn_k12 | [
"Mathematics -> Calculus -> Differential Calculus -> Applications of Derivatives"
] | 0.984375 |
Tom's age is T years, which is also the sum of the ages of his three children. His age N years ago was twice the sum of their ages then. Find the ratio T/N. Express your answer as a single numerical value. | 5 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 |
If the point $(a,9)$ is on the graph of the function $y=3^{x}$, then the value of $\tan \frac{a\pi }{6}$ is. | \sqrt{3} | cn_k12 | [
"Mathematics -> Precalculus -> Functions",
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.828125 |
Jungkook owns 3 balls and Yoongi owns 2 balls. How many balls are the two players owning? | 5 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1 |
Given that $x_{1}$ and $x_{2}$ are the two roots of the quadratic equation $-2x^{2}+x+5=0$, find the value of the algebraic expression $x_{1}^{2}x_{2}+x_{1}x_{2}^{2}$. | -\frac{5}{4} | cn_k12 | [
"Mathematics -> Algebra -> Equations and Inequalities -> Quadratic Functions"
] | 0.9375 |
What angle is formed by the minute and hour hands at 3:05? | 62.5^
| olympiads | [
"Mathematics -> Geometry -> Plane Geometry -> Angles"
] | 0.078125 |
Given that the complex number $z$ satisfies the equation $$\frac{z}{1+i}$$=-3i, where $i$ is the imaginary unit, find the modulus of the complex number $z$. | 3\sqrt{2} | cn_k12 | [
"Mathematics -> Algebra -> Intermediate Algebra -> Complex Numbers"
] | 0.953125 |
The position relationships between two straight lines in space are ____. | intersection, parallel, and skew | cn_k12 | [
"Mathematics -> Geometry -> Other"
] | 0.109375 |
There were 25,000 spectators at a basketball match. Out of these, 15,320 were men. Of the remaining spectators, the ratio of children to women was 7:3. How many children were there? | 6,776 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.8125 |
In $\triangle ABC$, $BD$ is a median. $CF$ intersects $BD$ at $E$ so that $\overline{BE}=\overline{ED}$. Point $F$ is on $AB$. Then, if $\overline{BF}=5$, $\overline{BA}$ equals: | $15$ | harp | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0 |
What is the side length of the larger square if a small square is drawn inside a larger square, and the area of the shaded region and the area of the unshaded region are each $18 \mathrm{~cm}^{2}$? | 6 \mathrm{~cm} | omnimath | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 0 |
Cut a rectangle (parallelogram) into two parts, from which a rhombus can be formed. | Solution is correct | olympiads | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.578125 |
Given sets M=\left\{x\mid\sqrt{x+1}\geqslant 0\right\} and N=\left\{x\mid x^{2}+x-2 \lt 0\right\}, describe the intersection M \cap N as a set in the format {x | conditions}. | \left\{x\mid -1\leqslant x \lt 1\right\} | big_math | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.078125 |
Find the smallest four-digit number that is equal to the square of the sum of the numbers formed by its first two digits and its last two digits. | 2025 | olympiads | [
"Mathematics -> Algebra -> Other"
] | 0.015625 |
a and b can do a piece of work in 2 days and 6 days respectively. Both work for 1 day and then a goes away. How long will b take to complete the remaining work? | 2 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.796875 |
When $3z^3-4z^2-14z+3$ is divided by $3z+5$, the quotient is $z^2-3z+\frac{1}{3}$. What is the remainder? | \frac{4}{3} | math | [
"Mathematics -> Algebra -> Polynomial Operations -> Other"
] | 0.5625 |
Find the amount of BaO that is required to react with 3 moles of H2O to form 3 moles of Ba(OH)2 | 3 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.71875 |
Jar X is 1/2 full of water. Jar Y, which has a certain ratio of the capacity of Jar X, is 1/2 full of water. If the water in Jar Y is poured into Jar X, then Jar X will be filled to 0.75 of its capacity. What is the ratio of Jar Y's capacity to Jar X's capacity? | 1:2 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.78125 |
Given that $|\vec{a}|=1$, $|\vec{b}|=2$, $\angle AOB = \frac{2\pi}{3}$, and $\vec{c} = \frac{1}{2}\vec{a} + \frac{1}{4}\vec{b}$, find the angle between $\vec{c}$ and $\vec{b}$. | 60^ullet | cn_k12 | [
"Mathematics -> Algebra -> Linear Algebra -> Vectors"
] | 0.09375 |
Given the equation sin(α)sin(β) + cos(α)cos(β) = 0, calculate the value of sin(2α) + sin(2β). Express your answer as a single number. | 0 | big_math | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.734375 |
Factorize: $3m^{2}-12=$____. | 3\left(m+2\right)\left(m-2\right) | cn_k12 | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.984375 |
In Ilwoong's drive, there are folders numbered 1 to 25, and each folder has 10 subfolders. If there are 8 files in each subfolder, how many files are there in Ilwoong's drive? | 2000 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 |
Let proposition $p$: The equation $x^{2}+mx+1=0$ has two distinct real roots, and proposition $q$: The equation $4x^{2}+4(m+2)x+1=0$ has no real roots. If the proposition "$p$ and $q$" is true, find the range of values for $m$. | (-3,-2) | cn_k12 | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.09375 |
Haley has 20 marbles. In her class 2 boys love to play marbles. If she distributes her marbles equally. How many will each of the boys receive? | 10 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 |
What is the smallest three-digit number in Pascal's triangle? | 100 | math | [
"Mathematics -> Combinatorics -> Other"
] | 0.703125 |
An example that proves the proposition "If $a \gt b$, then $a^{2} \gt b^{2}$" is false is ____. | a=1, b=-2 | cn_k12 | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.046875 |
Christmas came and as usual she received 77 gifts to put under the Christmas tree. She wanted to make other kids happy so she sent some of her gifts to the orphanage downtown. There were 11 gifts left under their Christmas tree. How many gifts did she send to the orphanage? | 66 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.96875 |
A riot occurred in the kingdom of natural numbers. In the digits ranging from $0$ to $9$, the larger digits felt superior and refused to be placed behind smaller digits when appearing together, causing natural numbers like 36, 121, and 1234 to disappear. Only numbers like 2, 55, 433, and 4321 remain. How many digits ar... | 3 | olympiads | [
"Mathematics -> Discrete Mathematics -> Combinatorics"
] | 0.1875 |
Arrange 6 volunteers $A$, $B$, $C$, $D$, $E$, $F$ to take care of 3 elderly people $X$, $Y$, $Z$. Each pair of volunteers takes care of one elderly person. Considering the distance between the volunteers' and the elderly people's residences, volunteer $A$ cannot be assigned to take care of elderly person $X$, and volun... | 42 | big_math | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Permutations"
] | 0 |
Kaleb has some shirts. If he got rid of seven of them, he would have 10 shirts. How many shirts does Kaleb have initially? | 17 | orca_math | [
"Mathematics -> Algebra -> Simple Equations"
] | 1 |
If the cost price is 96% of the selling price, what is the profit percentage? | 4.17\% | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.890625 |
Given $\|\mathbf{v}\| = 5$ and $\|\mathbf{w}\| = 8,$ find the largest possible value of
\[\|\operatorname{proj}_{\mathbf{w}} \mathbf{v}\|.\] | 5 | math | [
"Mathematics -> Algebra -> Linear Algebra -> Other"
] | 0.90625 |
Find the area of a trapezium whose parallel sides are 30 cm and 12 cm long, and the distance between them is some length. The area of the trapezium is 336 square centimeters. What is the distance between the parallel sides? | 16 | orca_math | [
"Mathematics -> Geometry -> Plane Geometry -> Area"
] | 1 |
What is the ones digit of $7^{35}$ when written as an integer? | 3 | math | [
"Mathematics -> Number Theory -> Other"
] | 1 |
In the polar coordinate system, find the distance from the point $(2, \frac{\pi}{3})$ to the line $\rho(\cos \theta + \sqrt{3}\sin \theta) = 6$. Express your answer as a single number. | 1 | big_math | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.109375 |
bhatia , ashtikar and singh begin to play with rs 70 each . at the end the ratio of the amounts left with ashtikar and singh is 1 : 2 and of those with singh and bhatia is 4 : 1 . what is singh ' s gain ( in rs ) ? | 50 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.171875 |
Given that the roots of the odd-degree real-coefficient equation \(f(x)=a_{0} x^{n}+a_{1} x^{n-1}+\cdots+a_{n-1} x+a_{n}=0\) all lie on the unit circle, and \(-a_{n}=a_{0} \neq 0\), find \(a_{0}+a_{1}+\cdots+a_{n}\). | a_0 + a_1 + imes + a_n = 0 | olympiads | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.03125 |
A pair of articles was bought for some amount at a discount of 40%. The marked price of each of the articles is $15. What was the total amount paid for the pair of articles? | $18 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.953125 |
Factor the expression: $x^2 - 4y^2 - z^2 + 4yz$. | (x + 2y - z)(x - 2y + z)) | cn_k12 | [
"Mathematics -> Algebra -> Algebraic Expressions -> Other"
] | 0.125 |
Let $f(n)$ and $g(n)$ be polynomials of degree 2014 such that $f(n)+(-1)^{n} g(n)=2^{n}$ for $n=1,2, \ldots, 4030$. Find the coefficient of $x^{2014}$ in $g(x)$. | \frac{3^{2014}}{2^{2014} \cdot 2014!} | omnimath | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0 |
Given an arc length of π cm with a corresponding central angle of $\frac{π}{4}$, the area of the sector containing this arc is __________ cm². | 2π ext{ cm}^2 | cn_k12 | [
"Mathematics -> Geometry -> Plane Geometry -> Circles"
] | 0.078125 |
Given that $\overrightarrow a$ and $\overrightarrow b$ are both unit vectors, and $(3\overrightarrow a+\overrightarrow b)⊥(\overrightarrow a-2\overrightarrow b)$, find the cosine value of the angle between vectors $\overrightarrow a$ and $\overrightarrow b$. | \dfrac{1}{5} | cn_k12 | [
"Mathematics -> Algebra -> Other"
] | 0.421875 |
If $m$ is a positive integer, and both $m-4$ and $m+5$ are perfect squares, then the value of $m$ is . | 20 \text{ or } 4 | cn_k12 | [
"Mathematics -> Algebra -> Equations and Inequalities -> Other"
] | 0.515625 |
Mike has earned a total of $160 in wages this week. He received $52 from his first job, then later received the wages from his second job where he works a certain number of hours a week and his second job pays $9 per hour. How many hours does Mike work at his second job? | 12 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1 |
Simplify $\frac{\sqrt{2}}{\sqrt{3}} \cdot \frac{\sqrt{4}}{\sqrt{5}} \cdot \frac{\sqrt{6}}{\sqrt{7}}$ and rationalize the denominator of the resulting fraction. | \frac{4\sqrt{35}}{35} | math | [
"Mathematics -> Algebra -> Intermediate Algebra -> Other"
] | 0.796875 |
In the triangle \( T_{a} = \triangle A_{1}A_{2}A_{3} \), there is an inscribed triangle \( T_{b} = \triangle B_{1}B_{2}B_{3} \), and in the triangle \( T_{b} \) there is an inscribed triangle \( T_{c} = \triangle C_{1}C_{2}C_{3} \), such that the sides of the triangles \( T_{a} \) and \( T_{c} \) are parallel. Express ... | b = \sqrt{a c} | olympiads | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.140625 |
Thus, potassium dichromate can be used as an oxidizing agent only for the reactions:
$$
2 \mathrm{Br}^{-} - 2 e^{-} = \mathrm{Br}_{2}
$$
and
$$
2 \mathrm{I}^{-} - 2 e^{-} = \mathrm{I}_{2}
$$ | \text{K}_{2} \text{Cr}_{2} \text{O}_{7} \text{ can be used as an oxidizing agent for the oxidation of bromide and iodide ions.} | olympiads | [
"Mathematics -> Other"
] | 0.015625 |
A jar contains some blue pens, 21 black pens, and 6 red pens. Four blue pens are removed and then seven black pens are removed. There are 25 pens left in the jar. How many blue pens were initially in the jar? | 9 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.65625 |
If $\cos (\alpha+\beta)= \frac {1}{3}$ and $\cos (\alpha-\beta)= \frac {1}{5}$, then $\tan \alpha \cdot \tan \beta=$ \_\_\_\_\_\_. | - \frac {1}{4} | cn_k12 | [
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.296875 |
The coordinates of point $A$ are $(-3,4)$. The distance from point $A$ to the $y$-axis is ____. | 3 | cn_k12 | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.984375 |
Paul got a box of 531 crayons and 38 erasers for his birthday. At the end of the school year, he had some crayons left while not having lost a single eraser. He had 353 more crayons than erasers left. How many crayons did he have left at the end of the school year? | 391 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.6875 |
A car company produced some cars in North America and 2,871 cars in Europe. The company produced 6,755 cars in all. How many cars were produced in North America? | 3,884 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 |
Given the initial values a = 3, b = "-5", and c = 8, and the sequence of assignments a = b, b = c, c = a, determine the final values of a, b, and c after these assignments are executed. Express your answer as a comma-separated list of values. | -5, 8, -5 | big_math | [
"Mathematics -> Algebra -> Other"
] | 0.515625 |
Every bedtime, Juwella reads a book. Three nights ago, she read 15 pages. Two nights ago she read twice that many pages, while last night she read 5 pages more than the previous night. She promised to read the remaining pages of the book tonight. If the book has 100 pages, how many pages will she read tonight? | 20 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.984375 |
Ajay bought 15 kg of dal at the rate of Rs 14.50 per kg and 10 kg at the rate of Rs 13 per kg. He mixed the two and sold the mixture at the rate of Rs 15 per kg. What was his total gain in this transaction? | 27.50 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1 |
In a certain junior high school graduating class, each student gave a photo of themselves to every other student in the class as a memento. In total, 2550 photos were given out. If there are $x$ students in the class, write an equation based on the given information. | x(x-1) = 2550 | cn_k12 | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0.484375 |
How many ways can one color the squares of a $6 \times 6$ grid red and blue such that the number of red squares in each row and column is exactly 2? | 67950 | omnimath | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability -> Counting Methods -> Combinations"
] | 0 |
Given vector $\overrightarrow{a}=(\sin α, \cos α - 2\sin α)$ and vector $\overrightarrow{b}=(1, 2)$, and if $\overrightarrow{a} \parallel \overrightarrow{b}$, find the value of $\tan α$. | \frac{1}{4} | cn_k12 | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Geometry -> Other",
"Mathematics -> Precalculus -> Trigonometric Functions"
] | 0.546875 |
Select 4 people from 6 to visit the USA Pavilion, the UK Pavilion, the France Pavilion, and the Saudi Pavilion at the Shanghai World Expo, with the requirement that each pavilion is visited by one person, each person visits only one pavilion, and among these 6 people, person A and person B will not visit the France Pav... | 240 | cn_k12 | [
"Mathematics -> Applied Mathematics -> Probability -> Counting Methods -> Combinations"
] | 0.09375 |
In the geometric sequence $\{a_n\}$, the sum of the first $n$ terms is given by $S_n=5^{n+1}+a$. Find the value of $a$. Express your answer as a single integer. | -5 | big_math | [
"Mathematics -> Algebra -> Sequences and Series"
] | 0.046875 |
Aida has twice as many dolls as Sophie, and Sophie has twice as many dolls as Vera. Aida, Sophie, and Vera have combined 140 dolls. How many dolls does Vera have? | 20 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1 |
Analyze the flowchart of a program that calculates the sum of S using a while-type loop structure. The loop starts with S = 0 and n = 1, and in each iteration, S is incremented by n and n is incremented by 2. The loop continues until n exceeds 51. What is the final value of S after the loop structure is executed? | 625 | big_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.140625 |
There were a total of 6 soccer games this year. Jessica missed some of the games and went to 2 soccer games in all. How many soccer games did Jessica miss? | 4 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 1 |
The average (arithmetic mean) of two numbers a and b is 45, and the average of b and c is some value. The value of c - a is 50. What is the average of b and c? | 70 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.921875 |
What is the ratio of the area of the shaded part to the unshaded part? (The vertices of all squares, except the largest one, are located in the middle of the corresponding sides). | 5:3 | olympiads | [
"Mathematics -> Geometry -> Plane Geometry -> Other"
] | 0.046875 |
One eighth of three fifths of four sevenths of five eleventh of a number minus one ninth of two thirds of three fourths of five eighths of that same number is 30. What will be 75% of that number? | -1476 | orca_math | [
"Mathematics -> Algebra -> Prealgebra -> Fractions"
] | 0.015625 |
How many of the numbers from the set $\{1,\ 2,\ 3,\ldots,\ 50\}$ have a perfect square factor other than one? | 19 | math | [
"Mathematics -> Number Theory -> Other"
] | 0.09375 |
Find the number of moles of NaNO3 formed when a certain number of moles of HNO3 is combined with 1 mole of NaHCO3, and the result is 1 mole of NaNO3. How many moles of HNO3 were combined? | 1 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.9375 |
4 men can check exam papers in 8 days working 5 hours regularly . what is the total hours when 2 men will check the double of the papers in 20 days ? | 320 | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.15625 |
A certain percentage of a number is 10 less than 3/5th of that number. The number is 100. What is the percentage? | 50\% | orca_math | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.75 |
Let $x, y$ be real numbers. If $4x^2 + y^2 + xy = 5$, then the maximum value of $2x + y$ is \_\_\_\_\_\_. | 2\sqrt{2} | cn_k12 | [
"Mathematics -> Algebra -> Other",
"Mathematics -> Calculus -> Other"
] | 0.0625 |
In the country of Francisca, there are 2010 cities, some of which are connected by roads. Between any two cities, there is a unique path which runs along the roads and which does not pass through any city twice. What is the maximum possible number of cities in Francisca which have at least 3 roads running out of them? | 1004 | olympiads | [
"Mathematics -> Discrete Mathematics -> Graph Theory"
] | 0.078125 |
How many kilograms of water need to be evaporated from 0.5 tons of cellulose mass containing $85\%$ water to obtain a mass with $75\%$ water content? | 200 \, \text{kg} | olympiads | [
"Mathematics -> Applied Mathematics -> Math Word Problems"
] | 0.09375 |
An island in the Indian Ocean was 4 miles wide and 7 miles long. What is the perimeter of the island? | 22 | orca_math | [
"Mathematics -> Geometry -> Plane Geometry -> Perimeter"
] | 1 |
Find all sets of four positive real numbers \((a, b, c, d)\) satisfying
\[
\begin{array}{l}
a + b + c + d = 1, \\
\text{and} \\
\max \left\{\frac{a^{2}}{b}, \frac{b^{2}}{a}\right\} \max \left\{\frac{c^{2}}{d}, \frac{d^{2}}{c}\right\} = (\min \{a+b, c+d\})^{4}.
\end{array}
\] | \left(\frac{1}{4}, \frac{1}{4}, \frac{1}{4}, \frac{1}{4}\right) | olympiads | [
"Mathematics -> Algebra -> Other"
] | 0.171875 |
Given P(B) = 0.3, P(B|A) = 0.9, P(B|\\overline{A}) = 0.2, calculate P(\\overline{A}). Express your answer as a fraction. | \\frac{6}{7} | big_math | [
"Mathematics -> Applied Mathematics -> Statistics -> Probability"
] | 0.671875 |
Given \((x+1)^{2017} = a_{0} x^{2017} + a_{1} x^{2016} + a_{2} x^{2015} + \cdots + a_{2016} x + a_{2017}\), find the value of \(a_{1} + a_{5} + a_{9} + \cdots + a_{2017}\). | 2^{2015} | olympiads | [
"Mathematics -> Algebra -> Polynomial Operations"
] | 0.015625 |
Find the number of dissimilar terms in the expansion of (a + b + c)^10. Express your answer as a whole number. | 66 | big_math | [
"Mathematics -> Applied Mathematics -> Combinatorics"
] | 0.171875 |
The ellipse $mx^2 + ny^2 = 1$ intersects with the line $x + y = 1$ at points M and N, and the midpoint of MN is P. Given that the slope of OP is $\frac{\sqrt{2}}{2}$, find the value of $\frac{m}{n}$. | \frac{2\sqrt{2}}{2} - 1 | cn_k12 | [
"Mathematics -> Geometry -> Other"
] | 0.125 |
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