problem stringlengths 10 5.15k | answer dict |
|---|---|
Let \( a \) and \( b \) be integers such that \( ab = 72 \). Find the minimum value of \( a + b \). | {
"answer": "-17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute $\sin(-30^\circ)$ and verify by finding $\cos(-30^\circ)$, noticing the relationship, and confirming with the unit circle properties. | {
"answer": "\\frac{\\sqrt{3}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many pairs of integers $a$ and $b$ are there such that $a$ and $b$ are between $1$ and $42$ and $a^9 = b^7 \mod 43$ ? | {
"answer": "42",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the value of $V_3$ in Horner's method (also known as Qin Jiushao algorithm) for finding the value of the polynomial $f(x) = 4x^6 + 3x^5 + 4x^4 + 2x^3 + 5x^2 - 7x + 9$ when $x = 4$. | {
"answer": "80",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest natural number written in the decimal system with the product of the digits equal to $10! = 1 \cdot 2 \cdot 3\cdot ... \cdot9\cdot10$ . | {
"answer": "45578899",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A circle with center $O$ has radius $5,$ and has two points $A,B$ on the circle such that $\angle AOB = 90^{\circ}.$ Rays $OA$ and $OB$ are extended to points $C$ and $D,$ respectively, such that $AB$ is parallel to $CD,$ and the length of $CD$ is $200\%$ more than the radius of circle $O.$ Determine the length of $AC.$ | {
"answer": "5 + \\frac{15\\sqrt{2}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, \(\angle AFC = 90^\circ\), \(D\) is on \(AC\), \(\angle EDC = 90^\circ\), \(CF = 21\), \(AF = 20\), and \(ED = 6\). Determine the total area of quadrilateral \(AFCE\). | {
"answer": "297",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that Ms. Demeanor's class consists of 50 students, more than half of her students bought crayons from the school bookstore, each buying the same number of crayons, with each crayon costing more than the number of crayons bought by each student, and the total cost for all crayons was $19.98, determine the cost of each crayon in cents. | {
"answer": "37",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangular pyramid \(ABCD\) with base \(ABC\), the lateral edges are pairwise perpendicular, \(DA = DB = 5, DC = 1\). A ray of light is emitted from a point on the base. After reflecting exactly once from each lateral face (the ray does not reflect from the edges), the ray hits a point on the pyramid's base. What is the minimum distance the ray could travel? | {
"answer": "\\frac{10 \\sqrt{3}}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many four-digit numbers are composed of four distinct digits such that one digit is the average of any two other digits? | {
"answer": "216",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ivan Tsarevich is fighting the Dragon Gorynych on the Kalinov Bridge. The Dragon has 198 heads. With one swing of his sword, Ivan Tsarevich can cut off five heads. However, new heads immediately grow back, the number of which is equal to the remainder when the number of heads left after Ivan's swing is divided by 9. If the remaining number of heads is divisible by 9, no new heads grow. If the Dragon has five or fewer heads before the swing, Ivan Tsarevich can kill the Dragon with one swing. How many sword swings does Ivan Tsarevich need to defeat the Dragon Gorynych? | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a$, $b$, and $c$ are integers, and $a-2b=4$, $ab+c^2-1=0$, find the value of $a+b+c$. | {
"answer": "-3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On an island, there are knights who always tell the truth and liars who always lie. At the main celebration, 100 islanders sat around a large round table. Half of the attendees said the phrase: "both my neighbors are liars," while the remaining said: "among my neighbors, there is exactly one liar." What is the maximum number of knights that can sit at this table? | {
"answer": "67",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many multiples of 4 are between 100 and 350? | {
"answer": "62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that \( x \) and \( y \) are positive numbers, determine the minimum value of \(\left(x+\frac{1}{y}\right)^{2}+\left(y+\frac{1}{2x}\right)^{2}\). | {
"answer": "3 + 2 \\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Allison's birthday cake is in the form of a $5 \times 5 \times 3$ inch rectangular prism with icing on the top, front, and back sides but not on the sides or bottom, calculate the number of $1 \times 1 \times 1$ inch smaller prisms that will have icing on exactly two sides. | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The base of the quadrilateral prism \( A B C D A_{1} B_{1} C_{1} D_{1} \) is a rhombus \( A B C D \) with \( B D = 12 \) and \( \angle B A C = 60^{\circ} \). A sphere passes through the vertices \( D, A, B, B_{1}, C_{1}, D_{1} \).
a) Find the area of the circle obtained in the cross section of the sphere by the plane passing through points \( A_{1}, B_{1} \), and \( C_{1} \).
b) Find the angle \( A_{1} C B \).
c) Given that the radius of the sphere is 8, find the volume of the prism. | {
"answer": "192\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of functions of the form \( f(x) = ax^3 + bx^2 + cx + d \) such that
\[ f(x)f(-x) = f(x^3). \] | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Pablo is solving a quadratic equation and notices that the ink has smeared over the coefficient of $x$, making it unreadable. He recalls that the equation has two distinct negative, integer solutions. He needs to calculate the sum of all possible coefficients that were under the smeared ink.
Given the equation form:
\[ x^2 + ?x + 24 = 0 \]
What is the sum of all distinct coefficients that could complete this equation? | {
"answer": "-60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Tom paid $180, Dorothy paid $200, Sammy paid $240, and Alice paid $280, and they agreed to split the costs evenly, calculate the amount that Tom gave to Sammy minus the amount that Dorothy gave to Alice. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An acute isosceles triangle, $ABC$, is inscribed in a circle. Through $B$ and $C$, tangents to the circle are drawn, meeting at point $D$. If $\angle ABC = \angle ACB = 3 \angle D$ and $\angle BAC = k \pi$ in radians, then find $k$. | {
"answer": "\\frac{1}{13}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $ABC$ is isosceles with $AC = BC$ and $\angle ACB = 106^\circ.$ Point $M$ is in the interior of the triangle so that $\angle MAC = 7^\circ$ and $\angle MCA = 23^\circ.$ Find the number of degrees in $\angle CMB.$ | {
"answer": "83",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ that satisfy the conditions: $|\overrightarrow{a}| = 2$, $|\overrightarrow{b}| = \sqrt{2}$, and $\overrightarrow{a}$ is perpendicular to $(2\overrightarrow{b} - \overrightarrow{a})$, find the angle between $\overrightarrow{a}$ and $\overrightarrow{b}$. | {
"answer": "\\frac{\\pi}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The perimeter of the quadrilateral formed by the four vertices of the ellipse $C: \frac {x^{2}}{4}+ \frac {y^{2}}{16}=1$ is equal to _____. | {
"answer": "8 \\sqrt {5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The road from Petya's house to the school takes 20 minutes. One day, on his way to school, he remembered that he forgot his pen at home. If he continues his journey at the same speed, he will arrive at school 3 minutes before the bell rings. However, if he returns home to get the pen and then goes to school at the same speed, he will be 7 minutes late for the start of the class. What fraction of the way had he traveled when he remembered about the pen? | {
"answer": "\\frac{7}{20}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are several pairs of integers $ (a, b) $ satisfying $ a^2 - 4a + b^2 - 8b = 30 $ . Find the sum of the sum of the coordinates of all such points. | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a triangle $\triangle ABC$, where the lengths of the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. It is known that vector $\overrightarrow{m} = (\sin A + \sin C, \sin B - \sin A)$ and vector $\overrightarrow{n} = (\sin A - \sin C, \sin B)$ are orthogonal.
1. Find the measure of angle $C$.
2. If $a^2 = b^2 + \frac{1}{2}c^2$, find the value of $\sin(A - B)$. | {
"answer": "\\frac{\\sqrt{3}}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Four identical point charges are initially placed at the corners of a square, storing a total energy of 20 Joules. Determine the total amount of energy stored if one of these charges is moved to the center of the square. | {
"answer": "10\\sqrt{2} + 10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two rectangles, each measuring 7 cm in length and 3 cm in width, overlap to form the shape shown on the right. What is the perimeter of this shape in centimeters? | {
"answer": "28",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An equilateral triangle $ABC$ has an area of $27\sqrt{3}$. The rays trisecting $\angle BAC$ intersect side $BC$ at points $D$ and $E$. Find the area of $\triangle ADE$. | {
"answer": "3\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the greatest prime factor of $15! + 18!$? | {
"answer": "17",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given sets $M=\{1, 2, a^2 - 3a - 1 \}$ and $N=\{-1, a, 3\}$, and the intersection of $M$ and $N$ is $M \cap N = \{3\}$, find the set of all possible real values for $a$. | {
"answer": "\\{4\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The tetrahedron $ S.ABC$ has the faces $ SBC$ and $ ABC$ perpendicular. The three angles at $ S$ are all $ 60^{\circ}$ and $ SB \equal{} SC \equal{} 1$ . Find the volume of the tetrahedron. | {
"answer": "1/8",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
An airline company is planning to introduce a network of connections between the ten different airports of Sawubonia. The airports are ranked by priority from first to last (with no ties). We call such a network *feasible* if it satisfies the following conditions:
- All connections operate in both directions
- If there is a direct connection between two airports A and B, and C has higher priority than B, then there must also be a direct connection between A and C.
Some of the airports may not be served, and even the empty network (no connections at all) is allowed. How many feasible networks are there? | {
"answer": "512",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Tom's favorite number is between $100$ and $150$. It is a multiple of $13$, but not a multiple of $3$. The sum of its digits is a multiple of $4$. What is Tom's favorite number? | {
"answer": "143",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If the chord cut by the line $ax-by+2=0$ on the circle $x^{2}+y^{2}+2x-4y+1=0$ is $4$, find the minimum value of $\dfrac{2}{a}+\dfrac{3}{b}$. | {
"answer": "4+2\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the graph of a power function passes through the points $(2, 16)$ and $\left( \frac{1}{2}, m \right)$, then $m = \_\_\_\_\_\_$. | {
"answer": "\\frac{1}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Use all digits from 1 to 9 to form three three-digit numbers such that their product is:
a) the smallest;
b) the largest. | {
"answer": "941 \\times 852 \\times 763",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a regular pentagon \(ABCDE\). Point \(K\) is marked on side \(AE\), and point \(L\) is marked on side \(CD\). It is known that \(\angle LAE + \angle KCD = 108^\circ\) and \(AK: KE = 3:7\). Find \(CL: AB\).
A regular pentagon is a pentagon where all sides and all angles are equal. | {
"answer": "0.7",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a 6 by 6 grid, each of the 36 small squares measures 3 cm by 3 cm and is initially shaded. Five unshaded squares of side 1.5 cm are placed on the grid in such a way that some may overlap each other and/or the edges of the grid. If such overlap occurs, the overlapping parts are not visible as shaded. The area of the visible shaded region can be written in the form $A-B\phi$. What is the value $A+B$? | {
"answer": "335.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
If: (1) \(a, b, c, d\) are all elements of the set \(\{1,2,3,4\}\); (2) \(a \neq b\), \(b \neq c\), \(c \neq d\), \(d \neq a\); (3) \(a\) is the smallest among \(a, b, c, d\). Then, how many different four-digit numbers \(\overline{abcd}\) can be formed? | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the parametric equation of line $l$ as $\begin{cases}x=-\frac{1}{2}+\frac{\sqrt{2}}{2}t \\ y=\frac{1}{2}+\frac{\sqrt{2}}{2}t\end{cases}$ and the parametric equation of ellipse $C$ as $\begin{cases}x=2\cos\theta \\ y=\sqrt{3}\sin\theta\end{cases}$, in a polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the polar coordinates of point $A$ are $(\frac{\sqrt{2}}{2},\frac{3}{4}\pi)$.
(1) Convert the coordinates of point $A$ to rectangular coordinates and convert the parametric equation of the ellipse to a standard form.
(2) Line $l$ intersects ellipse $C$ at points $P$ and $Q$. Find the value of $|AP|\cdot|AQ|$. | {
"answer": "\\frac{41}{14}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The function \( f(n) \) is defined on the set of natural numbers \( N \) as follows:
\[ f(n) = \begin{cases}
n - 3, & \text{ if } n \geqslant 1000, \\
f[f(n + 7)], & \text{ if } n < 1000.
\end{cases} \]
What is the value of \( f(90) \)? | {
"answer": "999",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In right triangle $ABC$ with $\angle BAC = 90^\circ$, we have $AB = 15$ and $BC = 17$. Find $\tan A$ and $\sin A$. | {
"answer": "\\frac{8}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Square $IJKL$ has one vertex on each side of square $WXYZ$. Point $I$ is on $WZ$ such that $WI = 9 \cdot IZ$. Determine the ratio of the area of square $IJKL$ to the area of square $WXYZ$.
A) $\frac{1}{20}$
B) $\frac{1}{50}$
C) $\frac{1}{40}$
D) $\frac{1}{64}$
E) $\frac{1}{80}$ | {
"answer": "\\frac{1}{50}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
I have three 30-sided dice that each have 6 purple sides, 8 green sides, 10 blue sides, and 6 silver sides. If I roll all three dice, what is the probability that they all show the same color? | {
"answer": "\\frac{2}{25}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiaoming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, due to some reasons, Xiaoming first took the subway and then transferred to the bus, taking 40 minutes to reach the school. The transfer process took 6 minutes. Calculate the time Xiaoming spent on the bus that day. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two numbers are independently selected from the set of positive integers less than or equal to 6. What is the probability that the sum of the two numbers is less than their product? Express your answer as a common fraction. | {
"answer": "\\frac{4}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The first and twentieth terms of an arithmetic sequence are 3 and 63, respectively. What is the fortieth term? | {
"answer": "126",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Starting at $(0,0),$ an object moves in the coordinate plane via a sequence of steps, each of length one. Each step is left, right, up, or down, all four equally likely. Find the probability $p$ that the object reaches $(3,1)$ exactly in four steps. | {
"answer": "\\frac{1}{32}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum number of squares that need to be colored in a 65x65 grid (totaling 4,225 squares) so that among any four cells forming an "L" shape, there is at least one colored square?
| {
"answer": "1408",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular garden 60 feet long and 15 feet wide is enclosed by a fence. To utilize the same fence but change the shape, the garden is altered to an equilateral triangle. By how many square feet does this change the area of the garden? | {
"answer": "182.53",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the volume of the region enclosed by \[|x + y + z| + |x + y - z| + |x - y + z| + |-x + y + z| \le 6.\] | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given \(\alpha, \beta \geq 0\) and \(\alpha + \beta \leq 2\pi\), find the minimum value of \(\sin \alpha + 2 \cos \beta\). | {
"answer": "-\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $T$ denote the sum of all three-digit positive integers where each digit is different and none of the digits are 5. Calculate the remainder when $T$ is divided by $1000$. | {
"answer": "840",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the distance between the foci of the ellipse
\[\frac{x^2}{36} + \frac{y^2}{16} = 8.\] | {
"answer": "8\\sqrt{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
When three standard dice are tossed, the numbers $x, y, z$ are obtained. Find the probability that $xyz = 72$. | {
"answer": "\\frac{1}{36}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that triangle XYZ is a right triangle with two altitudes of lengths 6 and 18, determine the largest possible integer length for the third altitude. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A certain item has a cost price of $4$ yuan and is sold at a price of $5$ yuan. The merchant is preparing to offer a discount on the selling price, but the profit margin must not be less than $10\%$. Find the maximum discount rate that can be offered. | {
"answer": "12\\%",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The least positive angle $\alpha$ for which $$ \left(\frac34-\sin^2(\alpha)\right)\left(\frac34-\sin^2(3\alpha)\right)\left(\frac34-\sin^2(3^2\alpha)\right)\left(\frac34-\sin^2(3^3\alpha)\right)=\frac1{256} $$ has a degree measure of $\tfrac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Find $m+n$ . | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In trapezoid $KLMN$ with bases $KN$ and $LN$, the angle $\angle LMN$ is $60^{\circ}$. A circle is circumscribed around triangle $KLN$ and is tangent to the lines $LM$ and $MN$. Find the radius of the circle, given that the perimeter of triangle $KLN$ is 12. | {
"answer": "\\frac{4 \\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The organizing committee of the sports meeting needs to select four volunteers from Xiao Zhang, Xiao Zhao, Xiao Li, Xiao Luo, and Xiao Wang to take on four different tasks: translation, tour guide, etiquette, and driver. If Xiao Zhang and Xiao Zhao can only take on the first two tasks, while the other three can take on any of the four tasks, then the total number of different dispatch plans is \_\_\_\_\_\_ (The result should be expressed in numbers). | {
"answer": "36",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A hyperbola in the coordinate plane passing through the points $(2,5)$ , $(7,3)$ , $(1,1)$ , and $(10,10)$ has an asymptote of slope $\frac{20}{17}$ . The slope of its other asymptote can be expressed in the form $-\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers. Compute $100m+n$ .
*Proposed by Michael Ren* | {
"answer": "1720",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Twelve tiles numbered $1$ through $12$ are turned up at random, and an eight-sided die is rolled. Calculate the probability that the product of the numbers on the tile and the die will be a perfect square. | {
"answer": "\\frac{13}{96}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
For each positive integer $p$, let $c(p)$ denote the unique positive integer $k$ such that $|k - \sqrt[3]{p}| < \frac{1}{2}$. For example, $c(8)=2$ and $c(27)=3$. Find $T = \sum_{p=1}^{1728} c(p)$. | {
"answer": "18252",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In Nevada, 580 people were asked what they call soft drinks. The results of the survey are shown in the pie chart. The central angle of the "Soda" sector of the graph is $198^\circ$, to the nearest whole degree. How many of the people surveyed chose "Soda"? Express your answer as a whole number. | {
"answer": "321",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A high school's second-year students are studying the relationship between students' math and Chinese scores. They conducted a simple random sampling with replacement and obtained a sample of size $200$ from the second-year students. The sample observation data of math scores and Chinese scores are organized as follows:
| | Chinese Score | | | Total |
|----------|---------------|---------|---------|-------|
| Math Score | Excellent | Not Excellent | | |
| Excellent | $45$ | $35$ | | $80$ |
| Not Excellent | $45$ | $75$ | | $120$ |
| Total | $90$ | $110$ | | $200$ |
$(1)$ According to the independence test with $\alpha = 0.01$, can it be concluded that there is an association between math scores and Chinese scores?
$(2)$ In artificial intelligence, $L(B|A)=\frac{{P(B|A)}}{{P(\overline{B}|A)}}$ is commonly used to represent the odds of event $B$ occurring given that event $A$ has occurred, which is called likelihood ratio in statistics. Now, randomly select a student from the school, let $A=$"the selected student has a non-excellent Chinese score" and $B=$"the selected student has a non-excellent math score". Please estimate the value of $L(B|A)$ using the sample data.
Given: ${\chi^2}=\frac{{n{{(ad-bc)}^2}}}{{({a+b})({c+d})({a+c})({b+d})}}$
| $\alpha $ | $0.05$ | $0.01$ | $0.001$ |
|-----------|--------|--------|---------|
| $x_{a}$ | $3.841$| $6.635$| $10.828$| | {
"answer": "\\frac{15}{7}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The total GDP of the capital city in 2022 is 41600 billion yuan, express this number in scientific notation. | {
"answer": "4.16 \\times 10^{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A rectangular garden that is $14$ feet wide and $19$ feet long is paved with $2$-foot square pavers. Given that a bug walks from one corner to the opposite corner in a straight line, determine the total number of pavers the bug visits, including the first and the last paver. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A square with side length \(10 \text{ cm}\) is drawn on a piece of paper. How many points on the paper are exactly \(10 \text{ cm}\) away from two of the vertices of this square? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Paul needs to save 40 files onto flash drives, each with 2.0 MB space. 4 of the files take up 1.2 MB each, 16 of the files take up 0.9 MB each, and the rest take up 0.6 MB each. Determine the smallest number of flash drives needed to store all 40 files. | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the maximum value of $\lambda ,$ such that for $\forall x,y\in\mathbb R_+$ satisfying $2x-y=2x^3+y^3,x^2+\lambda y^2\leqslant 1.$ | {
"answer": "\\frac{\\sqrt{5} + 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the puzzle:
$$
\text { SI } \cdot \text { SI } = \text { SALT. }
$$ | {
"answer": "98",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of positive integers $n$ such that
\[(n - 2)(n - 4)(n - 6) \dotsm (n - 98) > 0.\] | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a table, there are 2 candles, each 20 cm long, but of different diameters. The candles burn evenly, with the thin candle burning completely in 4 hours and the thick candle in 5 hours. After how much time will the thin candle become twice as short as the thick candle if they are lit simultaneously? | {
"answer": "20/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $F\_1$ and $F\_2$ are the left and right foci of the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 (a > 0, b > 0)$, and there exists a point $P$ on the hyperbola such that $\angle F\_1PF\_2 = 60^{\circ}$ and $|OP| = 3b$ (where $O$ is the origin), find the eccentricity of the hyperbola.
A) $\frac{4}{3}$
B) $\frac{2\sqrt{3}}{3}$
C) $\frac{7}{6}$
D) $\frac{\sqrt{42}}{6}$ | {
"answer": "\\frac{\\sqrt{42}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Xiao Ming's home is 30 minutes away from school by subway and 50 minutes by bus. One day, Xiao Ming took the subway first and then transferred to the bus, taking a total of 40 minutes to reach school, with the transfer process taking 6 minutes. How many minutes did Xiao Ming take the bus that day? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $b_n$ be the integer obtained by writing the integers from $5$ to $n+4$ from left to right. For example, $b_2 = 567$, and $b_{10} = 567891011121314$. Compute the remainder when $b_{25}$ is divided by $55$ (which is the product of $5$ and $11$ for the application of the Chinese Remainder Theorem). | {
"answer": "39",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a basketball tournament every two teams play two matches. As usual, the winner of a match gets $2$ points, the loser gets $0$ , and there are no draws. A single team wins the tournament with $26$ points and exactly two teams share the last position with $20$ points. How many teams participated in the tournament? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that point \( F \) is the right focus of the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) (\(a > b > 0\)), and the eccentricity of the ellipse is \(\frac{\sqrt{3}}{2}\), a line \( l \) passing through point \( F \) intersects the ellipse at points \( A \) and \( B \) (point \( A \) is above the \( x \)-axis), and \(\overrightarrow{A F} = 3 \overrightarrow{F B}\). Find the slope of the line \( l \). | {
"answer": "-\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let α and β be acute angles, and cos α = 1/7, sin(α + β) = 5√3/14. Find β. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Each of $2011$ boxes in a line contains two red marbles, and for $1 \le k \le 2011$, the box in the $k\text{th}$ position also contains $k+1$ white marbles. Liam begins at the first box and successively draws a single marble at random from each box, in order. He stops when he first draws a red marble. Let $Q(n)$ be the probability that Liam stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $Q(n) < \frac{1}{4022}$?
A) 60
B) 61
C) 62
D) 63
E) 64 | {
"answer": "62",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate \(\sqrt{114 + 44\sqrt{6}}\) and express it in the form \(x + y\sqrt{z}\), where \(x\), \(y\), and \(z\) are integers and \(z\) has no square factors other than 1. Find \(x + y + z\). | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\underline{xyz}$ represent the three-digit number with hundreds digit $x$ , tens digit $y$ , and units digit $z$ , and similarly let $\underline{yz}$ represent the two-digit number with tens digit $y$ and units digit $z$ . How many three-digit numbers $\underline{abc}$ , none of whose digits are 0, are there such that $\underline{ab}>\underline{bc}>\underline{ca}$ ? | {
"answer": "120",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given $|\vec{a}| = |\vec{b}| = 2$, and $(\vec{a} + 2\vec{b}) \cdot (\vec{a} - \vec{b}) = -2$, find the angle between $\vec{a}$ and $\vec{b}$. | {
"answer": "\\frac{\\pi}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
At exactly noon, Anna Kuzminichna looked out the window and saw Klava, the village shop clerk, going on a break. Two minutes past twelve, Anna Kuzminichna looked out the window again, and no one was at the closed store. Klava was absent for exactly 10 minutes, and when she returned, she found that Ivan and Foma were waiting at the door, with Foma evidently arriving after Ivan. Find the probability that Foma had to wait no more than 4 minutes for the store to open. | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The first 14 terms of the sequence $\{a_n\}$ are 4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38. According to this pattern, find $a_{16}$. | {
"answer": "46",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that the product of Kiana's age and the ages of her two older siblings is 256, and that they have distinct ages, determine the sum of their ages. | {
"answer": "38",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $\mathcal{F}$ be a family of subsets of $\{1,2,\ldots, 2017\}$ with the following property: if $S_1$ and $S_2$ are two elements of $\mathcal{F}$ with $S_1\subsetneq S_2$ , then $|S_2\setminus S_1|$ is odd. Compute the largest number of subsets $\mathcal{F}$ may contain. | {
"answer": "2 \\binom{2017}{1008}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the function $f(x) = \left( \frac{1}{3}\right)^{ax^2-4x+3}$,
$(1)$ If $a=-1$, find the intervals of monotonicity for $f(x)$;
$(2)$ If $f(x)$ has a maximum value of $3$, find the value of $a$;
$(3)$ If the range of $f(x)$ is $(0,+\infty)$, find the range of values for $a$. | {
"answer": "\\{0\\}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the area of the portion of the circle defined by \(x^2 - 10x + y^2 = 9\) that lies above the \(x\)-axis and to the left of the line \(y = x-5\)? | {
"answer": "4.25\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the least positive integer with exactly $12$ positive factors? | {
"answer": "150",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given Karl's rectangular garden measures \(30\) feet by \(50\) feet with a \(2\)-feet wide uniformly distributed pathway and Makenna's garden measures \(35\) feet by \(55\) feet with a \(3\)-feet wide pathway, compare the areas of their gardens, assuming the pathways take up gardening space. | {
"answer": "225",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the sum of the digits of the greatest prime number that is a divisor of 8,191? | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
All the complex roots of $(z - 2)^6 = 64z^6$ when plotted in the complex plane, lie on a circle. Find the radius of this circle. | {
"answer": "\\frac{2\\sqrt{3}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find \(AX\) in the diagram where \(AC = 27\) units, \(BC = 36\) units, and \(BX = 30\) units. | {
"answer": "22.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$, respectively. Given that $(\sqrt{3}\cos10°-\sin10°)\cos(B+35°)=\sin80°$.
$(1)$ Find angle $B$.
$(2)$ If $2b\cos \angle BAC=c-b$, the angle bisector of $\angle BAC$ intersects $BC$ at point $D$, and $AD=2$, find $c$. | {
"answer": "\\sqrt{6}+\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two right triangles, $ABC$ and $ACD$, are joined at side $AC$. Squares are drawn on four of the sides. The areas of three of the squares are 25, 49, and 64 square units. Determine the number of square units in the area of the fourth square. | {
"answer": "138",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $ABCDEF$ be a regular hexagon. Let $G, H, I, J, K,$ and $L$ be the centers, respectively, of equilateral triangles with bases $\overline{AB}, \overline{BC}, \overline{CD}, \overline{DE}, \overline{EF},$ and $\overline{FA},$ each exterior to the hexagon. What is the ratio of the area of hexagon $GHIJKL$ to the area of hexagon $ABCDEF$?
A) $\frac{7}{6}$
B) $\sqrt{3}$
C) $2\sqrt{3}$
D) $\frac{10}{3}$
E) $\sqrt{2}$ | {
"answer": "\\frac{10}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A graduating high school class has $45$ people. Each student writes a graduation message to every other student, with each pair writing only one message between them. How many graduation messages are written in total? (Answer with a number) | {
"answer": "1980",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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