problem stringlengths 10 5.15k | answer dict |
|---|---|
Two circular tracks $\alpha$ and $\beta$ of the same radius are tangent to each other. A car $A$ travels clockwise on track $\alpha$ and a car $B$ travels counterclockwise on track $\beta$. At the start, cars $A$ and $B$ are on the same line with the center of track $\alpha$, and this line is tangent to track $\beta$. After the start, the cars begin to approach the point of tangency of the tracks. Each car completes one full lap on its track in one hour (and never switches to the other track). For how much time during this hour will the distance between the cars be at least the diameter of each track? | {
"answer": "1/2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the minimum value of the constant \(\lambda\) such that for all positive integers \(n\) and any positive real numbers \(x_{k}\) \((1 \leq k \leq n)\) summing to 1, the following inequality holds:
$$
\lambda \prod_{k=1}^{n}\left(1-x_{k}\right) \geq 1-\sum_{k=1}^{n} x_{k}^{2}.
$$ | {
"answer": "e",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Ella adds up all the odd integers from 1 to 499, inclusive. Mike adds up all the integers from 1 to 500, inclusive. What is Ella's sum divided by Mike's sum? | {
"answer": "\\frac{500}{1001}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), it is known that \(AB = 3\), \(AC = 3\sqrt{7}\), and \(\angle ABC = 60^\circ\). The bisector of angle \(ABC\) is extended to intersect at point \(D\) with the circle circumscribed around the triangle. Find \(BD\). | {
"answer": "4\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the Cartesian coordinate system, define $d(P, Q) = |x_1 - x_2| + |y_1 - y_2|$ as the "polyline distance" between two points $P(x_1, y_1)$ and $Q(x_2, y_2)$. Then, the minimum "polyline distance" between a point on the circle $x^2 + y^2 = 1$ and a point on the line $2x + y - 2 \sqrt{5} = 0$ is __________. | {
"answer": "\\frac{\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Triangle $DEF$ has vertices $D(0,10)$, $E(4,0)$, $F(10,0)$. A vertical line intersects $DF$ at $P$ and $\overline{EF}$ at $Q$, forming triangle $PQF$. If the area of $\triangle PQF$ is 16, determine the positive difference of the $x$ and $y$ coordinates of point $P$. | {
"answer": "8\\sqrt{2}-10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In triangle \(ABC\), the side lengths are 4, 5, and \(\sqrt{17}\). Find the area of the region consisting of those and only those points \(X\) inside triangle \(ABC\) for which the condition \(XA^{2} + XB^{2} + XC^{2} \leq 21\) is satisfied. | {
"answer": "\\frac{5 \\pi}{9}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a tetrahedron \( A B C D \) with side lengths \( A B = 41 \), \( A C = 7 \), \( A D = 18 \), \( B C = 36 \), \( B D = 27 \), and \( C D = 13 \), let \( d \) be the distance between the midpoints of edges \( A B \) and \( C D \). Find the value of \( d^{2} \). | {
"answer": "137",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On a number line, there are three points A, B, and C which represent the numbers -24, -10, and 10, respectively. Two electronic ants, named Alpha and Beta, start moving towards each other from points A and C, respectively. Alpha moves at a speed of 4 units per second, while Beta moves at a speed of 6 units per second.
(1) At which point on the number line do Alpha and Beta meet?
(2) After how many seconds will the sum of Alpha's distances to points A, B, and C be 40 units? If at that moment Alpha turns back, will Alpha and Beta meet again on the number line? If they can meet, find the meeting point; if they cannot, explain why. | {
"answer": "-44",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( f(n) \) be the number of 0's in the decimal representation of the positive integer \( n \). For example, \( f(10001123) = 3 \) and \( f(1234567) = 0 \). Find the value of
\[ f(1) + f(2) + f(3) + \ldots + f(99999) \] | {
"answer": "38889",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two ferries leave simultaneously from opposite shores of a river and cross it perpendicularly to the banks. The speeds of the ferries are constant. The ferries meet each other 720 meters from the nearest shore. Upon reaching the shore, they immediately depart back. On the return trip, they meet 400 meters from the other shore. What is the width of the river? | {
"answer": "1280",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A chest of gold coins is divided among 10 pirates where the kth pirate takes k/10 of the remaining coins. Determine the smallest number of coins initially in the chest such that each pirate gets a positive whole number of coins and find the number of coins the 10th pirate receives. | {
"answer": "362880",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a_{1}, a_{2}, b_{1}, b_{2}, \cdots, b_{k}$ are vectors in the plane that are pairwise non-parallel, with $\left|a_{1}-a_{2}\right|=1$, and $\left|a_{i}-b_{j}\right| \in\{1,2,3\} (i=1,2; j=1,2, \cdots, k)$, determine the maximum value of $k$. | {
"answer": "10",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangular prism \( P-ABC \),
\[
\begin{array}{l}
\angle APB = \angle BPC = \angle CPA = 90^{\circ}, \\
PA = 4, \, PB = PC = 3.
\end{array}
\]
Find the minimum sum of the squares of the distances from any point on the base \( ABC \) to the three lateral faces. | {
"answer": "\\frac{144}{41}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A 60-degree angle contains five circles, where each subsequent circle (starting from the second) touches the previous one. By how many times is the sum of the areas of all five circles greater than the area of the smallest circle? | {
"answer": "7381",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the minimum number of sides of a regular polygon that approximates the area of its circumscribed circle with an error of less than 1 per thousand (0.1%)? | {
"answer": "82",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A man, standing on a lawn, is wearing a circular sombrero of radius 3 feet. The hat blocks the sunlight, causing the grass directly under it to die instantly. If the man walks in a circle of radius 5 feet, what area of dead grass will result? | {
"answer": "60\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( \triangle ABC \) be a triangle such that \( AB = 7 \), and let the angle bisector of \( \angle BAC \) intersect line \( BC \) at \( D \). If there exist points \( E \) and \( F \) on sides \( AC \) and \( BC \), respectively, such that lines \( AD \) and \( EF \) are parallel and divide triangle \( ABC \) into three parts of equal area, determine the number of possible integral values for \( BC \). | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are three pastures full of grass. The first pasture is 33 acres and can feed 22 cows for 27 days. The second pasture is 28 acres and can feed 17 cows for 42 days. How many cows can the third pasture, which is 10 acres, feed for 3 days (assuming the grass grows at a uniform rate and each acre produces the same amount of grass)? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two brothers had tickets to a stadium located 10 km from their home. Initially, they planned to walk to the stadium. However, they changed their plan and decided to use a bicycle. They agreed that one would start on the bicycle and the other would walk simultaneously. After covering part of the distance, the first brother would leave the bicycle, and the second brother would ride the bicycle after reaching it, continuing until he caught up with the first brother at the entrance of the stadium. How much time do the brothers save compared to their initial plan to walk the entire way, given that each brother covers each kilometer 12 minutes faster on the bicycle than on foot? | {
"answer": "60",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We inscribe spheres with a radius of \(\frac{1}{2}\) around the vertices of a cube with edge length 1. There are two spheres that touch each of these eight spheres. Calculate the difference in volume between these two spheres. | {
"answer": "\\frac{10}{3} \\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{0}^{2 \sqrt{2}} \frac{x^{4} \, dx}{\left(16-x^{2}\right) \sqrt{16-x^{2}}}
$$ | {
"answer": "20 - 6\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers from 1 to 9 are placed in the cells of a \(3 \times 3\) table such that the sum of the numbers on one diagonal equals 7, and the sum on the other diagonal equals 21. What is the sum of the numbers in the five shaded cells? | {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The circles $k_{1}$ and $k_{2}$, both with unit radius, touch each other at point $P$. One of their common tangents that does not pass through $P$ is the line $e$. For $i>2$, let $k_{i}$ be the circle different from $k_{i-2}$ that touches $k_{1}$, $k_{i-1}$, and $e$. Determine the radius of $k_{1999}$. | {
"answer": "\\frac{1}{1998^2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Suppose \( x_{1}, x_{2}, \ldots, x_{2011} \) are positive integers satisfying
\[ x_{1} + x_{2} + \cdots + x_{2011} = x_{1} x_{2} \cdots x_{2011} \]
Find the maximum value of \( x_{1} + x_{2} + \cdots + x_{2011} \). | {
"answer": "4022",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Which five-digit numbers are greater in quantity: those not divisible by 5 or those whose first and second digits from the left are not a five? | {
"answer": "72000",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A school table tennis championship was held using the Olympic system. The winner won 6 matches. How many participants in the championship won more matches than they lost? (In the first round of the championship, conducted using the Olympic system, participants are divided into pairs. Those who lost the first match are eliminated from the championship, and those who won in the first round are again divided into pairs for the second round. The losers are again eliminated, and winners are divided into pairs for the third round, and so on, until one champion remains. It is known that in each round of our championship, every participant had a pair.) | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the manufacturing of a steel cable, it was found that the cable has the same length as the curve defined by the system of equations:
$$
\left\{\begin{array}{l}
x+y+z=10 \\
x y+y z+x z=18
\end{array}\right.
$$
Find the length of the cable.
| {
"answer": "4 \\pi \\sqrt{\\frac{23}{3}}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a circle $O: x^2 + y^2 = 1$ and a point $A(-2, 0)$, if there exists a fixed point $B(b, 0)$ ($b \neq -2$) and a constant $\lambda$ such that for any point $M$ on the circle $O$, it holds that $|MB| = \lambda|MA|$. The maximum distance from point $P(b, \lambda)$ to the line $(m+n)x + ny - 2n - m = 0$ is ______. | {
"answer": "\\frac{\\sqrt{10}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a rhombus \( ABCD \), the angle at vertex \( A \) is \( 60^\circ \). Point \( N \) divides side \( AB \) in the ratio \( AN:BN = 2:1 \). Find the tangent of angle \( DNC \). | {
"answer": "\\frac{\\sqrt{243}}{17}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Calculate the definite integral:
$$
\int_{\pi / 2}^{2 \pi} 2^{8} \cdot \cos ^{8} x \, dx
$$ | {
"answer": "105\\pi",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A gives B as many cents as B has and C as many cents as C has. Similarly, B then gives A and C as many cents as each then has. C, similarly, then gives A and B as many cents as each then has. After this, each person gives half of what they have to each other person. If each finally has 24 cents, calculate the number of cents A starts with. | {
"answer": "24",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(p\), \(q\), \(r\), \(s\), and \(t\) be positive integers such that \(p+q+r+s+t=3015\) and let \(N\) be the largest of the sums \(p+q\), \(q+r\), \(r+s\), and \(s+t\). What is the smallest possible value of \(N\)? | {
"answer": "1508",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \(a\), \(b\), and \(c\) be positive real numbers. Find the minimum value of
\[
\frac{5c}{a+b} + \frac{5a}{b+c} + \frac{3b}{a+c} + 1.
\] | {
"answer": "7.25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the nearest integer to \((3+\sqrt{2})^6\)? | {
"answer": "7414",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Arrange 6 volunteers for 3 different tasks, each task requires 2 people. Due to the work requirements, A and B must work on the same task, and C and D cannot work on the same task. How many different arrangements are there? | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Solve the equations:
(1) 2x^2 - 5x + 1 = 0;
(2) 3x(x - 2) = 2(2 - x). | {
"answer": "-\\frac{2}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the largest constant $C$ so that for all real numbers $x$, $y$, and $z$,
\[x^2 + y^2 + z^3 + 1 \ge C(x + y + z).\] | {
"answer": "\\sqrt{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A five-digit number \(abcde\) satisfies:
\[ a < b, \, b > c > d, \, d < e, \, \text{and} \, a > d, \, b > e. \]
For example, 34 201, 49 412. If the digit order's pattern follows a variation similar to the monotonicity of a sine function over one period, then the five-digit number is said to follow the "sine rule." Find the total number of five-digit numbers that follow the sine rule.
Note: Please disregard any references or examples provided within the original problem if they are not part of the actual problem statement. | {
"answer": "2892",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The fractional part of a positive number, its integer part, and the number itself form an increasing geometric progression. Find all such numbers. | {
"answer": "\\frac{\\sqrt{5} + 1}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find the number of integers between 1 and 2013 with the property that the sum of its digits equals 9. | {
"answer": "101",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Vasya and Petya simultaneously started running from the starting point of a circular track in opposite directions at constant speeds. At some point, they met. Vasya completed a full lap and, continuing to run in the same direction, reached the point of their first meeting at the same moment Petya completed a full lap. Find the ratio of Vasya's speed to Petya's speed. | {
"answer": "\\frac{1+\\sqrt{5}}{2}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Numbers from 1 to 9 are arranged in the cells of a \(3 \times 3\) table such that the sum of the numbers on one diagonal is 7, and the sum on the other diagonal is 21. What is the sum of the numbers in the five shaded cells?
| {
"answer": "25",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a geometric sequence $\\{a\_n\\}$, $a\_n > 0 (n \in \mathbb{N}^*)$, the common ratio $q \in (0, 1)$, and $a\_1a\_5 + 2aa\_5 + a\_2a\_8 = 25$, and the geometric mean of $a\_3$ and $a\_5$ is $2$.
(1) Find the general term formula of the sequence $\\{a\_n\\}$;
(2) If $b\_n = \log_2 a\_n$, find the sum of the first $n$ terms $S\_n$ of the sequence $\\{b\_n\\}$;
(3) Determine whether there exists $k \in \mathbb{N}^*$ such that $\frac{S\_1}{1} + \frac{S\_2}{2} + \dots + \frac{S\_n}{n} < k$ holds for any $n \in \mathbb{N}^*$. If so, find the minimum value of $k$; if not, explain the reason. | {
"answer": "19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the triangle \(A B C\), angle \(C\) is a right angle, and \(AC: AB = 3: 5\). A circle with its center on the extension of leg \(AC\) beyond point \(C\) is tangent to the extension of hypotenuse \(AB\) beyond point \(B\) and intersects leg \(BC\) at point \(P\), with \(BP: PC = 1: 4\). Find the ratio of the radius of the circle to leg \(BC\). | {
"answer": "37/15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given a rectangular parallelepiped with edge lengths AB = 5, BC = 3, and CG = 4, and point M as the midpoint of diagonal AG, determine the volume of the rectangular pyramid with base BCHE and apex M. | {
"answer": "2\\sqrt{34}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the hyperbola $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$, let $F_1$ and $F_2$ be the left and right foci, and let $P$ and $Q$ be two points on the right branch. If $\overrightarrow{PF_2} = 2\overrightarrow{F_2Q}$ and $\overrightarrow{F_1Q} \cdot \overrightarrow{PQ} = 0$, determine the eccentricity of this hyperbola. | {
"answer": "\\frac{\\sqrt{17}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
There are eight points on a circle that divide the circumference equally. Count the number of acute-angled triangles or obtuse-angled triangles that can be formed with these division points as vertices. | {
"answer": "32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Select 3 numbers from the set $\{0,1,2,3,4,5,6,7,8,9\}$ such that their sum is an even number not less than 10. How many different ways are there to achieve this? | {
"answer": "51",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A five-digit number is called irreducible if it cannot be expressed as a product of two three-digit numbers.
What is the greatest number of consecutive irreducible five-digit numbers? | {
"answer": "99",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In a 24-hour format digital watch that displays hours and minutes, calculate the largest possible sum of the digits in the display if the sum of the hour digits must be even. | {
"answer": "22",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Mom asks Xiao Ming to boil water and make tea for guests. Washing the kettle takes 1 minute, boiling water takes 15 minutes, washing the teapot takes 1 minute, washing the teacups takes 1 minute, and getting the tea leaves takes 2 minutes. Xiao Ming estimates that it will take 20 minutes to complete these tasks. According to the most efficient arrangement you think of, how many minutes will it take to make the tea? | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many three-digit numbers are there in which any two adjacent digits differ by 3? | {
"answer": "20",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
\(ABCD\) is a convex quadrilateral where \(AB = 7\), \(BC = 4\), and \(AD = DC\). Also, \(\angle ABD = \angle DBC\). Point \(E\) is on segment \(AB\) such that \(\angle DEB = 90^\circ\). Find the length of segment \(AE\). | {
"answer": "1.5",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
What is the maximum area that a rectangle can have if the coordinates of its vertices satisfy the equation \( |y-x| = (y+x+1)(5-x-y) \), and its sides are parallel to the lines \( y = x \) and \( y = -x \)? Give the square of the value of the maximum area found as the answer. (12 points) | {
"answer": "432",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Travis is hopping around on the vertices of a cube. Each minute he hops from the vertex he's currently on to the other vertex of an edge that he is next to. After four minutes, what is the probability that he is back where he started? | {
"answer": "7/27",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A reader mentioned that his friend's house in location $A$, where he was invited for lunch at 1 PM, is located 1 km from his own house in location $B$. At 12 PM, he left $B$ in his wheelchair heading towards location $C$ for a stroll. His friend, intending to join him and help him reach on time for lunch, left $A$ at 12:15 PM heading towards $C$ at a speed of 5 km/h. They met and then proceeded to $A$ together at a speed of 4 km/h, arriving exactly at 1 PM.
How much distance did our reader cover in the direction of $C$? | {
"answer": "2/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Whole numbers whose decimal representation reads the same from left to right as from right to left are called symmetrical. For example, the number 5134315 is symmetrical, while 5134415 is not. How many seven-digit symmetrical numbers exist such that adding 1100 to them leaves them unchanged as symmetrical numbers? | {
"answer": "810",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A four-digit palindrome between $1000$ and $10,000$ is chosen at random. What is the probability that it is divisible by $9$? | {
"answer": "\\frac{1}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A triangle \(A B C\) is considered. Point \(F\) is the midpoint of side \(A B\). Point \(S\) lies on the ray \(A C\) such that \(C S = 2 A C\). In what ratio does the line \(S F\) divide side \(B C\)? | {
"answer": "2:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On Namek, each hour has 100 minutes. On the planet's clocks, the hour hand completes one full circle in 20 hours and the minute hand completes one full circle in 100 minutes. How many hours have passed on Namek from 0:00 until the next time the hour hand and the minute hand overlap? | {
"answer": "20/19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
How many five-character license plates consist of a consonant, followed by a vowel, followed by a consonant, a digit, and then a special character from the set {$, #, @}? (For this problem, consider Y as both a consonant and a vowel.) | {
"answer": "79,380",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One of the angles of a trapezoid is $60^{\circ}$. Find the ratio of its bases if it is known that a circle can be inscribed in this trapezoid and also circumscribed around this trapezoid. | {
"answer": "1:3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Find all \( x \in [1,2) \) such that for any positive integer \( n \), the value of \( \left\lfloor 2^n x \right\rfloor \mod 4 \) is either 1 or 2. | {
"answer": "4/3",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the 100th year of his reign, the Immortal Treasurer decided to start issuing new coins. This year, he issued an unlimited supply of coins with a denomination of \(2^{100} - 1\), next year with a denomination of \(2^{101} - 1\), and so on. As soon as the denomination of a new coin can be obtained without change using previously issued new coins, the Treasurer will be removed from office. In which year of his reign will this happen? | {
"answer": "200",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the probability of failing on the first attempt and succeeding on the second attempt to guess the last digit of a phone number is requested, calculate the probability of this event. | {
"answer": "\\frac{1}{10}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the number of 0-1 binary sequences of ten 0's and ten 1's which do not contain three 0's together. | {
"answer": "24068",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given the vector $$\overrightarrow {a_{k}} = (\cos \frac {k\pi}{6}, \sin \frac {k\pi}{6} + \cos \frac {k\pi}{6})$$ for k=0, 1, 2, …, 12, find the value of $$\sum\limits_{k=0}^{11} (\overrightarrow {a_{k}} \cdot \overrightarrow {a_{k+1}})$$. | {
"answer": "9\\sqrt{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Compute the value of \(1^{25}+2^{24}+3^{23}+\ldots+24^{2}+25^{1}\). | {
"answer": "66071772829247409",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine the smallest positive integer $m$ such that $11m-3$ and $8m + 5$ have a common factor greater than $1$. | {
"answer": "108",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Evaluate \[\frac{3}{\log_8{5000^4}} + \frac{2}{\log_9{5000^4}},\] giving your answer as a fraction in lowest terms. | {
"answer": "\\frac{1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The lengths of the edges of a regular tetrahedron \(ABCD\) are 1. \(G\) is the center of the base \(ABC\). Point \(M\) is on line segment \(DG\) such that \(\angle AMB = 90^\circ\). Find the length of \(DM\). | {
"answer": "\\frac{\\sqrt{6}}{6}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $[a]$ represent the greatest integer less than or equal to $a$. Determine the largest positive integer solution to the equation $\left[\frac{x}{7}\right]=\left[\frac{x}{8}\right]+1$. | {
"answer": "104",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In an album, there is a rectangular grid of size $3 \times 7$. Igor's robot was asked to trace all the lines with a marker, and it took 26 minutes (the robot draws lines at a constant speed). How many minutes will it take to trace all the lines of a $5 \times 5$ grid? | {
"answer": "30",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Anička received a rectangular cake for her birthday. She cut the cake with two straight cuts. The first cut was made such that it intersected both longer sides of the rectangle at one-third of their length. The second cut was made such that it intersected both shorter sides of the rectangle at one-fifth of their length. Neither cut was parallel to the sides of the rectangle, and at each corner of the rectangle, there were either two shorter segments or two longer segments of the divided sides joined.
Anička ate the piece of cake marked in grey. Determine what portion of the cake this was. | {
"answer": "2/15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Two spheres touch the plane of triangle \(ABC\) at points \(B\) and \(C\) and are located on opposite sides of this plane. The sum of the radii of these spheres is 12, and the distance between their centers is \(4 \sqrt{29}\). The center of a third sphere with radius 8 is at point \(A\), and it touches each of the first two spheres externally. Find the radius of the circumcircle of triangle \(ABC\). | {
"answer": "4\\sqrt{5}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Three male students and three female students, a total of six students, stand in a row. If female students do not stand at the end of the row, and female students A and B are not adjacent to female student C, then find the number of different arrangements. | {
"answer": "144",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The numbers \( a, b, c, d \) belong to the interval \([-4 ; 4]\). Find the maximum value of the expression \( a + 2b + c + 2d - ab - bc - cd - da \). | {
"answer": "72",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A and B play a repeated game. In each game, they take turns tossing a fair coin, and the player who first gets heads wins the game. In the first game, A tosses first. For subsequent games, the loser of the previous game starts first. What is the probability that A wins the 6th game? | {
"answer": "\\frac{1}{2}\\left(1 - \\frac{1}{729}\\right)",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Some nodes (vertices of unit squares) in a $6 \times 6$ grid are painted red such that on the boundary of any $k \times k$ subgrid ($1 \leqslant k \leqslant 6$) there is at least one red point. Find the minimum number of red points required to satisfy this condition. | {
"answer": "16",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Using a permutation of the numbers $10, 20, 30, 40$ for $A, B, C, D$, maximize the value of the expression $\frac{1}{A-\frac{1}{B+\frac{1}{C-\frac{1}{D}}}}$. Then, find the value of $A+2B+3C+4D$. | {
"answer": "290",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The four hydrogen atoms in the methane molecule $\mathrm{CH}_{4}$ are located at the vertices of a regular tetrahedron with edge length 1. The carbon atom $C$ is located at the center of the tetrahedron $C_{0}$. Let the four hydrogen atoms be $H_{1}, H_{2}, H_{3}, H_{4}$. Then $\sum_{1 \leq i < j \leq 4} \overrightarrow{C_{0} \vec{H}_{i}} \cdot \overrightarrow{C_{0} \vec{H}_{j}} = \quad$ . | {
"answer": "-2",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
The sum of the house numbers on one side of a street from corner to corner is 117. What is the house number of the fifth house from the beginning of this section? | {
"answer": "13",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
We divide the height of a cone into three equal parts, and through the division points, we lay planes parallel to the base. How do the volumes of the resulting solids compare to each other? | {
"answer": "1:7:19",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the diagram, $\triangle PQR$ is right-angled at $P$ and $\angle PRQ=\theta$. A circle with center $P$ is drawn passing through $Q$. The circle intersects $PR$ at $S$ and $QR$ at $T$. If $QT=8$ and $TR=10$, determine the value of $\cos \theta$. | {
"answer": "\\frac{\\sqrt{7}}{3}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Cheburashka spent his money to buy as many mirrors from Galya's store as Gena bought from Shapoklyak's store. If Gena were buying from Galya, he would have 27 mirrors, and if Cheburashka were buying from Shapoklyak, he would have 3 mirrors. How many mirrors would Gena and Cheburashka have bought together if Galya and Shapoklyak agreed to set a price for the mirrors equal to the average of their current prices? (The average of two numbers is half of their sum, for example, for the numbers 22 and 28, the average is 25.) | {
"answer": "18",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the 2011 Shanghai Spring College Entrance Examination, there were 8 colleges recruiting students. If exactly 3 students were admitted by 2 of these colleges, then the number of ways this could happen is __________. | {
"answer": "168",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Petya wrote a natural number \( A \) on the board. If you multiply it by 8, you get the square of a natural number. How many such three-digit numbers \( B \) exist for which \( A \cdot B \) is also a square of a natural number? | {
"answer": "15",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A small class of nine boys are to change their seating arrangement by drawing their new seat numbers from a box. After the seat change, what is the probability that there is only one pair of boys who have switched seats with each other and only three boys who have unchanged seats? | {
"answer": "1/32",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
On sides \( BC \) and \( AC \) of triangle \( ABC \), points \( D \) and \( E \) are chosen respectively such that \( \angle BAD = 50^\circ \) and \( \angle ABE = 30^\circ \). Find \( \angle BED \) if \( \angle ABC = \angle ACB = 50^\circ \). | {
"answer": "40",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
In the isosceles trapezoid \( KLMN \), the base \( KN \) is equal to 9, and the base \( LM \) is equal to 5. Points \( P \) and \( Q \) lie on the diagonal \( LN \), with point \( P \) located between points \( L \) and \( Q \), and segments \( KP \) and \( MQ \) perpendicular to the diagonal \( LN \). Find the area of trapezoid \( KLMN \) if \( \frac{QN}{LP} = 5 \). | {
"answer": "7\\sqrt{21}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let $p$, $q$, and $r$ be constants, and suppose that the inequality \[\frac{(x-p)(x-q)}{x-r} \le 0\] is true if and only if $x > 2$ or $3 \le x \le 5$. Given that $p < q$, find the value of $p + q + 2r$. | {
"answer": "12",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
You are given 10 numbers - one one and nine zeros. You are allowed to select two numbers and replace each of them with their arithmetic mean.
What is the smallest number that can end up in the place of the one? | {
"answer": "\\frac{1}{512}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Determine how many roots of the equation
$$
4 \sin 2x + 3 \cos 2x - 2 \sin x - 4 \cos x + 1 = 0
$$
are located in the interval \(\left[10^{2014!} \pi, 10^{2014!+2015} \pi\right]\). In the answer, write down the sum of all the digits of the found number. | {
"answer": "18135",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A fair 6-sided die and a fair 8-sided die are rolled once each. Calculate the probability that the sum of the numbers rolled is greater than 10. | {
"answer": "\\frac{3}{16}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Kevin writes a nonempty subset of $S = \{ 1, 2, \dots 41 \}$ on a board. Each day, Evan takes the set last written on the board and decreases each integer in it by $1.$ He calls the result $R.$ If $R$ does not contain $0$ he writes $R$ on the board. If $R$ contains $0$ he writes the set containing all elements of $S$ not in $R$ . On Evan's $n$ th day, he sees that he has written Kevin's original subset for the $1$ st time. Find the sum of all possible $n.$ | {
"answer": "94",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
A point \( A \) in the plane with integer coordinates is said to be visible from the origin \( O \) if the open segment \( ] O A[ \) contains no point with integer coordinates. How many such visible points are there in \( [0,25]^{2} \setminus \{(0,0)\} \)? | {
"answer": "399",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Given that $a, b \in \mathbb{R}$, and $a^2 + 2ab - 3b^2 = 1$, find the minimum value of $a^2 + b^2$. | {
"answer": "\\frac{\\sqrt{5} + 1}{4}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
One way to pack a 100 by 100 square with 10000 circles, each of diameter 1, is to put them in 100 rows with 100 circles in each row. If the circles are repacked so that the centers of any three tangent circles form an equilateral triangle, what is the maximum number of additional circles that can be packed? | {
"answer": "1443",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
Let \( x, y, z, w \in [0,1] \). Find the maximum value of \( S = x^2 y + y^2 z + z^2 w + w^2 x - xy^2 - yz^2 - zw^2 - wx^2 \). | {
"answer": "\\frac{8}{27}",
"ground_truth": null,
"style": null,
"task_type": "math"
} |
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