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stringlengths 18
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|---|---|---|
Three tired cowboys entered a saloon and hung their hats on a bison horn at the entrance. Late at night, when the cowboys were leaving, they were unable to distinguish one hat from another and randomly took three hats. Find the probability that none of them took their own hat.
|
\frac{1}{3}
| 0.875 |
Given \( x \in \mathbf{R} \), find the maximum value of \( y = \frac{\sin x}{2 - \cos x} \).
|
\frac{\sqrt{3}}{3}
| 0.5 |
In a meadow, ladybugs have gathered. If a ladybug has 6 spots on its back, it always tells the truth; if it has 4 spots, it always lies. There were no other types of ladybugs in the meadow. The first ladybug said, "Each of us has the same number of spots on our backs." The second said, "Altogether, we have 30 spots on our backs." The third countered, "No, altogether we have 26 spots on our backs." Each of the remaining ladybugs declared, "Exactly one of the three statements is true." How many ladybugs are in the meadow in total?
|
5
| 0.875 |
On the Cartesian plane, find the number of integer coordinate points (points where both x and y are integers) that satisfy the following system of inequalities:
\[
\begin{cases}
y \leq 3x, \\
y \geq \frac{1}{3}x, \\
x + y \leq 100.
\end{cases}
\]
|
2551
| 0.5 |
How many three-digit natural numbers are there such that the sum of their digits is equal to 24?
|
10
| 0.625 |
Given the set
$$
M=\{1,2, \cdots, 2020\},
$$
for any non-empty subset $A$ of $M$, let $\lambda_{A}$ be the sum of the maximum and minimum numbers in the subset $A$. What is the arithmetic mean of all such $\lambda_{A}$?
|
2021
| 0.25 |
In triangle \(ABC\), the angle bisector \(AL\) is drawn. Points \(E\) and \(D\) are marked on segments \(AB\) and \(BL\) respectively such that \(DL = LC\) and \(ED \parallel AC\). Find the length of segment \(ED\) if it is known that \(AE = 15\) and \(AC = 12\).
|
3
| 0.125 |
On the leg \(BC\) of the right triangle \(ABC\), a circle is constructed with \(BC\) as its diameter. This circle intersects the hypotenuse \(AB\) at point \(P\). Chord \(PQ\) is parallel to leg \(BC\). Line \(BQ\) intersects leg \(AC\) at point \(D\). Given that \(AC = b\) and \(DC = d\), find \(BC\).
|
\sqrt{bd}
| 0.625 |
Does there exist a fraction equivalent to $\frac{7}{13}$ such that the difference between the denominator and the numerator is 24?
|
\frac{28}{52}
| 0.375 |
All digits in the 6-digit natural numbers $a$ and $b$ are even, and any number between them contains at least one odd digit. Find the largest possible value of the difference $b-a$.
|
111112
| 0.75 |
There is one three-digit number and two two-digit numbers written on the board. The sum of the numbers containing the digit seven is 208. The sum of the numbers containing the digit three is 76. Find the sum of all three numbers.
|
247
| 0.5 |
Let \( a \) and \( b \) be real numbers. The function \( f(x)=a x+b \) satisfies: For any \( x \in[0,1] \), \( |f(x)| \leqslant 1 \). Determine the maximum value of \( ab \).
|
\frac{1}{4}
| 0.5 |
At a certain point on the bank of a wide and turbulent river, 100 meters away from a bridge, a siren is installed that emits sound signals at regular intervals. Another identical siren was taken by Glafira, who got on a bike and positioned herself at the beginning of the bridge on the same bank. Gavrila got into a motorboat, located on the bank halfway between the first siren and the beginning of the bridge. The experimenters start simultaneously, and the speeds of the bicycle and the motorboat relative to the water are both 20 km/h and directed perpendicular to the bank. It turned out that the sound signals from both sirens reach Gavrila simultaneously. Determine the distance from the starting point where Gavrila will be when he is 40 meters away from the bank. Round your answer to the nearest whole number of meters. The riverbank is straight, and the current speed at each point is directed along the bank.
|
41
| 0.125 |
A convex $n$-gon $P$, where $n>3$, is divided into equal triangles by diagonals that do not intersect inside it. What are the possible values of $n$ if the $n$-gon is described?
|
n = 4
| 0.5 |
Two positive integers \( x \) and \( y \) are such that:
\[ \frac{2010}{2011} < \frac{x}{y} < \frac{2011}{2012} \]
Find the smallest possible value for the sum \( x + y \).
|
8044
| 0.625 |
A gardener is preparing to plant a row of 20 trees, with the choice between two types of trees: maple trees and sycamore trees. The number of trees between any two maple trees (excluding these two maple trees) cannot be equal to 3. What is the maximum number of maple trees that can be planted in these 20 trees?
|
12
| 0.25 |
What is the smallest two-digit positive integer \( k \) for which the product \( 45k \) is a perfect square?
|
20
| 0.5 |
In triangle \(ABC\), whose area is \(S\), the side lengths \(AB\) and \(AC\) are in the ratio \(1:5\). The midline parallel to \(BC\) intersects the median and bisector drawn from point \(A\) at points \(M\) and \(N\) respectively. Find the area of triangle \(AMN\).
|
\frac{S}{12}
| 0.5 |
Let the equation \(x y = 6(x + y)\) have all positive integer solutions \(\left(x_{1}, y_{1}\right), \left(x_{2}, y_{2}\right), \cdots, \left(x_{n}, y_{n}\right)\). Then find
$$
\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)=
$$
|
290
| 0.5 |
What is the smallest number of kings that must be taken so that, after being placed arbitrarily on an $8 \times 8$ chessboard, there will necessarily be two kings attacking the same square?
|
10
| 0.25 |
On side \( AB \) of parallelogram \( ABCD \), point \( F \) is selected, and on the extension of side \( BC \) beyond vertex \( B \), point \( H \) is chosen such that \( \frac{AB}{BF} = \frac{BC}{BH} = 5 \). Point \( G \) is selected so that \( BFGH \) is a parallelogram. Line \( GD \) intersects \( AC \) at point \( X \). Find \( AX \), if \( AC = 100 \).
|
40
| 0.125 |
How many pairs of positive integers \((m, n)\) are there such that
\[ 7m + 3n = 10^{2004} \]
and \( m \mid n \)?
|
2010011
| 0.125 |
The sides of triangle \(ABC\) are divided by points \(M, N\), and \(P\) such that \(AM : MB = BN : NC = CP : PA = 1 : 4\). Find the ratio of the area of the triangle bounded by lines \(AN, BP\), and \(CM\) to the area of triangle \(ABC\).
|
\frac{3}{7}
| 0.75 |
For a natural number $n$, let $S_{n}$ denote the least common multiple of all the numbers $1, 2, \ldots, n$. Does there exist a natural number $m$ such that $S_{m+1} = 4S_{m}$?
|
\text{No}
| 0.75 |
There are 101 numbers written in a sequence (the numbers are not necessarily integers). The arithmetic mean of all the numbers except the first one is 2022, the arithmetic mean of all the numbers except the last one is 2023, and the arithmetic mean of the first and last numbers is 51. What is the sum of all the written numbers?
|
202301
| 0.875 |
The planar vectors $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ satisfy $|\vec{a}|=3$, $|\vec{b}|=4$, and $|\vec{a}-\vec{b}|=2 \sqrt{3}$. Find the minimum value of the function $\mathrm{f}(\mathrm{x})=\left|\mathrm{x} \overrightarrow{\mathrm{a}}+\frac{1}{\mathrm{x}} \overrightarrow{\mathrm{b}}\right|$ $(x \neq 0)$.
|
\sqrt{37}
| 0.75 |
Given an equilateral triangle \( \triangle ABC \) with side length 1 and \( PA \perp \) plane \( ABC \), and \( PA = \frac{\sqrt{6}}{4} \). Let \( A' \) be the reflection of point \( A \) over plane \( PBC \). Determine the angle formed between line \( A'C \) and \( AB \).
|
90^\circ
| 0.875 |
Consider a set of \( n \) lines drawn in the plane, no two of which are parallel and no three of which are concurrent. These lines divide the plane into a certain number of regions, some finite and some infinite. Show that it is possible to color at least \( \sqrt{\frac{n}{2}} \) of these lines in blue in such a way that there is no finite region whose perimeter is entirely blue.
|
\sqrt{\frac{n}{2}}
| 0.25 |
A six-digit number 111aaa is the product of two consecutive positive integers \( b \) and \( b+1 \). Find the value of \( b \).
|
333
| 0.75 |
For \( x, y, z \in (0,1] \), find the maximum value of the expression
$$
A = \frac{\sqrt{8 x^{4} + y} + \sqrt{8 y^{4} + z} + \sqrt{8 z^{4} + x} - 3}{x + y + z}
$$
|
2
| 0.625 |
\( 3 \cdot 4^{x} + \frac{1}{3} \cdot 9^{x+2} = 6 \cdot 4^{x+1} - \frac{1}{2} \cdot 9^{x+1} \).
|
x = -\frac{1}{2}
| 0.375 |
Given the ellipse \( C: \frac{x^{2}}{3} + y^{2} = 1 \) with the upper vertex as \( A \), a line \( l \) that does not pass through \( A \) intersects the ellipse \( C \) at points \( P \) and \( Q \). Additionally, \( A P \perp A Q \). Find the maximum area of triangle \( \triangle A P Q \).
|
\frac{9}{4}
| 0.875 |
Point \( D \) lies on the extension of side \( AC \) of triangle \( ABC \), whose area is \( S \); point \( A \) is between \( D \) and \( C \). Let \( O \) be the centroid of triangle \( ABC \). It is known that the area of triangle \( DOC \) is \( S_l \). Express the area of triangle \( DOB \) in terms of \( S \) and \( S_l \).
|
2S_l - \frac{S}{3}
| 0.5 |
On a plane, there are \( N \) points. We mark the midpoints of all possible line segments with endpoints at these points. What is the minimum number of marked points that can be obtained?
|
2N-3
| 0.5 |
Initially, the screen of a computer displays a prime number. Each second, the number on the screen is replaced by a new number obtained by adding the last digit of the previous number increased by 1. After how much time, at maximum, will a composite number appear on the screen?
|
5 \text{ seconds}
| 0.25 |
Given prime numbers \( p \) and \( q \) such that \( p^{2} + 3pq + q^{2} \) is a perfect square, what is the maximum possible value of \( p+q \)?
|
10
| 0.75 |
Calculate the limit of the function:
$\lim _{x \rightarrow 0}\left(1-\sin ^{2} \frac{x}{2}\right)^{\frac{1}{\ln \left(1+\operatorname{tg}^{2} 3 x\right)}}$
|
e^{-\frac{1}{36}}
| 0.625 |
In triangle \( \triangle ABC \), \( AB = AC \) and \( \angle A = 100^\circ \). \( I \) is the incenter, and \( D \) is a point on \( AB \) such that \( BD = BI \). Find the measure of \( \angle BCD \).
|
30^\circ
| 0.875 |
On an island, there are 100 knights and 100 liars. Each of them has at least one friend. One day, exactly 100 people said: "All my friends are knights," and exactly 100 people said: "All my friends are liars." What is the smallest possible number of pairs of friends where one is a knight and the other is a liar?
|
50
| 0.375 |
As shown in the figure, \(ABCD\) is a rectangle and \(AEFG\) is a square. If \(AB = 6\), \(AD = 4\), and the area of \(\triangle ADE\) is 2, find the area of \(\triangle ABG\).
|
3
| 0.75 |
I have chosen five of the numbers $\{1,2,3,4,5,6,7\}. If I told you what their product was, that would not be enough information for you to figure out whether their sum was even or odd. What is their product?
|
420
| 0.375 |
The sequence $\left\{a_{n}\right\}$ is defined such that $a_{n}$ is the last digit of the sum $1 + 2 + \cdots + n$. Let $S_{n}$ be the sum of the first $n$ terms of the sequence $\left\{a_{n}\right\}$. Calculate $S_{2016}$.
|
7066
| 0.25 |
Let \( N \) be a given natural number. Express \( N \) as the sum of \( k \) natural numbers \( x_{1}, x_{2}, \cdots, x_{k} \). If \( N = kt + r \) (where \( t \) and \( r \) are non-negative integers), \( 0 \leq r < k \) and \( 1 < k \leq N \), then the maximum value of the product \( x_{1} x_{2} \cdots x_{k} \) is \( t^{k-r} (t+1)^{r} \).
|
t^{k-r}(t+1)^r
| 0.625 |
You are the owner of a company that sells Tesla Model S electric cars. The purchase price of one car is 2.5 million rubles. To bring one car across the border, you need to pay customs duties and fees amounting to 2 million rubles per car. The monthly office rent is 50,000 rubles, the total monthly salary of employees is 370,000 rubles, and other monthly expenses, including car import, office rental for placement and storage of imported cars, are fixed and amount to 180,000 rubles. Your store received an order for 30 Tesla Model S cars.
a) Determine the minimum possible price per car (in million rubles) at which you are willing to accept this order. Assume that all operations related to fulfilling this order can be completed within one month and no other car orders will be received this month.
b) A company has appeared in the city selling these cars for 5.3 million rubles, providing a charging station worth 400,000 rubles for free with each purchased car. What is the minimum order volume at which you could compete with this company if charging stations can be freely purchased on the competitive market?
|
2
| 0.625 |
In triangle \(ABC\) with \(BC = 4\) and \(AB = 2\sqrt{19}\), it is known that the center of the circle passing through the midpoints of the sides of the triangle lies on the bisector of angle \(C\). Find \(AC\).
|
10
| 0.375 |
A four-digit number $a b c d$ is called balanced if
$$
a+b=c+d
$$
Calculate the following quantities:
a) How many numbers $a b c d$ are there such that $a+b=c+d=8$?
b) How many numbers $a b c d$ are there such that $a+b=c+d=16$?
c) How many balanced numbers are there?
|
615
| 0.625 |
On the board, all natural numbers from 3 to 223 inclusive that leave a remainder of 3 when divided by 4 are written. Every minute, Borya erases any two of the written numbers and instead writes their sum, reduced by 2. In the end, only one number remains on the board. What could this number be?
|
6218
| 0.875 |
A triangle has sides of length 888, 925, and \( x > 0 \). Find the value of \( x \) that minimizes the area of the circle circumscribed about the triangle.
|
259
| 0.25 |
Construct a square such that two adjacent vertices lie on a circle with a unit radius, and the side connecting the other two vertices is tangent to the circle. Calculate the sides of the square!
|
\frac{8}{5}
| 0.125 |
What digit should be placed instead of the question mark in the number 888...88?99...999 (with the eights and nines written 50 times each) so that it is divisible by 7?
|
5
| 0.5 |
The set \( A = \left\{ x \left\lvert\, x = \left\lfloor \frac{5k}{6} \right\rfloor \right., k \in \mathbb{Z}, 100 \leqslant k \leqslant 999 \right\} \), where \(\left\lfloor x \right\rfloor\) denotes the greatest integer less than or equal to \( x \). Determine the number of elements in set \( A \).
|
750
| 0.25 |
On the board, the number 27 is written. Every minute, the number is erased from the board and replaced with the product of its digits increased by 12. For example, after one minute, the number on the board will be $2 \cdot 7 + 12 = 26$. What number will be on the board after an hour?
|
14
| 0.25 |
In the right triangle \(ABC\) as shown in the figure, the length of \(AB\) is 12 cm, and the length of \(AC\) is 24 cm. Points \(D\) and \(E\) are on \(AC\) and \(BC\), respectively. Determine the area of the isosceles right triangle \(BDE\) in square centimeters.
|
80
| 0.25 |
The numbers \( x \) and \( y \) are such that the equations \(\operatorname{tg} x + \operatorname{tg} y = 4\) and \(3 \sin (2x + 2y) = \sin 2x \sin 2y\) are satisfied. Find \(\operatorname{ctg} x \operatorname{ctg} y\).
|
\frac{7}{6}
| 0.75 |
Determine all polynomials \( p(x) \) with real coefficients such that
\[ p\left((x+1)^{3}\right) = (p(x) + 1)^{3} \]
and
\[ p(0) = 0. \]
|
p(x) = x
| 0.875 |
For any integer $x$, the function $f(x)$ satisfies $f(x+1)=\frac{1+f(x)}{1-f(x)}$. If $f(1)=2$, then find the value of $f(1993)$.
|
2
| 0.625 |
The rules of the game are as follows: from 64 different items, on each turn, a player needs to form a set of items that has not been mentioned in the game before, in which the number of items equals the player's age in years. Players take turns making moves; any player can start the game. The player who cannot make a move loses. Sets of items are considered different if they differ by at least one item or if they contain a different number of items. The game involves Vasily and Fyodor; each player has the opportunity to make at least one move. It is known that: a) Vasily is 2 years older than Fyodor; b) Fyodor is at least 5 years old; c) Fyodor always wins. What is the minimum age of Vasily?
|
34
| 0.375 |
Let \( S = \{1, 2, 3, \ldots, 500\} \). Select 4 different numbers from \( S \), and arrange them in ascending order to form a geometric sequence with a positive integer common ratio. Find the number of such geometric sequences.
|
94
| 0.625 |
Let the set \( M = \{1, 2, \cdots, 1000\} \). For any non-empty subset \( X \) of \( M \), let \( a_X \) denote the sum of the maximum and minimum numbers in \( X \). What is the arithmetic mean of all such values \( a_X \)?
|
1001
| 0.75 |
How many different four-digit numbers, divisible by 4, can be made from the digits 1, 2, 3, and 4,
a) if each digit can be used only once?
b) if each digit can be used multiple times?
|
64
| 0.125 |
Given the equation \(x^{5}-x^{2}+5=0\) with five roots \(x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\), and \(f(x)=x^{2}+1\), compute \(\prod_{k=1}^{5} f\left(x_{k}\right)\).
|
37
| 0.5 |
Esquecinaldo has a poor memory for storing numbers but an excellent memory for remembering sequences of operations. To remember his five-digit bank code, he can recall that:
1. The code has no repeated digits.
2. None of the digits are zero.
3. The first two digits form a power of 5.
4. The last two digits form a power of 2.
5. The middle digit is a multiple of 3.
6. The sum of all the digits is an odd number.
Now he doesn't need to memorize the number because he knows his code is the largest number that satisfies these conditions. What is this code?
|
25916
| 0.875 |
If three numbers $a_{1}, a_{2}, a_{3}$ are chosen from the numbers $1,2, \cdots, 14$ in increasing order such that they satisfy the conditions $a_{2} - a_{1} \geqslant 3$ and $a_{3} - a_{2} \geqslant 3$, how many different ways are there to choose such numbers?
|
120
| 0.375 |
Consider the quadratic polynomial \( P(x) = ax^2 + bx + c \) with distinct positive roots. Vasya wrote four numbers on the board: the roots of \( P(x) \) and the roots of another polynomial \( Q(x) = cx^2 + bx + a \) multiplied by 4. What is the smallest integer value that the sum of the written numbers can have?
|
9
| 0.375 |
How many functions \( f : \{1,2, \ldots, 2013\} \rightarrow \{1,2, \ldots, 2013\} \) satisfy \( f(j) < f(i) + j - i \) for all integers \( i, j \) such that \( 1 \leq i < j \leq 2013 \) ?
|
\binom{4025}{2013}
| 0.625 |
Alice and Bob are playing a game with dice. They each roll a die six times and take the sums of the outcomes of their own rolls. The player with the higher sum wins. If both players have the same sum, then nobody wins. Alice's first three rolls are 6, 5, and 6, while Bob's first three rolls are 2, 1, and 3. The probability that Bob wins can be written as a fraction \( \frac{a}{b} \) in lowest terms. What is \( a + b \)?
|
3895
| 0.875 |
Given that \( x, y, z \) are real numbers not all equal to 0, find the maximum value of \( f(x, y, z) = \frac{xy + 2yz}{x^2 + y^2 + z^2} \).
|
\frac{\sqrt{5}}{2}
| 0.75 |
Given $0 \leq a_k \leq 1$ for $k=1,2,\ldots,2020$, and defining $a_{2021}=a_1, a_{2022}=a_2$, find the maximum value of $\sum_{k=1}^{2020}\left(a_{k}-a_{k+1} a_{k+2}\right)$.
|
1010
| 0.625 |
a) Given four distinct real numbers \( a_{1} < a_{2} < a_{3} < a_{4} \), arrange these numbers in such an order \( a_{i_{1}}, a_{i_{2}}, a_{i_{3}}, a_{i_{4}} \) (where \( i_{1}, i_{2}, i_{3}, i_{4} \) are the numbers 1, 2, 3, 4 in some permutation) so that the sum
\[
\Phi = (a_{i_{1}} - a_{i_{2}})^{2} + (a_{i_{2}} - a_{i_{3}})^{2} + (a_{i_{3}} - a_{i_{4}})^{2} + (a_{i_{4}} - a_{i_{1}})^{2}
\]
is as small as possible.
b) Given \( n \) distinct real numbers \( a_{1}, a_{2}, a_{3}, \ldots, a_{n} \), arrange these numbers in such an order \( a_{i_{1}}, a_{i_{2}}, a_{i_{3}}, \ldots, a_{i_{n}} \) so that the sum
\[
\Phi = (a_{i_{1}} - a_{i_{2}})^{2} + (a_{i_{2}} - a_{i_{3}})^{2} + \ldots + (a_{i_{n-1}} - a_{i_{n}})^{2} + (a_{i_{n}} - a_{i_{1}})^{2}
\]
is as small as possible.
|
a_1, a_2, a_4, a_3
| 0.25 |
Calculate the definite integral:
$$
\int_{\pi / 2}^{\pi} 2^{8} \cdot \sin ^{2} x \cos ^{6} x \, dx
$$
|
5 \pi
| 0.125 |
A large rectangle consists of three identical squares and three identical small rectangles. The perimeter of a square is 24, and the perimeter of a small rectangle is 16. What is the perimeter of the large rectangle?
The perimeter of a figure is the sum of the lengths of all its sides.
|
52
| 0.625 |
There are $12k$ people attending a meeting, and each person has shaken hands with exactly $3k + 6$ other people. Additionally, for any two individuals, the number of people who have shaken hands with both of them is the same. How many people are attending the meeting? Provide a proof of your conclusion.
|
36
| 0.625 |
Find the number of 5-digit numbers where the ten-thousands place is not 5, the units place is not 2, and all digits are distinct.
|
21840
| 0.25 |
A high-rise building has 7 elevators, but each one only stops on 6 floors. Nevertheless, there is always one elevator for every two floors, which connects the two floors directly.
Show that the skyscraper can have a maximum of 14 storeys and that such a skyscraper with 14 storeys is actually feasible.
|
14
| 0.25 |
Simplify the expression \( \sin (2 x-\pi) \cos (x-3 \pi)+\sin \left(2 x-\frac{9 \pi}{2}\right) \cos \left(x+\frac{\pi}{2}\right) \).
|
\sin(3x)
| 0.125 |
Let \( x \) be a positive real number. What is the maximum value of \( \frac{2022 x^{2} \log (x + 2022)}{(\log (x + 2022))^{3} + 2 x^{3}} \)?
|
674
| 0.75 |
In the product $2 \times 3 \times 5 \times 7 \times 11 \times 13 \times 17$, what is the sum of all the digits in the resulting number?
|
12
| 0.75 |
There are 36 criminal gangs operating in Chicago, some of which are hostile toward each other. Each gangster belongs to several gangs, with no two gangsters belonging to the same set of gangs. It is known that no gangster belongs to two gangs that are hostile toward each other. Additionally, it turns out that each gang, which a particular gangster does not belong to, is hostile to at least one gang that the gangster does belong to. What is the maximum number of gangsters that can be in Chicago?
|
531441
| 0.5 |
A problem-solving robot with an initial IQ balance of 25 solves a set of questions with point values ranging from $1$ to $10$. Each time it correctly solves a problem, the robot’s IQ balance decreases by the point value of the problem and then increases by 1. If the robot's IQ balance becomes less than the point value of a problem, it fails to solve that problem. What is the maximum score the robot can achieve?
|
31
| 0.375 |
A car was traveling at a speed of \( V \). Upon entering a city, the driver reduced the speed by \( x \% \), and upon leaving the city, increased it by \( 0.5 x \% \). It turned out that this new speed is \( 0.6 x \% \) less than the speed \( V \). Find the value of \( x \).
|
20
| 0.875 |
Given that \(a, b \in \mathbb{R}\), and the equation \(x^{4} + a x^{3} + 2 x^{2} + b x + 1 = 0\) has a real root, find the minimum possible value of \(a^2 + b^2\).
|
8
| 0.75 |
Let \( C \) be the circle with radius 1 and center at the origin. A point \( P \) is chosen at random on the circumference of \( C \), and another point \( Q \) is chosen at random in the interior of \( C \). What is the probability that the rectangle with diagonal \( PQ \), and sides parallel to the x-axis and y-axis, lies entirely inside (or on) \( C \)?
|
\frac{4}{\pi^2}
| 0.625 |
Find the number of different ways to arrange all natural numbers from 1 to 9 inclusively, one in each cell of a 3x3 grid, such that the sums of the numbers in each row and each column are equal. The table cannot be rotated or reflected.
|
72
| 0.125 |
Plot on the (x, y) plane the set of points whose coordinates satisfy the system of inequalities:
\[
\left\{
\begin{array}{l}
(|x|-x)^{2}+(|y|-y)^{2} \leq 16, \\
2y + x \leq 0
\end{array}
\right.
\]
and find the area of the resulting figure.
|
5 + \pi
| 0.5 |
A gardener wants to plant three maple trees, four oak trees, and five birch trees in a row. He randomly determines the arrangement of these trees, and each possible arrangement is equally likely. Let the probability that no two birch trees are adjacent be represented by \(\frac{m}{n}\) in simplest form. Find \(m+n\).
|
106
| 0.875 |
The sum of sides \( AB \) and \( BC \) of triangle \( ABC \) is 11, angle \( B \) is \( 60^\circ \), and the radius of the inscribed circle is \(\frac{2}{\sqrt{3}}\). It is also known that side \( AB \) is longer than side \( BC \). Find the height of the triangle dropped from vertex \( A \).
|
4\sqrt{3}
| 0.75 |
In the figure, \( L_{1} \) and \( L_{2} \) are tangents to the three circles. If the radius of the largest circle is 18 and the radius of the smallest circle is \( 4b \), find \( c \), where \( c \) is the radius of circle \( W \).
|
12
| 0.875 |
The number of eggs in a basket was \( a \). Eggs were given out in three rounds. In the first round, half of the eggs plus half an egg were given out. In the second round, half of the remaining eggs plus half an egg were given out. In the third round, again, half of the remaining eggs plus half an egg were given out. The basket then became empty. Find \( a \).
|
7
| 0.875 |
There are four points \( A, B, C, D \) on the surface of a sphere with radius \( R \), such that \( AB = BC = CA = 3 \). If the maximum volume of the tetrahedron \( ABCD \) is \( \frac{9 \sqrt{3}}{4} \), what is the surface area of the sphere?
|
16\pi
| 0.625 |
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 \(AFCE\).
|
297
| 0.25 |
The Planar National Park is an undirected 3-regular planar graph (i.e., all vertices have degree 3). A visitor walks through the park as follows: she begins at a vertex and starts walking along an edge. When she reaches the other endpoint, she turns left. On the next vertex, she turns right, and so on, alternating left and right turns at each vertex. She does this until she gets back to the vertex where she started. What is the largest possible number of times she could have entered any vertex during her walk, over all possible layouts of the park?
|
3
| 0.25 |
What is the maximum number of kings that can be placed on a chessboard such that no two kings can attack each other?
|
16
| 0.125 |
In an \( n \times n \) checkerboard, the rows are numbered 1 to \( n \) from top to bottom, and the columns are numbered 1 to \( n \) from left to right. Chips are to be placed on this board so that each square has a number of chips equal to the absolute value of the difference of the row and column numbers. If the total number of chips placed on the board is 2660, find \( n \).
|
20
| 0.625 |
Find the area of a trapezoid with bases 4 and 7 and side lengths 4 and 5.
|
22
| 0.625 |
In an equilateral triangle \(ABC\) with side length \(a\), points \(M, N, P, Q\) are located as shown in the figure. It is known that \(MA + AN = PC + CQ = a\). Find the measure of angle \(NOQ\).
|
60^\circ
| 0.375 |
On the side AB of square ABCD, an equilateral triangle AKB is constructed (externally). Find the radius of the circumscribed circle around triangle CKD if $\mathrm{AB}=1$.
|
1
| 0.875 |
Given an isosceles trapezoid \(ABCD\) with \(AD = 10\), \(BC = 2\), and \(AB = CD = 5\). The angle bisector of \(\angle BAD\) intersects the extension of the base \(BC\) at point \(K\). Find the angle bisector of \(\angle ABK\) in triangle \(ABK\).
|
\frac{\sqrt{10}}{2}
| 0.875 |
Hooligan Vasya likes to run on the escalator in the metro. He runs downward twice as fast as upward. If the escalator is not working, it will take Vasya 6 minutes to run up and down. If the escalator is moving downward, it will take Vasya 13.5 minutes to run up and down. How many seconds will it take Vasya to run up and down the escalator if it is moving upward? (The escalator always moves at a constant speed.)
|
324
| 0.75 |
What is the coefficient of \(x^{4}\) in the product
$$
\left(1-2x+4x^{2}-8x^{3}+16x^{4}\right) \cdot \left(1+2x+4x^{2}+8x^{3}+16x^{4}\right)?
$$
|
16
| 0.625 |
In a round-robin tournament, 23 teams participated. Each team played exactly one match with every other team. We say that 3 teams form a "cycle of victories" if, considering only the matches between them, each team won exactly once. What is the maximum number of such cycles that could have occurred during the tournament?
|
506
| 0.125 |
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