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.


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?

Solution: \boxed{n = 4}


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 \( x + y \).


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?


What is the smallest two-digit positive integer \( k \) for which the product \( 45k \) is a perfect square?


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 \(AMN\).


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
$$
\sum_{k=1}^{n}\left(x_{k}+y_{k}\right)=
$$


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?


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 \( AC = 100 \).


How many pairs of positive integers \((m, n)\) are there such that
\[ 7m + 3n = 10^{2004} \]
and \( m \mid n \)?


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 \(ABC\).


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 $S_{m+1} = 4S_{m}$?

Solution: \boxed{\text{No}}


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?


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 $\mathrm{f}(\mathrm{x})=\left|\mathrm{x} \overrightarrow{\mathrm{a}}+\frac{1}{\mathrm{x}} \overrightarrow{\mathrm{b}}\right|$ $(x \neq 0)$.


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 \( AB \).


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.


A six-digit number 111aaa is the product of two consecutive positive integers \( b \) and \( b+1 \). Find the value of \( b \).


For \( x, y, z \in (0,1] \), find the maximum value of the expression

$$
A = \frac{\sqrt{8 x^{4} + y} + \sqrt{8 y^{4} + z} + \sqrt{8 z^{4} + x} - 3}{x + y + z}
$$


Solution: \boxed{x = -\frac{1}{2}}


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 \( \triangle A P Q \).


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 \( S_l \).

Solution: \boxed{2S_l - \frac{S}{3}}


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?


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?

Solution: \boxed{5 \text{ seconds}}


Given prime numbers \( p \) and \( q \) such that \( p^{2} + 3pq + q^{2} \) is a perfect square, what is the maximum possible value of \( p+q \)?


Calculate the limit of the function:

$\lim _{x \rightarrow 0}\left(1-\sin ^{2} \frac{x}{2}\right)^{\frac{1}{\ln \left(1+\operatorname{tg}^{2} 3 x\right)}}$


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 \( \angle BCD \).


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?


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 \(\triangle ABG\).


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?


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 $S_{2016}$.


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 \( t^{k-r} (t+1)^{r} \).


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?


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 \(AC\).


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?


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?


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.


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!


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?


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 \( A \).


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?


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.


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 \(\operatorname{ctg} x \operatorname{ctg} y\).


Determine all polynomials \( p(x) \) with real coefficients such that

\[ p\left((x+1)^{3}\right) = (p(x) + 1)^{3} \]

and

\[ p(0) = 0. \]

Solution: \boxed{p(x) = x}


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 $f(1993)$.


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?


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.


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 \( a_X \)?


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?


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?


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?


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?


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 \( 1 \leq i < j \leq 2013 \) ?


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 \( a + b \)?


Given that \( x, y, z \) are real numbers not all equal to 0, find the maximum value of \( f(x, y, z) = \frac{xy + 2yz}{x^2 + y^2 + z^2} \).


Solution: \boxed{a_1, a_2, a_4, a_3}


Calculate the definite integral:

$$
\int_{\pi / 2}^{\pi} 2^{8} \cdot \sin ^{2} x \cos ^{6} x \, dx
$$

Solution: \boxed{5 \pi}


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.


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.


Divide a cube with edge length of $3 \mathrm{~cm}$ into $n$ smaller cubes, where the smaller cubes do not all have the same volume and have edge lengths that are whole centimeters. Determine the value of $n$:
(A) 4
(B) 8
(C) 12
(D) 16
(E) 20

(From the 7th Annual American Mathematics Contest, 1991)


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.


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.


Simplify the expression \( \sin (2 x-\pi) \cos (x-3 \pi)+\sin \left(2 x-\frac{9 \pi}{2}\right) \cos \left(x+\frac{\pi}{2}\right) \).


Let \( x \) be a positive real number. What is the maximum value of \( \frac{2022 x^{2} \log (x + 2022)}{(\log (x + 2022))^{3} + 2 x^{3}} \)?


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?


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?


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?


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 \( x \).


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 \(a^2 + b^2\).


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) \( C \)?


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.


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.

Solution: \boxed{5 + \pi}


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 \(m+n\).


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 \( A \).


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 \( W \).


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 \( a \).


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?


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 \(AFCE\).


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?


What is the maximum number of kings that can be placed on a chessboard such that no two kings can attack each other?


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 \( n \).


Find the area of a trapezoid with bases 4 and 7 and side lengths 4 and 5.


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 \(NOQ\).


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 $\mathrm{AB}=1$.


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 \(ABK\).


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.)


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?


Given sets \( M = \left\{ x \mid x^{2} - 9 > 0 \right\} \) and \( N = \left\{ x \in \mathbf{Z} \mid x^{2} - 8x + a < 0 \right\} \). If the number of subsets of \( M \cap N \) is 4, then the range of the real number \( a \) is \(\qquad\).

Solution: \boxed{[12, 15)}


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 \( \angle BCD \).


Five friends, one of whom had a monkey, bought a bag of nuts, which they planned to divide amongst themselves the next morning. However, during the night, one friend woke up and wanted some nuts. He divided all the nuts in the bag into five equal parts, with one nut left over, which he gave to the monkey, and took his fifth part. Following him, another friend woke up; unaware that someone had already taken nuts, he divided the remaining nuts in the bag again into five parts, with one nut left over, which he gave to the monkey, and took his fifth part. Subsequently, the remaining three friends woke up one after another, each performing the same operation: dividing the remaining nuts into five parts, taking their fifth part, with one nut left over each time, which was given to the monkey. Finally, in the morning, all five friends took out the bag, divided the remaining nuts into five parts, and the one nut left over was once again given to the monkey. Determine the smallest number of nuts in the bag such that this division is possible.


Let the vertices of a regular 10-gon be \( A_1, A_2, \ldots, A_{10} \) in that order. Show that the length of the segment \( A_1A_4 - A_1A_2 \) is equal to the radius of the circumcircle.

Solution: \boxed{A_1A_4 - A_1A_2 = R}


\( f : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) (where \( \mathbb{R} \) is the real line) is defined by \( f(x, y) = \left( -\frac{y}{x^2 + 4y^2}, \frac{x}{x^2 + 4y^2}, 0 \right) \). Can we find \( F : \mathbb{R}^3 \rightarrow \mathbb{R}^3 \), such that:
1. If \( F = (F_1, F_2, F_3) \), then \( F_i \) all have continuous partial derivatives for all \( (x, y, z) \neq (0, 0, 0) \);
2. \( \nabla \times F = 0 \) for all \( (x, y, z) \neq 0 \);
3. \( F(x, y, 0) = f(x, y) \)?

Solution: \boxed{\text{No}}


Points \( A \) and \( B \) lie on a circle with center \( O \) and radius 6, and point \( C \) is equidistant from points \( A, B, \) and \( O \). Another circle with center \( Q \) and radius 8 is circumscribed around triangle \( A C O \). Find \( B Q \).


The diagonals of a trapezoid are perpendicular to each other, its height is 4 units, and one of its diagonals is 5 units. What is the area of the trapezoid?


Given a polynomial \( P(x) \) with integer coefficients and known values \( P(2) = 3 \) and \( P(3) = 2 \), what is the maximum number of integer solutions that the equation \( P(x) = x \) can have?


If \( c \) boys were all born in June 1990 and the probability that their birthdays are all different is \( \frac{d}{225} \), find \( d \).


14. \( l \) is a tangent line to the hyperbola \( a: \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \quad (a > 0, b > 0) \) at the point \( T \). The line \( l \) intersects the asymptotes of hyperbola \( a \) at points \( A \) and \( B \). The midpoint of the line segment \( AB \) is \( M(2,1) \).

An ellipse \(\frac{x^2}{a_0^2} + \frac{y^2}{b_0^2} = 1 \quad \left(a_0 > 0, b_0 > 0\right)\) has points \( C(0, b_0) \) and \( D(a_0, 0) \), with \( O \) and \( T \) located on opposite sides of the line segment \( CD \). The internal and external angle bisectors of \(\angle TCD\) intersect the \( x \)-axis at points \( E \) and \( F \), respectively. The area of triangle \( \triangle TEF \) is denoted by \( S \).

Find:
(1) The coordinates of point \( T \).
(2) The locus of the midpoint of line segment \( CD \) that minimizes \( S \).


A line of soldiers 1 mile long is jogging. The drill sergeant, in a car, moving at twice their speed, repeatedly drives from the back of the line to the front of the line and back again. When each soldier has marched 15 miles, how much mileage has been added to the car, to the nearest mile?


We say that a number is ascending if each of its digits is greater than the digit to its left. For example, 2568 is ascending, and 175 is not. How many ascending numbers are there between 400 and 600?


Given that the sum of $n$ positive integers $x_{1}, x_{2}, \cdots, x_{n}$ is 2016. If these $n$ numbers can be divided into 32 groups with equal sums as well as into 63 groups with equal sums, find the minimum value of $n$.


You are trapped in a room with only one exit, a long hallway with a series of doors and land mines. To get out, you must open all the doors and disarm all the mines. In the room is a panel with 3 buttons, which contains an instruction manual. The red button arms a mine, the yellow button disarms two mines and closes a door, and the green button opens two doors. Initially, 3 doors are closed, and 3 mines are armed. The manual warns that attempting to disarm two mines or open two doors when only one is armed/closed will reset the system to its initial state. What is the minimum number of buttons you must push to get out?


How many positive numbers are there among the 2014 terms of the sequence: \( \sin 1^{\circ}, \sin 10^{\circ}, \sin 100^{\circ}, \sin 1000^{\circ}, \ldots ? \)


Find the intersection point of the line and the plane.

$$
\begin{aligned}
& \frac{x-1}{-1}=\frac{y+5}{4}=\frac{z-1}{2} \\
& x-3y+7z-24=0
\end{aligned}
$$

Solution: \boxed{(0, -1, 3)}


Given \( z_{1}=x+\sqrt{5}+y i \) and \( z_{2}=x-\sqrt{5}+y i \), where \( x, y \in \mathbf{R} \) and \(\left|z_{1}\right|+\left|z_{2}\right|=6 \), find the product of the maximum and minimum values of \( f(x, y)=|2 x-3 y-12| \).


In a plane, 4 points \( A_{1}, A_{2}, A_{3}, A_{4} \) are given, and the distance between any two of them is at least 1. What is the maximum possible number of line segments \( A_{i}A_{j} \) with a length of 1 that can be drawn between these points?


50 students with blond hair, brown hair, and red hair are sitting around a round table. It is known that in any group of students sitting consecutively, there is at least one student with brown hair between any two students with blond hair, and at least one student with red hair between any two students with brown hair. What is the minimum number of students with red hair that can be sitting at this table?


Petya's favorite TV game is called "Lottery on the Couch." During the game, viewers can send SMS messages with three-digit numbers containing only the digits 1, 2, 3, and 4. At the end of the game, the host announces a three-digit number, also consisting only of these digits. An SMS is considered a winning one if the number in it differs from the host's number by no more than one digit (for example, if the host announces the number 423, then messages 443 and 123 are winning, but 243 and 224 are not).

Petya wants to send as few messages as possible so that at least one is guaranteed to be winning. How many SMS messages does he need to send?


In a circle \(\omega\) centered at \(O\), \(AA'\) and \(BB'\) are diameters perpendicular to each other such that the points \(A, B, A', B'\) are arranged in an anticlockwise direction in this order. Let \(P\) be a point on the minor arc \(A'B'\) such that \(AP\) intersects \(BB'\) at \(D\) and \(BP\) intersects \(AA'\) at \(C\). Suppose the area of the quadrilateral \(ABCD\) is 100. Find the radius of \(\omega\).


Let \( n \in \mathbb{N}^{*} \). Show that for all \( a, b > 0 \), we have:

\[
\left(1+\frac{a}{b}\right)^{n}+\left(1+\frac{b}{a}\right)^{n} \geqslant 2^{n+1}
\]

When does equality hold?

Solution: \boxed{a=b}


Compute the definite integral:

$$
\int_{0}^{2 \pi} \sin ^{6}\left(\frac{x}{4}\right) \cos ^{2}\left(\frac{x}{4}\right) d x
$$


It is known that the lengths of the sides of a triangle are consecutive natural numbers, and the radius of its inscribed circle is 4. Find the radius of the circumcircle of this triangle.

Solution: \boxed{ \frac{65}{8}}


In triangle \(ABC\), point \(N\) is taken on side \(AB\), and point \(M\) is taken on side \(AC\). Segments \(CN\) and \(BM\) intersect at point \(O\). The ratio \(AN:NB = 2:3\) and \(BO:OM = 5:2\). Find the ratio \(CO:ON\).


A team of five people is laying a 30-meter pipe into the ground. To prepare the trench, they decided to dig in turns with the condition that each person digs for the amount of time it takes the other four together to dig one-third of the trench. The 30-meter trench was completed when the last team member had worked their allotted time. How many meters of the trench could they have dug in the same time if they all worked together?

Solution: \boxed{80 \text{m}}


Given a segment \( AB \) of fixed length 3 with endpoints moving on the parabola \( y^2 = x \), find the shortest distance from the midpoint \( M \) of segment \( AB \) to the y-axis.


Calculate the lengths of the arcs of the curves given by the equations in polar coordinates.

$$
\rho=8 \cos \varphi, 0 \leq \varphi \leq \frac{\pi}{4}
$$


The sports event lasted for \( n \) days; \( N \) sets of medals were awarded; on the 1st day, 1 set of medals and \( \frac{1}{7} \) of the remaining sets were awarded; on the 2nd day, 2 sets of medals and \( \frac{1}{7} \) of the remaining sets were awarded; on the next to last, \( (n-1) \)th day, \( (n-1) \) sets of medals and \( \frac{1}{7} \) of all remaining sets were awarded; finally, on the last day, the remaining \( n \) sets of medals were awarded. How many days did the sports event last and how many sets of medals were awarded?


The minimum positive period of the function \( f(x) = |\sin(2x) + \sin(3x) + \sin(4x)| \) is equal to...

Solution: \boxed{2 \pi}


Let \( p = a^b + b^a \). If \( a \), \( b \), and \( p \) are all prime, what is the value of \( p \)?


Find the area of the figure defined on the coordinate plane by the system

$$
\left\{\begin{array}{l}
\sqrt{1-x}+2 x \geqslant 0 \\
-1-x^{2} \leqslant y \leq 2+\sqrt{x}
\end{array}\right.
$$


To reach the Solovyov family's dacha from the station, one must first travel 3 km on the highway and then 2 km on a path. Upon arriving at the station, the mother called her son Vasya at the dacha and asked him to meet her on his bicycle. They started moving towards each other at the same time. The mother walks at a constant speed of 4 km/h, while Vasya rides at a speed of 20 km/h on the path and 22 km/h on the highway. At what distance from the station did Vasya meet his mother? Give the answer in meters.

Solution: \boxed{800 \text{ meters}}


In a tournament, each participant plays a match against every other participant. The winner of a match earns 1 point, the loser 0 points, and if the match is a draw, both players earn half a point. At the end of the tournament, the participants are ranked according to their score (in the case of a tie, the order is arbitrary). It is noted that each participant won half of their points against the last ten ranked players. How many people participated in the tournament?


The figure $ABCDE$ is a star pentagon (each line segment of the pentagon has 2 vertices on one side and 1 vertex on the other side). What is the sum of the angles $A B C, B C D, C D E, D E A, E A B$?


What are the last three digits of \(2003^N\), where \(N = 2002^{2001}\)?


Consider the equation

$$
(x-1)(x-2) \ldots(x-2016) = (x-1)(x-2) \ldots(x-2016)
$$

written on a board. What is the smallest integer $k$ such that we can erase $k$ factors among these 4032 factors so that there is at least one factor remaining on each side of the equation and the equation no longer has any real solutions?


Let the real numbers \(a\) and \(b\) such that the equation \(a x^{3} - x^{2} + b x - 1 = 0\) has three positive real roots. For all real numbers \(a\) and \(b\) that meet the conditions, find the minimum value of \(P = \frac{5 a^{2} - 3 a b + 2}{a^{2}(b - a)}\).

Solution: \boxed{12 \sqrt{3}}


Different three-digit numbers are formed using the digits 1, 3, and 5, where all the digits in each number are distinct. Find the sum of all such three-digit numbers.


The side of a triangle is \(a\). Find the segment that connects the midpoints of the medians drawn to the two other sides.


Find all values of the parameters \(a, b, c\) for which the system of equations
\[
\left\{
\begin{array}{l}
a x + b y = c \\
b x + c y = a \\
c x + a y = b
\end{array}
\right\}
\]
has at least one negative solution (where \(x, y < 0\)).

Solution: \boxed{a + b + c = 0}


In a certain year, a specific date was never a Sunday in any month. Determine this date.


A digit is inserted between the digits of a two-digit number to form a three-digit number. Some two-digit numbers, when a certain digit is inserted in between, become three-digit numbers that are $k$ times the original two-digit number (where $k$ is a positive integer). What is the maximum value of $k$?


Given a tetrahedron \( S-ABC \), point \( A_1 \) is the centroid of \( \triangle SBC \). Point \( G \) is on the segment \( AA_1 \) such that \(\frac{|AG|}{|GA_1|}=3\). Line \( SG \) intersects the plane of \( \triangle ABC \) at point \( M \). Determine \(\frac{|A_1 M|}{|AS|} = \quad \).


This century will mark the 200th anniversary of the birth of the famous Russian mathematician Pafnuty Lvovich Chebyshev, a native of Kaluga province. The sum of the digits in the hundreds and thousands places of the year he was born is 3 times the sum of the digits in the units and tens places, and the digit in the tens place is greater than the digit in the units place. Determine the year of birth of P.L. Chebyshev, given that he was born and died in the same century and lived for 73 years.


Two players are playing a game. One of them thinks of a sequence of integers ($x_{1}, x_{2}, \ldots, x_{n}$), which are single-digit numbers, both positive and negative. The other player is allowed to ask for the value of the sum $a_{1} x_{1}+\ldots+a_{n} x_{n}$, where $(a_{1}, \ldots, a_{n})$ is any chosen set of coefficients. What is the minimum number of questions needed for the guesser to determine the thought-of sequence?


Arrange the numbers 1 to 9 in a row from left to right such that every three consecutive numbers form a three-digit number that is a multiple of 3. How many arrangements are there?


In a right triangle, the sine of the smaller angle is $\frac{1}{3}$. A line is drawn perpendicular to the hypotenuse, dividing the triangle into two equal areas. In what ratio does this line divide the hypotenuse?


Find the minimum value of the function \( f(x)=\cos 3x + 4 \cos 2x + 8 \cos x \) for \( x \in \mathbb{R} \).


Construct the cross-section of a triangular pyramid \( A B C D \) with a plane passing through the midpoints \( M \) and \( N \) of edges \( A C \) and \( B D \) and the point \( K \) on edge \( C D \), for which \( C K: K D = 1: 2 \). In what ratio does this plane divide edge \( A B \)?


A permutation of $\{1, 2, \ldots, n\}$ is chosen at random. What is the average number of fixed points?


Let \( A, B, C \) be distinct points on a circle \( \odot O \) such that \(\angle AOB = 120^\circ\). Point \( C \) lies on the minor arc \( \overparen{AB} \) (and \( C \) is not coincident with \( A \) or \( B \)). If \(\overrightarrow{OC} = \lambda \overrightarrow{OA} + \mu \overrightarrow{OB}\) (\( \lambda, \mu \in \mathbb{R} \)), what is the range of values for \( \lambda + \mu \)?

Solution: \boxed{(1, 2]}


Given the function \( f(x)=\{\begin{array}{ll}x+\frac{1}{2} & 0 \leqslant x \leqslant \frac{1}{2}, \\ 2(1-x) & \frac{1}{2}<x \leqslant 1,\end{array} \), define \( f_{n}(x)=\underbrace{f(f(\cdots f}_{n \uparrow 1}(x) \cdots)), n \in \mathbf{N}^{*} \). Find the value of \( f_{2006}\left(\frac{2}{15}\right) \).


Find the range of the function \( f(x) = g(g^2(x)) \), where \( g(x) = \frac{3}{x^2 - 4x + 5} \).

Solution: \boxed{[ \frac{3}{50}, 3]}


Laura has 2010 lights and 2010 switches, with each switch controlling a different light. She wants to determine which switch corresponds to which light. To achieve this, Charlie will operate the switches. Each time, Charlie presses some switches simultaneously, and the number of lit lights is the same as the number of switches pressed (note: after each operation, the switches return to their original state).

Questions:
(1) What is the maximum number of different operations Charlie can perform so that Laura can accurately determine the correspondence between the switches and lights?
(2) If Laura operates the switches herself, what is the minimum number of operations needed to determine the correspondence between the switches and lights?


A line passing through the origin with a positive slope intersects the ellipse \(\frac{x^{2}}{4} + y^{2} = 1\) at points \(E\) and \(F\). Given points \(A(2,0)\) and \(B(0,1)\), find the maximum area of the quadrilateral \(AEBF\).


In a city, there are 9 bus stops and several buses. Any two buses have at most one common stop. Each bus stops at exactly three stops. What is the maximum number of buses that can be in the city?


Solve the inequality in integers:

$$
\frac{1}{\sqrt{x-2y+z+1}} + \frac{2}{\sqrt{2x-y+3z-1}} + \frac{3}{\sqrt{3y-3x-4z+3}} > x^2 - 4x + 3
$$

Solution: \boxed{(3, 1, -1)}


The sequence \(101, 104, 116, \cdots\) is given by the general term \(a_{n} = 100 + n^{2}\), where \(n = 1, 2, 3, \cdots\). For each \(n\), let \(d_{n}\) denote the greatest common divisor of \(a_{n}\) and \(a_{n+1}\). Find the maximum value of \(d_{n}\) for all positive integers \(n\).


Cat food is sold in large and small packages (with more food in the large package than in the small one). One large package and four small packages are enough to feed a cat for exactly two weeks. Is one large package and three small packages necessarily enough to feed the cat for 11 days?

Solution: \boxed{Yes}


In front of an elevator are people weighing 130, 60, 61, 65, 68, 70, 79, 81, 83, 87, 90, 91, and 95 kg. The elevator has a capacity of 175 kg. What is the minimum number of trips the elevator must make so that all the people can be transported?


Calculate \(\sin (\alpha+\beta)\), given \(\sin \alpha+\cos \beta=\frac{1}{4}\) and \(\cos \alpha+\sin \beta=-\frac{8}{5}\).


Anya calls a date beautiful if all 6 digits in its recording are different. For example, 19.04.23 is a beautiful date, but 19.02.23 and 01.06.23 are not.

a) How many beautiful dates will there be in April 2023?

b) How many beautiful dates will there be in the entire year of 2023?


Consider the equation \( x^{2}-2 y^{2}=1 \), which we label (1). There are many pairs of positive integers \( (x, y) \) that satisfy equation (1).

(a) Determine a pair of positive integers \( (x, y) \) with \( x \leq 5 \) that satisfies equation (1).

(b) Determine a pair of positive integers \( (u, v) \) such that
\[ (3+2 \sqrt{2})^{2}=u+v \sqrt{2} \]
and show that \( (u, v) \) satisfies equation (1).

(c) Suppose that \( (a, b) \) is a pair of positive integers that satisfies equation (1). Suppose also that \( (c, d) \) is a pair of positive integers such that
\[ (a+b \sqrt{2})(3+2 \sqrt{2})=c+d \sqrt{2} \]
Show that \( (c, d) \) satisfies equation (1).

(d) Determine a pair of positive integers \( (x, y) \) with \( y>100 \) that satisfies equation (1).

Solution: \boxed{(577, 408)}


Find all real numbers \( a \) such that there exist non-negative real numbers \( x_1, x_2, x_3, x_4, x_5 \) satisfying
\[
\sum_{k=1}^{5} k x_k = a, \quad \sum_{k=1}^{5} k^3 \cdot x_k = a^2, \quad \sum_{k=1}^{5} k^5 x_k = a^3.
\]

Solution: \boxed{0, 1, 4, 9, 16, 25}


What is the smallest eight-digit positive integer that has exactly four digits which are 4?


The house number. A person mentioned that his friend's house is located on a long street (where the houses on the side of the street with his friend's house are numbered consecutively: $1, 2, 3, \ldots$), and that the sum of the house numbers from the beginning of the street to his friend's house matches the sum of the house numbers from his friend's house to the end of the street. It is also known that on the side of the street where his friend's house is located, there are more than 50 but fewer than 500 houses.

What is the house number where the storyteller's friend lives?


In the diagram, point \( P \) is inside quadrilateral \( ABCD \). Also, \( DA = DP = DC \) and \( AP = AB \). If \(\angle ADP = \angle CDP = 2x^\circ\), \(\angle BAP = (x+5)^\circ\), and \(\angle BPC = (10x-5)^\circ\), what is the value of \( x \)?


On a board, the numbers $1, 2, 3, \ldots, 235$ were written. Petya erased several of them. It turned out that among the remaining numbers, no number is divisible by the difference of any two others. What is the maximum number of numbers that could remain on the board?


The four-digit number $\overline{a b c d}$ is divisible by 3, and $a, b, c$ are permutations of three consecutive integers. How many such four-digit numbers are there?


In a right-angled triangle $ABC$ with the right angle at $C$, the angle bisector $BD$ and the altitude $CH$ are drawn. A perpendicular $CK$ is dropped from vertex $C$ to the angle bisector $BD$. Find the angle $HCK$ if $BK: KD = 3:1$.


If three numbers are chosen simultaneously from the integers $1,2,\dots,14$ in such a way that the absolute value of the difference between any two numbers is not less than 3, how many different ways can this be done?


In a group of 8 people, each person knows exactly 6 others. In how many ways can you choose four people such that every pair among the four knows each other? (Assume that if person A knows person B, then person B also knows person A, and people do not know themselves.)


Given the numbers \(\log _{\left(\frac{x}{2}-1\right)^{2}}\left(\frac{x}{2}-\frac{1}{4}\right)\), \(\log _{\sqrt{x-\frac{11}{4}}}\left(\frac{x}{2}-1\right)\), \(\log _{\frac{x}{2}-\frac{1}{4}}\left(x-\frac{11}{4}\right)^{2}\). For which \(x\) are two of these numbers equal, and the third one greater than them by 1?


Consider the line \( l: y = kx + m \) (where \( k \) and \( m \) are integers) which intersects the ellipse \( \frac{x^2}{16} + \frac{y^2}{12} = 1 \) at two distinct points \( A \) and \( B \), and the hyperbola \( \frac{x^2}{4} - \frac{y^2}{12} = 1 \) at two distinct points \( C \) and \( D \). Determine whether there exists a line \( l \) such that the vector \( \overrightarrow{AC} + \overrightarrow{BD} = 0 \). If such a line exists, how many such lines are there? If not, provide an explanation.


Let \( p \) be a prime number and \( q \) a prime divisor of the number \( 1 + p + \ldots + p^{p-1} \). Show that \( q \equiv 1 \pmod{p} \).

Solution: \boxed{q \equiv 1 \pmod{p}}


Find a number of the form \(2^{l} 3^{m}\) if the sum of all its divisors is 403.


A heap of stones has a total weight of 100 kg, where the weight of each stone does not exceed 2 kg. Choose some stones from this heap in any combination and find the difference between the sum of these chosen stones' weights and 10 kg. Denote by $d$ the minimum value of the absolute value of these differences. Find the maximum value of $d$ among all heaps of stones satisfying the above conditions.


Calculate the area of the figure bounded by the ellipse given by the parametric equations \(x=4 \cos t\) and \(y=3 \sin t\).


The domain of the function \( f(x) \) is \( (0,1) \), and the function is defined as follows:
\[
f(x)=\begin{cases}
x, & \text{if } x \text{ is an irrational number}, \\
\frac{p+1}{q}, & \text{if } x=\frac{p}{q}, \; p, q \in \mathbf{N}^{*}, \; (p, q) = 1, \; p < q.
\end{cases}
\]

Find the maximum value of \( f(x) \) in the interval \(\left( \frac{7}{8}, \frac{8}{9} \right) \).


Let \( p \) and \( q \) be two prime numbers such that \( q \) divides \( 3^p - 2^p \). Show that \( p \) divides \( q - 1 \).

Solution: \boxed{p \mid q - 1}


Cat food is sold in large and small packages (with more food in the large package than in the small one). One large package and four small packages are enough to feed a cat for exactly two weeks. Is one large package and three small packages necessarily enough to feed the cat for 11 days?

Solution: \boxed{Yes}


Given that \( T = \sin 50^{\circ} \times \left(S + \sqrt{3} \times \tan 10^{\circ}\right) \), find the value of \( T \).


Find the maximum possible volume of a cylinder inscribed in a cone with a height of 27 and a base radius of 9.


Anya, Borya, and Vasya took the same 6-question test, where each question could be answered with "yes" or "no". The answers are shown in the table:

| Question # | 1 | 2 | 3 | 4 | 5 | 6 |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| Anya | no | yes | no | yes | no | yes |
| Borya | no | no | yes | yes | no | yes |
| Vasya | yes | no | no | no | yes | no |

It turned out that Anya had two incorrect answers, and Borya had only two correct answers. How many incorrect answers does Vasya have?


A bag contains 15 balls, marked with the 15 numbers $2^{0}, 2^{1}, 2^{2}, \ldots, 2^{14}$ respectively. Each ball is either red or blue, and there is at least one ball of each color. Let $a$ be the sum of the numbers on all red balls, $b$ be the sum of the numbers on all blue balls, and $d$ be the H.C.F. of $a$ and $b$. Find the greatest possible value of $d$.


The random variable \( X \) is defined by the distribution function

$$
F(x)= \begin{cases}
0 & \text{if } x \leq 0 \\
\frac{(1-\cos x)}{2} & \text{if } 0 < x \leq \pi \\
1 & \text{if } x > \pi
\end{cases}
$$

Find the probability density function of the variable \( X \). Calculate the probability that the random variable \( X \) takes values in the interval \( (\pi / 3, \pi / 2) \).


Let \( n \) be a given integer such that \( n \geq 2 \).

(a) Determine the smallest constant \( C \) such that the following inequality holds for all real numbers \( x_1, x_2, \ldots, x_n \geq 0 \):

\[
\sum_{1 \leq i<j \leq n} x_i x_j (x_i^2 + x_j^2) \leq C \left( \sum_{1 \leq i \leq n} x_i \right)^4
\]

(b) Determine under what conditions equality holds for this constant \( C \).


Two motorcyclists start racing from the same point and at the same time, one with a speed of 80 km/h and the other with a speed of 60 km/h. Half an hour later, a third motorcyclist starts from the same point in the same direction. Find the speed of the third motorcyclist, given that he catches up with the first motorcyclist 1 hour and 15 minutes later than he catches up with the second motorcyclist.

Solution: \boxed{100 \text{ km/h}}


Find all values of the parameter \(a\) for which the quadratic trinomial \(\frac{1}{3} x^2 + \left(a+\frac{1}{2}\right) x + \left(a^2 + a\right)\) has two roots, the sum of the cubes of which is exactly 3 times their product. In your answer, specify the largest of such \(a\).


What is the minimum number of points that must be marked inside a convex $n$-gon such that any triangle whose vertices are vertices of the $n$-gon contains at least one marked point inside it?


In the diagram, the area of triangle $\triangle ABC$ is 100 square centimeters, and the area of triangle $\triangle ABD$ is 72 square centimeters. Point $M$ is the midpoint of side $CD$, and $\angle MHB = 90^{\circ}$. Given that $AB = 20$ centimeters, find the length of $MH$ in centimeters.


In a convex quadrilateral \(ABCD\), side \(AB\) is equal to diagonal \(BD\), \(\angle A=65^\circ\), \(\angle B=80^\circ\), and \(\angle C=75^\circ\). What is \(\angle CAD\) (in degrees)?


Given that \( m \in \{11, 13, 15, 17, 19\} \) and \( n \in \{2000, 2001, \cdots, 2019\} \), what is the probability that the units digit of \( m^n \) is 1?


By definition, a polygon is regular if all its angles and sides are equal. Points \( A, B, C, D \) are consecutive vertices of a regular polygon (in that order). It is known that the angle \( ABD = 135^\circ \). How many vertices does this polygon have?


One mole of an ideal gas undergoes a closed cycle in which:

$1-2$ - isobaric process, where the volume increases by 4 times;

$2-3$ - isothermal process, where the pressure increases;

$3-1$ - process in which the gas is compressed according to the law $T=\gamma V^{2}$.

Determine by what factor the volume in state 3 exceeds the initial volume in state 1.


For which values of \(a\) does the equation \(|x-3| = a x - 1\) have two solutions? Enter the midpoint of the interval of parameter \(a\) in the provided field. Round the answer to three significant digits according to rounding rules and enter it in the provided field.


Let's call a number greater than 25 semi-prime if it is the sum of two distinct prime numbers. What is the largest number of consecutive natural numbers that can be semi-prime?


If \( P \) is the circumcenter of \( \triangle ABC \), and
\[
\overrightarrow{P A}+\overrightarrow{P B}+\lambda \overrightarrow{P C}=\mathbf{0}, \quad \text{with} \quad \angle C=120^{\circ},
\]
determine the value of the real number \( \lambda \).


There are 36 students at the Multiples Obfuscation Program, including a singleton, a pair of identical twins, a set of identical triplets, a set of identical quadruplets, and so on, up to a set of identical octuplets. Two students look the same if and only if they are from the same identical multiple. Nithya the teaching assistant encounters a random student in the morning and a random student in the afternoon (both chosen uniformly and independently), and the two look the same. What is the probability that they are actually the same person?


Compute the limit of the function:

$$
\lim _{x \rightarrow \pi} \frac{\ln (2+\cos x)}{\left(3^{\sin x}-1\right)^{2}}
$$

Solution: \boxed{\frac{1}{2 \ln^2 3}}


Suppose that a function \( M(n) \), where \( n \) is a positive integer, is defined by
\[
M(n)=\left\{
\begin{array}{ll}
n - 10 & \text{if } n > 100 \\
M(M(n + 11)) & \text{if } n \leq 100
\end{array}
\right.
\]

How many solutions does the equation \( M(n) = 91 \) have?


Two friends agree to meet at a specific place between 12:00 PM and 12:30 PM. The first one to arrive waits for the other for 20 minutes before leaving. Find the probability that the friends will meet, assuming each chooses their arrival time randomly (between 12:00 PM and 12:30 PM) and independently.


A reservoir has 400 tons of water. At midnight each day, the inlet and outlet gates are opened simultaneously. The amount of water $w$ (tons) flowing out through the outlet gate is a function of time $t$ (hours): $w = 120 \sqrt{6t} \text{ for } (0 \leq t \leq 24)$.

(1) In order to ensure that the reservoir still contains 400 tons of water by midnight the next day, how many tons of water need to be added to the reservoir per hour (assuming the rate of inflow is constant each hour)?

(2) Under the conditions of part (1), at what time of day will the amount of water in the reservoir be at its minimum, and how many tons of water will be in the reservoir at that time?


Eight people sit around a circular table, each of whom is either a knight or a liar. When asked about their neighbors, each of them answered: "My neighbors are a liar and a knight." How many liars are among them? How would the answer change if there were nine people at the table?


Sasa wants to make a pair of butterfly wings for her Barbie doll. As shown in the picture, she first drew a trapezoid and then drew two diagonals, which divided the trapezoid into four triangles. She cut off the top and bottom triangles, and the remaining two triangles are exactly a pair of beautiful wings. If the areas of the two triangles that she cut off are 4 square centimeters and 9 square centimeters respectively, then the area of the wings that Sasa made is $\qquad$ square centimeters.


If the function \( f(x) = x^3 - 6x^2 + 9x \) has a maximum value in the interval \( \left(3-a^2, a\right) \), then the range of the real number \( a \) is \(\qquad\)

Solution: \boxed{ (\sqrt{2}, 4] }


Let \( P \) and \( Q \) be points on line \( l \) with \( PQ = 12 \). Two circles, \( \omega \) and \( \Omega \), are both tangent to \( l \) at \( P \) and are externally tangent to each other. A line through \( Q \) intersects \( \omega \) at \( A \) and \( B \), with \( A \) closer to \( Q \) than \( B \), such that \( AB = 10 \). Similarly, another line through \( Q \) intersects \( \Omega \) at \( C \) and \( D \), with \( C \) closer to \( Q \) than \( D \), such that \( CD = 7 \). Find the ratio \( AD / BC \).


How many ordered triples \((x, y, z)\) satisfy the following conditions:
\[ x^2 + y^2 + z^2 = 9, \]
\[ x^4 + y^4 + z^4 = 33, \]
\[ xyz = -4? \]


Find all pairs of natural numbers \( x \) and \( y \) such that the ratio \(\frac{x y^{3}}{x+y}\) is a prime number.

Solution: \boxed{(14, 2)}


Given the complex number \( z_{1}=\mathrm{i}(1-\mathrm{i})^{3} \), when the complex number \( z \) satisfies \( |z|=1 \), find the maximum value of \( \left|z-z_{1}\right| \).

Solution: \boxed{1 + 2\sqrt{2}}


For which values of the parameter \(a\) does the equation \(x^{3} + 16x^{2} + ax + 64 = 0\) have three distinct real roots that form a geometric progression?


In a notebook, there is a grid rectangle of size $3 \times 7$. Igor's robot was asked to trace all the lines with a marker, and it took him 26 minutes (the robot draws lines at a constant speed). How many minutes will it take him to trace all the lines in a $5 \times 5$ grid square with the marker?

Solution: \boxed{30 \text{ minutes}}


For what smallest natural number \( k \) does the quadratic trinomial

\[ y = kx^2 - px + q \]

with natural coefficients \( p \) and \( q \) have two distinct positive roots less than 1?


In figure 1, \( \triangle ABC \) and \( \triangle EBC \) are two right-angled triangles with \( \angle BAC = \angle BEC = 90^\circ \). Given that \( AB = AC \) and \( EDB \) is the angle bisector of \( \angle ABC \), find the value of \( \frac{BD}{CE} \).


The bases \(AB\) and \(CD\) of the trapezoid \(ABCD\) are 41 and 24 respectively, and its diagonals are mutually perpendicular. Find the dot product of the vectors \(\overrightarrow{AD}\) and \(\overrightarrow{BC}\).


A roulette can land on any number from 0 to 2007 with equal probability. The roulette is spun repeatedly. Let $P_{k}$ be the probability that at some point the sum of the numbers that have appeared in all spins equals $k$. Which number is greater: $P_{2007}$ or $P_{2008}$?


In the coordinate plane, draw the following lines \( y = k \), \( y = \sqrt{3} x + 2k \), and \( y = -\sqrt{3} x + 2k \), where \( k = 0, \pm 1, \pm 2, \ldots, \pm 10 \). These 63 lines divide the plane into several equilateral triangles. Find the number of equilateral triangles with a side length of \( \frac{2}{\sqrt{3}} \).


Given a regular triangular pyramid $P-ABC$ with the side length of the base equilateral triangle being 1, the circumsphere center $O$ satisfies $\overrightarrow{OA} + \overrightarrow{OB} + \overrightarrow{OC} = \overrightarrow{0}$. Find the volume of this regular triangular pyramid.


Find the square numbers such that when divided by 11, the quotient is a prime number and the remainder is 4.

Solution: \boxed{n^2 = 81}


In triangle $ABC$, the angles $A$ and $C$ at the base are $20^{\circ}$ and $40^{\circ}$, respectively. It is given that $AC - AB = 5$ cm. Find the length of the angle bisector of angle $B$.

Solution: \boxed{5 \text{ cm}}


Given the ellipse \(\frac{x^{2}}{4}+\frac{y^{2}}{3}=1\) with the left focus \( F \), and a line \( l \) passing through \((1,1)\) that intersects the ellipse at points \( A \) and \( B \). When the perimeter of triangle \( \triangle FAB \) is at its maximum, what is the area of triangle \( \triangle FAB \)?


In the drawing, there is a grid consisting of 25 small equilateral triangles.

How many rhombuses can be formed from two adjacent small triangles?


Given that the graph of the function \( y=\frac{\left|x^{2}-1\right|}{x-1} \) intersects with the graph of the function \( y=kx-2 \) at two points. Determine the range of the real number \( k \).


By calculating the limit $\lim _{\Delta x \rightarrow 0} \frac{\Delta y}{\Delta x}$, find the derivatives of the following functions:
1) $y = 3x^2 - 4x$
2) $y = \frac{1}{x}$
3) $y = \sqrt{x}$
4) $y = \cos 3x$

Solution: \boxed{f'(x) = -3\sin (3x)}


Let the random variables $\xi$ and $\eta$ denote the lifetimes of the blue and red light bulbs, respectively. The lifetime of the flashlight is equal to the minimum of these two values. Clearly, $\min (\xi, \eta) \leq \xi$. Let's move to the expected values: $\operatorname{E} \min (\xi, \eta) \leq \mathrm{E} \xi=2$. Therefore, the expected lifetime of the flashlight is no more than 2 years.


In the coordinate plane, a point whose x-coordinate and y-coordinate are both integers is called a lattice point. For any natural number \( n \), let us connect the origin \( O \) with the point \( A_{n}(n, n+3) \). Let \( f(n) \) denote the number of lattice points on the segment \( OA_{n} \) excluding the endpoints. Find the value of \( f(1) + f(2) + \cdots + f(1990) \).


There are 90 players in a league. Each of the 90 players plays exactly one match against each of the other 89 players. Each match ends with a win for one player and a loss for the other player, or with a tie for both players. Each player earns 1 point for a win, 0 points for a loss, and 0.5 points for a tie. After all matches have been played, the points earned by each player are added up. What is the greatest possible number of players whose total score can be at least 54 points?


Calculate the volumes of solids formed by rotating the regions bounded by the graphs of the functions around the y-axis.

$$
y = (x-1)^{2}, x=0, x=2, y=0
$$


Seventeen people dine every Saturday night around a round table. How many times is it possible for them to dine if each person wants to have two new neighbors each time? What is the result for eighteen people?


Let \( f(x) = x^2 + ax + b \) have two real roots in the interval \([0,1]\). Then, find the range of values of \( a^2 - 2b \).

Solution: \boxed{[0, 2]}


Given a cyclic quadrilateral \(A B C D\) with side lengths \(AB = 1\), \(BC = 3\), \(CD = DA = 2\), find the area of quadrilateral \(A B C D\).


Let \( x > 0 \), plane vectors \( \overrightarrow{A B} = (0,1), \overrightarrow{B C} = (1,2) \), and \( \overrightarrow{C D} = \left(\log_{3} x, \log_{9} x\right) \). If \( \overrightarrow{A C} \cdot \overrightarrow{B D} = 2 \), then find the value of \( x \).


Numbers \( p \) and \( q \) are such that the parabolas \( y = -2x^2 \) and \( y = x^2 + px + q \) intersect at two points, enclosing a certain region.

Find the equation of the vertical line that divides the area of this region in half.

Solution: \boxed{x = -\frac{p}{6}}


Arrange the numbers

$$
\begin{gathered}
x=(a+b)(c+d) \\
y=(a+c)(b+d) \\
z=(a+d)(b+c)
\end{gathered}
$$

in ascending order, given that \(a < b < c < d\).

Solution: \boxed{x < y < z}


Kiril Konstantinovich's age is 48 years, 48 months, 48 weeks, 48 days, and 48 hours. How many full years old is Kiril Konstantinovich?


Given that the real numbers \(a\), \(b\), and \(c\) satisfy \(\left|a x^{2} + b x + c\right|\) with a maximum value of 1 on the interval \(x \in [-1,1]\), what is the maximum possible value of \(\left|c x^{2} + b x + a\right|\) on the interval \(x \in [-1,1]\)?


Given the set \( A = \{x_{1}, x_{2}, x_{3}, x_{4}, x_{5}\} \), calculate the sums of the two-element subsets of \( A \) to form the set \( B = \{4, 5, 6, 7, 8, 9, 10, 12, 13, 14\} \). Determine the set \( A \).

Solution: \boxed{\{1, 3, 4, 5, 9\}}


Find the point \( M^{\prime} \) that is symmetric to the point \( M \) with respect to the plane.

\[ M(2, -1, 1) \]

\[ x - y + 2z - 2 = 0 \]

Solution: \boxed{(1, 0, -1)}


Does there exist a function \( f: \mathbb{N} \rightarrow \mathbb{N} \) such that \( f(f(n)) = n + 1987 \) for every natural number \( n \)?

Solution: \boxed{\text{No}}


Using 125 small cube blocks to form a larger cube, some of the small cube blocks on the surface of the larger cube are painted. The painting on the opposite pairs of faces - top and bottom, left and right, front and back - corresponds to each other. Among these 125 small cube blocks, there are $\qquad$ blocks that have none of their faces painted.


There are some stamps with denominations of 0.5 yuan, 0.8 yuan, and 1.2 yuan, with a total value of 60 yuan. The number of 0.8 yuan stamps is four times the number of 0.5 yuan stamps. How many 1.2 yuan stamps are there?


Diligent Masha wrote out all natural numbers from 372 to 506 inclusive. Then she calculated two sums: first the sum of all odd numbers in this range, and then the sum of all even numbers. After that, she subtracted the smaller sum from the larger sum. What result did she get?


Find the maximum value of the expression for \( a, b > 0 \):

$$
\frac{|4a - 10b| + |2(a - b\sqrt{3}) - 5(a\sqrt{3} + b)|}{\sqrt{a^2 + b^2}}
$$

Solution: \boxed{2 \sqrt{87}}


Find the smallest natural number that starts with the digit 5, which, when this 5 is removed from the beginning of its decimal representation and appended to its end, becomes four times smaller.


Toward the end of a game of Fish, the 2 through 7 of spades, inclusive, remain in the hands of three distinguishable players: DBR, RB, and DB, such that each player has at least one card. If it is known that DBR either has more than one card or has an even-numbered spade, or both, in how many ways can the players' hands be distributed?


The coefficients \( p \) and \( q \) of the quadratic equation \( x^{2}+p x+q=0 \) are chosen at random in the interval \( (0, 2) \). What is the probability that the roots of this equation will be real numbers?


What is the smallest \( n > 1 \) for which the average of the first \( n \) (non-zero) squares is a square?


Find the number of solutions to $\sin x = \lg x$.


Vehicle A and Vehicle B start from points A and B, respectively, at the same time and travel towards each other. They meet after 3 hours, at which point Vehicle A turns back towards point A, and Vehicle B continues forward. After Vehicle A reaches point A, it turns around and heads towards point B. Half an hour later, it meets Vehicle B again. How many hours does it take for Vehicle B to travel from A to B?


A point inside an equilateral triangle is at distances of 3, 4, and 5 units from the sides. What is the length of the side of the triangle?

Solution: \boxed{8 \sqrt{3}}


There is a cube placed on stands and six different paints. How many ways can you paint all the faces of the cube (one color for each face, not necessarily using all the paints) so that adjacent faces (sharing an edge) are painted in different colors? (16 points)


Let $PA$, $PB$, and $PC$ be three non-coplanar rays originating from point $P$, with each pair of rays forming a $60^\circ$ angle. A sphere with a radius of 1 is tangent to each of these three rays. Find the distance from the center of the sphere $O$ to point $P$.


a) In a three-digit number, the first digit on the left was erased. Then, the resulting two-digit number was multiplied by 7, and the original three-digit number was obtained. Find such a number.

b) In a three-digit number, the middle digit was erased, and the resulting number is 6 times smaller than the original. Find such a three-digit number.


Given the parabola \( y^{2} = 2 p x \) with focus \( F \) and directrix \( l \), a line passing through \( F \) intersects the parabola at points \( A \) and \( B \) such that \( |AB| = 3p \). Let \( A' \) and \( B' \) be the projections of \( A \) and \( B \) onto \( l \), respectively. If a point \( M \) is randomly chosen inside the quadrilateral \( AA'B'B \), what is the probability that \( M \) lies inside the triangle \( FA'B' \)?


A set of distinct positive integers has a sum of 1987. What is the maximum possible value for three times the total number of integers plus the number of odd integers?


A rectangular grid is given. Let's call two cells neighboring if they share a common side. Count the number of cells that have exactly four neighboring cells. There are 23 such cells. How many cells have three neighboring cells?


In the side face $A A^{\prime} B^{\prime} B$ of a unit cube $A B C D - A^{\prime} B^{\prime} C^{\prime} D^{\prime}$, there is a point $M$ such that its distances to the two lines $A B$ and $B^{\prime} C^{\prime}$ are equal. What is the minimum distance from a point on the trajectory of $M$ to $C^{\prime}$?


In trapezoid \(ABCD\), the side \(BC\) is equal to the diagonal \(BD\). On the smaller arc \(AB\) of the circumcircle of triangle \(ABC\), a point \(E\) is chosen such that \(BC = BE\). Find the angle \(\angle AED\).


Calculate the limit of the function:

$$
\lim _{x \rightarrow 0}\left(\frac{e^{3 x}-1}{x}\right)^{\cos ^{2}\left(\frac{\pi}{4}+x\right)}
$$


Out of 60 right-angled triangles with legs of 2 and 3, a rectangle was formed. What can be the maximum perimeter of this rectangle?


The sequence \(\left\{a_{n}\right\}\) satisfies: \(a_1 = 1\), and for each \(n \in \mathbf{N}^{*}\), \(a_n\) and \(a_{n+1}\) are the roots of the equation \(x^2 + 3nx + b_n = 0\). Find \(\sum_{k=1}^{20} b_k\).


There are 5 black and white pieces arranged in a circle. The rule is: place a white piece between two adjacent pieces of the same color, and place a black piece between two adjacent pieces of different colors, then remove the original 5 pieces. Starting from the initial state as shown in figure (1), and following the above rule, determine the maximum number of black pieces that can be present on the circle.


Find the sum of the squares of the roots of the equation \(\left(x^{2}+6x\right)^{2}-1580\left(x^{2}+6x\right)+1581=0\).


There are 4 balls of different masses. How many weighings on a balance scale without weights are needed to arrange these balls in descending order of mass?


In the cube \( A C_{1} \), determine the magnitude of the dihedral angle \( C_{1}-D_{1} B-C \).


A deck of cards contains 52 cards, with each suit ("Diamonds", "Clubs", "Hearts", "Spades") having 13 cards. The ranks are $2, 3, \ldots, 10, J, Q, K, A$. Two cards of the same suit with consecutive ranks are called "straight flush" cards, and $A$ with 2 is also considered as consecutive (i.e., $A$ can be used as 1). Determine the number of ways to select 13 cards from this deck such that every rank appears and no "straight flush" cards are included.


Find the number of ordered integer pairs \((m, n)\), where \(1 \leqslant m \leqslant 99\), \(1 \leqslant n \leqslant 99\), such that \((m+n)^2 + 3m + n\) is a perfect square.


In $\triangle ABC$ shown in Figure 2-13, points $D$ and $E$ lie on the extensions of $AC$ and $AB$ respectively. $BD$ and $CE$ intersect at $P$, and $BD = CE$. If point $P$ satisfies $\angle AEP - \angle ADP = k^2(\angle PED - \angle PDE)$ where $k$ is a constant, then $AB = AC$.

Solution: \boxed{AB = AC}


What is the maximum value of the sum of the cosines of all the angles of an isosceles triangle?


Find the maximum number of elements in a set $S$ that satisfies the following conditions:
(1) Every element in $S$ is a positive integer not exceeding 100.
(2) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the greatest common divisor (gcd) of $a + b$ and $c$ is 1.
(3) For any two distinct elements $a$ and $b$ in $S$, there exists another element $c$ in $S$ such that the gcd of $a + b$ and $c$ is greater than 1.


The box contains 120 colored pencils: 35 red, 23 green, 14 yellow, 26 blue, 11 brown, and 11 black. What is the minimum number of pencils one needs to take from the box in the dark (without seeing the pencils) to ensure that there are at least 18 pencils of one color among them?


How many pairs of positive integers $(x, y)$ are solutions to the equation $3x + 5y = 501?$


Inflation over two years will be:

$$
\left((1+0,025)^{\wedge 2-1}\right)^{*} 100 \%=5,0625 \%
$$

The real interest rate of a bank deposit with reinvestment for the second year will be $(1.06 * 1.06 /(1+0,050625)-1) * 100=6,95 \%$


Let \( m = 76^{2006} - 76 \). Find the remainder when \( m \) is divided by 100.


Let \( O \) be the vertex of the parabola and \( F \) be the focus, and let \( PQ \) be a chord passing through \( F \). Given \( |OF| = a \) and \( |PQ| = b \), find the area of triangle \( OPQ \).

Solution: \boxed{a \sqrt{ab}}


For any positive integers \( x \), \( y \), and \( z \), if \( x \mid y^3 \), \( y \mid z^3 \), \( z \mid x^3 \), and \( xyz \mid (x + y + z)^n \), find the smallest positive integer \( n \) that satisfies these conditions.


Find the sum of all three-digit natural numbers that do not contain the digits 0 or 5.


In the square \( ABCD \) with side length 1, point \( F \) is the midpoint of side \( BC \), and \( E \) is the foot of the perpendicular dropped from vertex \( A \) to \( DF \).

Find the length of \( BE \).


Find the number of consecutive 0's at the end of the base 10 representation of 2006!.


Vasya needs to write one digit on each face of several cubes in such a way that any ordered combination of three digits from 000 to 999 inclusive can be obtained by selecting some three different cubes and placing them with suitable faces up in the correct order. Note that the digits 6 and 9 are not considered interchangeable when rotated by 180 degrees. What is the minimum number of cubes that Vasya must use?


What is the largest area that a figure in the $xy$-plane can have, located between the lines $x = -3$ and $x = 1$, and bounded below by the $x$-axis, and above by a tangent to the graph of the function $y = x^2 + 16$ at the point of tangency with the abscissa $x_0$ in the interval $-3 \leq x_0 \leq 1$?


Let \(ABC\) be a triangle with area 1. Let points \(D\) and \(E\) lie on \(AB\) and \(AC\), respectively, such that \(DE\) is parallel to \(BC\) and \(\frac{DE}{BC} = \frac{1}{3}\). If \(F\) is the reflection of \(A\) across \(DE\), find the area of triangle \(FBC\).


Let \( F \) be the left focus of the ellipse \( E: \frac{x^{2}}{3}+y^{2}=1 \). A line \( l \) with a positive slope passes through point \( F \) and intersects \( E \) at points \( A \) and \( B \). Through points \( A \) and \( B \), lines \( AM \) and \( BN \) are drawn such that \( AM \perp l \) and \( BN \perp l \), each intersecting the x-axis at points \( M \) and \( N \), respectively. Find the minimum value of \( |MN| \).


Let \( x * y = x + y - xy \), where \( x \) and \( y \) are real numbers. If \( a = 1 * (0 * 1) \), find the value of \( a \).


In how many ways can you place two knights - one white and one black - on a $16 \times 16$ chessboard so that they threaten each other? (A knight moves in an "L" shape, i.e., it moves to one of the nearest squares that is not on the same rank, file, or diagonal as its current position.)


Given $\left\{\begin{array}{l}\sin \alpha+\sin \beta=1 \\ \cos \alpha+\cos \beta=0\end{array}\right.$, find $\cos 2 \alpha+\cos 2 \beta$.


An isosceles trapezoid with a $30^{\circ}$ angle is circumscribed around a circle. Its midline is equal to 10. Find the radius of the circle.


Given 10 distinct points on a plane, consider the midpoints of all segments connecting all pairs of points. What is the minimum number of such midpoints that could result?


Two vertical chords are drawn in a circle, dividing the circle into 3 distinct regions. Two horizontal chords are added in such a way that there are now 9 regions in the circle. A fifth chord is added that does not lie on top of one of the previous four chords. The maximum possible number of resulting regions is \( M \) and the minimum possible number of resulting regions is \( m \). What is \( M^{2} + m^{2} \)?


In triangle ABC with sides AB = 5, BC = √17, and AC = 4, point M is taken on side AC such that CM = 1. Find the distance between the centers of the circumcircles of triangles ABM and BCM.


On the sides $AB$ and $CD$ of rectangle $ABCD$, points $E$ and $F$ are marked such that $AFCE$ forms a rhombus. It is known that $AB = 16$ and $BC = 12$. Find $EF$.


In a swamp, there are 64 tufts of grass arranged in an \(8 \times 8\) square. On each tuft, there is either a frog or a toad. Frogs always tell the truth, while toads always lie. Each of them, both the frogs and the toads, proclaimed: "At least one of the neighboring tufts has a toad." What is the maximum number of toads that could be on these tufts?

Tufts are considered neighbors if they are horizontally or vertically adjacent in the rows and columns of the square; diagonal tufts are not considered neighbors.


Let the set \( M = \{1, 2, \cdots, 2020\} \). For any non-empty subset \( X \) of \( M \), let \( \alpha_X \) represent the sum of the maximum and minimum numbers in \( X \). Find the arithmetic mean of all such \( \alpha_X \).


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?

Solution: \boxed{28 \text{ meters}}


Find the sum of all fractions in lowest terms with value greater than 10 but smaller than 100 and with the denominator equal to 3.


Given a quadratic polynomial \( f(x) \) such that the equation \( (f(x))^3 - 4f(x) = 0 \) has exactly three solutions. How many solutions does the equation \( (f(x))^2 = 1 \) have?


A scatterbrained scientist in his laboratory has developed a unicellular organism, which has a probability of 0.6 of dividing into two identical organisms and a probability of 0.4 of dying without leaving any offspring. Find the probability that after some time the scatterbrained scientist will have no such organisms left.


There are 300 black and white pieces in total. The black crow divides the black and white pieces into 100 piles, each containing 3 pieces. There are 27 piles with exactly $l$ white pieces, 42 piles with 2 or 3 black pieces, and the number of piles with 3 white pieces is equal to the number of piles with 3 black pieces. How many white pieces are there in total?


A natural number \( 1 \leq n \leq 221 \) is called lucky if, when dividing 221 by \( n \), the remainder is wholly divisible by the incomplete quotient (the remainder can be equal to 0). How many lucky numbers are there?


A $4 \times 8$ paper rectangle is folded along the diagonal as shown in the figure. What is the area of the triangle that is covered twice?


The last two digits of the decimal representation of the square of a natural number are the same and are not zero. What are these digits? Find all solutions.


The largest prime factor of 101101101101 is a four-digit number \(N\). Compute \(N\).


There are 23 socks in a drawer: 8 white and 15 black. Every minute, Marina goes to the drawer and pulls out a sock. If at any moment Marina has pulled out more black socks than white ones, she exclaims, "Finally!" and stops the process.

What is the maximum number of socks Marina can pull out before she exclaims, "Finally!"? The last sock Marina pulled out is included in the count.


The quadrilateral $A B C D$ is inscribed in a circle with center $O$. The diagonals $A C$ and $B D$ are perpendicular. Show that the distance from $O$ to the line $(A D)$ is equal to half the length of the segment $[B C]$.


The RSA Factoring Challenge, which ended in 2007, involved factoring extremely large numbers that were the product of two prime numbers. The largest number successfully factored in this challenge was RSA-640, which has 193 decimal digits and carried a prize of $20,000. The next challenge number carried a prize of $30,000, and contains $N$ decimal digits. Your task is to submit a guess for $N`. Only the team(s) that have the closest guess(es) receives points. If $k$ teams all have the closest guesses, then each of them receives $\left\lceil\frac{20}{k}\right\rceil$ points.


In which cases does a month have the largest number of Saturdays? What is this number?


Solve the equation \(\frac{x}{2+\frac{x}{2+\frac{x}{2+\ddots \frac{x}{2+\frac{x}{1+\sqrt{1+x}}}}}}=1\)
(where there are 1985 terms of 2 in the expression).


The polynomial $a x^{2} + b x + c$ has integer coefficients and two distinct roots that are greater than 0 but less than 1. How small can $|a|$ be?


Given \( x = \frac{\pi}{12} \), find the value of the function \( f(x) = \sqrt{\sin^4{x} + 4\cos^2{x}} - \sqrt{\cos^4{x} + 4\sin^2{x}} \).


Let complex numbers \(z_{1}\) and \(z_{2}\) satisfy \(\operatorname{Re}(z_{1}) > 0\) and \(\operatorname{Re}(z_{2}) > 0\) and \(\operatorname{Re}(z_{1}^2) = \operatorname{Re}(z_{2}^2) = 2\) (where \(\operatorname{Re}(z)\) denotes the real part of the complex number \(z\)).

(1) Find the minimum value of \(\operatorname{Re}(z_{1} z_{2})\).

(2) Find the minimum value of \(\left|z_{1} + 2\right| + \left|\overline{z_{2}} + 2\right| - \left|\overline{z_{1}} - z_{2}\right|\).


Find the intersection point of a line and a plane.

$$
\begin{aligned}
& \frac{x-1}{2}=\frac{y-2}{0}=\frac{z-4}{1} \\
& x-2 y+4 z-19=0
\end{aligned}
$$

Solution: \boxed{(3, 2, 5)}


The numbers from 1 to 10 were divided into two groups such that the product of the numbers in the first group is divisible by the product of the numbers in the second group.

What is the smallest possible value of the quotient obtained by dividing the product of the first group by the product of the second group?


Let \(a\), \(b\), and \(c\) be the lengths of the three sides of a triangle. Suppose \(a\) and \(b\) are the roots of the equation
\[
x^2 + 4(c + 2) = (c + 4)x,
\]
and the largest angle of the triangle is \(x^\circ\). Find the value of \(x\).


Let \( p_{1}, p_{2}, \ldots, p_{97} \) be prime numbers (not necessarily distinct). What is the greatest integer value that the expression

\[ \sum_{i=1}^{97} \frac{p_{i}}{p_{i}^{2}+1}=\frac{p_{1}}{p_{1}^{2}+1}+\frac{p_{2}}{p_{2}^{2}+1}+\ldots+\frac{p_{97}}{p_{97}^{2}+1} \]

can take?


The Dorokhov family plans to purchase a vacation package to Crimea. The vacation will include the mother, father, and their eldest daughter Polina, who is 5 years old. They have chosen to stay at the "Bristol" hotel and have reached out to two travel agencies, "Globus" and "Around the World," to calculate the cost of the tour from July 10 to July 18, 2021.

The deals from each agency are as follows:
- At "Globus":
  - 11,200 rubles per person for those under 5 years old.
  - 25,400 rubles per person for those above 5 years old.
  - A 2% discount on the total cost as regular customers.

- At "Around the World":
  - 11,400 rubles per person for those under 6 years old.
  - 23,500 rubles per person for those above 6 years old.
  - A 1% commission fee is applied to the total cost.

Determine which travel agency offers the best deal for the Dorokhov family and identify the minimum cost for their vacation in Crimea. Provide only the number in your answer, without units of measurement.


Let the sequences \(\left\{x_{n}\right\}\) and \(\left\{y_{n}\right\}\) satisfy \( x_{n} + i y_{n} = \left(\frac{\sqrt{3}i - 1}{2}\right)^{n} \). Find the sum of the first 1994 terms of the sequence \(\left\{x_{n}\right\}\), denoted by \(S_{1994}\).


A cylinder has a height that is 3 times its base radius. If it is divided into a large and a small cylinder, the surface area of the large cylinder is 3 times the surface area of the small cylinder. Find the ratio of the volume of the large cylinder to the volume of the small cylinder.


For what value of \( k \) do we have the identity (true for any \( a, b, c \)):

$$
(a+b)(b+c)(c+a) = (a+b+c)(ab + bc + ca) + k \cdot abc?
$$


A clock chimes 2 times at 2 o'clock, taking 2 seconds to complete. How many seconds does it take to chime 12 times at 12 o'clock?


Find the sum of all possible distinct two-digit numbers, all of whose digits are odd.


During the draw before the math marathon, the team captains were asked to name the smallest possible sum of the digits in the decimal representation of the number \( n+1 \), given that the sum of the digits of the number \( n \) is 2017. What answer did the captain of the winning team give?


In the expression \( S = \sqrt{x_{1} - x_{2} + x_{3} - x_{4}} \), \( x_{1}, x_{2}, x_{3}, x_{4} \) are a permutation of 1, 2, 3, and 4. Determine the number of distinct permutations that make \( S \) a real number.


How many times should two dice be rolled so that the probability of getting two sixes at least once is greater than $1/2$?


Given natural numbers \( k \) and \( n \) such that \( 1 < k \leq n \). What is the smallest \( m \) for which the following statement holds: for any placement of \( m \) rooks on an \( n \times n \) board, it is always possible to select \( k \) rooks from these \( m \) such that no two of the selected rooks can attack each other?

Solution: \boxed{n(k-1) + 1}


On a rectangular table of size $x$ cm $\times 80$ cm, identical sheets of paper of size 5 cm $\times 8$ cm are placed. The first sheet is placed in the bottom left corner, and each subsequent sheet is placed one centimeter above and one centimeter to the right of the previous one. The last sheet touches the top right corner. What is the length of $x$ in centimeters?

Solution: \boxed{77 \text{ cm}}


How many pairs of integers \(x, y\) between 1 and 1000 are there such that \(x^{2} + y^{2}\) is divisible by 7?


How many pairs of natural numbers \(a\) and \(b\) exist such that \(a \geq b\) and the equation

\[ \frac{1}{a} + \frac{1}{b} = \frac{1}{6} \]

is satisfied?


Given the sequence \(\{a_n\}\):
\[ a_n = 2^n + 3^n + 6^n + 1 \text{ for } n \in \mathbb{Z}_+ \]

Is there an integer \( k \geq 2 \) such that \( k \) is coprime with all numbers in the sequence \(\{a_n\}\)? If it exists, find the smallest integer \( k \); if it does not exist, explain why.


Let point \( C \) be a moving point on the parabola \( y^2 = 2x \). From \( C \), two tangent lines \( CA \) and \( CB \) are drawn to the circle \((x-1)^2 + y^2 = 1\), intersecting the negative half of the \( y \)-axis at \( A \) and the positive half of the \( y \)-axis at \( B \). Find the minimum area of triangle \( \triangle ABC \).


Solution: \boxed{\{4, 5\}}


Find the number of natural numbers that do not exceed 2016 and are coprime with it. Recall that two integers are called coprime if they have no common natural divisors other than one.


Two girls knit at constant, but different speeds. The first girl takes a tea break every 5 minutes, and the second girl every 7 minutes. Each tea break lasts exactly 1 minute. When the girls went for a tea break together, it turned out that they had knitted the same amount. By what percentage is the first girl's productivity higher if they started knitting at the same time?


On the side \( BC \) of an equilateral triangle \( ABC \), points \( K \) and \( L \) are marked such that \( BK = KL = LC \). On the side \( AC \), point \( M \) is marked such that \( AM = \frac{1}{3} AC \). Find the sum of the angles \( \angle AKM \) and \( \angle ALM \).


Find the length of the segment of the line that is parallel to the bases of a trapezoid and passes through the intersection point of the diagonals, if the bases of the trapezoid are equal to \( a \) and \( b \).

Solution: \boxed{ \frac{2ab}{a+b} }


Given a monotonically increasing sequence of positive integers $\left\{a_{n}\right\}$ that satisfies the recurrence relation $a_{n+2}=3 a_{n+1}-a_{n}$, with $a_{6}=280$, find the value of $a_{7}$.


Find all positive integers \( n \) for which we can find one or more integers \( m_1, m_2, ..., m_k \), each at least 4, such that:
1. \( n = m_1 m_2 ... m_k \)
2. \( n = 2^M - 1 \), where \( M = \frac{(m_1 - 1)(m_2 - 1) ... (m_k - 1)}{2^k} \)


Can a rectangle of size $55 \times 39$ be divided into rectangles of size $5 \times 11$?

Solution: \boxed{\text{No}}


Among all proper fractions where both the numerator and the denominator are two-digit numbers, find the smallest fraction that is greater than $\frac{5}{6}$. Provide the numerator of this fraction in your answer.


There is a type of cell that dies at a rate of 2 cells per hour, and each remaining cell divides into 2. If the number of cells is 1284 after 8 hours, how many cells were there initially?


Given the numbers $\log _{\sqrt{29-x}}\left(\frac{x}{7}+7\right), \log _{(x+1)^{2}}(29-x), \log _{\sqrt{\frac{x}{7}+7}}(-x-1)$. For what values of $x$ are two of these numbers equal, and the third is greater than them by 1?


The radius of the circle circumscribed around the triangle $K L M$ is $R$. A line is drawn through vertex $L$, perpendicular to the side $K M$. This line intersects at points $A$ and $B$ the perpendicular bisectors of the sides $K L$ and $L M$. It is known that $A L = a$. Find $B L$.


The plane vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ have an included angle of $\frac{\pi}{3}$. If $|\boldsymbol{a}|$, $|\boldsymbol{b}|$, and $|\boldsymbol{a}+\boldsymbol{b}|$ form an arithmetic sequence, find the ratio $|\boldsymbol{a}| : |\boldsymbol{b}| : |\boldsymbol{a} + \boldsymbol{b}|$.

Solution: \boxed{3 : 5 : 7}


How many triples of non-negative integers \((x, y, z)\) satisfy the equation
\[ x y z + x y + y z + z x + x + y + z = 2012 \]


In \( \triangle ABC \), \( AB = 4 \), \( BC = 7 \), \( CA = 5 \). Let \(\angle BAC = \alpha\). Find the value of \( \sin^6 \frac{\alpha}{2} + \cos^6 \frac{\alpha}{2} \).


An ant starts at vertex \( A \) of a cube \( ABCD-EFGH \). Each time, it crawls with equal probability to one of the three neighboring vertices. What is the probability that the ant is back at vertex \( A \) after six moves?


A circular cone has vertex \( I \), a base with radius 1, and a slant height of 4. Point \( A \) is on the circumference of the base and point \( R \) is on the line segment \( IA \) with \( IR = 3 \). Shahid draws the shortest possible path starting at \( R \), traveling once around the cone, and ending at \( A \). If \( P \) is the point on this path that is closest to \( I \), what is the length \( IP \)?


An arbitrary finite sequence composed of 0s and 1s is called a word. The word \(A A A\) obtained by repeating word \(A\) three times is called the triplet of \(A\). For example, if \(A = 101\), its triplet is 101101101. For a word, the following two operations are allowed:
(1) Adding the triplet of any word at any position (including the beginning and the end) of the word;
(2) Removing the triplet of any word from it.
For example, from the word 0001, we can get 0111001 or 1, and so on. Is it possible, through a number of operations, to transform word 10 into word 01?

Solution: \boxed{\text{No}}


Joe knows that to convert from pounds to kilograms, you need to divide the mass in pounds by 2 and then decrease the resulting number by 10%. From this, Joe concluded that to convert from kilograms to pounds, you need to multiply the mass in kilograms by 2 and then increase the resulting number by 10%. By what percentage does he err from the correct mass in pounds?

Solution: \boxed{1\% \text{ less}}


In the sequence of positive integers, starting with \(2018, 121, 16, \ldots\), each term is the square of the sum of digits of the previous term. What is the \(2018^{\text{th}}\) term of the sequence?


Given complex numbers \( z_{1}, z_{2}, z_{3} \) such that \( \frac{z_{1}}{z_{2}} \) is a purely imaginary number, \( \left|z_{1}\right|=\left|z_{2}\right|=1 \), and \( \left|z_{1}+z_{2}+z_{3}\right|=1 \), find the minimum value of \( \left|z_{3}\right| \).

Solution: \boxed{\sqrt{2} - 1}


Damao, Ermao, and Sanmao are three brothers. Damao tells Sanmao: “When Dad was 36 years old, my age was 4 times yours, and Ermao’s age was 3 times yours.” Ermao adds: “Yes, at that time, the sum of our three ages was exactly half of Dad’s current age.” Sanmao says: “Now, the total age of all four of us adds up to 108 years.” How old is Sanmao this year?


Let's call a natural number semi-prime if it is greater than 25 and is the sum of two distinct prime numbers. What is the largest number of consecutive natural numbers that can be semi-prime? Justify your answer.


Let \( A B C D E F \) be a regular hexagon. Let \( G \) be a point on \( E D \) such that \( E G = 3 G D \). If the area of \( A G E F \) is 100, find the area of the hexagon \( A B C D E F \).


A Vandal and a Moderator are editing a Wikipedia article. The article originally is error-free. Each day, the Vandal introduces one new error into the Wikipedia article. At the end of the day, the moderator checks the article and has a \( \frac{2}{3} \) chance of catching each individual error still in the article. After 3 days, what is the probability that the article is error-free?


Let \( x_{1}, x_{2}, \ldots, x_{n} \) be strictly positive real numbers. Show that

\[ \frac{x_{1}}{x_{n}} + \frac{x_{2}}{x_{n-1}} + \ldots + \frac{x_{n}}{x_{1}} \geq n. \]

Solution: \boxed{n}


Consider a table with $m$ rows and $n$ columns. In how many ways can this table be filled with all zeros and ones so that there is an even number of ones in every row and every column?


Given four points \( A, B, C, D \) in space, with at most one of the segments \( AB, AC, AD, BC, BD, CD \) having a length greater than 1, find the maximum sum of the lengths of these six segments.

Solution: \boxed{5 + \sqrt{3}}


Let \([x]\) be the largest integer not greater than \(x\). If \(B = [10 + \sqrt{10 + \sqrt{10 + \sqrt{10 + \cdots}}}]\), find the value of \(B\).


Write the result of the following expressions in decimal form:
(a) \(7 \times \frac{2}{3} + 16 \times \frac{5}{12}\)
(b) \(5 - \left(2 \div \frac{5}{3}\right)\)
(c) \(1 + \frac{2}{1 + \frac{3}{1 + 4}}\)


Given positive integer \( n = abc < 10000 \), where \( a \), \( b \), and \( c \) are prime numbers, and the equations \( 2a + 3b = c \) and \( 4a + c + 1 = 4b \) hold, find the value of \( n \).


Let $F_{1}$ and $F_{2}$ be the two foci of an ellipse $C$. Let $AB$ be a chord of the ellipse passing through the point $F_{2}$. In the triangle $\triangle F_{1}AB$, the lengths are given as follows:
$$
|F_{1}A| = 3, \; |AB| = 4, \; |BF_{1}| = 5.
$$

Find $\tan \angle F_{2}F_{1}B$.


Let \( A \) and \( B \) be two sets, and \((A, B)\) be called a "pair". If \( A \neq B \), then \((A, B)\) and \((B, A)\) are considered different "pairs". Find the number of different pairs \((A, B)\) that satisfy the condition \( A \cup B = \{1,2,3,4\} \).


Given \( x, y \in \mathbf{R} \) and \( 2 x^{2} + 3 y^{2} \leq 12 \), find the maximum value of \( |x + 2y| \).


Find the smallest natural number \( N \) such that \( N+2 \) is divisible by 2, \( N+3 \) by 3, ..., \( N+10 \) by 10.


A group of 8 boys and 8 girls was paired up randomly. Find the probability that there is at least one pair with two girls. Round your answer to the nearest hundredth.


In the captain's log, we found the formula $d = p \sqrt{h}$ for determining the distance to the horizon, but the number in place of $p$ is not legible. In the formula, $d$ represents the distance to the horizon in kilometers, and $h$ is the observer's height above sea level in meters. Determine the value of $p$ so that we obtain a usable formula. (Consider the Earth's radius to be 6370 km.)


Two granaries, A and B, originally each stored whole bags of grain. If 90 bags are transferred from granary A to granary B, the number of bags in granary B will be twice the number of bags in granary A. If a certain number of bags are transferred from granary B to granary A, then the number of bags in granary A will be six times the number of bags in granary B. What is the minimum number of bags originally stored in granary A?


Let \( ABC \) be a triangle such that \( AB = 2 \), \( CA = 3 \), and \( BC = 4 \). A semicircle with its diameter on \(\overline{BC}\) is tangent to \(\overline{AB}\) and \(\overline{AC}\). Compute the area of the semicircle.

Solution: \boxed{\frac{27 \pi}{40}}


Inside a convex 13-sided polygon, there are 200 points such that no three of these 213 points (including the vertices of the polygon) lie on the same line. The polygon is divided into triangles, each vertex of which is any three of the given 213 points. What is the maximum number of triangles that could result?


Let \( P \) be a polynomial with \( P(1) = P(2) = \cdots = P(2007) = 0 \) and \( P(0) = 2009! \). \( P(x) \) has leading coefficient 1 and degree 2008. Find the largest root of \( P(x) \).


Find the possible value of \(x+y\) if it is known that \(x^{3}-6x^{2}+15x=12\) and \(y^{3}-6y^{2}+15y=16\).


Find the sum of the coefficients of the polynomial \( P(x) = x^4 - 29x^3 + ax^2 + bx + c \), given that \( P(5) = 11 \), \( P(11) = 17 \), and \( P(17) = 23 \).


Let the function \( f_{0}(x) = |x| \), \( f_{1}(x) = \left| f_{0}(x) - 1 \right| \), \( f_{2}(x) = \left| f_{1}(x) - 2 \right| \). Determine the area of the enclosed region formed by the graph of \( f_{2}(x) \) and the x-axis.


It is easy to see for $n=1,2,3$ that $n$ circles divide the plane into at most $2^n$ parts. Into how many parts can the plane be divided by drawing four circles, by suitably choosing the sizes and mutual positions of the circles?


There are 1987 sets, each with 45 elements. The union of any two sets has 89 elements. How many elements are there in the union of all 1987 sets?


There are 2011 street lamps numbered \(1, 2, 3, \ldots, 2011\). For the sake of saving electricity, it is required to turn off 300 of these lamps. However, the conditions are that no two adjacent lamps can be turned off simultaneously, and the lamps at both ends cannot be turned off. How many ways are there to turn off the lamps under these conditions? (Express your answer in terms of binomial coefficients).


In an isosceles triangle \( ABC \) where \( AB = BC \), side \( AC = 10 \). A circle with a diameter of 15 is inscribed in the angle \( ABC \) such that it touches the side \( AC \) at its midpoint. Find the radius of the circle inscribed in the triangle \( ABC \).


Given an acute-angled triangle \(ABC\). Point \(M\) is the intersection point of its altitudes. Find angle \(A\) if it is known that \(AM = BC\).


Let \( A B C D E F \) be a regular hexagon of area 1. Let \( M \) be the midpoint of \( D E \). Let \( X \) be the intersection of \( A C \) and \( B M \), let \( Y \) be the intersection of \( B F \) and \( A M \), and let \( Z \) be the intersection of \( A C \) and \( B F \). If \([P]\) denotes the area of polygon \( P \) for any polygon \( P \) in the plane, evaluate \([B X C] + [A Y F] + [A B Z] - [M X Z Y]\).


In a pasture where the grass grows evenly every day, the pasture can feed 10 sheep for 20 days, or 14 sheep for 12 days. How many days of grass growth each day is enough to feed 2 sheep?

Solution: \boxed{2 \text{ days}}


In a certain meeting, there are 30 participants. Each person knows at most 5 others among the remaining participants, and in any group of 5 people, there are always two who do not know each other. Find the largest positive integer \( k \) such that there exists a subset of \( k \) people from these 30 people who do not know each other.


We will call a natural number interesting if all its digits are different, and the sum of any two adjacent digits is the square of a natural number. Find the largest interesting number.


How many ordered pairs of positive integers \((x, y)\) satisfy the equation
\[
x \sqrt{y} + y \sqrt{x} + \sqrt{2006 x y} - \sqrt{2006 x} - \sqrt{2006 y} - 2006 = 0 ?
\]


Let \(ABCDEF\) be a regular hexagon. A frog starts at vertex \(A\). Each time, it can jump to one of the two adjacent vertices. If the frog reaches point \(D\) within 5 jumps, it stops jumping; if it does not reach point \(D\) within 5 jumps, it stops after completing 5 jumps. How many different ways can the frog jump from the start until it stops?


Dima needed to arrive at the station at 18:00, where his father was supposed to pick him up by car. However, Dima caught an earlier train and arrived at the station at 17:05. Instead of waiting for his father, he started walking towards him. They met on the way, Dima got into the car, and they arrived home 10 minutes earlier than the calculated time. What was Dima's walking speed before he met his father, if the car's speed was $60 \ \text{km/h}$?

Solution: \boxed{6 \text{ km/h}}