Alice chooses three primes \( p, q, r \) independently and uniformly at random from the set of primes of at most 30. She then calculates the roots of \( p x^{2} + q x + r \). What is the probability that at least one of her roots is an integer?


The function \( f(x) \) satisfies the condition: for any real numbers \( a \) and \( b \), the equality \( f\left(\frac{a+2b}{3}\right) = \frac{f(a)+2f(b)}{3} \) holds. Find the value of the function \( f(2022) \), if \( f(1) = 1 \), \( f(4) = 7 \).


On the \(xy\)-plane, let \(S\) denote the region consisting of all points \((x, y)\) for which
\[
\left|x+\frac{1}{2} y\right| \leq 10, \quad |x| \leq 10, \quad \text{and} \quad |y| \leq 10.
\]

The largest circle centered at \((0,0)\) that can be fitted in the region \(S\) has area \(k \pi\). Find the value of \(k\).


Two natural numbers \(a\) and \(b\) have a sum of 100. Additionally, when \(a\) is divided by 5, the remainder is 2; when \(b\) is divided by 6, the remainder is 3. Find the maximum product of \(a\) and \(b\).


The edge length of cube \( ABCD-A_{1}B_{1}C_{1}D_{1} \) is 1. What is the distance between the lines \( A_{1}C_{1} \) and \( BD_{1} \)?


In a square $\mathrm{ABCD}$, point $\mathrm{E}$ is on $\mathrm{BC}$ with $\mathrm{BE} = 2$ and $\mathrm{CE} = 1$. Point $\mathrm{P}$ moves along $\mathrm{BD}$. What is the minimum value of $\mathrm{PE} + \mathrm{PC}$?


Points \( A \) and \( C \) lie on the circumference of a circle with radius \(\sqrt{50}\). \( B \) is a point inside the circle such that \(\angle A B C = 90^{\circ}\). If \( A B = 6 \) and \( B C = 2\), find the distance from \( B \) to the centre of the circle.


Find the smallest positive integer \( k \) such that \( z^{10} + z^{9} + z^{6} + z^{5} + z^{4} + z + 1 \) divides \( z^k - 1 \).


Let the strictly increasing sequence $\left\{a_{n}\right\}$ consist of positive integers with $a_{7}=120$ and $a_{n+2}=a_{n}+a_{n+1}$ for $n \in \mathbf{Z}_{+}$. Find $a_{8}=$.


Vanya received three sets of candies for New Year. Each set contains three types of candies: hard candies, chocolates, and gummy candies. The total number of hard candies in all three sets is equal to the total number of chocolates in all three sets, and also to the total number of gummy candies in all three sets. In the first set, there are equal numbers of chocolates and gummy candies, and 7 more hard candies than chocolates. In the second set, there are equal numbers of hard candies and chocolates, and 15 fewer gummy candies than hard candies. How many candies are in the third set if it is known that there are no hard candies in it?


From the similarity of the right triangles \(A K C\) and \(B L C\), formed by the segments of the tangent, line \(l\), and the radii \(A K\) and \(B L\) drawn to the points of tangency at time \(T\), we obtain

$$
\frac{a T}{a T - R} = \frac{L + x}{x}
$$

where \(x\) represents the distance \(B C\).

From the given equation, we find \( x = a T \frac{L}{R} - L \). Hence, the speed of the moving point \(C\) at the intersection of the lines is \(a \frac{L}{R}\).

Solution: \boxed{a \frac{L}{R}}


Given the function \( f(x) = \frac{1}{2} + \log_{2} \frac{x}{1-x} \), define \( S_{n} = f\left(\frac{1}{n}\right) + f\left(\frac{2}{n}\right) + f\left(\frac{3}{n}\right) + \cdots + f\left(\frac{n-1}{n}\right) \), where \( n \in \mathbb{N}^* (n \geq 2) \). Find \( S_{n} \).


Two circles with radii $\sqrt{5}$ and $\sqrt{2}$ intersect at point $A$. The distance between the centers of the circles is 3. A line through point $A$ intersects the circles at points $B$ and $C$ such that $A B = A C$ (point $B$ does not coincide with $C$). Find $A B$.


How many times does 5 occur as a divisor in the numbers from 1 to 50000?


Increasing sequence of positive integers \( a_1, a_2, a_3, \ldots \) satisfies \( a_{n+2} = a_n + a_{n+1} \) (for \( n \geq 1 \)). If \( a_7 = 120 \), what is \( a_8 \) equal to?


In trapezoid \(ABCD\), side \(AB\) is perpendicular to the bases \(AD\) and \(BC\). Point \(E\) is the midpoint of side \(CD\).

Find the ratio \(AD : BC\) if \(AE = 2AB\) and \(AE \perp CD\).


Find the following limits as \(n\) approaches infinity:
1. \(\lim_{n \to \infty} \sum_{1 \le i \le n} \frac{1}{\sqrt{n^2 + i^2}}\)
2. \(\lim_{n \to \infty} \sum_{1 \le i \le n} \frac{1}{\sqrt{n^2 + i}}\)
3. \(\lim_{n \to \infty} \sum_{1 \le i \le n^2} \frac{1}{\sqrt{n^2 + i}}\)

Solution: \boxed{\infty}


Alice has an equilateral triangle \(ABC\) with an area of 1. Point \(D\) is on \(BC\) such that \(BD = DC\), point \(E\) is on \(CA\) such that \(CE = 2EA\), and point \(F\) is on \(AB\) such that \(2AF = FB\). Line segments \(AD\), \(BE\), and \(CF\) intersect at a single point \(M\). What is the area of triangle \(EMC\)?


A truck and a car are moving in the same direction on adjacent lanes at speeds of 65 km/h and 85 km/h respectively. How far apart will they be 3 minutes after they are even with each other?

Solution: \boxed{1 \text{ km}}


Let \( a \), \( b \), and \( c \) be the side lengths of a triangle, and assume that \( a \leq b \) and \( a \leq c \). Let \( x = \frac{b + c - a}{2} \). If \( r \) and \( R \) denote the inradius and circumradius, respectively, find the minimum value of \( \frac{a x}{r R} \).


Given a sequence $\left\{a_{n}\right\}$ that satisfies the conditions: $a_{1}=1$, $a_{2}=2$, and for any three consecutive terms $a_{n-1}, a_{n}, a_{n+1}$, the relation $a_{n}=\frac{(n-1) a_{n-1}+(n+1) a_{n+1}}{2 n}$ holds. Determine the general term $a_{n}$.

Solution: \boxed{3 - \frac{2}{n}}


A circle is divided into seven arcs such that the sum of any two adjacent arcs does not exceed $103^\circ$. Determine the largest possible value of $A$ such that, in any such division, each of the seven arcs contains at least $A^\circ$.


The altitudes of an acute isosceles triangle, where \(AB = BC\), intersect at point \(H\). Find the area of triangle \(ABC\), given \(AH = 5\) and the altitude \(AD\) is 8.


Vovochka approached an arcade machine which displayed the number 0 on the screen. The rules of the game stated: "The screen shows the number of points. If you insert a 1 ruble coin, the number of points increases by 1. If you insert a 2 ruble coin, the number of points doubles. If you reach 50 points, the machine gives out a prize. If you get a number greater than 50, all the points are lost." What is the minimum amount of rubles Vovochka needs to get the prize? Answer: 11 rubles.


Given 6 digits: \(0, 1, 2, 3, 4, 5\). Find the sum of all four-digit even numbers that can be written using these digits (the same digit can be repeated in a number).


Show that in any connected planar graph, denoting $s$ as the number of vertices, $f$ as the number of faces, and $a$ as the number of edges, we have: $f = a - s + 2$.

Solution: \boxed{f = a - s + 2}


From the set \( \{1, 2, 3, \ldots, 999, 1000\} \), select \( k \) numbers. If among the selected numbers, there are always three numbers that can form the side lengths of a triangle, what is the smallest value of \( k \)? Explain why.


Fill each of the boxes and triangles in the following equation with a natural number, so that the equation holds:
\[ \square^{2} + 12 = \triangle^{2} \]
Then find: \(\square + \triangle = 6\).


Oleg has four cards, each with a natural number on each side (a total of 8 numbers). He considers all possible sets of four numbers where the first number is written on the first card, the second number on the second card, the third number on the third card, and the fourth number on the fourth card. For each set of four numbers, he writes the product of the numbers in his notebook. What is the sum of the eight numbers on the cards if the sum of the sixteen numbers in Oleg’s notebook is $330?


In an acute-angled triangle \( \triangle ABC \) with sides \( a \), \( b \), and \( c \) opposite the angles \( A \), \( B \), and \( C \) respectively, if \( 2a^{2} = 2b^{2} + c^{2} \), what is the minimum value of \( \tan A + \tan B + \tan C \)?


Alice and Bob are independently trying to figure out a secret password to Cathy's bitcoin wallet. They know:
- It is a 4-digit number whose first digit is 5;
- It is a multiple of 9;
- The larger number is more likely to be the password than a smaller number.

Alice knows the second and third digits, and Bob knows the third and fourth digits. Initially:
Alice: "I have no idea what the number is."
Bob: "I have no idea too."

After this conversation, they both knew which number they should try first. Identify this number.


How many positive integers less than 500 have exactly 15 positive integer factors?


Let \( f(x) \) be a function defined on \(\mathbf{R}\), such that \( f(0)=1008 \), and for any \( x \in \mathbf{R} \), it holds that:
\[ f(x+4) - f(x) \leq 2(x+1) \]
\[ f(x+12) - f(x) \geq 6(x+5) \]
Find \( \frac{f(2016)}{2016} \).


Erin lists all three-digit primes that are 21 less than a square. What is the mean of the numbers in Erin's list?


Using the digits 1 through 9 to form three three-digit numbers \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) (each digit can be used only once), what is the maximum value of \( a + b - c \)?


Given that \( f(x) \) and \( g(x) \) are two quadratic functions both with a leading coefficient of 1, where \( g(6) = 35 \) and \( \frac{f(-1)}{g(-1)} = \frac{f(1)}{g(1)} = \frac{21}{20} \), what is \( f(6) \)?


\(ABCD\) is a convex quadrilateral. It is known that \(\angle CAD = \angle DBA = 40^\circ\), \(\angle CAB = 60^\circ\), and \(\angle CBD = 20^\circ\). Find the angle \(\angle CDB\).


Huanhuan and Lele are playing a game together. In the first round, they both gain the same amount of gold coins, and in the second round, they again gain the same amount of gold coins.

At the beginning, Huanhuan says: "The number of my gold coins is 7 times the number of your gold coins."

At the end of the first round, Lele says: "The number of your gold coins is 6 times the number of my gold coins now."

At the end of the second round, Huanhuan says: "The number of my gold coins is 5 times the number of your gold coins now."

Find the minimum number of gold coins Huanhuan had at the beginning.


Six IMO competitions are hosted sequentially by two Asian countries, two European countries, and two African countries, where each country hosts once but no continent can host consecutively. How many such arrangements are possible?


Two points, A and B, are 30 cm apart. Two strings, A and B, start simultaneously from A and B respectively and move towards each other. String A is 151 cm long and moves at 2 cm per second; String B is 187 cm long and moves at 3 cm per second. If both strings are ignited at their tail ends upon departure, string A burns at a rate of 1 cm per second and string B burns at a rate of 2 cm per second. How many seconds will it take for the two strings to completely pass each other from the moment they meet?

Solution: \boxed{40 \ \text{s}}


Find all triples of natural numbers \( n \), \( p \), and \( q \) such that \( n^p + n^q = n^{2010} \).

Solution: \boxed{(2, 2009, 2009)}


What work is required to stretch a spring by \(0.06 \, \text{m}\), if a force of \(1 \, \text{N}\) stretches it by \(0.01 \, \text{m}\)?

Solution: \boxed{0.18 \, \text{J}}


A settlement is built in the shape of a square consisting of 3 blocks by 3 blocks (each block is a square with a side length $b$, for a total of 9 blocks). What is the shortest path that a paver must take to pave all the streets if they start and end their path at a corner point $A$? (The sides of the square are also streets).


Solve the inequality \(\frac{\sqrt{\frac{x}{\gamma}+(\alpha+2)}-\frac{x}{\gamma}-\alpha}{x^{2}+a x+b} \geqslant 0\).

Indicate the number of integer roots of this inequality in the answer. If there are no integer roots or there are infinitely many roots, indicate the number 0 on the answer sheet.

Given:
\[
\alpha = 3, \gamma = 1, a = -15, b = 54.
\]


The equation \( x^{4} - 7x - 3 = 0 \) has exactly two real roots \( a \) and \( b \), where \( a > b \). Find the value of the expression \( \frac{a - b}{a^{4} - b^{4}} \).


Find the number of 9-digit numbers in which each digit from 1 to 9 occurs exactly once, the digits 1, 2, 3, 4, 5 are in ascending order, and the digit 6 is placed before the digit 1 (for example, 916238457).


The plane is covered by a grid of squares with a side length of 1. Is it possible to construct an equilateral triangle with vertices at the grid points?

Solution: \boxed{\text{No}}


Let \([x]\) denote the greatest integer less than or equal to the real number \(x\). Define
\[ A = \left[\frac{19}{20}\right] + \left[\frac{19^2}{20}\right] + \cdots + \left[\frac{19^{2020}}{20}\right]. \]

Find the remainder when \(A\) is divided by 11.


Team A and Team B each field 7 players who will compete in a Go match according to a predetermined order. Initially, the 1st player from each team competes. The loser is eliminated, and the winner competes with the next player from the losing team. This continues until all players from one team are eliminated. The other team is declared the winner, forming a complete match process. What is the total number of possible match processes?


The numbers from 1 to 37 are written in a sequence such that the sum of any initial segment of numbers is divisible by the next number following that segment.

What number is in the third position if the first number is 37 and the second number is 1?


Among all possible triangles \(ABC\) such that \(BC = 2 \sqrt[4]{3}\) and \(\angle BAC = \frac{\pi}{3}\), find the one with the maximum area. What is the area of this triangle?


A square and an equilateral triangle together have the property that the area of each is the perimeter of the other. Find the square's area.

Solution: \boxed{12 \sqrt[3]{4}}


Let \( a \) and \( b \) be real numbers such that there exist distinct real numbers \( m \), \( n \), and \( p \) satisfying the following equations:

$$
\left\{
\begin{array}{l}
m^{3} + a m + b = 0 \\
n^{3} + a n + b = 0 \\
p^{3} + a p + b = 0
\end{array}
\right.
$$

Show that \( m + n + p = 0 \).


ABCD is a convex quadrilateral with AB = BC, ∠CBD = 2 ∠ADB, and ∠ABD = 2 ∠CDB. Show that AD = DC.

Solution: \boxed{AD = DC}


In $\triangle ABC$, point $O$ is the midpoint of $BC$. A line passing through point $O$ intersects line segments $AB$ and $AC$ at different points $M$ and $N$, respectively. If $\overrightarrow{AB} = m \overrightarrow{AM}$ and $\overrightarrow{AC} = n \overrightarrow{AN}$, find the value of $m + n$.


Let \( p \) and \( q \) be integers such that \( p + q = 2010 \). If both roots of the equation \( 10x^{2} + px + q = 0 \) are positive integers, find the sum of all possible values of \( p \).


\(ABCD\) is a rectangle with \(AB = 2\) and \(BC = 1\). A point \(P\) is randomly selected on \(CD\). Find the probability that \(\angle APB\) is the largest among the three interior angles of \(\triangle PAB\).


Five cards labeled A, B, C, D, and E are placed consecutively in a row. How many ways can they be re-arranged so that no card is moved more than one position away from where it started? (Not moving the cards at all counts as a valid re-arrangement.)


Let \( a, b, c, d \) be any positive real numbers. Determine the range of the sum
$$
S = \frac{a}{a+b+d} + \frac{b}{a+b+c} + \frac{c}{b+c+d} + \frac{d}{a+c+d}.
$$

Solution: \boxed{(1, 2)}


Calculate the area of the figure bounded by the graphs of the functions:

$$
x=4-(y-1)^{2}, \quad x=y^{2}-4 y+3
$$


Given three positive numbers \(a, b, c\) satisfying \(a \leqslant b+c \leqslant 3a\), \(3b^{2} \leqslant a(a+c) \leqslant 5b^{2}\), find the minimum value of \(\frac{b-2c}{a}\).


How many integers from 1 to 33000 are there that are neither divisible by 3 nor by 5, but are divisible by 11?


Solve the inequality

$$
\sqrt{3 x-7}-\sqrt{3 x^{2}-13 x+13} \geqslant 3 x^{2}-16 x+20
$$

In the answer, indicate the sum of all integer values \( x \) that satisfy the inequality.


The fishermen caught several crucian carps and pikes. Each fisherman caught as many crucian carps as the total number of pikes caught by all the other fishermen. How many fishermen were there if the total number of crucian carps caught is 10 times the total number of pikes caught? Justify your answer.


Find all prime numbers \( p \) such that both \( 4p^2 + 1 \) and \( 6p^2 + 1 \) are also prime numbers.


Anton thought of a three-digit number, and Alex is trying to guess it. Alex successively guessed the numbers 109, 704, and 124. Anton observed that each of these numbers matches the thought number exactly in one digit place. What number did Anton think of?


Let \( S(n) \) represent the sum of the digits of the non-negative integer \( n \). For example, \( S(1997) = 1 + 9 + 9 + 7 = 26 \). Find the value of \( S(1) + S(2) + \cdots + S(2012) \).


A three-digit number \( X \) was composed of three different digits, \( A, B, \) and \( C \). Four students made the following statements:

- Petya: "The largest digit in the number \( X \) is \( B \)."
- Vasya: "\( C = 8 \)."
- Tolya: "The largest digit is \( C \)."
- Dima: "\( C \) is the arithmetic mean of the digits \( A \) and \( B \)."

Find the number \( X \), given that exactly one of the students was mistaken.


A boy named Vasya wrote down the non-zero coefficients of a 9th degree polynomial \( P(x) \) in a notebook. Then, he calculated the derivative of the resulting polynomial and wrote down its non-zero coefficients, continuing this process until he obtained a constant, which he also wrote down.

What is the minimum number of different numbers that he could have written down?

Coefficients are recorded with their signs, and constant terms are written down as well. If there is a monomial of the form \( \pm x^n \), it is written as \( \pm 1 \).


A stalker throws a small nut from the Earth's surface at an angle of $\alpha=30^{\circ}$ to the horizontal with an initial speed $v_{0}=10 \, \mathrm{m}/\mathrm{s}$. The normal acceleration due to gravity is $g=10 \, \mathrm{m}/\mathrm{s}^{2}$. At the highest point of its trajectory, the nut enters a zone of gravitational anomaly (an area where the acceleration due to gravity changes its magnitude sharply), and continues to move within it. As a result, the nut falls to the ground at a distance $S=3 \sqrt{3} \, \mathrm{m}$ from the stalker. Determine the acceleration due to gravity inside the anomaly. (15 points)

Solution: \boxed{250 \, \text{m/s}^2}


Find the largest positive integer \( n \) such that the inequality \(\frac{8}{15} < \frac{n}{n+k} < \frac{7}{13}\) holds for exactly one integer \( k \). (5th American Mathematics Invitational, 1987)


Given that $\left\{a_{n}\right\}$ is a geometric series and $a_{1} a_{2017}=1$. If $f(x)=\frac{2}{1+x^{2}}$, then find the value of $f\left(a_{1}\right)+f\left(a_{2}\right)+f\left(a_{3}\right)+\cdots+f\left(a_{2017}\right)$.


In a tournament, any two players play against each other. Each player earns one point for a victory, 1/2 for a draw, and 0 points for a loss. Let \( S \) be the set of the 10 lowest scores. We know that each player obtained half of their score by playing against players in \( S \).

a) What is the sum of the scores of the players in \( S \)?

b) Determine how many participants are in the tournament.

Note: Each player plays only once with each opponent.


Select $k$ edges and diagonals from the faces of a cube such that any two chosen line segments lie on skew lines. What is the maximum value of $k$?


The sequence \(101, 104, 109, 116, \cdots\) has the general term \(a_{n} = 100 + n^{2}\). For each \(n\), let \(d_{n}\) represent the greatest common divisor (GCD) of \(a_{n}\) and \(a_{n+1}\). Determine the maximum value of \(d_{n}\).


Ksyusha runs twice as fast as she walks (both speeds are constant).

On Tuesday, when she left home for school, she first walked, and then, when she realized she was late, she started running. The distance Ksyusha walked was twice the distance she ran. As a result, she reached the school from home in exactly 30 minutes.

On Wednesday, Ksyusha left home even later, so she had to run twice the distance she walked. How many minutes did it take her to get from home to school on Wednesday?


In how many ways can a rectangular board of size \(2 \times 13\) be covered with rectangular tiles of size \(1 \times 2\)? (The tiles are placed such that they do not overlap and completely fit on the board.)


The base of the quadrilateral prism $A B C D A_{1} B_{1} C_{1} D_{1}$ is a rhombus $A B C D$, where $B D=3$ and $\angle A D C=60^{\circ}$. A sphere passes through the vertices $D, C, B, B_{1}, A_{1}, D_{1}$.

a) Find the area of the circle obtained in the cross-section of the sphere by the plane passing through the points $A_{1}, C_{1}$, and $D_{1}$.

b) Find the angle $B_{1} C_{1} A$.

c) It is additionally known that the radius of the sphere is 2. Find the volume of the prism.


Let \( w, x, y, \) and \( z \) be positive real numbers such that
\[
\begin{aligned}
0 & \neq \cos w \cos x \cos y \cos z \\
2 \pi & = w + x + y + z \\
3 \tan w & = k(1 + \sec w) \\
4 \tan x & = k(1 + \sec x) \\
5 \tan y & = k(1 + \sec y) \\
6 \tan z & = k(1 + \sec z)
\end{aligned}
\]
(Here \( \sec t \) denotes \( \frac{1}{\cos t} \) when \( \cos t \neq 0 \).) Find \( k \).


If the value of the 5th term in the expansion of $\left(x \sqrt{x}-\frac{1}{x}\right)^{6}$ is $\frac{15}{2}$, then $\lim _{n \rightarrow \infty}\left(x^{-1}+x^{-2}+\cdots+x^{-n}\right)=$


The probability of an event occurring in each of 900 independent trials is 0.5. Find the probability that the relative frequency of the event will deviate from its probability by no more than 0.02.


In triangle \(ABC\), it is known that \(AB = c\), \(BC = a\), \(AC = b\). Point \(O\) is the center of the circle that touches side \(AB\) and the extensions of sides \(AC\) and \(BC\). Point \(D\) is the intersection of ray \(CO\) with side \(AB\). Find the ratio \(\frac{CO}{OD}\).

Solution: \boxed{\frac{a+b}{c}}


For any positive integer \( n \), let \( f(n) \) denote the index of the highest power of 2 which divides \( n! \). For example, \( f(10) = 8 \) since \( 10! = 2^8 \times 3^4 \times 5^2 \times 7 \). Find the value of \( f(1) + f(2) + \cdots + f(1023) \).


Given a positive integer \( n \leq 500 \) with the following property: If one randomly selects an element \( m \) from the set \(\{1, 2, \cdots, 500\}\), the probability that \( m \) divides \( n \) is \(\frac{1}{100}\). Determine the maximum value of \( n \).


In a math competition consisting of problems $A$, $B$, and $C$, among the 39 participants, each person answered at least one problem correctly. Among the people who answered $A$ correctly, the number of people who only answered $A$ is 5 more than the number of people who answered other problems as well. Among the people who did not answer $A$ correctly, the number of people who answered $B$ is twice the number of people who answered $C$. Additionally, the number of people who only answered $A$ equals the sum of the number of people who only answered $B$ and the number of people who only answered $C$. What is the maximum number of people who answered $A$ correctly?


Given the ellipse \(\frac{y}{4} + x^{2} = 1\), let \(P\) be an arbitrary point on the ellipse. Draw lines through point \(P\) that are parallel to \(l_1: y=2x\) and \(l_2: y=-2x\). These lines intersect the lines \(l_2\) and \(l_1\) at points \(M\) and \(N\) respectively. Find the maximum value of \(|MN|\).


Given the sequence $\left\{a_{n}\right\}$ where $a_{1}=1, a_{2}=2,$ and the recurrence relation $a_{n} a_{n+1} a_{n+2}=a_{n}+a_{n+1}+a_{n+2}$ with the condition $a_{n+1} a_{n+2} \neq 1$, find $S_{1999}=\sum_{n=1}^{1999} a_{n}$.


In two weeks three cows eat all the grass on two hectares of land, together with all the grass that regrows there during the two weeks. In four weeks, two cows eat all the grass on two hectares of land, together with all the grass that regrows there during the four weeks.

How many cows will eat all the grass on six hectares of land in six weeks, together with all the grass that regrows there over the six weeks?

(Assume: - the quantity of grass on each hectare is the same when the cows begin to graze,
- the rate of growth of the grass is uniform during the time of grazing,
- and the cows eat the same amount of grass each week.)


Let circle $\Gamma_{1}$ be the circumscribed circle of $\triangle ABC$. Points $D$ and $E$ are the midpoints of arcs $\overparen{AB}$ (excluding point $C$) and $\overparen{AC}$ (excluding point $B$), respectively. The incircle of $\triangle ABC$ touches sides $AB$ and $AC$ at points $F$ and $G$, respectively. The lines $EG$ and $DF$ intersect at point $X$. If $X D = X E$, show that $AB = AC$.

Solution: \boxed{AB = AC}


Determine the share of the Japanese yen in the currency structure of the National Wealth Fund (NWF) as of 01.12.2022 using one of the following methods:

First method:

a) Find the total amount of NWF funds placed in Japanese yen as of 01.12.2022:

\[ J P Y_{22} = 1388.01 - 41.89 - 2.77 - 309.72 - 554.91 - 0.24 = 478.48 \text{ (billion rubles)} \]

b) Determine the share of the Japanese yen in the currency structure of NWF funds as of 01.12.2022:

\[ \alpha_{22}^{J P Y} = \frac{478.48}{1388.01} \approx 34.47\% \]

c) Calculate by how many percentage points and in what direction the share of the Japanese yen in the currency structure of NWF funds changed during the period considered in the table:

\[ \Delta \alpha^{J P Y} = \alpha_{22}^{J P Y} - \alpha_{21}^{J P Y} = 34.47 - 47.06 = -12.59 \approx -12.6 \text{ (percentage points)} \]

Second method:

a) Determine the share of the euro in the currency structure of NWF funds as of 01.12.2022:

\[ \alpha_{22}^{E U R} = \frac{41.89}{1388.01} \approx 3.02\% \]

b) Determine the share of the Japanese yen in the currency structure of NWF funds as of 01.12.2022:

\[ \alpha_{22}^{J P Y} = 100 - 3.02 - 0.2 - 22.31 - 39.98 - 0.02 = 34.47\% \]

c) Calculate by how many percentage points and in what direction the share of the Japanese yen in the currency structure of NWF funds changed during the period considered in the table:

\[ \Delta \alpha^{J P Y} = \alpha_{22}^{J P Y} - \alpha_{21}^{J P Y} = 34.47 - 47.06 = -12.59 \approx -12.6 \text{ (percentage points)} \]


Calculate the value of the total differential of the function \( z = \operatorname{arcctg} \frac{x}{y} \) at \( x = 1, y = 3, dx = 0.01, dy = -0.05 \).


Let \( f: \mathbb{Z}_{>0} \rightarrow \mathbb{Z} \) be a function with the following properties:

(i) \( f(1) = 0 \),

(ii) \( f(p) = 1 \) for all prime numbers \( p \),

(iii) \( f(x y) = y f(x) + x f(y) \) for all \( x, y \) in \( \mathbb{Z}_{>0} \).

Determine the smallest integer \( n \geq 2015 \) that satisfies \( f(n) = n \).


Since \(\left(1-\frac{1}{k^2}\right)=\left(1-\frac{1}{k}\right)\left(1+\frac{1}{k}\right)=\frac{k-1}{k} \cdot \frac{k+1}{k}\), find the value of the product

\[ \left(1-\frac{1}{4}\right)\left(1-\frac{1}{9}\right)\left(1-\frac{1}{16}\right) \cdots \left(1-\frac{1}{2021^2}\right)\left(1-\frac{1}{2022^2}\right). \]


A positive integer sequence has its first term as 8 and its second term as 1. From the third term onwards, each term is the sum of the two preceding terms. What is the remainder when the 2013th term in this sequence is divided by 105?


Find the area enclosed by the graph \( x^{2}+y^{2}=|x|+|y| \) on the \( xy \)-plane.

Solution: \boxed{\pi + 2}


Simplify the expression \(\frac{\sin 4\alpha + \sin 5\alpha + \sin 6\alpha}{\cos 4\alpha + \cos 5\alpha + \cos 6\alpha}\).

Solution: \boxed{\tan 5\alpha}


Given two circles, we want to draw a common tangent - a line that touches both circles. How many such lines exist? (The answer depends on the ratio of the circles' radii and the distance between their centers.)


A gear mechanism has a smaller gear with 12 teeth and a larger gear with 32 teeth. Due to a manufacturing defect, one tooth on the smaller gear and one slot on the larger gear stick and make noise upon contact. At what intervals do we hear the squeaks if the larger gear takes 3 seconds for one full revolution? How can the squeaking be eliminated, assuming all other tooth-slot pairings are flawless and silent?

Solution: \boxed{9 \text{ seconds}}


The sum of the integer parts of all positive real solutions \( x \) that satisfy \( x^{4}-x^{3}-2 \sqrt{5} x^{2}-7 x^{2}+\sqrt{5} x+3 x+7 \sqrt{5}+17=0 \) is


In the diagram, \(\triangle ABC\) is right-angled at \(C\). Point \(D\) is on \(AC\) so that \(\angle ABC = 2 \angle DBC\). If \(DC = 1\) and \(BD = 3\), determine the length of \(AD\).


What will the inflation be over two years:
$$
\left((1+0,015)^{\wedge} 2-1\right)^{*} 100 \%=3,0225 \%
$$

What will be the real yield of a bank deposit with an extension for the second year:
$$
(1,07 * 1,07 /(1+0,030225)-1) * 100 \%=11,13 \%
$$


Calculate the line integral

$$
\int_{L}(x-y) d x+d y+z d z
$$

from the point \( M(2,0,4) \) to the point \( N(-2,0,4) \) (where \( y \geq 0 \)) along the curve \( L \), formed by the intersection of the paraboloid \( z=x^{2}+y^{2} \) and the plane \( z=4 \).


A sculpture in the Guggenheim Museum in New York is shaped like a cube. A bug that has landed on one of the vertices wants to inspect the sculpture as quickly as possible so it can move on to other exhibits (for this, it only needs to reach the opposite vertex of the cube). Which path should it choose?


Let the function \( f(x) = x - \ln(a x + 2a + 1) + 2 \). If for any \( x \geq -2 \), \( f(x) \geq 0 \) always holds, then the range of the real number \( a \) is \(\quad\).

Solution: \boxed{[0, 1]}


Let the function \( f(x) \) have a derivative \( f'(x) \) on \( \mathbf{R} \), such that for any \( x \in \mathbf{R} \), \( f(x) + f(-x) = x^2 \). In the interval \( (0, +\infty) \), \( f'(x) > x \). If \( f(1+a) - f(1-a) \geq 2a \), then the range of the real number \( a \) is ______.


In the unit cube \(ABCD-A_1B_1C_1D_1\), points \(E\) and \(F\) are the midpoints of edges \(AB\) and \(BC\) respectively. Find the distance from point \(D\) to the plane \(B_1EF\).


An integer with four digits is a multiple of 5. When this integer is divided by 11, 7, and 9, the remainders are 7, 4, and 4 respectively. What is the smallest such integer?


The cross-section of a regular triangular pyramid passes through the midline of the base and is perpendicular to the base. Find the area of the cross-section if the side of the base is 6 and the height of the pyramid is 8.


On the ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$, different points $A\left(x_{1}, y_{1}\right), B\left(4, \frac{9}{5}\right), C\left(x_{2}, y_{2}\right)$ are such that their distances to the focus $F(4,0)$ form an arithmetic sequence. If the perpendicular bisector of segment $A C$ intersects the $x$-axis at point $T$, find the slope $k$ of the line $BT$. $\qquad$ .


On the side $BC$ of an acute-angled triangle $ABC$ ($AB \neq AC$), a semicircle is constructed with $BC$ as its diameter. The semicircle intersects the altitude $AD$ at point $M$. Given $AD = a$, $MD = b$, and $H$ is the orthocenter of the triangle $ABC$, find $AH$.

Solution: \boxed{\frac{a^2 - b^2}{a}}


Six pirates - a captain and five crew members - are sitting around a campfire facing the center. They need to divide a treasure of 180 gold coins. The captain proposes a way to divide the coins (i.e., how many coins each pirate should receive: each pirate must receive a whole non-negative number of coins; different pirates may receive different amounts of coins). After this, the other five pirates vote on the captain's proposal. A pirate will vote "yes" only if he receives more coins than each of his two neighbors. The proposal is accepted if at least three out of the five crew members vote "yes".

What is the maximum number of coins the captain can receive under these rules?


A function \( f: A \rightarrow A \) is called idempotent if \( f(f(x)) = f(x) \) for all \( x \in A \). Let \( I_{n} \) be the number of idempotent functions from \(\{1, 2, \ldots, n\}\) to itself. Compute
\[
\sum_{n=1}^{\infty} \frac{I_{n}}{n!}.
\]

Solution: \boxed{e^e - 1}


In the figure given, there is a point \( P \) within \(\angle M A N\), and it is known that \( \tan \angle M A N = 3 \). The distance from point \( P \) to the line \( A N \) is \( P D = 12 \) and \( A D = 30 \). A line passing through \( P \) intersects \( A N \) and \( A M \) at points \( B \) and \( C \) respectively. Find the minimum area of \(\triangle A B C\).


A first-grader has one hundred cards with natural numbers from 1 to 100 written on them, as well as a large supply of "+" and "=" signs. What is the maximum number of correct equations they can form? (Each card can be used no more than once, each equation can have only one "=" sign, and cards cannot be flipped or combined to create new numbers.)


Given the set $\{1,2,3,4,5,6,7,8,9,10\}$, determine the number of subsets of this set that contain at least 2 elements, such that the absolute difference between any two elements in each subset is greater than 1.


If a complex number \( z \) satisfies \( |z|=1 \), and \( z^2 = a + bi \), where \( a \) and \( b \) are real numbers, then the maximum value of \( a + b \) is ______.


Find the natural number that has six natural divisors (including one and the number itself), two of which are prime, and the sum of all its natural divisors is equal to 78.


A \(4 \times 4\) Sudoku grid is filled with digits so that each column, each row, and each of the four \(2 \times 2\) sub-grids that compose the grid contains all of the digits from 1 to 4.

Find the total number of possible \(4 \times 4\) Sudoku grids.


In the diagram below, points $E$ and $F$ are located on sides $AB$ and $BD$ of triangle $\triangle ABD$ such that $AE = AC$ and $CD = FD$. If $\angle ABD = 60^\circ$, determine the measure of angle $\angle ECF$.


Julia possesses a deck of 18 cards, numbered from 1 to 18. After shuffling, she distributes the cards face down in 3 rows and 6 columns.

Julia chooses a sum \(X\) and flips two cards. If the sum is \(X\), she removes the pair from the table, but if it is not \(X\), she returns the cards to their original positions. She repeats this process until all pairs with a sum equal to \(X\) are flipped.

a) If she chooses \(X=8\), which pairs will be removed from the table?

b) For which value of \(X\) will all the cards be removed from the table?

c) For a specific \(X\), exactly 2 cards remain on the table. How many possible values of \(X\) are there?


Given the equation:

\[ A=\operatorname{tg}\left(\frac{7 \pi}{4}+\frac{1}{2} \arccos \frac{2 a}{b}\right)+\operatorname{tg}\left(\frac{7 \pi}{4}-\frac{1}{2} \arccos \frac{2 a}{b}\right) \]

Find the value of \(A\).

Solution: \boxed{A = -\frac{b}{a}}


Somewhere in the universe, \(n\) students are taking a 10-question math competition. Their collective performance is called laughable if, for some pair of questions, there exist 57 students such that either all of them answered both questions correctly or none of them answered both questions correctly. Compute the smallest \(n\) such that the performance is necessarily laughable.


A convex polyhedron has 32 faces, each of which is either a triangle or a pentagon. For the \( V \) vertices, each vertex has \( T \) triangular faces and \( P \) pentagonal faces intersecting. Find the value of \( P + T + V \).


Integers \( n \) and \( m \) satisfy the inequalities \( 3n - m < 5 \), \( n + m > 26 \), and \( 3m - 2n < 46 \). What values can \( 2n + m \) take? List all possible options.


Fill in the numbers 1, 2, 3, ..., 15, 16 into the 16 squares in the table below, and satisfy the following conditions.

\[
\begin{array}{|c|c|c|c|}
\hline
A & B & C & D \\
\hline
E & F & G & H \\
\hline
J & K & L & M \\
\hline
N & P & Q & R \\
\hline
\end{array}
\]

1. \( A + C + F = 10 \)
2. \( B + H = R \)
3. \( D - C = 13 \)
4. \( E \times M = 126 \)
5. \( F + G = 21 \)
6. \( G \div J = 2 \)
7. \( H \times M = 36 \)
8. \( J \times P = 80 \)
9. \( K - N = Q \)

What is \( L \)?


Find all functions \( f \) from the reals to the reals such that
\[ f(f(x) + y) = 2x + f(f(y) - x) \]
for all real \( x \) and \( y \).

Solution: \boxed{f(x) = x + c}


Each of \( A \) and \( B \) is a four-digit palindromic integer, \( C \) is a three-digit palindromic integer, and \( A - B = C \). What are the possible values of \( C \)? [A palindromic integer reads the same 'forwards' and 'backwards'.]


Given that point \( O \) is the origin of coordinates, the curves \( C_{1}: x^{2} - y^{2} = 1 \) and \( C_{2}: y^{2} = 2px \) intersect at points \( M \) and \( N \). If the circumcircle of \( \triangle OMN \) passes through the point \( P\left(\frac{7}{2}, 0\right) \), then what is the equation of the curve \( C_{2} \)?

Solution: \boxed{y^2 = \frac{3}{2}x}


Consider the sequence of Fibonacci numbers defined by the recursion $f_{1}=f_{2}=1, f_{n}=f_{n-1}+f_{n-2} \text{ for } n \geq 3$. Suppose that for positive integers $a$ and $b$, the fraction $\frac{a}{b}$ is smaller than one of the fractions $\frac{f_{n}}{f_{n-1}}$ or $\frac{f_{n+1}}{f_{n}}$ and larger than the other. Show that $b \geq f_{n+1}$.

Solution: \boxed{b \geq f_{n + 1}}


ABC is a triangle. The bisector of ∠C meets AB at D. Show that \(CD^2 < CA \cdot CB\).

Solution: \boxed{CD^2 < CA \cdot CB}


In a country, some cities are connected by direct bus routes. The residents of this country consider an odd natural number \( n \) to be unlucky, so there does not exist a cyclic route consisting of exactly \( n \) trips (and not entering any city twice). Nonetheless, a path containing exactly 100 trips (and not passing through any city twice) exists from any city to any other city. For which odd \( n \) is this possible?

Solution: \boxed{n = 101}


A sealed horizontal cylindrical vessel of length \( l \) is divided into two parts by a movable partition. On one side of the partition, there is 1 mole of oxygen, and on the other side, there is 1 mole of helium and 1 mole of oxygen, with the partition being initially in equilibrium. At a certain moment, the partition becomes permeable to helium but remains impermeable to oxygen. Find the displacement of the partition. The temperatures of the gases are the same and do not change during the process.


In trapezoid \( ABCD \), point \( X \) is taken on the base \( BC \) such that segments \( XA \) and \( XD \) divide the trapezoid into three similar but pairwise unequal, non-isosceles triangles. The side \( AB \) has a length of 5. Find \( XC \cdot BX \).


The pages in a book are numbered as follows: the first sheet contains two pages (numbered 1 and 2), the second sheet contains the next two pages (numbered 3 and 4), and so on. A mischievous boy named Petya tore out several consecutive sheets: the first torn-out page has the number 185, and the number of the last torn-out page consists of the same digits but in a different order. How many sheets did Petya tear out?


Let \( a \) and \( b \) be real numbers, and there exists a complex number \( z \) such that \( |z| \leq 1 \), and \( z + \bar{z}|z| = a + b \mathrm{i} \). Find the maximum value of \( ab \).


Let us denote the product of all natural numbers from 1 to $m$ by $m!$, i.e., $m! = 1 \cdot 2 \cdot 3 \cdot \ldots \cdot (m-1) \cdot m$. Find all values of $m$ for which the sum $1! + 2! + 3! + \ldots + m!$ is a perfect square.

Solution: \boxed{m=1, \, m=3}


For a natural number \(a\), \(S(a)\) represents the sum of the digits of \(a\) (for example, \(S(123) = 1 + 2 + 3 = 6\)). If the digits of a natural number \(n\) are all different and \(S(3n) = 3S(n)\), what is the maximum value of \(n\)?


Let \([x]\) be the largest integer not greater than \(x\). If \( A = \left\lfloor \frac{2008 \times 80 + 2009 \times 130 + 2010 \times 180}{2008 \times 15 + 2009 \times 25 + 2010 \times 35} \right\rfloor \), find the value of \(A\).


Find the possible value of $x + y$ given that $x^3 + 6x^2 + 16x = -15$ and $y^3 + 6y^2 + 16y = -17$.


Find the largest natural number \( n \) that satisfies the inequality \( n^{300} < 3^{500} \).


Let \(ABCDE\) be a convex pentagon such that \(\angle ABC = \angle ACD = \angle ADE = 90^\circ\) and \(AB = BC = CD = DE = 1\). Compute the length \(AE\).


The teacher wrote a two-digit number on the board. Each of the three boys made two statements.

- Andrey: "This number ends in the digit 6" and "This number is divisible by 7."
- Borya: "This number is greater than 26" and "This number ends in the digit 8."
- Sasha: "This number is divisible by 13" and "This number is less than 27."

It is known that each of the boys told the truth once and made a mistake once. What number could have been written on the board? List all possible options.


There are 200 candies. What is the minimum number of schoolchildren that these candies can be distributed to so that, no matter how the candies are distributed, there are always at least two schoolchildren who receive the same number of candies (possibly none)?


Let \( X = \{1,2, \cdots, 100\} \). For any non-empty subset \( M \) of \( X \), define the characteristic of \( M \), denoted as \( m(M) \), as the sum of the maximum and minimum elements of \( M \). Find the average value of the characteristics of all non-empty subsets of \( X \).


Given $m$ and $n$ are non-negative integers, the sets $A=\{1, n\}$ and $B=\{2, 4, m\}$, and the set $C=\{c \mid c=xy, x \in A, y \in B\}$. If $|C|=6$ and the sum of all elements in $C$ is 42, find the value of $m+n$.


Given a positive integer \( N \) (written in base 10), define its integer substrings to be integers that are equal to strings of one or more consecutive digits from \( N \), including \( N \) itself. For example, the integer substrings of 3208 are \( 3, 2, 0, 8, 32, 20, 320, 208 \), and 3208. (The substring 08 is omitted from this list because it is the same integer as the substring 8, which is already listed.) What is the greatest integer \( N \) such that no integer substring of \( N \) is a multiple of 9? (Note: 0 is a multiple of 9.)


Let \( O \) be the circumcenter of triangle \( ABC \), and let us draw the reflections of \( O \) with respect to the sides of the triangle. Show that the resulting triangle \( A_1 B_1 C_1 \) is congruent to triangle \( ABC \). Determine the orthocenter of triangle \( A_1 B_1 C_1 \).

Solution: \boxed{O}


Calculate

$$
\operatorname{tg} \frac{\pi}{47} \cdot \operatorname{tg} \frac{2 \pi}{47} + \operatorname{tg} \frac{2 \pi}{47} \cdot \operatorname{tg} \frac{3 \pi}{47} + \ldots + \operatorname{tg} \frac{k \pi}{47} \cdot \operatorname{tg} \frac{(k+1) \pi}{47} + \ldots + \operatorname{tg} \frac{2021 \pi}{47} \cdot \operatorname{tg} \frac{2022 \pi}{47}
$$


The function \( y = f(x) \) is defined on the set \( (0, +\infty) \) and takes positive values on it. It is known that for any points \( A \) and \( B \) on the graph of the function, the areas of the triangle \( AOB \) and the trapezoid \( ABH_BH_A \) are equal (\( H_A, H_B \) are the bases of the perpendiculars dropped from points \( A \) and \( B \) to the x-axis; \( O \) is the origin). Find all such functions. Given that \( f(1) = 4 \), find the value of \( f(4) \).


Consider the set \( S = \{1, 2, 3, \cdots, 2010, 2011\} \). A subset \( T \) of \( S \) is said to be a \( k \)-element RP-subset if \( T \) has exactly \( k \) elements and every pair of elements of \( T \) are relatively prime. Find the smallest positive integer \( k \) such that every \( k \)-element RP-subset of \( S \) contains at least one prime number.


Among the unseen beasts that left tracks on unknown paths, there was a herd of one-headed 34-legged creatures and three-headed Dragons. There are a total of 286 legs and 31 heads in the herd. How many legs does a three-headed Dragon have?


As shown in the figure, in a trapezoid $ABCD$, $AD$ is parallel to $BC$, and the ratio of $BC$ to $AD$ is 5:7. Point $F$ is on segment $AD$, and point $E$ is on segment $CD$, satisfying the ratios $AF:FD=4:3$ and $CE:ED=2:3$. If the area of quadrilateral $ABEF$ is 123, what is the area of $ABCD$?


Alice encounters one of the brothers in the forest who says: "The true owner of the rattle is telling the truth today." Humpty Dumpty, nearby, claims that the chances the speaker is the true owner of the rattle are exactly 13 out of 14. How did Humpty Dumpty arrive at these numbers?


Let \( A \) be the sum of the digits of \( 2012^{2012} \). Let \( B \) be the sum of the digits of \( A \), and \( C \) the sum of the digits of \( B \). Determine \( C \).


In triangle $ABC$, the internal angle bisector from $C$ intersects the median from $B$ at point $P$ and side $AB$ at point $T$. Show that

$$
\frac{CP}{PT} - \frac{AC}{BC} = 1
$$


In the parallelogram \(KLMN\), side \(KL\) is equal to 8. A circle tangent to sides \(NK\) and \(NM\) passes through point \(L\) and intersects sides \(KL\) and \(ML\) at points \(C\) and \(D\) respectively. It is known that \(KC : LC = 4 : 5\) and \(LD : MD = 8 : 1\). Find the side \(KN\).


In a town of \( n \) people, a governing council is elected as follows: each person casts one vote for some person in the town, and anyone that receives at least five votes is elected to council. Let \( c(n) \) denote the average number of people elected to council if everyone votes randomly. Find \( \lim _{n \rightarrow \infty} \frac{c(n)}{n} \).

Solution: \boxed{1 - \frac{65}{24e}}


Calculate the mass of the tetrahedron bounded by the planes \(x=0\), \(y=0\), \(z=0\), and \(\frac{x}{10} + \frac{y}{8} + \frac{z}{3} = 1\), if the mass density at each point is given by the function \(\rho = \left(1 + \frac{x}{10} + \frac{y}{8} + \frac{z}{3}\right)^{-6}\).

Solution: \boxed{m = 2}


Given that \( A \) is a two-digit number and the remainder when \( A^2 \) is divided by 15 is 1, find the number of such \( A \). ( )


Given a natural number \( x = 9^n - 1 \), where \( n \) is a natural number. It is known that \( x \) has exactly three distinct prime divisors, one of which is 11. Find \( x \).


Given the equation \(x + 11y + 11z = n\) where \(n \in \mathbf{Z}_{+}\), there are 16,653 sets of positive integer solutions \((x, y, z)\). Find the minimum value of \(n\).


Assume that Olga Pavlovna has \( x \) liters and Maria Petrovna has \( y \) liters of jam left. The numbers \( x \) and \( y \) are randomly and independently chosen from the interval from 0 to 1. A random point with coordinates \((x, y)\) is selected from the unit square \( F \). The event \( A \) "in total, Olga Pavlovna and Maria Petrovna have at least 1 liter but less than 1.5 liters of jam remaining" is recorded as the inequality \( 1 \leq x + y < 1.5 \) and is represented by the trapezoid bounded between the lines \( x + y = 1 \) and \( x + y = 1.5 \). Then

\[
\mathrm{P}(A) = \frac{S_A}{S_F} = \frac{3}{8} = 0.375
\]


The points \( S, T, U \) lie on the sides of the triangle \( PQR \) such that \( QS = QU \) and \( RS = RT \). Given that \(\angle TSU = 40^\circ\), what is \(\angle TPU\)?

A) \(60^\circ\)
B) \(70^\circ\)
C) \(80^\circ\)
D) \(90^\circ\)
E) \(100^\circ\)


A rectangular table of size \( x \) cm \( \times 80 \) cm is covered with identical sheets of paper measuring 5 cm \( \times 8 \) cm. The first sheet is placed at the bottom-left corner, and each subsequent sheet is placed one centimeter higher and one centimeter to the right of the previous one. The last sheet touches the top-right corner. What is the length \( x \) in centimeters?


20 phones are connected with wires such that each wire connects two phones, each pair of phones is connected by at most one wire, and each phone has at most two wires connected to it. We need to paint the wires (each wire entirely with one color) so that the wires connected to each phone are of different colors. What is the minimum number of colors needed for such painting?


How many integer solutions does the inequality

$$
|x| + |y| < 1000
$$

have, where \( x \) and \( y \) are integers?


Find all continuous functions on the entire number line that satisfy the identity \(2 f(x+y) = f(x) f(y)\) and the condition \(f(1) = 10\).

Solution: \boxed{f(x) = 2 \cdot 5^x}


After walking \( \frac{2}{5} \) of the length of a narrow bridge, a pedestrian noticed that a car was approaching the bridge from behind. He then turned back and met the car at the beginning of the bridge. If the pedestrian had continued walking forward, the car would have caught up with him at the end of the bridge. Find the ratio of the car's speed to the pedestrian's speed.


In a convex quadrilateral \(ABCD\), angles \(A\) and \(C\) are both equal to \(100^\circ\). Points \(X\) and \(Y\) are chosen on sides \(AB\) and \(BC\) respectively such that \(AX=CY\). It is found that line \(YD\) is parallel to the bisector of angle \(ABC\). Find angle \(AXY\).


Find all prime numbers \( p \) and \( q \) such that \( p + q = (p - q)^3 \).

Solution: \boxed{p = 5, \, q = 3}


A number, when added to 3, is a multiple of 7, and when subtracted by 5, is a multiple of 8. What is the smallest such number?


Find the maximum value of $S$ such that any finite number of small squares with a total area of $S$ can be placed inside a unit square $T$ with side length 1, in such a way that no two squares overlap.


127 is the number of non-empty sets of natural numbers \( S \) that satisfy the condition "if \( x \in S \), then \( 14-x \in S \)". The number of such sets \( S \) is \(\qquad \).


Three positive integers differ from each other by at most 6. The product of these three integers is 2808. What is the smallest integer among them?


An irrigation canal has the shape of an isosceles trapezoid, where the legs are equal to the smaller base. At what angle of inclination of the legs is the cross-sectional area of the canal the greatest?


In triangle $ABC$, a median $BM$ is drawn. It is given that $\angle ABM = 40^\circ$, and $\angle MBC = 70^\circ$. Find the ratio $AB:BM$. Justify your answer.


Investigate the convergence of the functional series \(\sum_{n=1}^{\infty} \frac{1}{n^{2}+3} \cdot\left(\frac{x+1}{x-1}\right)^{n}\).

Solution: \boxed{(-\infty, 0]}


Given a square \(ABCD\) with point \(P\) inside such that \(PA = 1\), \(PB = 2\), and \(PC = 3\), calculate the angle \(\widehat{APB}\).


Let \( a < b < c < d < e \) be real numbers. Among the 10 sums of the pairs of these numbers, the least three are 32, 36, and 37, while the largest two are 48 and 51. Find all possible values of \( e \).


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}}


Given that \( P_{0}(1, f(1)) \) is a point on the curve \( C: f(x) = \mathrm{e}^{x} \), the tangent line \( l_{1} \) to \( C \) at \( P_{0} \) intersects the \( x \)-axis at point \( Q_{1}(x_{1}, 0) \). A vertical line through \( Q_{1} \) intersects \( C \) at \( P_{1} \), and the tangent line \( l_{2} \) to \( C \) at \( P_{1} \) intersects the \( x \)-axis at \( Q_{2}(x_{2}, 0) \). A vertical line through \( Q_{2} \) intersects \( C \) at \( P_{2} \), and the tangent line \( l_{3} \) to \( C \) at \( P_{2} \) intersects the \( x \)-axis at \( Q_{3}(x_{3}, 0) \). By repeating this operation, points \( Q_{4}(x_{4}, 0), Q_{5}(x_{5}, 0), \cdots \) are obtained in succession. Find \( x_{2023} \).


Calculate the limit of the function:

\[
\lim _{x \rightarrow \frac{\pi}{2}} \frac{e^{\operatorname{tg} 2 x} - e^{-\sin 2 x}}{\sin x - 1}
\]


In the United States, dates are written as: month number, day number, and year. In Europe, however, the format is day number, month number, and year. How many days in a year are there where the date cannot be read unambiguously without knowing which format is being used?


If the integers \( a, b, \) and \( c \) satisfy:
\[
a + b + c = 3, \quad a^3 + b^3 + c^3 = 3,
\]
then what is the maximum value of \( a^2 + b^2 + c^2 \)?


On the diagonals $AC$ and $CE$ of a regular hexagon $ABCDEF$, points $M$ and $N$ are taken respectively, such that $\frac{AM}{AC} = \frac{CN}{CE} = \lambda$. It is known that points $B, M$, and $N$ lie on one line. Find $\lambda$.


Given the system of equations for positive numbers \( x, y, z \):

\[
\left\{
\begin{array}{l}
x^{2}+x y+y^{2}=48 \\
y^{2}+y z+z^{2}=9 \\
z^{2}+x z+x^{2}=57
\end{array}
\right.
\]

Find the value of the expression \( x y + y z + x z \).


Given the equation of circle $C_{0}$ as $x^{2}+y^{2}=r^{2}$, find the equation of the tangent line passing through a point $M\left(x_{0}, y_{0}\right)$ on circle $C_{0}$.

Solution: \boxed{x_0 x + y_0 y = r^2}


Find the smallest natural number that is divisible by $48^{2}$ and contains only the digits 0 and 1.


A homeowner currently pays a 10% income tax on rental income. By what percentage should the homeowner increase the rent to maintain the same income if the income tax is raised to 20%?


On a \(3 \times 3\) grid of 9 squares, each square is to be painted with either Red or Blue. If \(\alpha\) is the total number of possible colorings in which no \(2 \times 2\) grid consists of only Red squares, determine the value of \(\alpha\).


Given the sets
$$
\begin{array}{l}
A=\left\{(x, y) \mid x=m, y=-3m+2, m \in \mathbf{Z}_{+}\right\}, \\
B=\left\{(x, y) \mid x=n, y=a\left(a^{2}-n+1\right), n \in \mathbf{Z}_{+}\right\},
\end{array}
$$

find the total number of integers $a$ such that $A \cap B \neq \varnothing$.


Calculate the area of the figure bounded by the graphs of the following functions:

$$
y=\frac{e^{1 / x}}{x^{2}}, y=0, x=2, x=1
$$

Solution: \boxed{e - \sqrt{e}}


How many roots does the equation \(\sqrt{14-x^{2}}(\sin x-\cos 2x)=0\) have?


There is a reservoir A and a town B connected by a river. When the reservoir does not release water, the water in the river is stationary; when the reservoir releases water, the water in the river flows at a constant speed. When the reservoir was not releasing water, speedboat M traveled for 50 minutes from A towards B and covered $\frac{1}{3}$ of the river's length. At this moment, the reservoir started releasing water, and the speedboat took only 20 minutes to travel another $\frac{1}{3}$ of the river's length. The driver then turned off the speedboat's engine and allowed it to drift with the current, taking $\quad$ minutes for the speedboat to reach B.


How many integer solutions \((x, y)\) does the inequality \(|x| + |y| < n\) have?

Solution: \boxed{2n^2 - 2n + 1}


Compute the sum of all positive integers \( a \leq 26 \) for which there exist integers \( b \) and \( c \) such that \( a + 23b + 15c - 2 \) and \( 2a + 5b + 14c - 8 \) are both multiples of 26.


An easel in a corner hosts three $30 \text{ cm} \times 40 \text{ cm}$ shelves, with equal distances between neighboring shelves. Three spiders resided where the two walls and the middle shelf meet. One spider climbed diagonally up to the corner of the top shelf on one wall, another climbed diagonally down to the corner of the lower shelf on the other wall. The third spider stayed in place and observed that from its position, the other two spiders appeared at an angle of $120^\circ$. What is the distance between the shelves? (The distance between neighboring shelves is the same.)

Solution: \boxed{35 \text{ cm}}


Given the function \( f(x) = -x^2 + x + m + 2 \), if the inequality \( f(x) \geq |x| \) has exactly one integer in its solution set, determine the range of the real number \( m \).

Solution: \boxed{[-2, -1)}


Petya chooses non-negative numbers \(x_{1}, x_{2}, \ldots, x_{11}\) such that their sum is 1. Vasya arranges them in a row at his discretion, calculates the products of adjacent numbers, and writes on the board the largest of the resulting ten products. Petya wants the number on the board to be as large as possible, while Vasya wants it to be as small as possible. What number will be on the board with the best play from both Petya and Vasya?


Let $P$ be a moving point on the parabola $y^2 = 2x$. A tangent line is drawn at $P$ to the parabola, which intersects the circle $x^2 + y^2 = 1$ at points $M$ and $N$. The tangents to the circle at points $M$ and $N$ intersect at point $Q$. Find the equation of the locus of point $Q$.

Solution: \boxed{y^2 = -2 x}


In $\triangle ABC$, \(BC = 5\), \(AC = 12\), \(AB = 13\). Points \(D\) and \(E\) are located on sides \(AB\) and \(AC\) respectively, such that line segment \(DE\) divides $\triangle ABC$ into two regions of equal area. Determine the minimum length of such a line segment \(DE\).


Two-headed and seven-headed dragons came to a meeting. At the very beginning of the meeting, one of the heads of one of the seven-headed dragons counted all the other heads. There were 25 of them. How many dragons in total came to the meeting?


There is a magical tree with 63 fruits. On the first day, 1 fruit will fall from the tree. Starting from the second day, the number of fruits falling each day increases by 1 compared to the previous day. However, if the number of fruits on the tree is less than the number that should fall on that day, then the sequence resets and starts falling 1 fruit again each day, following the original pattern. On which day will all the fruits be gone from the tree?


Among the 200 natural numbers from 1 to 200, list the numbers that are neither multiples of 3 nor multiples of 5 in ascending order. What is the 100th number in this list?


Construct a point \( O \) in the scalene triangle \( ABC \) such that the angles \( AOB \), \( BOC \), and \( COA \) are equal.

Solution: \boxed{O}


Calculate the areas of the figures bounded by the lines given in polar coordinates.

$$
r=\cos \phi, \quad r=2 \cos \phi
$$


Let $\mathcal{P}$ and $\mathcal{P}^{\prime}$ be two convex quadrilaterals with $\mathcal{P}^{\prime}$ inside $\mathcal{P}$, and let $d$ (resp. $d^{\prime}$) be the sum of the lengths of the diagonals of $\mathcal{P}$ (resp. $\mathcal{P}^{\prime}$). Show that:

$$
d^{\prime} < 2d
$$

Solution: \boxed{d' < 2d}


Let \( AB \) be a focal chord of the parabola \( y^2 = 2px \) (where \( p > 0 \)), and let \( O \) be the origin. Then:
1. The minimum area of triangle \( OAB \) is \( \frac{p^2}{2} \).
2. The tangents at points \( A \) and \( B \) intersect at point \( M \), and the minimum area of triangle \( MAB \) is \( p^2 \).


There are $n$ balls that look identical, among which one ball is lighter than the others (all other balls have equal weight). If using an unweighted balance scale as a tool, it takes at least 5 weighings to find the lighter ball, then the maximum value of $n$ is ___.


Given \(\alpha, \beta \in \left(0, \frac{\pi}{2}\right)\), find the maximum value of \(A = \frac{\left(1 - \sqrt{\tan \frac{\alpha}{2} \tan \frac{\beta}{2}}\right)^2}{\cot \alpha + \cot \beta}\).

Solution: \boxed{3 - 2\sqrt{2}}


Three cones are standing on their bases on a table, touching each other. The radii of their bases are 23, 46, and 69. A truncated cone is placed on the table with its smaller base down, sharing a common slant height with each of the other cones. Find the radius of the smaller base of the truncated cone.


Find a nine-digit number in which all the digits are different and do not include zero, and the square root of this number has the form $\overline{a b a b c}$, where $\overline{a b} = c^3$.


Given a convex quadrilateral with area \( S \). A point inside it is chosen and reflected symmetrically with respect to the midpoints of its sides. This results in four vertices of a new quadrilateral. Find the area of the new quadrilateral.


A positive integer has exactly 8 divisors. The sum of its smallest 3 divisors is 15. This four-digit number has a prime factor such that the prime factor minus 5 times another prime factor equals twice the third prime factor. What is this number?


The plane figure $W$ consists of all points whose coordinates $(x, y)$ satisfy the inequality $|2 - |x| - ||y| - 2|| \leqslant 1$. Draw the figure $W$ and find its area.


The plane vectors $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ satisfy $|\boldsymbol{a}|=1, \boldsymbol{a} \cdot \boldsymbol{b}=\boldsymbol{b} \cdot \boldsymbol{c}=1,$ and $|\boldsymbol{a}-\boldsymbol{b}+\boldsymbol{c}| \leqslant 2 \sqrt{2}$. Find the maximum value of $\boldsymbol{a} \cdot \boldsymbol{c}$.


The elements of the set $\{1,2,3,4,5,6,7,8,9,10,11\}$ can be separated into the sets $\{3,9,10,11\}$ and $\{1,2,4,5,6,7,8\}$ so that each has the same sum of elements equal to 33.

(a) Describe a way to separate the elements of the set $\{1,2,3,4,5,6,7,8\}$ into three subsets such that the sum of the elements in each subset is the same.
(b) Explain why it is not possible to separate the numbers of the set $\{1,2,3,4,5,6,7,8,9,10\}$ into two subsets with the same sum.

(c) For each positive integer $n \geq 2$, determine the smallest positive integer $N$ such that the set $\{1,2, \ldots, N\}$ can be separated into exactly $n$ subsets with the same sum.


On the hypotenuse \( A B \) of an isosceles right triangle \( A B C \), points \( K \) and \( L \) are marked such that \( A K : K L : L B = 1 : 2 : \sqrt{3} \). Determine \( \angle K C L \).


There is an oak tree growing by the road from Yolkino to Palkino, and it is twice as close to Yolkino as it is to Palkino. Fedya, traveling at a constant (and positive) speed from Yolkino to Palkino, was twice as close to the oak tree as he was to Yolkino at 12:00. At 12:40, it turned out again that Fedya was twice as close to the oak tree as he was to Yolkino. When will Fedya arrive in Palkino?

Solution: \boxed{13:10 }


Find all positive integers \( n \) such that \( 3^n + 5^n \) is a multiple of \( 3^{n-1} + 5^{n-1} \).


Assume there is a bag with red, yellow, blue, and green balls. Each time, one ball is taken from the bag, the color is confirmed, and then the ball is put back into the bag. This process continues until two consecutive red balls are drawn. Let $\zeta$ be the total number of draws. If each ball is equally likely to be drawn, what is the expected value of $\zeta$?


ABC is a triangle with M the midpoint of BC. The segment XY with X on AB and Y on AC has its midpoint on AM. Show that it is parallel to BC.

Solution: \boxed{XY \parallel BC}


How many Pythagorean triangles are there in which one of the legs is equal to 2013? (A Pythagorean triangle is a right triangle with integer sides. Identical triangles count as one.).


Can we find a line normal to the curves \( y = \cosh x \) and \( y = \sinh x \)?

Solution: \boxed{\text{No}}


Let the parabola \( C: y^2 = 2x \) have its directrix intersect the \( x \)-axis at point \( A \). A line \( l \) through point \( B(-1, 0) \) is tangent to the parabola \( C \) at point \( K \). Draw a line through point \( A \) parallel to \( l \), which intersects the parabola \( C \) at points \( M \) and \( N \). Find the area of triangle \( \triangle KMN \).


Connecting point \( M \) with the vertices of triangle \( ABC \). Show that the resultant of the forces \( MA, MB \), and \( MC \) passes through the centroid \( G \) of the triangle and is equal to \( 3MG \).

Solution: \boxed{MH = 3 MG}


The UEFA Champions League playoffs is a 16-team soccer tournament in which Spanish teams always win against non-Spanish teams. In each of 4 rounds, each remaining team is randomly paired against one other team; the winner advances to the next round, and the loser is permanently knocked out of the tournament. If 3 of the 16 teams are Spanish, what is the probability that there are 2 Spanish teams in the final round?


Factory A and Factory B both produce the same type of clothing. Factory A produces 2700 sets of clothing per month, with the time ratio of producing tops to pants being 2:1. Factory B produces 3600 sets of clothing per month, with the time ratio of producing tops to pants being 3:2. If the two factories work together for one month, what is the maximum number of sets of clothing they can produce?


Let point \( O \) be a point inside triangle \( ABC \) that satisfies the equation
\[
\overrightarrow{OA} + 2 \overrightarrow{OB} + 3 \overrightarrow{OC} = 3 \overrightarrow{AB} + 2 \overrightarrow{BC} + \overrightarrow{CA}.
\]

Then, find the value of \(\frac{S_{\triangle AOB} + 2 S_{\triangle BOC} + 3 S_{\triangle COA}}{S_{\triangle ABC}}\).


Among all fractions of the form \(\frac{m}{n}\), where \(m\) and \(n\) are four-digit numbers with the same sum of digits, the largest one was chosen. Find it and write your answer as an improper fraction (do not round or simplify it!).


Let \(AB\) be the diameter of a circle, and \(CD\) a chord that is not perpendicular to it. If perpendiculars \(AE\) and \(BF\) are dropped from the ends of the diameter onto the chord, then the segments \(CF\) and \(DE\) are equal.

Solution: \boxed{CF = DE}


How many integers (1) have 5 decimal digits, (2) have the last digit 6, and (3) are divisible by 3?


Let {a_n} be a sequence of positive numbers with the sum of its first n terms being b_n. Let {b_n} be such that the product of its first n terms is c_n, and b_n + c_n = 1. The number in the sequence {1/a_n} that is closest to 2002 is ___ .


After the teacher Maria Ivanovna moved Vovochka from the first row to the second, Vanechka from the second row to the third, and Masha from the third row to the first, the average age of students in the first row increased by one week, in the second row increased by two weeks, and in the third row decreased by four weeks. It is known that there are 12 students each in the first and second rows. How many students are sitting in the third row?


Given that \(a, b, c\) are positive integers such that the roots of the three quadratic equations
\[
x^2 - 2ax + b = 0, \quad x^2 - 2bx + c = 0, \quad x^2 - 2cx + a = 0
\]
are all positive integers. Determine the maximum value of the product \(abc\).


Complex numbers \(a, b, c\) form an equilateral triangle with side length 18 in the complex plane. If \(|a+b+c|=36\), find \(|bc + ca + ab|\).


A cyclist traveled from point A to point B, stayed there for 30 minutes, and then returned to A. On the way to B, he overtook a pedestrian, and met him again 2 hours later on his way back. The pedestrian arrived at point B at the same time the cyclist returned to point A. How much time did it take the pedestrian to travel from A to B if his speed is four times less than the speed of the cyclist?

Solution: \boxed{10 \text{ hours}}


A hollow glass sphere with uniform wall thickness and an outer diameter of $16 \mathrm{~cm}$ floats in water in such a way that $\frac{3}{8}$ of its surface remains dry. What is the wall thickness, given that the specific gravity of the glass is $s = 2.523$?

Solution: \boxed{0.8 \, \text{cm}}


Find a six-digit number where the first digit is 6 times less than the sum of all the digits to its right, and the second digit is 6 times less than the sum of all the digits to its right.


In space, there are \( n \) (\( n \geqslant 3 \)) planes, where any three planes do not share a common perpendicular plane. There are the following four assertions:
(1) No two planes are parallel to each other;
(2) No three planes intersect in a single line;
(3) Any two intersection lines between the planes are not parallel;
(4) Each intersection line between the planes intersects with \( n-2 \) other planes.

Determine how many of these assertions are correct.


Let \( m=30030=2 \cdot 3 \cdot 5 \cdot 7 \cdot 11 \cdot 13 \) and let \( M \) be the set of its positive divisors which have exactly two prime factors. Determine the minimal integer \( n \) with the following property: for any choice of \( n \) numbers from \( M \), there exist three numbers \( a, b, c \) among them satisfying \( a \cdot b \cdot c = m \).


Six members of a study group decide on a real number \( b \), which their incoming seventh member needs to guess. Upon entering, the following statements are provided sequentially:

I. \( b \) is an even number.

II. Adding the cube of \( b \) to \( b \) results in a positive number smaller than 8000.

III. \( b \) is a real number whose square is 13.

IV. \( b \) is an integer divisible by 7.

V. \( b \) is a rational number.

VI. \( b \) is an integer divisible by 14.

Additionally, it is stated that out of I and II, III and IV, V and VI, one statement is always true and the other is always false. - How did the seventh group member figure out the number?


A gardener wants to plant 3 maple trees, 4 oak trees, and 5 birch trees in a row. He will randomly determine the order of these trees. What is the probability that no two birch trees are adjacent?


Find the angle $B$ of triangle $ABC$ if the length of the altitude $CH$ is half the length of side $AB$, and $\angle BAC = 75^\circ$.


Given that the positive real number \(a\) satisfies \(a^{a}=(9 a)^{8 a}\), find the value of \(\log _{a}(3 a)\).


Let's call a natural number "remarkable" if it is the smallest among all natural numbers with the same sum of digits as it.

How many three-digit remarkable numbers exist?


Two scientific production enterprises supply substrates for growing orchids to the market. In the "Orchid-1" substrate, there is three times more pine bark than sand, and twice as much peat as sand. In the "Orchid-2" substrate, there is half as much bark as there is peat, and one and a half times more sand than peat. In what ratio should the substrates be mixed so that the new mixed composition has equal parts of bark, peat, and sand?


Let \( T = \left\{9^{k} \mid k \right. \) be integers, \(\left.0 \leqslant k \leqslant 4000\right\} \). Given that \(9^{4000}\) has 3817 digits and its most significant digit is 9, how many elements in \(T\) have 9 as their most significant digit?
(The 8th American Mathematics Invitational, 1990)


In $\triangle ABC$, the sides opposite to angles $A$, $B$, and $C$ are denoted as $a$, $b$, and $c$ respectively, and $\cos A=\frac{1}{3}$.
1. Find the value of $\sin^{2} \frac{B+C}{2}+\cos 2A$.
2. If $a=\sqrt{3}$, find the maximum value of $b \cdot c$.


The sides of triangle $ABC$ are extended as shown in Figure 5, with $AA' = 3AB$, $BB' = 5BC$, and $CC' = 8CA$. By what factor is the area of triangle $ABC$ smaller than the area of triangle $A'B'C'$?


Given a cyclic quadrilateral \(A B C D\), the rays \(A B\) and \(D C\) intersect at point \(K\). It is found that the points \(B\), \(D\), and the midpoints of segments \(A C\) and \(K C\) lie on the same circle. What values can the angle \(A D C\) take?


In a $10 \times 10$ grid square with the centers of all unit squares marked (a total of 100 points), what is the minimum number of lines, not parallel to the sides of the square, that need to be drawn to cross out all the marked points?


Initially, the number 1 and two positive numbers \( x \) and \( y \) are written on a blackboard. In each move, a player can choose any two numbers on the board, not necessarily distinct, and write their sum or their difference on the board. Additionally, they can choose any non-zero number on the board and write its inverse. After a finite number of moves, describe how we can obtain the following numbers:

a) \( x^{2} \).

b) \( x y \).

Solution: \boxed{xy}


Find the smallest positive integer \( n \) such that there exists a sequence of \( n+1 \) terms \( a_0, a_1, \ldots, a_n \) with the properties \( a_0 = 0 \), \( a_n = 2008 \), and \( |a_i - a_{i-1}| = i^2 \) for \( i = 1, 2, \ldots, n \).


Find the number of solutions in natural numbers for the equation \((x-4)^{2}-35=(y-3)^{2}\).


Find the sum of the first 10 elements that appear in both the arithmetic progression $\{4,7,10,13, \ldots\}$ and the geometric progression $\{20,40,80,160, \ldots\}$.


In a castle, there are 16 identical square rooms arranged in a $4 \times 4$ grid. Each of these rooms is occupied by one person, making a total of 16 people, who are either liars or knights (liars always lie, knights always tell the truth). Each of these 16 people said: "There is at least one liar in one of the rooms adjacent to mine." Two rooms are considered adjacent if they share a wall. What is the maximum number of liars that could be among these 16 people?


On a road, there are three locations $A$, $O$, and $B$. $O$ is between $A$ and $B$, and $A$ is 1360 meters away from $O$. Two individuals, Jia and Yi, start simultaneously from points $A$ and $O$ towards point $B$. At the 10th minute after departure, both Jia and Yi are equidistant from point $O$. In the 40th minute, Jia and Yi meet at point $B$. What is the distance between points $O$ and $B$ in meters?

Solution: \boxed{2040 \text{ meters}}


A tetrahedron \(ABCD\) has edge lengths 7, 13, 18, 27, 36, 41, with \(AB = 41\). Determine the length of \(CD\).


The notebook lists all the irreducible fractions with numerator 15 that are greater than $\frac{1}{16}$ and less than $\frac{1}{15}$. How many such fractions are listed in the notebook?


Find the last non-zero digit of \(50! = 1 \times 2 \times \cdots \times 50\).


In the sequence \(\left\{a_{n}\right\}\), for \(1 \leqslant n \leqslant 5\), we have \(a_{n}=n^{2}\). Additionally, for all positive integers \(n\), the following holds:
\[ a_{n+5} + a_{n+1} = a_{n+4} + a_{n}. \]
Determine the value of \(a_{2023}\).


In an isosceles trapezoid \(ABCD\), the side \(AB\) and the shorter base \(BC\) are both equal to 2, and \(BD\) is perpendicular to \(AB\). Find the area of this trapezoid.


The distance between the midpoints of mutually perpendicular chords \(AC\) and \(BC\) of a circle is 10. Find the diameter of the circle.


Solve in integers: \(6x^{2} + 5xy + y^{2} = 6x + 2y + 7\)

Indicate the answer for which the value \(|x| + |y|\) is the largest. Write the answer in the form \((x; y)\).

Solution: \boxed{(-8; 25)}


In a six-digit number, the first digit, which is 2, was moved to the last position while leaving the other digits in the same order. The resulting number turned out to be three times the original number. Find the original number.


In the middle of a square lake with a side length of 10 feet, there is a reed extending 1 foot above the water. If the reed is bent, its tip reaches the shore. How deep is the lake?


Two people agree to meet between 7:00 AM and 8:00 AM, and the first person to arrive will wait for the other person for 20 minutes. The times at which they arrive at the designated meeting place between 7:00 AM and 8:00 AM are random and independent. What is the probability that they will meet?


On a blackboard, there are 1989 consecutive natural numbers written from $1, 2, 3, \ldots, 1989$. A transformation is performed as follows: erase any two numbers on the blackboard and replace them with the remainder when the sum of the two erased numbers is divided by 19. After performing this transformation multiple times, there are two numbers left on the blackboard: one is 89 and the other is a single-digit number. What is this single-digit number?


What is the minimum number of different numbers that must be chosen from $1, 2, 3, \ldots, 1000$ to ensure that among the chosen numbers, there exist 3 different numbers that can form the side lengths of a triangle?


2011 is a four-digit number such that the sum of its digits is 4. How many four-digit numbers have a digit sum of 4?


Based on the definition of the derivative, find \( f^{\prime}(0) \):

\[
f(x) = \begin{cases}
\frac{\cos x - \cos 3x}{x}, & x \neq 0 \\
0, & x = 0
\end{cases}
\]


Find the sum of the digits of all numbers in the sequence \(1, 2, 3, \ldots, 99, 100\).


The owner of an apartment insured it for 3,750,000 rubles (the actual cost of the apartment is 7,500,000 rubles). The actual damage amounted to 2,750,000 rubles. The insurance compensation under the proportional liability system was 1,350,000 rubles. Determine the amount of the deductible provided for in the contract.


Find the limits of the following functions:
1) \( f(x) = 2x - 3 - \frac{1}{x} \) as \( x \rightarrow 1 \):
2) \( y = \frac{x^3 - 3x^2 + 2x - 5}{x^2 + 2} \) as \( x \rightarrow -1 \):
3) \( y = x \sin \frac{1}{x} \) as \( x \rightarrow 0 \).


Let \( n \) be a positive integer, and \( d \) be a digit in decimal notation. If \(\frac{n}{810} = \overline{0.d25d25d25\cdots}\), find \( n \).


The recruits stood in a row one behind the other, all facing the same direction. Among them were three brothers: Peter, Nikolai, and Denis. There were 50 people ahead of Peter, 100 people ahead of Nikolai, and 170 people ahead of Denis. On the command "About-face!" everyone turned in the opposite direction. In doing so, it turned out that in front of one of the brothers now stood four times as many people as in front of another brother. How many recruits might there be in total, including the brothers? List all possible options.


Let \( m > 1 \). Under the constraints
\[
\begin{cases}
y \geqslant x, \\
y \leqslant mx, \\
x + y \leqslant 1
\end{cases}
\]

the maximum value of the objective function \( z = x + my \) is less than 2. Determine the range of \( m \).

Solution: \boxed{(1, 1 + \sqrt{2})}


Does there exist a polyhedron where all the faces are triangles except one which is a pentagon, and such that all vertices have an even degree?

Solution: \boxed{\text{No}}


The number 100 is represented as a sum of several two-digit numbers, and in each addend, the digits are swapped. What is the largest possible number that could be obtained in the new sum?


In triangle \(ABC\), the median \(BK\), the angle bisector \(BE\), and the altitude \(AD\) are given.

Find the side \(AC\), if it is known that the lines \(BK\) and \(BE\) divide the segment \(AD\) into three equal parts, and \(AB=4\).


Given \( S_{n}=1+2+3+\cdots+n \) for \( n \in \mathbf{N} \), find the maximum value of the function \( f(n)=\frac{S_{n}}{(n+32) S_{n+1}} \).


Show that the line
$$
\left(m^{2}+6m+3\right)x - \left(2m^{2}+18m+2\right)y - 3m + 2 = 0
$$
passes through a fixed point, regardless of the value assigned to \( m \).

Solution: \boxed{(-1, -\frac{1}{2})}


Given a triangle $ABC$, an inscribed circle touches the sides $AB, AC, BC$ at points $C_1, B_1, A_1$ respectively. Find the radius of the excircle $w$, which touches the side $AB$ at point $D$, the extension of side $BC$ at point $E$, and the extension of side $AC$ at point $G$. It is known that $CE = 6$, the radius of the inscribed circle is $1$, and $CB_1 = 1$.


A parallelepiped is composed of white and black unit cubes in a checkerboard pattern. It is known that there are $1 \frac{12}{13} \%$ more black cubes than white cubes. Find the surface area of the parallelepiped, given that each side of the parallelepiped is greater than 1.


As shown in the figure, an "L"-shaped paper with a perimeter of 52 cm can be split along the dashed line into two identical rectangles. If the longest side length is 16 cm, what is the area of the "L"-shaped paper in square centimeters?


Given the graph of the function \(y = x + \frac{x}{x}\) and the point \((1, 0)\), how many lines can be drawn through this point such that they do not intersect the given graph?


Determine whether there exists a number that begins with 2 having the property that, when the 2 is moved to the end, the number is

(a) doubled;

(b) tripled.


A shooter makes three attempts. Success (hitting the target) and failure (missing the target) of each attempt are independent of the outcomes of the other attempts, and the probability of successfully completing each attempt is constant and equal to \( p \). Find the probability of successfully completing two out of three attempts.


The triangle \(PQR\) is isosceles with \(PR = QR\). Angle \(PRQ = 90^\circ\) and length \(PQ = 2 \text{ cm}\). Two arcs of radius \(1 \text{ cm}\) are drawn inside triangle \(PQR\). One arc has its center at \(P\) and intersects \(PR\) and \(PQ\). The other arc has its center at \(Q\) and intersects \(QR\) and \(PQ\). What is the area of the shaded region, in \(\text{cm}^2\)?

Solution: \boxed{1 - \frac{\pi}{4}}


Using the greedy algorithm, propose a coloring of the first graph involved in exercise 6 (or rather its more readable planar version available in the solution to this exercise). Deduce its chromatic number.


As shown in the figure, in the right triangle $ABC$, point $F$ is on $AB$ such that $AF = 2FB$. The quadrilateral $EBCD$ is a parallelogram. What is the ratio $FD: EF$?


Through a point located inside a triangle, lines are drawn parallel to the sides of the triangle.

These lines divide the triangle into three smaller triangles and three quadrilaterals. Let \( a \), \( b \), and \( c \) be the heights of the three smaller triangles parallel to the sides of the original triangle. Find the height of the original triangle parallel to these.

Solution: \boxed{a + b + c}


It is known that the numbers \(x, y, z \) form an arithmetic progression in that order with a common difference \(\alpha = \arcsin \frac{\sqrt{7}}{4}\), and the numbers \(\frac{1}{\sin x}, \frac{4}{\sin y}, \frac{1}{\sin z}\) also form an arithmetic progression in that order. Find \(\sin ^{2} y\).


Let the function \( f(x) = ax^2 + bx + c \) where \( a \neq 0 \) satisfy \( |f(0)| \leq 2 \), \( |f(2)| \leq 2 \), and \( |f(-2)| \leq 2 \). Determine the maximum value of \( y = |f(x)| \) for \( x \in [-2, 2] \).


Fill the five numbers $2015, 2016, 2017, 2018, 2019$ into the five boxes labeled " $D, O, G, C, W$ " such that $D+O+G=C+O+W$. How many different ways can this be done?


In a convex 13-gon, all the diagonals are drawn. They divide it into polygons. Consider a polygon among them with the largest number of sides. What is the maximum number of sides it can have?


Let the real numbers \( x_{1}, x_{2}, \cdots, x_{1997} \) satisfy the following two conditions:
1. \( -\frac{1}{\sqrt{3}} \leq x_{i} \leq \sqrt{3} \) for \( i = 1, 2, \cdots, 1997 \);
2. \( x_{1} + x_{2} + \cdots + x_{1997} = -318 \sqrt{3} \).

Find the maximum value of \( x_{1}^{12} + x_{2}^{12} + \cdots + x_{1997}^{12} \) and justify your answer.


Leon has cards with digits from 1 to 7. How many ways are there to glue two three-digit numbers (one card will not be used) so that their product is divisible by 81, and their sum is divisible by 9?


The base of a right prism is an isosceles trapezoid \(ABCD\) with \(AB = CD = 13\), \(BC = 11\), and \(AD = 21\). The area of the diagonal cross-section of the prism is 180. Find the total surface area of the prism.


Given a regular 15-sided polygon with a side length of \(2a\). What is the area of the annulus bounded by the circles inscribed in and circumscribed around the 15-sided polygon?

Solution: \boxed{\pi a^2}


It is known that 999973 has exactly three distinct prime factors. Find the sum of these prime factors.


Calculate the limit of the function:

$$\lim _{x \rightarrow 0} \frac{e^{4 x}-e^{-2 x}}{2 \operatorname{arctg} x-\sin x}$$


Given that \( x \) is a four-digit number and the sum of its digits is \( y \). When the value of \( \frac{x}{y} \) is minimized, \( x = \) _______


In the expression \( S=\sqrt{x_{1}-x_{2}+x_{3}-x_{4}} \), where \( x_{1}, x_{2}, x_{3}, x_{4} \) is a permutation of the numbers \( 1, 2, 3, 4 \), find the number of different permutations that make \( S \) a real number.


There are four inequalities: $\sqrt{2 \sqrt{2 \sqrt{2 \sqrt{2}}}}<2$, $\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2}}}}<2$, $\sqrt{3 \sqrt{3 \sqrt{3 \sqrt{3}}}}<3$, $\sqrt{3+\sqrt{3+\sqrt{3+\sqrt{3}}}}<3$. The number of incorrect inequalities is ______.


The sequence \( a_{1}, a_{2}, a_{3}, \ldots \) is an arithmetic sequence with common difference 3 and \( a_{1}=1 \). The sequence \( b_{1}, b_{2}, b_{3}, \ldots \) is an arithmetic sequence with common difference 10 and \( b_{1}=2 \). What is the smallest integer larger than 2023 that appears in both sequences?


In the diagram below, $P$ is a point on the semi-circle with the diameter $AB$. The point $L$ is the foot of the perpendicular from $P$ onto $AB$, and $K$ is the midpoint of $PB$. The tangents to the semicircle at $A$ and at $P$ meet at the point $Q$. It is given that $PL$ intersects $QB$ at the point $M$, and $KL$ intersects $QB$ at the point $N$. Suppose $\frac{AQ}{AB}=\frac{5}{12}$, $QM=25$ cm and $MN = x$ cm. Find the value of $x$.


Given the complex numbers \( z_{1}=1+a i \) and \( z_{2}=2^{\frac{3}{4}}\left(\cos \frac{3 \pi}{8}+i \sin \frac{3 \pi}{8}\right) \), find all real values of \( a \) for which \( z_{1}^{3}=z_{2}^{2} \).


In a class, there are 15 boys and 15 girls. On Women's Day, some boys called some girls to congratulate them (no boy called the same girl more than once). It turned out that the children can be uniquely divided into 15 pairs, such that each pair consists of a boy and a girl whom he called. What is the maximum number of calls that could have been made?


Professor Antônio discovered an interesting property related to the integer $x$ that represents his age. He told his students that $x^{2}=\overline{a b a c}$ and that $x=\overline{a b}+\overline{a c}$. What is the professor's age?

Note: We are using a bar to distinguish the decimal representation of the four-digit number $\overline{a b c d}$ from the product $a \cdot b \cdot c \cdot d$. For example, if $\overline{a b c d}=1267$, then $a=1, b=2, c=6,$ and $d=7$. The notation is the same for numbers with other quantities of digits.


The diagram shows a triangle \(ABC\) with an area of \(12 \, \text{cm}^2\). The sides of the triangle are extended to points \(P, Q, R, S, T\) and \(U\) so that \(PA = AB = BS\), \(QA = AC = CT\), and \(RB = BC = CU\). What is the area (in \(\text{cm}^2\)) of hexagon \(PQRSTU\)?


Let $s(n)$ denote the sum of all odd digits of the number $n$. For example, $s(4)=0$, $s(173)=11$, and $s(1623)=4$. Calculate the value of the sum $s(1)+s(2)+s(3)+\ldots+s(321)$.


A pot contains $3 \pi$ liters of water taken at a temperature of $t=0{ }^{\circ} C$ and brought to a boil in 12 minutes. After this, without removing the pot from the stove, ice at a temperature of $t=0{ }^{\circ} C$ is added. The next time the water begins to boil is after 15 minutes. Determine the mass of the added ice. The specific heat capacity of water is $c_{B}=4200 \, \text{J} / \text{kg} \cdot { }^{\circ} \mathrm{C}$, the specific heat of fusion of ice is $\lambda=3.3 \times 10^{5} \, \text{J} / \text{kg}$, the density of water is $\rho = 1000 \, \text{kg} / \text{m}^{3}$.

Answer: 2.1 kg

Solution: \boxed{2.1 \, \text{kg}}


Let \( m \) and \( n \) be relatively prime positive integers. Determine all possible values of

\[ \operatorname{gcd}\left(2^{m}-2^{n}, 2^{m^{2}+mn+n^{2}}-1\right). \]

Solution: \boxed{1 \text{ and } 7}


Given a trapezoid \(A B C D\) with right angles at \(A\) and \(D\). Let \(A B = a\), \(C D = b\), and \(A D = h\).

1) What relationship must exist between \(a\), \(b\), and \(h\) for the diagonals \(A C\) and \(B D\) to be perpendicular?
2) Let \(M\) be the midpoint of \([A D]\). What relationship must exist between \(a\), \(b\), and \(h\) for the triangle \(B M C\) to be a right triangle at \(M\)?

Solution: \boxed{h^2 = 4ab}


In space, there are 4 pairwise skew lines \( l_{1}, l_{2}, l_{3}, l_{4} \), such that no three of them are parallel to the same plane. Draw a plane \( P \) such that the points \( A_{1}, A_{2}, A_{3}, A_{4} \) at the intersections of these lines with \( P \) form a parallelogram. How many lines do the centers of such parallelograms trace out?


A sequence \(\left(a_{n}\right)\) satisfies the conditions:

$$
a_{1}=3, \quad a_{2}=9, \quad a_{n+1}=4 a_{n}-3 a_{n-1} \quad (n > 1)
$$

Find the formula for the general term of the sequence.

Solution: \boxed{a_n = 3^n}


Find the real solution(s) to the equation \((x+y)^{2} = (x+1)(y-1)\).


a) What is the radius of the smallest circle that can contain $n$ points ($n=2,3,4,...,10,11$), one of which coincides with the center of the circle, so that the distance between any two points is at least 1?

b)* How many points can be placed inside a circle of radius 2 so that one of the points coincides with the center of the circle and the distance between any two points is at least 1?


Given an integer \( n \geqslant 2 \). Let \( a_{1}, a_{2}, \cdots, a_{n} \) and \( b_{1}, b_{2}, \cdots, b_{n} \) be positive numbers that satisfy
\[ a_{1} + a_{2} + \cdots + a_{n} = b_{1} + b_{2} + \cdots + b_{n}, \]
and for any \( i, j \) ( \( 1 \leqslant i < j \leqslant n \)), it holds that \( a_{i}a_{j} \geqslant b_{i} + b_{j} \). Find the minimum value of \( a_{1} + a_{2} + \cdots + a_{n} \).


In how many ways can a $2 \times n$ rectangle be tiled with $2 \times 1$ dominoes?


In the sequence $\{a_n\}$, $a_4 = 1$, $a_{11} = 9$, and the sum of any three consecutive terms is 15. What is $a_{2016}$?


On each edge of a regular tetrahedron of side 1, there is a sphere with that edge as diameter. S is the intersection of the spheres (so it is all points whose distance from the midpoint of every edge is at most 1/2). Show that the distance between any two points of S is at most 1/√6.


Find the sine of the angle at the vertex of an isosceles triangle, given that the perimeter of any inscribed rectangle, with two vertices lying on the base, is a constant value.


The number zero is written on a board. Peter is allowed to perform the following operations:

- Apply to one of the numbers written on the board a trigonometric (sin, $\cos$, $\operatorname{tg}$, or ctg) or inverse trigonometric (arcsin, arccos, $\operatorname{arctg}$, or arcctg) function and write the result on the board;
- Write on the board the quotient or product of two already written numbers.

Help Peter write $\sqrt{3}$ on the board.


The point \( P \) on the curve \( y = \frac{x+1}{x-1} \) (\( x \in \mathbf{R}, x \neq 1 \)) is at a minimum distance \( d \) from the origin \( O \). Find \( d \).

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


If the function \( f(x) = \frac{(\sqrt{1008} x + \sqrt{1009})^{2} + \sin 2018 x}{2016 x^{2} + 2018} \) has a maximum value of \( M \) and a minimum value of \( m \), then \( M + m \) equals \_\_\_\_.


In triangle \(ABC\), \(AB = (b^2 - 1)\) cm, \(BC = a^2\) cm, and \(AC = 2a\) cm, where \(a\) and \(b\) are positive integers greater than 1. Find the value of \(a - b\).


Solution: \boxed{n = 1}


Two players, \(A\) and \(B\), play rock-paper-scissors continuously until player \(A\) wins 2 consecutive games. Suppose each player is equally likely to use each hand sign in every game. What is the expected number of games they will play?


Point $P$ is located on side $AB$ of square $ABCD$ such that $AP : PB = 1 : 4$. Point $Q$ lies on side $BC$ of the square and divides it in the ratio $BQ : QC = 5$. Lines $DP$ and $AQ$ intersect at point $E$. Find the ratio of the lengths $AE : EQ$.


An ant starts from vertex \( A \) of rectangular prism \( ABCD-A_1B_1C_1D_1 \) and travels along the surface to reach vertex \( C_1 \) with the shortest distance being 6. What is the maximum volume of the rectangular prism?


In the pyramid $ABCD$, the area of the face $ABC$ is four times the area of the face $ABD$. Point $M$ is taken on the edge $CD$ such that $CM: MD = 2$. Through point $M$, planes parallel to the faces $ABC$ and $ABD$ are drawn. Find the ratio of the areas of the resulting sections.


Let \( A \) be a set of integers where the smallest element is 1 and the largest element is 100. Each element, except 1, is equal to the sum of two elements from \( A \) (it can be twice the same element). Find the minimum number of elements in set \( A \).


In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $c = 10$ and $\frac{\cos A}{\cos B} = \frac{b}{a} = \frac{4}{3}$, point $P$ is a point on the incircle of $\triangle ABC$. Let $d$ be the sum of the squares of the distances from $P$ to vertices $A$, $B$, and $C$. Find $d_{\min} + d_{\max}$.


Let \( n \) be an integer. Determine the remainder \( b \) of \( n^{a} - n \) divided by 30.


Henry starts to read a 290-page book on a Sunday. He reads four pages every day except on Sundays when he reads 25 pages. How many days does it take him to finish the book?


Let \(ABC\) be a triangle with \(AB = 3\), \(BC = 4\), and \(CA = 5\). What is the distance from \(B\) to line \(AC\)?


Calculate the circulation of the vector field given in spherical coordinates: \( \mathbf{F} = (2r) \mathbf{e}_{r} + (R + r) \sin \theta \mathbf{e}_{\varphi} \) along the circle \( L: \{ r = R, \theta = \frac{\pi}{2} \} \) in the positive direction of the angle \(\varphi\), directly and using Stokes' theorem.

Solution: \boxed{4\pi R^2}


For real number \( x \), let \( [x] \) denote the greatest integer less than or equal to \( x \). Find the positive integer \( n \) such that \(\left[\log _{2} 1\right] + \left[\log _{2} 2\right] + \left[\log _{2} 3\right] + \cdots + \left[\log _{2} n\right]=1994\).

Solution: \boxed{n = 312}


Two brothers sold a flock of sheep that belonged to both of them, receiving as many rubles for each sheep as there were sheep in the flock. The brothers divided the money received as follows: first the elder brother took ten rubles from the total amount, then the second brother took ten rubles, after which the first brother took another ten rubles, and so on. The younger brother lacked ten rubles in the end, so he took all the remaining small change, and in addition, to make the division fair, the elder brother gave the younger brother his pocket knife. What was the value of the pocket knife?


If the real number \( x \) satisfies \( \log _{2} x = 1 + \cos \theta \) where \( \theta \in \left[ -\frac{\pi}{2}, 0 \right] \), then the maximum value of the function \( f(x) = |x-1| + 2|x-3| \) is ____________.


There are 4 numbers written on a board. Vasya multiplied the first of these numbers by \(\sin \alpha\), the second by \(\cos \alpha\), the third by \(\operatorname{tg} \alpha\), and the fourth by \(\operatorname{ctg} \alpha\) (for some angle \(\alpha\)) and obtained a set of the same 4 numbers (possibly in a different order). What is the maximum number of distinct numbers that could have been written on the board?


A natural number is called 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.


A sequence of real numbers \( a_{0}, a_{1}, \ldots, a_{9} \) with \( a_{0}=0 \), \( a_{1}=1 \), and \( a_{2}>0 \) satisfies
\[ a_{n+2} a_{n} a_{n-1} = a_{n+2} + a_{n} + a_{n-1} \]
for all \( 1 \leq n \leq 7 \), but cannot be extended to \( a_{10} \). In other words, no values of \( a_{10} \in \mathbb{R} \) satisfy
\[ a_{10} a_{8} a_{7} = a_{10} + a_{8} + a_{7} .\]

Compute the smallest possible value of \( a_{2} \).


Ten Cs are written in a row. Some Cs are upper-case and some are lower-case, and each is written in one of two colors, green and yellow. It is given that there is at least one lower-case C, at least one green C, and at least one C that is both upper-case and yellow. Furthermore, no lower-case C can be followed by an upper-case C, and no yellow C can be followed by a green C. In how many ways can the Cs be written?


Let \( \triangle ABC \) be an equilateral triangle with side length 1. Points \( A_1 \) and \( A_2 \) are chosen on side \( BC \), points \( B_1 \) and \( B_2 \) are chosen on side \( CA \), and points \( C_1 \) and \( C_2 \) are chosen on side \( AB \) such that \( BA_1 < BA_2 \), \( CB_1 < CB_2 \), and \( AC_1 < AC_2 \).

Suppose that the three line segments \( B_1C_2 \), \( C_1A_2 \), and \( A_1B_2 \) are concurrent, and the perimeters of triangles \( AB_2C_1 \), \( BC_2A_1 \), and \( CA_2B_1 \) are all equal. Find all possible values of this common perimeter.


How many solutions does the equation

$$
15x + 6y + 10z = 1973
$$

have in integers that satisfy the following inequalities:

$$
x \geq 13, \quad y \geq -4, \quad z > -6
$$


Find the real coefficient polynomial \( f(x) \) that satisfies the following conditions:
(1) For any real number \( a \), it holds that \( f(a+1) = f(a) + f(1) \).
(2) There exists a real number \( k_{1} \neq 0 \) such that \( f(k_{1}) = k_{2}, f(k_{2}) = k_{3}, \dots, f(k_{n-1}) = k_{n}, f(k_{n}) = k_{1} \), where \( n \) is the degree of \( f(x) \).

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


Calculate the limit of the function:

$$\lim_{x \rightarrow \pi} (x + \sin x)^{\sin x + x}$$

Solution: \boxed{\pi^\pi}


Find the largest positive integer \( n \) such that \( n! \) ends with exactly 100 zeros.


Let \( n \) be a positive integer. If the sum \( 1 + 2 + \cdots + n \) is exactly equal to a three-digit number, and that three-digit number has all identical digits, what are all possible values of \( n \)?


In the first quadrant of the Cartesian coordinate plane, the points with integer coordinates are numbered as follows:
- (0,0) is point number 1
- (1,0) is point number 2
- (1,1) is point number 3
- (0,1) is point number 4
- (0,2) is point number 5
- (1,2) is point number 6
- (2,2) is point number 7
- (2,1) is point number 8
- (2,0) is point number 9

Following the sequence indicated by the arrows, what are the coordinates of the 2000th point?

Solution: \boxed{(44, 25)}


The height of the triangular pyramid \(ABCD\) dropped from vertex \(D\) passes through the point of intersection of the heights of triangle \(ABC\). Additionally, it is known that \(DB = 3\), \(DC = 2\), and \(\angle BDC = 90^{\circ}\). Find the ratio of the area of face \(ADB\) to the area of face \(ADC\).


When passengers boarded an empty tram, half of them took seats. How many passengers boarded initially if, after the first stop, their number increased by exactly $8 \%$ and it is known that the tram accommodates no more than 70 people?


The school table tennis championship was held in an Olympic system format. The winner won six matches. How many participants in the tournament won more games than they lost? (In an Olympic system tournament, participants are paired up. Those who lose a game in the first round are eliminated. Those who win in the first round are paired again. Those who lose in the second round are eliminated, and so on. In each round, a pair was found for every participant.)


Given vectors \(\vec{a} = (1, \sin \theta)\) and \(\vec{b} = (\cos \theta, \sqrt{3})\) where \(\theta \in \mathbf{R}\), find the range of the magnitude \(|\vec{a} - \vec{b}|\).

Solution: \boxed{[1, 3]}


A rectangular chessboard of size \( m \times n \) is composed of unit squares (where \( m \) and \( n \) are positive integers not exceeding 10). A piece is placed on the unit square in the lower-left corner. Players A and B take turns moving the piece. The rules are as follows: either move the piece any number of squares upward, or any number of squares to the right, but you cannot move off the board or stay in the same position. The player who cannot make a move loses (i.e., the player who first moves the piece to the upper-right corner wins). How many pairs of integers \( (m, n) \) are there such that the first player A has a winning strategy?


A sports team's members have unique numbers taken from the integers 1 to 100. If no member's number is the sum of the numbers of any two other members, nor is it twice the number of any other member, what is the maximum number of people this team can have?


Given a large cube of dimensions \( 4 \times 4 \times 4 \) composed of 64 unit cubes, select 16 of the unit cubes to be painted red, such that in every \( 1 \times 1 \times 4 \) rectangular prism within the large cube that is composed of 4 unit cubes, exactly 1 unit cube is red. How many different ways are there to select these 16 red unit cubes? Explain why.