Place \( n \) (\( n \geq 1 \)) distinct balls into 4 distinct boxes \( A_1, A_2, A_3, A_4 \) such that \( A_1 \) contains an odd number of balls and \( A_2 \) contains an even number of balls. Find the number of different methods \( g_n \) to distribute the balls.


Find the area of the region in the \(xy\)-plane satisfying \(x^{6} - x^{2} + y^{2} \leq 0\).


The sequence \(\left\{x_{n}\right\}\) satisfies \(x_{1}=\frac{1}{2}\) and \(x_{n+1}=x_{n}^{2}+x_{n}\). Let \([x]\) denote the greatest integer less than or equal to \(x\). Then find the value of \(\left[\frac{1}{1+x_{1}}+\frac{1}{1+x_{2}}+\cdots+\frac{1}{1+x_{2009}}\right]\).


Let the positive numbers \( x \) and \( y \) satisfy \( x^{3} + y^{3} = x - y \). Find the maximum value of the real number \( \lambda \) such that \( x^{2} + \lambda y^{2} \leq 1 \) always holds.

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


Let \( S^2 \) be the 2-sphere \( \{ (x, y, z) : x^2 + y^2 + z^2 = 1 \} \). Show that for any \( n \) points on \( S^2 \), the sum of the squares of the \( \frac{n(n - 1)}{2} \) distances between them (measured in space, not on \( S^2 \)) is at most \( n^2 \).


In a plane, there are 11 gears arranged in such a way that the first gear meshes with the second, the second with the third, and so on. Finally, the eleventh gear meshes with the first one. Can the gears in such a system rotate?

Solution: \boxed{\text{No}}


Let
$$
S=\sum_{r=0}^{n}\binom{3 n+r}{r}
$$

Evaluate \( S \div (23 \times 38 \times 41 \times 43 \times 47) \) when \( n = 12 \).


How many integer solutions does the equation

$$
\frac{1}{2022}=\frac{1}{x}+\frac{1}{y}
$$

have?


In the mystical mountain, there are only two types of legendary creatures: Nine-Headed Birds and Nine-Tailed Foxes. A Nine-Headed Bird has nine heads and one tail, while a Nine-Tailed Fox has nine tails and one head.

A Nine-Headed Bird discovers that, excluding itself, the total number of tails of the other creatures on the mountain is 4 times the number of heads. A Nine-Tailed Fox discovers that, excluding itself, the total number of tails of the other creatures on the mountain is 3 times the number of heads. How many Nine-Tailed Foxes are there on the mountain?


Katya has combined a square with a perimeter of 40 cm with a square with a perimeter of 100 cm as shown in the figure. What is the perimeter of the resulting figure in centimeters?


For which positive integers \( n \) does the number \( n^{2} + 1 \) divide \( n + 1 \)?


Anya wrote down 100 numbers in her notebook. Then, Sonya wrote down all the pairwise products of the numbers that Anya had written. Artem noticed that there were exactly 2000 negative numbers in Sonya's notebook. How many zeros did Anya initially write down in her notebook?


2011 warehouses are connected by roads such that each warehouse can be reached from any other, possibly by traveling through several roads. Each warehouse contains $x_{1}, \ldots, x_{2011}$ kg of cement, respectively. In one trip, any amount of cement can be transported from one warehouse to another along a connecting road. Ultimately, the plan is to have $y_{1}, \ldots, y_{2011}$ kg of cement at the warehouses, respectively, with the condition that $x_{1} + x_{2} + \ldots + x_{2011} = y_{1} + y_{2} + \ldots + y_{2011}$. What is the minimum number of trips required to execute this plan for any values of $x_{i}$ and $y_{i}$ and any road scheme?


In the figure below, \(ABCD\) is a rectangle, \(E\) is the midpoint of \(AD\), and \(F\) is the midpoint of \(CE\). If the area of triangle \(BDF\) is \(12 \, \text{cm}^2\), find the area of rectangle \(ABCD\) in \(\text{cm}^2\).

Solution: \boxed{96 \text{ cm}^2}


The line \( c \) is given by the equation \( y = x + 1 \). Points \( A \) and \( B \) have coordinates \( A(1, 0) \) and \( B(3, 0) \). On the line \( c \), find the point \( C \) from which the segment \( AB \) is seen at the largest angle.

Solution: \boxed{(1, 2)}


Divide the set of numbers \(\{1, 2, \cdots, 10\}\) into two groups such that the product of the numbers in the first group, \(p_{1}\), is divisible by the product of the numbers in the second group, \(p_{2}\). Find the minimum value of \(\frac{p_{1}}{p_{2}}\).


A city has 57 bus routes which satisfy the following conditions:
(1) From any station, it is possible to directly reach any other station without changing buses;
(2) Any two routes have exactly one shared station where passengers can transfer from one route to the other;
(3) Each route has at least 3 stations.

How many stations are there on each of the 57 routes?


Solve the following equation:

$$
\log \sqrt{7x+3} + \log \sqrt{4x+5} = \frac{1}{2} + \log 3
$$


Given that point \( P(x, y) \) satisfies \( |x| + |y| \leq 2 \), find the probability for point \( P \) to have a distance \( d \leq 1 \) from the \( x \)-axis.


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

What is the value of \( D \)?


If the minimum value of the function
$$
p(x)=2 x \ln x+x^{2}-a x+3 \quad (x > 0)
$$
is not less than 0, then the maximum value of the real number $a$ is $\qquad$ .


A circle with radius 5 is tangent to the \(x\)-axis, the \(y\)-axis, and the line \(4x - 3y + 10 = 0\). Find its center.

Solution: \boxed{(-5, 5)}


Teams A and B each have 7 players who will compete in a Go tournament in a predetermined order. The match starts with player 1 from each team competing against each other. The loser is eliminated, and the winner next competes against the loser’s teammate. This process continues until all players of one team are eliminated, and the other team wins. Determine the total number of possible sequences of matches.


The sequence \(\left\{a_{n}\right\}\) is defined by \(a_{1}=1.5\) and \(a_{n}=\frac{1}{n^{2}-1}\) for \(n \in \mathbb{N}, n>1\). Are there any values of \(n\) such that the sum of the first \(n\) terms of this sequence differs from 2.25 by less than 0.01? If yes, find the smallest such value of \(n\).

Solution: \boxed{n = 100}


There is a rectangular box with dimensions $m \times n \times p$ and another rectangular box with dimensions $(m+2) \times (n+2) \times (p+2)$, where $m$, $n$, and $p$ are positive integers and $m \leq n \leq p$. The volume of the former is half of the volume of the latter. Find the maximum value of $p$.


We place 2020 points on a circle, numbered randomly from 1 to 2020; all the numbers are different. For a chord connecting a vertex numbered \( a \) to a vertex numbered \( b \), the value associated with the chord is \( |\mathrm{b}-\mathrm{a}| \). Show that it is possible to connect the vertices with 1010 chords such that

$\triangleright$ the chords do not intersect,

$\triangleright$ exactly one chord starts from each vertex,

$\triangleright$ the sum of the values of the chords equals \( 1010^{2} \).


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

\[
\left\{\begin{array}{l}
x^{2} + xy + y^{2} = 12 \\
y^{2} + yz + z^{2} = 25 \\
z^{2} + xz + x^{2} = 37
\end{array}\right.
\]

find the value of the expression \(xy + yz + xz\).


To make the inequality \(0 \leq x^{2} + ax + 5 \leq 4\) have exactly one solution, what should \(a\) be?

Solution: \boxed{\pm 2}


If the numbers \( p = 3 \cdot 2^n - 1 \), \( q = 3 \cdot 2^{n-1} - 1 \), and \( r = 9 \cdot 2^{2n-1} - 1 \) are prime, show that the numbers \( A \) and \( B \) will be amicable if \( A = 2^n \cdot p \cdot q \) and \( B = 2^n \cdot r \). Apply this to the particular case \( n = 2 \).

Note: Amicable numbers are pairs of numbers such that the first is equal to the sum of all divisors of the second, and the second is equal to the sum of all divisors of the first.

Solution: \boxed{A = 220, B = 284}


A sequence \(a_{1}, a_{2}, a_{3}, \ldots\) of positive integers satisfies \(a_{1} > 5\) and \(a_{n+1} = 5 + 6 + \cdots + a_{n}\) for all positive integers \(n\). Determine all prime numbers \(p\) such that, regardless of the value of \(a_{1}\), this sequence must contain a multiple of \(p\).


In an ornithological park, there are birds of several species, with a total of 2021 individuals. The birds are seated in a row, and it turns out that between any two birds of the same species, there is an even number of birds. What is the smallest number of bird species that could be?


Given non-negative real numbers \( a \), \( b \), and \( c \) satisfying \( a + b + c = 1 \), find the maximum value of \( a^{2}(b-c) + b^{2}(c-a) + c^{2}(a-b) \).


An integer is square-free if it is not divisible by \( a^2 \) for any integer \( a > 1 \). Let \( S \) be the set of positive square-free integers. Determine, with justification, the value of
\[ \sum_{k \in S}\left\lfloor\sqrt{\frac{10^{10}}{k}}\right\rfloor \text{,} \]
where \(\lfloor x\rfloor\) denotes the greatest integer less than or equal to \( x \). For example, \(\lfloor 2.5\rfloor = 2 \).


A diagonal of a regular 2006-gon is called odd if its endpoints divide the boundary into two parts, each composed of an odd number of sides. Sides are also regarded as odd diagonals. Suppose the 2006-gon has been dissected into triangles by 2003 nonintersecting diagonals. Find the maximum possible number of isosceles triangles with two odd sides.


In the coordinate system, the vertices of triangle $ABC$ are: $A(0, 4)$, $B(3, 0)$, and $C(c, 6)$. The area of the triangle is 7. What is the value of $c$, if it is known that $0 < c < 3$?


Find the area of the figure defined on the coordinate plane by the inequalities \( |x| - 1 \leq y \leq \sqrt{1 - x^{2}} \).

Solution: \boxed{\frac{\pi}{2} + 1}


A three-digit number, all digits of which are different, is called balanced if it equals the sum of all possible two-digit numbers composed from its different digits. Provide an example of a balanced number. Justify your answer.


As shown in the figure, the area of the large square is 9, and the area of the small square in the center is 1. A, B, C, and D are four trapezoids. What is the sum of the areas of B and D? $\qquad$ .


The age of Ivan Ivanovich is 48 years, 48 months, 48 weeks, 48 days, and 48 hours. How many complete years old is Ivan Ivanovich? Remember to explain your answer.


A cafe has 3 tables and 5 individual counter seats. People enter in groups of size between 1 and 4, and groups never share a table. A group of more than 1 will always try to sit at a table, but will sit in counter seats if no tables are available. Conversely, a group of 1 will always try to sit at the counter first. One morning, \( M \) groups consisting of a total of \( N \) people enter and sit down. Then, a single person walks in and realizes that all the tables and counter seats are occupied by some person or group. What is the minimum possible value of \( M + N \)?


Among five externally identical coins, 3 are genuine and 2 are counterfeit, with the counterfeits weighing the same but it is unknown whether they are heavier or lighter than the genuine coins. How can you find at least one genuine coin with the fewest possible weighings?


There is a hot spring swimming pool with a continuous flow of spring water at the bottom. To drain the full pool, 10 pumps need to work for 8 hours, and 9 pumps need to work for 9 hours. To ensure that the water level in the pool remains unchanged (neither decreasing nor increasing), how many pumps are needed to pump out the water?


Find the smallest natural number that is greater than the sum of its digits by 1755.


Through the midpoints $M$ and $N$ of the edges $AD$ and $CC_1$ of the parallelepiped $ABCD A_1 B_1 C_1 D_1$, a plane is drawn parallel to the diagonal $DB_1$. Construct the section of the parallelepiped by this plane. In what ratio does it divide the edge $BB_1$?


Compute the number of ordered pairs of positive integers \((a, b)\) satisfying the equation \(\operatorname{gcd}(a, b) \cdot a + b^2 = 10000\).


The sets \( A \) and \( B \) form a partition of positive integers if \( A \cap B = \emptyset \) and \( A \cup B = \mathbb{N} \). The set \( S \) is called prohibited for the partition if \( k + l \neq s \) for any \( k, l \in A, s \in S \) and any \( k, l \in B, s \in S \).

a) Define Fibonacci numbers \( f_{i} \) by letting \( f_{1} = 1, f_{2} = 2 \) and \( f_{i+1} = f_{i} + f_{i-1} \), so that \( f_{3} = 3, f_{4} = 5 \), etc. How many partitions for which the set \( F \) of all Fibonacci numbers is prohibited are there? (We count \( A, B \) and \( B, A \) as the same partition.)

b) How many partitions for which the set \( P \) of all powers of 2 is prohibited are there? What if we require in addition that \( P \subseteq A \)?


I1.1 Find \( a \) if \( 2t+1 \) is a factor of \( 4t^{2}+12t+a \).

I1.2 \( \sqrt{K} \) denotes the nonnegative square root of \( K \), where \( K \geq 0 \). If \( b \) is the root of the equation \( \sqrt{a-x} = x-3 \), find \( b \).

I1.3 If \( c \) is the greatest value of \( \frac{20}{b + 2 \cos \theta} \), find \( c \).

I1.4 A man drives a car at \( 3c \ \text{km/h} \) for 3 hours and then \( 4c \ \text{km/h} \) for 2 hours. If his average speed for the whole journey is \( d \ \text{km/h} \), find \( d \).

Solution: \boxed{d = 34}


One way to pack a 100 by 100 square with 10000 circles, each of diameter 1, is to put them in 100 rows with 100 circles in each row. If the circles are repacked so that the centers of any three tangent circles form an equilateral triangle, what is the maximum number of additional circles that can be packed?


The non-negative numbers \(a, b, c, d, e, f, g\) have a sum of 1. Select the largest value among the sums \(a+b+c, b+c+d, c+d+e, d+e+f, e+f+g\). What is the minimum value of this largest sum obtained?


Given a triangle \( \triangle ABC \) with interior angles \( \angle A, \angle B, \angle C \) and opposite sides \( a, b, c \) respectively, where \( \angle A - \angle C = \frac{\pi}{2} \) and \( a, b, c \) are in arithmetic progression, find the value of \( \cos B \).


In trapezoid $ABCD$, points $E$ and $F$ are marked on the bases $AD = 17$ and $BC = 9$ respectively, such that $MENF$ is a rectangle, where $M$ and $N$ are the midpoints of the diagonals of the trapezoid. Find the length of segment $EF$.


As shown in the figure, point $D$ is the midpoint of side $BC$ of $\triangle ABC$. Points $E$ and $F$ lie on $AB$, with $AE = \frac{1}{3} AB$ and $BF = \frac{1}{4} AB$. If the area of $\triangle ABC$ is $2018$, what is the area of $\triangle DEF$?


Calculate the area of the figure bounded by the curves given by the equations:

$$
\begin{aligned}
& \left\{\begin{array}{l}
x=8 \cos ^{3} t \\
y=8 \sin ^{3} t
\end{array}\right. \\
& x=1(x \geq 1)
\end{aligned}
$$


What is the maximum number of colors that can be used to color the cells of an 8x8 chessboard such that each cell shares a side with at least two cells of the same color?


\( N \) is an integer whose representation in base \( b \) is 777. Find the smallest positive integer \( b \) such that \( N \) is the fourth power of a decimal integer.


Two overlapping triangles $P O R$ and $Q O T$ are such that points $P, Q, R,$ and $T$ lie on the arc of a semicircle of center $O$ and diameter $P Q$. Lines $Q T$ and $P R$ intersect at the point $S$. Angle $T O P$ is $3 x^{\circ}$ and angle $R O Q$ is $5 x^{\circ}$. Show that angle $R S Q$ is $4 x^{\circ}$.


Let \( a \) and \( b \) be two real numbers. We set \( s = a + b \) and \( p = ab \). Express \( a^3 + b^3 \) in terms of \( s \) and \( p \) only.

Solution: \boxed{s^3 - 3sp}


At the whistle of the physical education teacher, all 10 boys and 7 girls line up in a row in a random order as they manage.

Find the expected value of the quantity "Number of girls standing to the left of all boys".


Gru and the Minions plan to make money through cryptocurrency mining. They chose Ethereum as one of the most stable and promising currencies. They bought a system unit for 9499 rubles and two graphics cards for 31431 rubles each. The power consumption of the system unit is 120 W, and for each graphics card, it is 125 W. The mining speed for one graphics card is 32 million hashes per second, allowing it to earn 0.00877 Ethereum per day. 1 Ethereum equals 27790.37 rubles. How many days will it take for the team's investment to pay off, considering electricity costs of 5.38 rubles per kWh? (20 points)


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


Given a natural number \( x = 8^n - 1 \), where \( n \) is a natural number. It is known that \( x \) has exactly three distinct prime divisors, one of which is 31. Find \( x \).


For each positive integer $n$, an associated non-negative integer $f(n)$ is defined to satisfy the following three rules:
i) $f(a b)=f(a)+f(b)$.

ii) $f(n)=0$ if $n$ is a prime greater than 10.

iii) $f(1)<f(243)<f(2)<11$.

Given that $f(2106)<11$, determine the value of $f(96)$.


Given a rhombus \(ABCD\), \(\Gamma_{B}\) and \(\Gamma_{C}\) are circles centered at \(B\) and \(C\) passing through \(C\) and \(B\) respectively. \(E\) is an intersection point of circles \(\Gamma_{B}\) and \(\Gamma_{C}\). The line \(ED\) intersects circle \(\Gamma_{B}\) at a second point \(F\). Find the measure of \(\angle AFB\).


Of the integers from 1 to \(8 \cdot 10^{20}\) (inclusive), which are more numerous, and by how much: those containing only even digits in their representation or those containing only odd digits?


In triangle \(ABC\), the median \(AD\) and the angle bisector \(BE\) are perpendicular and intersect at point \(F\). It is known that \(S_{DEF} = 5\). Find \(S_{ABC}\).


Given a positive geometric sequence $\left\{a_{n}\right\}$ satisfying $a_{7}=a_{6}+2a_{5}$, if there exist two terms $a_{m}$ and $a_{n}$ such that $\sqrt{a_{m} \cdot a_{n}}=4a_{1}$, find the minimum value of $\frac{1}{m}+\frac{4}{n}$.


Petya has four cards with the digits $1, 2, 3, 4$. Each digit appears exactly once. How many natural numbers greater than 2222 can Petya form using these cards?


In $\triangle ABC$, the sides opposite to $\angle A$, $\angle B$, and $\angle C$ are denoted as $a$, $b$, and $c$, respectively. Given that
$$
\begin{array}{l}
f(x) = \boldsymbol{m} \cdot \boldsymbol{n}, \boldsymbol{m} = (2 \cos x, 1), \\
\boldsymbol{n} = (\cos x, \sqrt{3} \sin 2x), \\
f(A) = 2, b = 1, S_{\triangle ABC} = \frac{\sqrt{3}}{2}.
\end{array}
$$
Determine the value of
$$
\frac{b + \boldsymbol{c}}{\sin B + \sin C}.
$$


The side of the base of a regular quadrilateral pyramid is equal to \( a \). A lateral face forms a 45° angle with the plane of the base. Find the lateral surface area of the pyramid.

Solution: \boxed{a^2 \sqrt{2}}


How many quadruples (i, j, k, h) of ordered integers satisfy the condition \(1 \leqslant i < j \leqslant k < h \leqslant n+1\)?


Given that the cube root of \( m \) is a number in the form \( n + r \), where \( n \) is a positive integer and \( r \) is a positive real number less than \(\frac{1}{1000}\). When \( m \) is the smallest positive integer satisfying the above condition, find the value of \( n \).


Let \( n \) be a natural number. We call a sequence consisting of \( 3n \) letters Romanian if the letters \( I \), \( M \), and \( O \) all occur exactly \( n \) times. A swap is an exchange of two neighboring letters. Show that for every Romanian sequence \( X \) there exists a Romanian sequence \( Y \), so that at least \( \frac{3n^2}{2} \) swaps are necessary to obtain the sequence \( Y \) from the sequence \( X \).


Let \( x, y, z \) be positive real numbers such that:

\[ \begin{aligned}
& x^2 + xy + y^2 = 2 \\
& y^2 + yz + z^2 = 5 \\
& z^2 + xz + x^2 = 3
\end{aligned} \]

Determine the value of \( xy + yz + xz \).

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


Solution: \boxed{\log_{a} b}


Let the set \(I = \{0, 1, 2, \ldots, 22\}\). Define \(A = \{(a, b, c, d) \mid a, b, c, d \in I, a + d \equiv 1 \pmod{23}, \text{ and } a d - b c \equiv 0 \pmod{23}\}\). Determine the number of elements in the set \(A\).


Given that bus types $A$, $B$, and $C$ all depart at 6:00 AM from a certain station, with departure intervals of 10 minutes, 12 minutes, and 15 minutes, respectively, if Xiaoming arrives at the station at some time between 8:00 AM and 12:00 PM to catch one of these buses, what is Xiaoming's average waiting time in minutes?


What is the maximum number of L-shaped figures, consisting of 5 squares of size \(1 \times 1\), that can be placed in a \(7 \times 7\) square? (The L-shaped figures can be rotated and flipped, but cannot overlap with each other.)


In a warehouse, there are 8 cabinets, each containing 4 boxes, and each box contains 10 mobile phones. The warehouse, each cabinet, and each box are locked with a key. The manager is tasked with retrieving 52 mobile phones. What is the minimum number of keys the manager must take with him?


A positive integer \( n \) is said to be increasing if, by reversing the digits of \( n \), we get an integer larger than \( n \). For example, 2003 is increasing because, by reversing the digits of 2003, we get 3002, which is larger than 2003. How many four-digit positive integers are increasing?


The area of a rectangle is 12 square metres. The lengths of the sides, in metres, are whole numbers. The greatest possible perimeter (in metres) is:
(A) 14
(B) 16
(C) 12
(D) 24
(E) 26


Vojta bought 24 identical square tiles. Each tile had a side length of $40 \mathrm{~cm}$. Vojta wanted to arrange them in front of a cabin to form a rectangular platform with the smallest possible perimeter. What was the perimeter of the tiled rectangle in meters, given that no tile was leftover and Vojta did not cut or break any tiles?


Find the product of two approximate numbers: $0.3862 \times 0.85$.


Determine all pairs of positive integers \((m, n)\) such that \(2^m + 1 = n^2\).

Solution: \boxed{(3, 3)}


Calculate the limit of the function:

\[
\lim _{x \rightarrow -2} \frac{\operatorname{tg}\left(e^{x+2}-e^{x^{2}-4}\right)}{\operatorname{tg} x + \operatorname{tg} 2}
\]

Solution: \boxed{5 \cos^2 2}


The transgalactic ship encountered an amazing meteor shower. One part of the meteors flies along a straight line with equal speeds, one after another, at equal distances from each other. Another part flies similarly but along another straight line, parallel to the first one, with the same speeds but in the opposite direction, at the same distance from each other. The ship flies parallel to these lines. Astronaut Gavrila recorded that the ship encounters meteors flying towards it every 7 seconds, and those flying in the same direction as the ship every 13 seconds. He wondered how often the meteors would pass by if the ship were stationary. He thought it necessary to take the arithmetic mean of the two given times. Is Gavrila right? If so, write this arithmetic mean as the answer. If not, indicate the correct time in seconds, rounded to one decimal place.


In the set of the first ten thousand positive integers $\{1, 2, \cdots, 10000\}$, how many elements leave a remainder of 2 when divided by 3, a remainder of 3 when divided by 5, and a remainder of 4 when divided by 7?


A \(101 \times 101\) grid is given, where all cells are initially colored white. You are allowed to choose several rows and paint all the cells in those rows black. Then, choose exactly the same number of columns and invert the color of all cells in those columns (i.e., change white cells to black and black cells to white). What is the maximum number of black cells that the grid can contain after this operation?


Given that complex numbers \( z_{1}, z_{2}, z_{3} \) satisfy \( \left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1 \) and \( \left|z_{1}+z_{2}+z_{3}\right|=r \), where \( r \) is a given real number, express the real part of \( \frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}} \) in terms of \( r \).


What is the minimum number of colors required to paint all the vertices, edges, and diagonals of a convex \( n \)-gon, given that two conditions must be met:

1) Every two segments emanating from the same vertex must be of different colors;
2) The color of any vertex must differ from the color of any segment emanating from it?

Solution: \boxed{n}


A school uses a sum of money to buy balls. If they only buy volleyballs, they can buy exactly 15 of them. If they only buy basketballs, they can buy exactly 12 of them. Now, with the same amount of money, they buy a total of 14 volleyballs and basketballs. How many more volleyballs than basketballs were bought?


In the Cartesian coordinate system \(xOy\), points \(A\) and \(B\) lie on the parabola \(y^2 = 2x\) and satisfy \(\overrightarrow{OA} \cdot \overrightarrow{OB} = -1\). Let \(F\) be the focus of the parabola. Find the minimum value of \(S_{\triangle OFA} + S_{\triangle OFB} \).


There are three types of people, A, B, and C, with a total of 25 people. Type A people always tell the truth, Type B people always lie, and Type C people alternate between telling the truth and lying (for example, if a Type C person tells the truth this time, their next statement will certainly be a lie, and the statement after that will be true again).

The priest asks each person, "Are you a Type A person?" 17 people answer "yes."
The priest then asks each person, "Are you a Type C person?" 12 people answer "yes."
The priest finally asks each person, "Are you a Type B person?" 8 people answer "yes."

How many of the 25 people are Type C people?


Arrange six positive integers \(a, b, c, d, e, f\) in a sequence in alphabetical order such that \(a=1\). If any positive integer is greater than 1, then the number that is one less than this integer must appear to its left. For example, \(1,1,2,1,3,2\) meets the requirement; \(1,2,3,1,4,1\) meets the requirement; \(1,2,2,4,3,2\) does not meet the requirement. Find the number of different sequences that meet the requirement.


The monkeys - Masha, Dasha, Glasha, and Natasha - ate 16 bowls of semolina porridge for lunch. Each monkey had some portion of it. Glasha and Natasha together ate 9 portions. Masha ate more than Dasha, more than Glasha, and more than Natasha. How many bowls of porridge did Dasha get?


a) Determine the number of digits in the product \(111111 \cdot 1111111111\), where the first factor has 6 digits and the second has 10 digits.

b) The numbers \(2^{2016}\) and \(5^{2016}\) are written side by side to form a single number \(N\) which has a number of digits that is the sum of the number of digits of the two numbers. For example, if we did this with \(2^{3}\) and \(5^{3}\), we would obtain the number 8125, which has 4 digits. Determine the number of digits of \(N\).


The sum of two nonzero natural numbers is 210, and their least common multiple is 1547. What is their product? $\qquad$


For which (real) values of the variable \( x \) are the following equalities valid:

a) \(\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=\sqrt{2}\),

b) \(\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=1\),

c) \(\sqrt{x+\sqrt{2x-1}}+\sqrt{x-\sqrt{2x-1}}=2\).

(Without sign, the square root always means the non-negative square root!)


In how many ways can two people divide 10 distinct items between themselves so that each person gets 5 items?


Given the equation \(x^{2} - 402x + k = 0\), one root plus 3 is equal to 80 times the other root. Determine the value of \(k\).


Given a parallelogram \(ABCD\) with sides \(AB=2\) and \(BC=3\), find the area of this parallelogram, given that the diagonal \(AC\) is perpendicular to the segment \(BE\), where \(E\) is the midpoint of side \(AD\).


In a cube \( ABCD-A_1 B_1 C_1 D_1 \) with edge length 1, points \( X \) and \( Y \) are the centers of the squares \( AA_1 B_1 B \) and \( BB_1 C_1 C \) respectively. Point \( Z \) lies on the diagonal \( BD \) such that \( DZ = 3 ZB \). Determine the area of the cross-section cut by the plane \( XYZ \) on the circumscribed sphere of the cube.

Solution: \boxed{ \frac{5 \pi}{8}}


Triangle \( \triangle ABC \) has area 1. Points \( E \) and \( F \) are on \( AB \) and \( AC \) respectively such that \( EF \parallel BC \). If \( \triangle AEF \) and \( \triangle EBC \) have equal areas, find the area of \( \triangle EFC \).

Solution: \boxed{\sqrt{5} - 2}


In $\triangle ABC$, $\angle ABC = 40^\circ$, $\angle ACB = 20^\circ$. Point $N$ is inside $\triangle ABC$, $\angle NBC = 30^\circ$, and $\angle NAB = 20^\circ$. Find the measure of $\angle NCB$.


At the first site, higher-class equipment was used, and at the second site, first-class equipment was used. There was less higher-class equipment than first-class equipment. First, 40% of the equipment from the first site was transferred to the second site. Then, 20% of the equipment at the second site was transferred back to the first site, with half of the transferred equipment being first-class. After this, the amount of higher-class equipment at the first site was 26 units more than at the second site, and the total amount of equipment at the second site increased by more than 5% compared to the original amount. Find the total amount of first-class equipment.


From the first 539 positive integers, we select some such that their sum is at least one-third of the sum of the original numbers. What is the minimum number of integers we need to select for this condition to be satisfied?


Solve the problem: Octopuses with an even number of legs always lie, while octopuses with an odd number always tell the truth. Five octopuses met, each having between 7 to 9 legs.

The first said, "Together we have 36 legs";
The second said, "Together we have 37 legs";
The third said, "Together we have 38 legs";
The fourth said, "Together we have 39 legs";
The fifth said, "Together we have 40 legs".

How many legs did they actually have?


Let \( a \) and \( b \) be integers such that the difference between \( a^2 \) and \( b^2 \) is 144. Determine the largest possible value of \( d = a + b \).


Given the non-negative real numbers \(x, y, z\) that satisfy \(x + y + z = 1\). Find the maximum and minimum values of \(x^3 + 2y^2 + \frac{10}{3}z\).


Find the largest integer \( n \) such that \(\frac{(n-2)^{2}(n+1)}{2n-1}\) is an integer.


Given real numbers \( x, y, z, w \) satisfying \( x + y + z + w = 1 \), find the maximum value of \( M = xw + 2yw + 3xy + 3zw + 4xz + 5yz \).


Find the measure of the angle

$$
\delta=\arccos \left(\left(\sin 2539^{\circ}+\sin 2540^{\circ}+\cdots+\sin 6139^{\circ}\right)^{\cos } 2520^{\circ}+\cos 2521^{\circ}+\cdots+\cos 6120^{\circ}\right)
$$


Find all values of the parameter \(a\) for which there exists a number \(b\) such that the system

$$
\left\{\begin{array}{l}
x^{2}+y^{2}+2 a(a-x-y)=64 \\
y=8 \sin (x-2 b)-6 \cos (x-2 b)
\end{array}\right.
$$

has at least one solution \((x, y)\).

Solution: \boxed{a \in [-18, 18]}


The number \( n \) is such that \( 8d \) is a 100-digit number, and \( 81n - 102 \) is a 102-digit number. What can be the second digit from the beginning of \( n \)?


Assume that \( a_{i} \in \{1, -1\} \) for all \( i=1, 2, \ldots, 2013 \). Find the least positive value of the following expression:
\[
\sum_{1 \leq i < j \leq 2013} a_{i} a_{j}
\]


One side of the cards is painted in some color, and the other has a smiley face. There are four cards in front of you: the first is yellow, the second is black, the third has a happy smiley, and the fourth has a sad smiley. You need to check the statement: "If there is a happy smiley on one side of a card, then the other side is painted yellow."

What is the minimum number of cards you need to turn over to check the truth of this statement? Which ones? (7 points)


A polygon is said to be friendly if it is regular and it also has angles that, when measured in degrees, are either integers or half-integers (i.e., have a decimal part of exactly 0.5). How many different friendly polygons are there?


How many students are there in our city? The number expressing the quantity of students is the largest of all numbers where any two adjacent digits form a number that is divisible by 23.


Calculate the limit of the function:

\[
\lim _{x \rightarrow 3}\left(2-\frac{x}{3}\right)^{\sin (\pi x)}
\]


Given six points $A$, $B$, $C$, $D$, $E$, $F$ in space with no four points coplanar. What is the maximum number of line segments that can be drawn such that no tetrahedron is formed in the figure?


In an acute-angled triangle \( ABC \), the altitude \( BH \) and the median \( AM \) are drawn. It is known that the angle \( MCA \) is twice the angle \( MAC \), and \( BC = 10 \).

Find \( AH \).


Petya plans to spend all 90 days of his vacation in the village. Every second day (i.e., every other day), he goes swimming in the lake. Every third day, he goes to the store for groceries. Every fifth day, he solves math problems. (On the first day, Petya did all three activities and was very tired.) How many "pleasant" days will Petya have, when he has to go swimming but does not need to go to the store or solve problems? How many "boring" days will there be when there are no activities at all?


Among all natural numbers that are multiples of 20, what is the sum of those that do not exceed 3000 and are also multiples of 14?


A semicircle is constructed over the line segment $AB$ with midpoint $M$. Let $P$ be a point on the semicircle other than $A$ and $B$, and let $Q$ be the midpoint of the arc $AP$. The intersection of the line $BP$ with the line parallel to $PQ$ through $M$ is denoted as $S$.

Show that $PM = PS$.

(Karl Czakler)

Solution: \boxed{PM = PS}


Given that \( n \) is a positive integer and \( n! \) denotes the factorial of \( n \), find all positive integer solutions \((x, y)\) to the equation
\[
20(x!) + 2(y!) = (2x + y)!
\]


In quadrilateral \(ABCD\), it is given that \(\angle DAB = 150^\circ\), \(\angle DAC + \angle ABD = 120^\circ\), and \(\angle DBC - \angle ABD = 60^\circ\). Find \(\angle BDC\).


On the radius \( AO \) of a circle with center \( O \), a point \( M \) is chosen. On the same side of \( AO \) on the circle, points \( B \) and \( C \) are chosen so that \( \angle AMB = \angle OMC = \alpha \). Find the length of \( BC \), given that the radius of the circle is 12 and \( \sin \alpha = \frac{\sqrt{11}}{6} \).


A group of monkeys is divided into three subgroups, with each subgroup having an equal number of monkeys. They go to the orchard to pick peaches. After finishing the picking, they combine all the peaches and distribute them equally among all the monkeys. If each monkey receives 5 peaches, there are 27 peaches left over; if each monkey receives 7 peaches, then at least one monkey receives fewer than 7 peaches. What is the total number of peaches this group of monkeys has picked?


Grisha wrote 100 numbers on the board. Then he increased each number by 1 and noticed that the product of all 100 numbers did not change. He increased each number by 1 again, and again the product of all the numbers did not change, and so on. Grisha repeated this procedure $k$ times, and each of the $k$ times the product of the numbers did not change. Find the largest possible value of $k$.


Given the equation
\[ x^{2} + ax + b - 3 = 0 \quad (a, b \in \mathbf{R}) \]
has a real root in the interval \([1, 2]\), find the minimum value of \( a^{2} + (b - 4)^{2} \).


Given \( x = -2272 \), \( y = 10^3 + 10^2 c + 10 b + a \), and \( z = 1 \), which satisfy the equation \( a x + b y + c z = 1 \), where \( a \), \( b \), \( c \) are positive integers and \( a < b < c \). Find \( y \).


There are 4 different digits that can form 18 different four-digit numbers arranged in ascending order. The first four-digit number is a perfect square, and the second-last four-digit number is also a perfect square. What is the sum of these two numbers?


Find the minimum value of \( a^{2} + b^{2} + c^{2} + d^{2} \) given that \( a + 2b + 3c + 4d = 12 \).


In a trapezoid, the diagonals intersect at a right angle, and one of them is equal to the midsegment. Determine the angle that this diagonal forms with the bases of the trapezoid.


Inside a square, point $P$ has distances $a, b, c$ from vertices $A, B, C$ respectively. What is the area of the square? What is it in the case when $a = b = c$?


When \((1+x)^{38}\) is expanded in ascending powers of \(x\), \(N_{1}\) of the coefficients leave a remainder of 1 when divided by 3, while \(N_{2}\) of the coefficients leave a remainder of 2 when divided by 3. Find \(N_{1} - N_{2}\).


On a plate, there are various pancakes with three different fillings: 2 with meat, 3 with cottage cheese, and 5 with strawberries. Sveta consecutively ate all of them, choosing each next pancake at random. Find the probability that the first and the last pancakes she ate had the same filling.


Given the sequence \(\left\{a_{n}\right\}\) defined by:
\[
\begin{array}{l}
a_{1}=2, a_{2}=6, \\
a_{n+1}=\frac{a_{n}^{2}-2 a_{n}}{a_{n-1}} \text{ for } n=2,3, \ldots
\end{array}
\]

Determine \(\lim _{n \rightarrow \infty}\left\{\sqrt{a_{n} + n}\right\} = \)


In the parallelogram \(ABCD\), point \(K\) is the midpoint of side \(BC\), and point \(M\) is the midpoint of side \(CD\). Find \(AD\) if \(AK = 6\) cm, \(AM = 3\) cm, and \(\angle KAM = 60^\circ\).


In a classroom, 34 students are seated in 5 rows of 7 chairs. The place at the center of the room is unoccupied. A teacher decides to reassign the seats such that each student will occupy a chair adjacent to his/her present one (i.e., move one desk forward, back, left, or right). In how many ways can this reassignment be made?


Given a point \( P \) inside \( \triangle ABC \), perpendiculars are drawn from \( P \) to \( BC, CA, \) and \( AB \) with feet \( D, E, \) and \( F \) respectively. Semicircles are constructed externally on diameters \( AF, BF, BD, CD, CE, \) and \( AE \). These six semicircles have areas denoted \( S_1, S_2, S_3, S_4, S_5, \) and \( S_6 \). Given that \( S_5 - S_6 = 2 \) and \( S_1 - S_2 = 1 \), find \( S_4 - S_3 \).


In a certain region are five towns: Freiburg, Göttingen, Hamburg, Ingolstadt, and Jena.

On a certain day, 40 trains each made a journey, leaving one of these towns and arriving at one of the other towns. Ten trains traveled either from or to Freiburg. Ten trains traveled either from or to Göttingen. Ten trains traveled either from or to Hamburg. Ten trains traveled either from or to Ingolstadt.

How many trains traveled from or to Jena?

A) 0
B) 10
C) 20
D) 30
E) 40


Find all functions \( f \) from the set of real numbers to the set of real numbers that satisfy the following conditions:
1. \( f(x) \) is strictly increasing;
2. For all real numbers \( x \), \( f(x) + g(x) = 2x \), where \( g(x) \) is the inverse function of \( f(x) \).

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


Palm oil has increased in price by 10%. As a result, the cheese produced by one of the manufacturers increased in price by 3%. What percentage of palm oil is in this manufacturer's cheese?


\(\cos^4 2\alpha - 6 \cos^2 2\alpha \sin^2 2\alpha + \sin^4 2\alpha\).

Solution: \boxed{\cos 8 \alpha}


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?


Points A, B, C, D are consecutive vertices of a regular n-gon. What is n if \(\angle \mathrm{ACD}=120^{\circ}\)?


Find the maximum value of the expression \( |\ldots|\left|x_{1}-x_{2}\right|-x_{3}|\ldots-x_{1990}\right| \), where \( x_{1}, x_{2}, \ldots, x_{1990} \) are distinct natural numbers from 1 to 1990.


Inside a large triangle with a perimeter of 120, several segments were drawn, dividing it into nine smaller triangles as shown in the image. It turned out that the perimeters of all nine small triangles are equal. What can they be equal to? Indicate all possible options.

Perimeter of a figure is the sum of the lengths of all its sides.


Given that the function \( y = f(x) \) is an even function and its graph intersects the x-axis at four points, find the sum of all real roots of the equation \( f(x) = 0 \).


A merchant sold goods for 39 K. How much did he buy the goods for if he made the same percentage profit as the cost of the goods?


ABCD is a convex quadrilateral with area 1. The lines AD and BC meet at X. The midpoints of the diagonals AC and BD are Y and Z, respectively. Find the area of the triangle XYZ.


Find the largest \( n \) such that the sum of the fourth powers of any \( n \) prime numbers greater than 10 is divisible by \( n \).


In an isosceles triangle \(ABC\) where \(AB = BC\) and \(\angle ABC = 80^\circ\), point \(M\) is inside the triangle such that \(\angle MAC = 30^\circ\) and \(\angle MCA = 10^\circ\). Find \(\angle BMC\).


Find the smallest natural number \( n \) which satisfies the inequality
\[ 2006^{1003} < n^{2006}. \]


Let $\alpha$ and $\beta$ be a pair of conjugate complex numbers. If $|\alpha - \beta| = 2 \sqrt{3}$ and $\frac{\alpha}{\beta^{2}}$ is a real number, then $|\alpha| = \quad$.


The geometric mean of a set of \(n\) positive numbers is defined as the \(n\)-th root of the product of those numbers.
Yasmeen writes down a set of four numbers which have a geometric mean of 2048.
Zak writes down a set of four numbers which have a geometric mean of 8.
What is the geometric mean of the combined set of the eight numbers written by Yasmeen and Zak?


Xiao Hua plays a certain game where each round can be played several times freely. Each score in a round is one of the numbers $8$, $a$ (a natural number), or $0$. The total score for a round is the sum of all individual scores in that round. Xiao Hua has achieved the following total scores in some rounds: $103, 104, 105, 106, 107, 108, 109, 110$. It is also known that he cannot achieve a total score of $83$. What is the value of $a$?


Compute the number of positive four-digit multiples of 11 whose sum of digits (in base ten) is divisible by 11.


We have 50 numbered chips from 1 to 50, and they need to be colored either red or blue. We know that chip 5 is blue. For the coloring of the remaining chips, the following rules are followed:

a) If the chip with number $x$ and the chip with number $y$ are of different colors, then the chip with number $|x-y|$ is painted red.

b) If the chip with number $x$ and the chip with number $y$ are of different colors and $x \cdot y$ is a number between 1 and 50 (inclusive), then the chip with number $x \cdot y$ is painted blue.

Determine how many different colorings can be made on the set of chips.


Let's 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. Consider a random point with coordinates \((x, y)\) from the unit square \( F \). The event \( A \) "Olga Pavlovna and Maria Petrovna together have at least 1 liter, but less than 1.5 liters of jam" is represented by the inequality \( 1 \leq x + y < 1.5 \) and is depicted as a trapezoid bounded by the lines \( x + y = 1 \) and \( x + y = 1.5 \). Thus,

$$
\mathrm{P}(A) = \frac{S_{A}}{S_{F}} = \frac{3}{8} = 0.375
$$


Let \( n \) be the integral part of \( \frac{1}{\frac{1}{1980}+\frac{1}{1981}+\cdots+\frac{1}{2009}} \). Find the value of \( n \).


If $\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^{n}}>\frac{315}{412}$ (where $n$ is a positive natural number), what is the smallest value of $n$ that satisfies the condition?


\[
\sum_{k=1}^{70} \frac{k}{x-k} \geq \frac{5}{4}
\]
is a union of disjoint intervals the sum of whose lengths is 1988.


One mole of an ideal monoatomic gas is first heated isobarically. In this process, it does 20 J of work. Then, it is heated isothermally, receiving the same amount of heat as in the first case. How much work (in joules) does the gas do in the second case?

Solution: \boxed{50 \text{ J}}


The line \( y = 2x + 4 \) intersects the \( y \)-axis at \( R \). A second line, parallel to the \( y \)-axis, is drawn through \( P(p, 0) \), with \( p > 0 \). These two lines intersect at \( Q \).

(a) Determine the length of \( OR \). (Note that \( O \) is the origin \( (0,0) \).)

(b) Determine the coordinates of point \( Q \) in terms of \( p \).

(c) If \( p = 8 \), determine the area of \( OPQR \).

(d) If the area of \( OPQR \) is 77, determine the value of \( p \).


Anton, Artem, and Vera solved 100 math problems together. Each of them solved 60 problems. We call a problem difficult if only one person solved it, and easy if all three solved it. How much does the number of difficult problems differ from the number of easy problems?


For a natural number $N$, if at least six of the nine natural numbers from $1$ to $9$ can divide $N$, then $N$ is called a "six-divisible number". Among the natural numbers greater than $2000$, what is the smallest "six-divisible number"?


What is the maximum number of \(4 \times 1\) rectangles that can be placed in a \(6 \times 6\) square (without crossing the cell boundaries)?


There is a caravan with 100 camels, consisting of both one-humped and two-humped camels, with at least one of each kind. If you take any 62 camels, they will have at least half of the total number of humps in the caravan. Let \( N \) be the number of two-humped camels. How many possible values can \( N \) take within the range from 1 to 99?


The city's bus network has the following characteristics:
1) From any stop to any other stop, you can get there without a transfer;
2) For any pair of routes, there is exactly one stop where you can transfer from one route to the other;
3) Each route has exactly three stops.

How many bus routes are there in the city?


It is known that all 'krakozyabrs' have either horns or wings (or both). From the results of a global census of 'krakozyabrs,' it was found that 20% of the 'krakozyabrs' with horns also have wings, and 25% of the 'krakozyabrs' with wings also have horns. How many 'krakozyabrs' are left in the world if it is known that their number is more than 25 but less than 35?


How many four-digit numbers exist where the thousands digit is greater than the hundreds digit?


Consider a cube where all edges are colored either red or black in such a way that each face of the cube has at least one black edge. What is the minimum number of black edges?


Given that the positive real numbers \( x \) and \( y \) satisfy the equation \( x^3 + y^3 = (4x - 5y)y \), find the maximum value of \( y \).


On the continuation of side \( BC \) of parallelogram \( ABCD \), a point \( F \) is taken beyond point \( C \). Segment \( AF \) intersects diagonal \( BD \) at point \( E \) and side \( CD \) at point \( G \), where \( GF=3 \) and \( AE \) is 1 more than \( EG \). What part of the area of parallelogram \( ABCD \) is the area of triangle \( ADE \)?


The line \( x - 2y - 1 = 0 \) intersects the parabola \( y^2 = 4x \) at points \( A \) and \( B \). Let \( C \) be a point on the parabola such that \(\angle ACB = 90^\circ\). What are the coordinates of point \( C \)?

Solution: \boxed{(9, -6)}


The base edge of a regular triangular pyramid (tetrahedron) is one unit, and its side faces form an angle of $120^{\circ}$ with each other. What are the lengths of the pyramid's side edges?


Equilateral triangle \( \triangle ABC \) has side length 6. Let \( \omega \) be the circle through \( A \) and \( B \) such that \( CA \) and \( CB \) are both tangent to \( \omega \). A point \( D \) on \( \omega \) satisfies \( CD = 4 \). Let \( E \) be the intersection of line \( CD \) with segment \( AB \). What is the length of segment \( DE \)?


Let $A B C$ be a triangle with its longest side being $B C$. Show that if $P$ is an interior point of the triangle, then

$$
P A < P B + P C
$$

Solution: \boxed{PA < PB + PC}


16. Variance of the number of matches. A deck of playing cards is laid out on a table (for example, in a row). On top of each card, a card from another deck is placed. Some cards may match. Find:

a) the expected number of matches;

b) the variance of the number of matches.


Let \( N \) be the set of natural numbers, \( R \) be the set of real numbers, and \( S \) be the set of functions \( f: N \rightarrow R \) that satisfy the following two conditions:
1. \( f(1) = 2 \);
2. \( f(n+1) \geq f(n) \geq \frac{n}{n+1} f(2n) \) for \( n = 1, 2, \ldots \)

Find the smallest natural number \( M \) such that for any \( f \in S \) and any \( n \in N \), \( f(n) < M \).


What is the maximum number of self-intersection points that a closed broken line consisting of 7 segments can have? (Common endpoints of the segments are not counted.)


Calculate the area under the graph of the function \( y = \sin x \) on the interval from 0 to \( \pi \).


The perimeter of a square inscribed in a circle is \( p \). What is the area of the square that circumscribes the circle?


All two-digit numbers from 19 to 80 inclusive are written down one after the other as a single number \( N = 192021...7980 \). Is \( N \) divisible by 1980?

Solution: \boxed{ Yes}


Find the minimum value of \((a+b)\left(\frac{1}{a}+\frac{4}{b}\right)\), where \(a\) and \(b\) range over all positive real numbers.


Masha and Alina play on a $5 \times 5$ board. Masha can place chips in some cells one by one. After that, Alina covers all these cells with L-shaped tiles composed of three cells (non-overlapping and not going beyond the borders of the square; L-shaped tiles are allowed to be placed only on the cells). If Alina manages to cover all the cells with chips, she wins; if not, Masha wins. What is the minimum number of chips Masha needs to place so that Alina cannot win?


Determine the area of a triangle if two of its sides are 35 cm and 14 cm, and the bisector of the angle between them is 12 cm.

Solution: \boxed{235.2 \text{ cm}^2}


Given a triangle \(ABC\) with \(I\) as the incenter, suppose: \(CA + AI = BC\). Determine the ratio of the angles: \(\frac{\angle BAC}{\angle CAA}\).


Let \( f(x) = x^2 + px + q \). It is known that the inequality \( |f(x)| > \frac{1}{2} \) has no solutions on the interval \([3, 5]\). Find \(\underbrace{f(f(\ldots f}_{2017}\left(\frac{7+\sqrt{15}}{2}\right)) \ldots)\). Round the answer to hundredths if necessary.


In 2014, during the Beijing APEC meeting, measures such as traffic restrictions and suspension of polluting industries were implemented in Beijing-Tianjin-Hebei to ensure good air quality. After experiencing three bouts of smog in a month, Beijing's air quality reached level one (excellent) on November 3rd, a phenomenon dubbed the "APEC Blue". In 2013, the number of days with good air quality in Beijing accounted for only 47.9\%. In the first half of 2014, with a 30\% reduction in emissions, the number of days with good air quality increased by 20 days compared to the same period in 2013. To achieve the target of increasing the number of days with good air quality by 20\% for the entire year, how many more days with good air quality are needed in the second half of the year compared to the same period in 2013?


There are 8 children, each wearing either a red hat or a blue hat. If a child sees 3 or more children wearing red hats, they will take a red balloon; otherwise, they will take a blue balloon. The result is that some children take red balloons and some take blue balloons. Given this, how many children are wearing red hats?


If from the numbers \(1, 2, \cdots, 14\), we select \(a_{1}, a_{2}, a_{3}\) in ascending order such that they satisfy
\[ a_{2} - a_{1} \geq 3 \text{ and } a_{3} - a_{2} \geq 3, \]
how many different selections are possible?


All three-digit numbers are written in a row: $100101102 \ldots 998999$. How many times in this row does a zero follow a two?


If the intended number is multiplied by 6, then 382 is added to the product, the result is the largest three-digit number written with two identical even digits and one odd digit. Find the intended number.


The function \( J(x) \) is defined by:
\[
J(x)=\left\{\begin{array}{ll}
4+x & \text{ for } x \leq -2, \\
-x & \text{ for }-2 < x \leq 0, \\
x & \text{ for } x > 0.
\end{array}\right.
\]

How many distinct real solutions does the equation \( J(J(J(x))) = 0 \) have?


A middle school is preparing to form an 18-person soccer team. These 18 people will be selected from students in the 10 classes of the first grade. Each class must have at least one student on the team. How many different ways can the team be composed?


Three fixed circles pass through the points A and B. X is a variable point on the first circle different from A and B. The line AX meets the other two circles at Y and Z (with Y between X and Z). Show that the ratio XY/YZ is independent of the position of X.

Solution: \boxed{\frac{XY}{YZ}}


Let \( f(x) \) be an even function defined on \(\mathbf{R}\) with a period of 2 and strictly decreasing on the interval \([0,1]\), such that \( f(\pi) = 1 \) and \( f(2\pi) = 2 \). Determine the solution set of the system of inequalities \(\left\{\begin{array}{l}1 \leqslant x \leqslant 2, \\ 1 \leqslant f(x) \leqslant 2\end{array}\right.\).

Solution: \boxed{[\pi - 2, 8 - 2\pi]}


Given a function \( f(x) \) defined on \( \mathbf{R}^{+} \) as
\[
f(x) = \begin{cases}
\left|\log_3 x - 1\right|, & 0 < x \leq 9, \\
4 - \sqrt{x}, & x > 9
\end{cases}
\]
and let \( a, b, c \) be three distinct real numbers that satisfy \( f(a) = f(b) = f(c) \). Determine the range of \( abc \).

Solution: \boxed{(81, 144)}


Chester is traveling from Hualien to Lugang, Changhua, to participate in the Hua Luogeng Golden Cup Mathematics Competition. Before setting off, his father checked the car's odometer, which read a palindromic number of 69,696 kilometers (a palindromic number remains the same when read forward or backward). After driving for 5 hours, they arrived at their destination, and the odometer displayed another palindromic number. During the journey, the father's driving speed never exceeded 85 kilometers per hour. What is the maximum average speed (in kilometers per hour) at which Chester's father could have driven?

Solution: \boxed{82.2 \text{ km/h}}


100 chips are lined up in a row. It is allowed to swap two chips that have one chip between them.

Is it possible to reverse the order of all the chips using such operations?

Solution: \boxed{\text{No}}


Let \( P \) be a point inside the isosceles trapezoid \( ABCD \) where \( AD \) is one of the bases, and let \( PA, PB, PC, \) and \( PD \) bisect angles \( A, B, C, \) and \( D \) respectively. If \( PA = 3 \) and \( \angle APD = 120^\circ \), find the area of trapezoid \( ABCD \).


It is known that the equations \(x^{2}+(2a-5)x+a^{2}+1=0\) and \(x^{3}+(2a-5)x^{2}+(a^{2}+1)x+a^{2}-4=0\) have common roots. Find the sum of these roots.


For each prime number \( p \), find the largest natural power of \( p \), denoted by \( p^k \), such that \( p^k \) divides \( (p^2)! \).


In the trapezoid \(ABCD\) (\(AD \parallel BC\)), a perpendicular \(EF\) is drawn from point \(E\) (the midpoint of \(CD\)) to line \(AB\). Find the area of the trapezoid if \(AB = 5\) and \(EF = 4\).


At exactly noon, a truck left a village and headed for a city, and at the same time, a car left the city and headed for the village. If the truck had left 45 minutes earlier, they would have met 18 kilometers closer to the city. If the car had left 20 minutes earlier, they would have met $k$ kilometers closer to the village. Find $k$.


Let \( P \) be a perfect square. If \( q-2 \) and \( q+2 \) are both prime numbers, then \( q \) is called a \( P \)-type perfect square. For example, 9 is a \( P \)-type perfect square.

What is the largest \( P \)-type perfect square less than 1000?


The pentagon \(ABCDE\) is inscribed around a circle. The angles at its vertices \(A\), \(C\), and \(E\) are \(100^{\circ}\). Find the angle \(ACE\).


Let \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a function that satisfies the functional equation
\[ f(x-y)=2009 f(x) f(y) \]
for all \( x, y \in \mathbb{R} \). If \( f(x) \) is never zero, what is \( f(\sqrt{2009}) \)?


Among some products, the share of top-quality products is 31%. What is the most probable number of top-quality products in a randomly selected batch of 75 products?


Find the coordinates of point $A$ which is equidistant from points $B$ and $C$.

$A(0 ; 0 ; z)$

$B(-1 ;-1 ;-6)$

$C(2 ; 3 ; 5)$

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


Car A and Car B start simultaneously from locations A and B, respectively, and they travel between these two locations at constant speeds. After the first time they meet, Car A takes 4 hours to reach B, while Car B takes 1 hour to reach A. How many hours have both cars traveled by the 15th time they meet (excluding meetings at locations A and B)?


Given a positive integer $n$ that is not divisible by 2 or 3, and there do not exist non-negative integers $a$ and $b$ such that $\left|2^{a}-3^{b}\right|=n$, find the smallest value of $n$.


Angry reviews about the work of an online store are left by $80\%$ of dissatisfied customers (those who were poorly served in the store). Of the satisfied customers, only $15\%$ leave a positive review. A certain online store earned 60 angry and 20 positive reviews. Using this statistic, estimate the probability that the next customer will be satisfied with the service in this online store.


Find all two-digit numbers, each of which is 13 more than the sum of the squares of its digits.


In triangle \(ABC\), the altitudes \(AH\) and \(CP\) are drawn. Find the measure of angle \(B\) if it is known that \(|AC| = 2|PH|\).


In a mathematics competition, there are 30 problems. Each correctly solved problem is worth 4 points, an incorrect solution results in a loss of 1 point, and no points are awarded for problems that are not attempted. How many different total scores can a contestant achieve?


Given 6 points on a plane arranged in the form of a rectangle as shown in Figure 1, how many triangles exist where one vertex is at point $A$ and the other two vertices are any of the remaining points? How many such triangles exist where one vertex is at point $F$?


In a unit cube \(ABCDA_1B_1C_1D_1\), eight planes \(AB_1C, BC_1D, CD_1A, DA_1B, A_1BC_1, B_1CD_1, C_1DA_1,\) and \(D_1AB_1\) intersect the cube. What is the volume of the part that contains the center of the cube?


Let \( S \) be the sum of all the real coefficients of the expansion of \( (1+ix)^{2009} \). What is \( \log_{2}(S) \) ?


Let \( a, b \in \{2, 3, \cdots, 8\} \). Find the maximum value of \(\frac{a}{10b + a} + \frac{b}{10a + b}\).


In the quadrilateral \(ABCD\), the following angles are known:

\[
\angle BAC = 50^\circ, \quad \angle ABD = 60^\circ, \quad \angle DBC = 20^\circ, \quad \angle BDC = 30^\circ.
\]

Is it true that \(\angle CAD = 30^\circ\)?


A magician and their assistant plan to perform a trick. The spectator writes a sequence of $N$ digits on a board. The magician's assistant then covers two adjacent digits with a black dot. Next, the magician enters and has to guess both covered digits (including the order in which they are arranged). What is the smallest $N$ for which the magician and the assistant can arrange the trick so that the magician can always correctly guess the covered digits?


The first and second terms of a sequence are 4 and 5, respectively. Each term after the second is determined by increasing the previous term by one and dividing the result by the term before that. For example, the third term equals \(\frac{5+1}{4}\). What is the 1234th term of this sequence?


Find all integers \(x\) and \(y\) for which \(x^2 + xy - y = 2\). Justify your answer.

Solution: \boxed{(2, -2), (0, -2)}


In an acute-angled triangle \( ABC \), the altitude \( CE \) on \( AB \) and the altitude \( BD \) on \( AC \) intersect at point \( H \). A circle with diameter \( DE \) intersects \( AB \) and \( AC \) at points \( F \) and \( G \) respectively. Line \( FG \) intersects \( AH \) at point \( K \). Given that \( BC = 25 \), \( BD = 20 \), and \( BE = 7 \), find the length of \( AK \).


The base of a triangle is \( a \), and its height is \( m \). Draw a square on the base of the triangle such that two of its vertices lie on the sides of the triangle. Subsequently, inscribe a square in the resulting smaller triangle in the same manner, and continue this process indefinitely. What is the sum of the areas of all the squares?

Solution: \boxed{\frac{a m^2}{a+2m}}


A directory has 710 pages. How many digits were needed to number its pages?


It is known that \(\frac{7}{13}+\sin \phi=\cos \phi\) for some real \(\phi\). What is \(\sin 2 \phi\)?


Alice has 8 keys strung on a ring. The keys are indistinguishable at a glance, and both sides of each key are identical. In order to differentiate the keys, Alice places a colored cap on each one. How many colors does she need?


A three-stage launch vehicle consists of cylindrical stages. All these cylinders are similar to each other. The length of the middle stage is half the sum of the lengths of the first and third stages. When fueled, the mass of the middle stage is $13 / 6$ times less than the combined mass of the fueled first and third stages. Find the ratio of the lengths of the first and third stages. The masses of the engines, instruments, and the shells of the stages can be neglected.


During a chess game, at some point, Hedgehog had twice fewer pieces on the board than Bear, and these pieces were five times fewer than the free squares on the board. How many of Bear's pieces were captured by this moment?


Let \( f(x) = \left\{ \begin{array}{cc} 1 & 1 \leqslant x \leqslant 2 \\ x-1 & 2 < x \leqslant 3 \end{array} \right. \). For any \( a \,(a \in \mathbb{R}) \), define \( v(a) = \max \{ f(x) - a x \mid x \in [1,3] \} - \min \{ f(x) - a x \mid x \in [1,3] \} \). Draw the graph of \( v(a) \) and find the minimum value of \( v(a) \).


Find the 6-digit number beginning and ending in the digit 2 that is the product of three consecutive even integers.


Given points \(A(-0.8, 4.132)\), \(B(1.2, -1.948)\), and \(C(2.8, -3.932)\) lie on the parabola
\[ y = ax^2 + bx + c. \]
When the x-coordinate of point \(D\) on the curve is \(x = 1.8\), find the y-coordinate of \(D\).


Let \( a, b > 1 \) be odd numbers such that \( a + b = 2^l \). Find \( k \in \mathbb{N}^{*} \) such that \( k^2 \mid a^k + b^k \).


In the figure shown, there are three concentric circles and two perpendicular diameters. The three shaded regions have equal area. The radius of the small circle is 2. The product of the three radii is \( Y \). What is the value of \( Y^{2} \)?


Diagonals of a convex quadrilateral \( ABCD \) intersect at point \( E \). Given \( AB = BC \), \( DB \) is the angle bisector of \( \angle D \), \(  \angle ABC = 100^\circ \), and \( \angle BEA = 70^\circ \). Find \( \angle CAD \).


The height of a trapezoid, whose diagonals are mutually perpendicular, is 4. Find the area of the trapezoid if one of its diagonals is 5.


Show that in any right triangle, the sum of the semiperimeter and the radius of the inscribed circle is equal to the sum of the legs.

Solution: \boxed{a + b}


In an isosceles triangle \(ABC\) (\(AB = BC\)), the medians \(AM\) and \(CN\) intersect at point \(D\) at a right angle. Find all the angles of triangle \(ABC\) and its base \(AC\) if the area of quadrilateral \(NBMD\) is 4.


A boy and a girl: We assume that the birth of a girl and a boy are equally probable. It is known that a certain family has two children.

a) What is the probability that one of them is a boy and the other is a girl?

b) Additionally, it is known that one of the children is a boy. What is the probability now that the family has one boy and one girl?

c) Additionally, it is known that the boy was born on Monday. What is the probability now that the family has one boy and one girl?


$ABCD$ is a right trapezoid with bases $AD=4$ and $BC=3$. $AB \perp BC$. A circle passing through $C$ and $D$ is tangent to $AB$ at point $E$. Find the distance from point $E$ to line $CD$.

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


Find the smallest positive integer \( n \) such that every \( n \)-element subset of \( S = \{1, 2, \ldots, 150\} \) contains 4 numbers that are pairwise coprime (it is known that there are 35 prime numbers in \( S \)).


Let $p$ be an odd prime number. Determine the number of tuples $\left(a_{1}, a_{2}, \ldots, a_{p}\right)$ of natural numbers with the following properties:
1) $1 \leq a_{i} \leq p$ for all $i=1, \ldots, p$.
2) $a_{1}+a_{2}+\cdots+a_{p}$ is not divisible by $p$.
3) $a_{1} a_{2}+a_{2} a_{3}+\ldots a_{p-1} a_{p}+a_{p} a_{1}$ is divisible by $p$.


There are 100 students who want to sign up for the class "Introduction to Acting." There are three class sections for "Introduction to Acting," each of which will fit exactly 20 students. The 100 students, including Alex and Zhu, are put in a lottery, and 60 of them are randomly selected to fill up the classes. What is the probability that Alex and Zhu end up getting into the same section for the class?


For the quadratic trinomial \( a x^{2} + b x + c \) where \( a > 100 \), what is the maximum number of integer values for \( x \) such that the absolute value of the quadratic trinomial does not exceed 50?


It is known that an arithmetic progression includes the terms \( a_{2n} \) and \( a_{2m} \) such that \(\frac{a_{2n}}{a_{2m}} = -1\). Is there a term in this progression that is equal to zero? If so, what is the position of this term?

Solution: \boxed{n+m}


64 people are lined up - each is either a knight, who always tells the truth, or a liar, who always lies. One of the knights standing in the line said that they stand next to a knight and a liar, and all the other 63 people repeated this phrase. Indicate how many of them are knights.


We write on the board the equation

$$
(x-1)(x-2) \cdots(x-2016)=(x-1)(x-2) \cdots(x-2016),
$$

where there are 2016 linear factors on each side. What is the smallest positive value of $k$ such that we can omit exactly $k$ of these 4032 linear factors in such a way that there is at least one linear factor on each side, and the resulting equation has no real roots?


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

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


Calculate the result of the expression \(1 \times 2 + 3 \times 4 + 5 \times 6 + 7 \times 8 + 9 \times 10\).


In triangle \( \triangle ABC \), if \( \tan A \cdot \tan B = \tan A \cdot \tan C + \tan C \cdot \tan B \), then find the value of \( \frac{a^{2} + b^{2}}{c^{2}} \).


Masha, Dasha, and Sasha are tasked with harvesting currants from all the bushes in the garden plot. Masha and Dasha can gather all the berries together in 7 hours and 30 minutes, Masha and Sasha in 6 hours, and Dasha and Sasha in 5 hours. How many hours will it take for the children to gather all the berries if they work together?


Inside an angle of $30^\circ$ with vertex $A$, a point $K$ is chosen such that its distances to the sides of the angle are 1 and 2. Through point $K$, all possible straight lines intersecting the sides of the angle are drawn. Find the minimum area of the triangle cut off from the angle by one of these lines.


In the number \(2 * 0 * 1 * 6 * 02 *\), each of the 5 asterisks needs to be replaced by any of the digits \(0, 2, 4, 7, 8, 9\) (digits can repeat) so that the resulting 11-digit number is divisible by 12. How many ways can this be done?


A square is divided into 8 smaller squares, with 7 of the smaller squares having a side length of 2. What is the side length of the original square? $\qquad$ .


Several points, including points \( A \) and \( B \), are marked on a line. All possible segments with endpoints at the marked points are considered. Vasya counted that point \( A \) is inside 50 of these segments, and point \( B \) is inside 56 segments. How many points were marked? (Endpoints of a segment are not considered as internal points.)


What is the largest number of integers that can be selected from the natural numbers from 1 to 3000 such that the difference between any two of them is different from 1, 4, and 5?


In the following diagram, \(ABCD\) is a square, and \(E\) is the center of the square \(ABCD\). \(P\) is a point on a semi-circle with diameter \(AB\). \(Q\) is a point on a semi-circle with diameter \(AD\). Moreover, \(Q, A,\) and \(P\) are collinear (that is, they are on the same line). Suppose \(QA = 14 \text{ cm}\), \(AP = 46 \text{ cm}\), and \(AE = x \text{ cm}\). Find the value of \(x\).


Find the smallest positive integer \( n \) that satisfies:
1. \( n \) has exactly 144 different positive divisors.
2. Among the positive divisors of \( n \), there are 10 consecutive integers.

(Note: 26th International Mathematical Olympiad shortlist, 1985)


Let \( p \) be a real number between 0 and 1. Jocelin has a coin that lands heads with probability \( p \) and tails with probability \( 1-p \); she also has a number written on a blackboard. Each minute, she flips the coin, and if it lands heads, she replaces the number \( x \) on the blackboard with \( 3x + 1 \); if it lands tails, she replaces it with \( \frac{x}{2} \). Given that there are constants \( a, b \) such that the expected value of the value written on the blackboard after \( t \) minutes can be written as \( at + b \) for all positive integers \( t \), compute \( p \).


Given \(\overrightarrow{AB} = 6 \vec{i} + \vec{j}\), \(\overrightarrow{BC} = x \vec{i} + y \vec{j}\), \(\overrightarrow{CD} = -2 \vec{i} - 3 \vec{j}\), and \(\overrightarrow{BC} \parallel \overrightarrow{DA}\). Find the relationship between \(x\) and \(y\).

Solution: \boxed{x + 2y = 0}


Given the acute angle \( x \) that satisfies the equation \( \sin^3 x + \cos^3 x = \frac{\sqrt{2}}{2} \), find \( x \).


We start with 5000 forints in our pocket to buy gifts, visiting three stores. In each store, we find a gift that we like and purchase it if we have enough money. The prices in each store are independently 1000, 1500, or 2000 forints, each with a probability of $\frac{1}{3}$. What is the probability that we are able to purchase gifts from all three stores and still have money left?


If $\otimes$ is a new operation defined as \( p \otimes q = p^2 - 2q \), what is the value of \( 7 \otimes 3 \)?


Determine the value of the expression

$$
1+2-3+4+5-6+7+8-9+10+11-12+\cdots+94+95-96+97+98-99
$$

(The expression consists of 99 terms. The operations alternate between two additions and one subtraction.)


Given that for any positive integer \( n \), \( 9^{2n} - 8^{2n} - 17 \) is always divisible by \( m \), find the largest positive integer \( m \).


Find the smallest positive integer \( k \) for which there exists a \( k \)-coloring of the positive integers and a function \( f: \mathbb{N}^{*} \rightarrow \mathbb{N}^{*} \) such that:

1. For all \( n, m \in \mathbb{N}^{*} \) of the same color, \( f(n+m) = f(n) + f(m) \)
2. There exist \( n, m \in \mathbb{N}^{*} \) such that \( f(n+m) \neq f(n) + f(m) \)


On the side \( AB \) of the parallelogram \( ABCD \), a point \( F \) is chosen, and on the extension of the side \( BC \) beyond the vertex \( B \), a point \( H \) is taken such that \( \frac{AB}{BF} = \frac{BC}{BH} = 5 \). The point \( G \) is chosen such that \( BFGH \) forms a parallelogram. \( GD \) intersects \( AC \) at point \( X \). Find \( AX \), if \( AC = 100 \).


In triangle \(ABC\), median \(BM\) is drawn. \(MD\) and \(ME\) are angle bisectors of triangles \(AMB\) and \(CMB\) respectively. Segments \(BM\) and \(DE\) intersect at point \(P\), with \(BP = 2\) and \(MP = 4\).

a) Find the length of segment \(DE\).

b) Additionally, it is known that a circle can be circumscribed around quadrilateral \(ADEC\). Find its radius.

Solution: \boxed{2 \sqrt{85}}


Given that \( x - \frac{1}{x} = 2b \) and \( x^3 - \frac{1}{x^3} = c \), find the value of \( c \).

Solution: \boxed{8b^3 + 6b}


Toss a die 6 times, and let the number obtained on the $i$-th toss be $a_{i}$. If there exists a positive integer $k$ such that $\sum_{i=1}^{k} a_{i}=6$, what is the probability $p=\frac{n}{m}$, where $m$ and $n$ are coprime positive integers. Find the value of $\log _{6} m - \log _{7} n$.


Let \( S = 2x^2 - xy + y^2 + 2x + 3y \), where \( x \) and \( y \) are real numbers. Find the minimum value of \( S \).


Determine the last digit of the product of all even natural numbers that are less than 100 and are not multiples of ten.


Oleg has 550 rubles and wants to gift his mom an odd number of tulips for International Women's Day, ensuring that each tulip is of a different color. In the store Oleg visits, one tulip costs 49 rubles and there are eleven different shades of tulips available. How many ways can Oleg gift his mom the flowers? (The answer should be a compact expression without summation signs, ellipses, etc.)


Divide the sequence $\{2n + 1\}$ into cycles by the length of the partitions as: $(3), (5,7), (9,11,13), (15,17,19,21), (23), (25,27), (29,31,33), (35,37,39,41), (43), \cdots$. Find the sum of the numbers inside the 100th bracket.


Find the value of \(\frac{6 \sin \alpha - 7 \cos \alpha + 1}{8 \sin \alpha + 9 \cos \alpha - 1}\) if \(\operatorname{tg} \frac{\alpha}{2} = 4\).


Three crazy painters started painting the floor, each in their own color. One managed to paint $75\%$ of the floor red, another $70\%$ green, and the third $65\%$ blue. What part of the floor is certainly painted with all three colors?


Under the action of the gravitational force $\dot{g}$, directed opposite to the $OZ$ axis, a unit mass body slides down a spiral:

$$
x=a \cos \varphi, \quad y=a \sin \varphi, \quad z=b(2 \pi-\varphi)
$$

from point $A(a, 0, 2 \pi b)$ to point $B(a, 0, 0)$.

Find the work done by the field during this movement.

Solution: \boxed{2\pi gb}


The figures \(\triangle, \boldsymbol{\Lambda}, \diamond, \boldsymbol{\uparrow}, \odot\) and \(\square\) are repeated indefinitely in the sequence:

\[ \triangle, \boldsymbol{\leftrightarrow}, \diamond, \boldsymbol{\uparrow}, \odot, \square, \triangle, \boldsymbol{\leftrightarrow}, \diamond, \boldsymbol{\phi}, \odot, \square, \ldots \]

(a) What figure will appear in the 1000th position of the sequence?

(b) In what position does the 1000th \(\diamond\) appear?


The midline of an isosceles trapezoid is equal to 5. It is known that a circle can be inscribed in the trapezoid.

The midline of the trapezoid divides it into two parts, the ratio of the areas of which is \(\frac{7}{13}\). Find the height of the trapezoid.

Solution: \boxed{ 4.}


Given positive real numbers \( a \) and \( b \) with \( a < b \), let \( x_1, x_2, \cdots, x_{2022} \in [a, b] \). Find the maximum value of \( \frac{\left|x_1 - x_2\right| + \left|x_2 - x_3\right| + \cdots + \left|x_{2021} - x_{2022}\right| + \left|x_{2022} - x_1\right|}{x_1 + x_2 + \cdots + x_{2022}} \).


Numbers \( C_n, n \geqslant 1 \) (Catalan numbers) have many interesting combinatorial interpretations. For example, consider the number \( N_n \) of different ways to count the sum of \( n \) numbers \( a, b, c, d, \ldots \) where only two numbers are added together at a time (changing the order of addition is not allowed). For \( n=3 \), the sum \( a+b+c \) can be computed by adding two numbers at a time, in the following ways: \( a+b+c=(a+(b+c))=((a+b)+c) \). The number of different ways is \( N_3 = 2 \). For \( n=4 \), there are already five possibilities \( N_4 = 5 \):

$$
\begin{aligned}
a+b+c+d & =((a+b)+(c+d))=(((a+b)+c)+d)= \\
& =((a+(b+c))+d)=(a+((b+c)+d))= \\
& =(a+(b+(c+d))) .
\end{aligned}
$$

(a) Show that for any \( n \geqslant 3 \), the number \( N_n \) of different ways to count sums is equal to \( C_n \).

(b) Consider diagonal triangulations of a regular \( n \)-gon, \( n \geqslant 4 \), i.e., partitions into triangles using non-intersecting diagonals (emanating from any vertex; it is clear that the number of such diagonals emanating from one vertex is \( n-3 \)). Show that the number \( N_n \) of such triangulations is equal to \( C_{n-1} \).

(c) Show that the Catalan numbers \( C_n, n>1 \), satisfy the recurrence relation

$$
C_n^* = \sum_{i=1}^{n-1} C_i^* C_{n-i}^*
$$

(with \( C_0^* = 0 \) and \( C_1^* = 1 \) ).

(d) Establish that the generating function \( F^*(x) = \sum_{n \geqslant 1} C_n^* x^n \) of the sequence \( \left(C_n^*\right)_{n \geqslant 1} \), defined by the recurrence relations $(*)$, satisfies the equation

$$
F^*(x) = x + \left( F^*(x) \right)^2 .
$$

(e) Show that (considering the condition \( F^*(0) = 0 \))

$$
F^*(x) = \frac{1}{2}\left(1-(1-4x)^{1/2}\right), \quad |x| \leqslant \frac{1}{4},
$$

and deduce from this that the coefficients \( C_n^* \) (for \( x^n \)) coincide, as expected, with the Catalan numbers \( C_n \):

$$
C_n^* = -\frac{1}{2} C_{1/2}^n (-4)^n = \frac{1}{n} C_{2(n-1)}^{n-1} = C_n
$$

Solution: \boxed{C_n^* = C_n}


There is a ten-digit number. From left to right:
- Its first digit indicates how many zeros are in the number.
- Its second digit indicates how many ones are in the number.
- Its third digit indicates how many twos are in the number.
- $\cdots \cdots$
- Its tenth digit indicates how many nines are in the number.

Find this ten-digit number.


Find the smallest positive integer \( k \) such that the equation \( 2xy - 3x - 5y = k \) has an odd number of positive integral solutions.


The coordinates \( (x ; y ; z) \) of point \( M \) are consecutive terms of a geometric progression, and the numbers \( x y, y z, x z \) in the given order are terms of an arithmetic progression, with \( z \geq 1 \) and \( x \neq y \neq z \). Find the smallest possible value of the square of the distance from point \( M \) to point \( N(1 ; 1 ; 1) \).


It is known that the values \( g(k) \) and \( g\left(\frac{1}{m}\right) \) of the quadratic polynomial \( g(x) = m x^2 + n x + k \) have different signs. Can the roots of the polynomial \( g(x) \) have the same signs?

Solution: \boxed{\text{No}}


A snail crawls from one tree to another. In half a day, it crawled \( l_1 = 4 \, \text{m} \). It got bored, turned back, and after covering \( l_2 = 3 \, \text{m} \), it got tired and went to sleep. The next day, it repeated the same pattern. This process continues every day. The distance between the trees is \( s = 40 \, \text{m} \). On which day of its journey will the snail reach the tree?


Find the number of all integer solutions to the inequality \(\sqrt{1+\sin \frac{\pi x}{4}-3 \cos \frac{\pi x}{2}}+\sqrt{6} \cdot \sin \frac{\pi x}{4} \geq 0\) that belong to the interval \([1991; 2013]\).


Find the smallest positive integer \( n \) such that if the first digit is moved to become the last digit, then the new number is \( \frac{7n}{2} \).


A five-digit number has one of its digits crossed out, and the resulting four-digit number is added to the original number. The sum is 54321. Find the original number.


Given that in triangle $\triangle ABC$, the two medians $BD$ and $CE$ intersect at point $G$. Points $A$, $D$, $G$, and $E$ are concyclic, and $BC = 6$. Find the length of $AG$.


Find all integers \( n \geq 1 \) such that \( n \) divides \( 2^{n} - 1 \).


Find all values of parameters \(a\) and \(b\) for which the polynomial

\[ f(x) = x^5 - 3x^4 + ax^3 + bx^2 - 5x - 5 \]

is divisible by \(x^2 - 1\) without remainder.

Solution: \boxed{a=4, b=8}


Let \( G = (V, E) \) be a graph and \( k \in \mathbb{N} \). If the chromatic number of every finite subgraph of \( G \) is at most \( k \), then the chromatic number of \( G \) is also at most \( k \).

Solution: \boxed{k}


Find the area of trapezoid \(ABCD (AD \| BC)\) if its bases are in the ratio \(5:3\), and the area of triangle \(ADM\) is 50, where \(M\) is the point of intersection of lines \(AB\) and \(CD\).


Ali Baba arrived at a cave where there is gold, diamonds, and a chest in which they can be carried. A full chest of gold weighs 200 kg, a full chest of diamonds weighs 40 kg, and an empty chest weighs nothing. A kilogram of gold costs 20 dinars in the market, and a kilogram of diamonds costs 60 dinars. Ali Baba can lift and carry no more than 100 kg. What is the maximum amount of money he can get for the treasures he brings from the cave in one trip?

Solution: \boxed{3000 \text{ dinars}}


Show that the sum of the squares of the elements of a row in Pascal's triangle is itself an element of Pascal's triangle.


A porter needs to transport 200 buns from the kitchen to the construction site (he is currently in the kitchen). Each time, he can carry 40 buns. However, he eats 1 bun each way, whether traveling from the kitchen to the construction site or from the construction site to the kitchen. How many buns can the porter transport to the construction site at most?


In triangle \(ABC\), angle \(A\) is \(40^\circ\). The triangle is randomly thrown onto a table. Find the probability that vertex \(A\) ends up east of the other two vertices.


Calculate the value of the expression \(\arccos \frac{\sqrt{6}+1}{2 \sqrt{3}} - \arccos \sqrt{\frac{2}{3}}\). Express the result in the form \(\frac{a \pi}{b}\), where \(a\) and \(b\) are integers that are coprime, and indicate the value of \(|a-b|\).


In a box, there are 7 red pencils and 5 blue pencils. In the dark, pencils are taken out. How many pencils need to be taken out to ensure there are at least two red ones and at least three blue ones among them?


Given real numbers $b$ and $c$, and the function $f(x) = x^2 + bx + c$, the equation $f(f(x)) = 0$ has exactly three different real roots. Find the maximum value of the sum of the roots of $f(x)$.


Let $\varphi(n)$ denote the number of positive integers not exceeding $n$ that are coprime to $n$. Given that $g(n)$ satisfies the condition $\sum_{d \mid n} g(d) = \varphi(n)$ for any positive integer $n$, where the summation $\sum_{d \mid n}$ indicates that $d$ ranges over all positive divisors of $n$, find $g(50)$.


An exam paper contains 4 multiple-choice questions, each with three options (A), (B), and (C). Several students take the exam, and it is found that for any group of 3 students, there is always at least one question where their answers are all different. Furthermore, each student answers every question. What is the maximum number of students who can take the exam?


50 schoolchildren and their parents are going on a tour to Nizhny Novgorod, some of whom drive cars. Each car can accommodate 6 people, including the driver. What is the minimum number of parents that need to be invited on the tour?


Given an ellipse \( C: \frac{x^{2}}{2}+y^{2}=1 \) with left and right foci \( F_{1}, F_{2} \) respectively, let \( P \) be a point on the ellipse \( C \) in the first quadrant. The extended lines \( P F_{1}, P F_{2} \) intersect the ellipse \( C \) at points \( Q_{1}, Q_{2} \) respectively. Find the maximum value of the difference of areas of triangles \( \triangle P F_{1} Q_{2} \) and \( \triangle P F_{2} Q_{1} \).


There are 5 paths leading to the top of a mountain. Two people ascend and then descend. How many different ways can this happen if the two people are not distinguished and

a) each path can be used by at most one person and at most once,

b) each path can be used twice but only in different directions,

c) there are no restrictions,

d), e), and f) are the same as a), b), and c), but the two people are distinguished.


A canteen bought 500 kilograms of rice and 200 kilograms of flour. After some time, they found that the amount of rice and flour consumed was the same, and the remaining rice was exactly 7 times the remaining flour. How many kilograms of rice and flour were consumed?


The Absent-Minded Scientist boards a bus with \( n \) seats. All \( n \) tickets have been sold to \( n \) passengers. The Absent-Minded Scientist is the first to enter the bus and, without looking at his ticket, takes a random seat. Subsequently, each passenger enters one-by-one. If a passenger finds their assigned seat free, they take it. If their seat is occupied, they take the first available free seat they find. What is the probability that the last passenger who boards the bus will sit in their assigned seat?


A natural number is called lucky if all its digits are equal to 7. For example, 7 and 7777 are lucky, but 767 is not. João wrote down the first twenty lucky numbers starting from 7, and then added them. What is the remainder of that sum when divided by 1000?


Determine all composite positive integers $n$ with the following property: Let $1 = d_1 < d_2 < \ldots < d_k = n$ be all the positive divisors of $n$. Then:

$$
\left(d_{2} - d_{1}\right) : \left(d_{3} - d_{2}\right) : \cdots : \left(d_{k} - d_{k-1}\right) = 1 : 2 : \cdots : (k - 1).
$$


A regular triangular prism $A B C A_{1} B_{1} C_{1}$ is inscribed in a sphere with base $A B C$ and lateral edges $A A_{1}$, $B B_{1}$, $C C_{1}$. Segment $C D$ is the diameter of this sphere, point $K$ is the midpoint of edge $A A_{1}$, and point $L$ is the midpoint of edge $A B$. Find the volume of the prism if $D L = \sqrt{6}$ and $D K = 3$.

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


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

$$
r = \sin 6 \phi
$$


Consider all possible broken lines that travel along the sides of the cells and connect two opposite corners of a square sheet of grid paper with dimensions $100 \times 100$ by the shortest path. What is the minimum number of such broken lines that need to be taken so that their union contains all the vertices of the cells?


On the sides \( BC \) and \( AC \) of triangle \( ABC \), points \( M \) and \( N \) are taken respectively such that \( CM:MB = 1:3 \) and \( AN:NC = 3:2 \). Segments \( AM \) and \( BN \) intersect at point \( K \). Find the area of quadrilateral \( CMKN \), given that the area of triangle \( ABC \) is 1.


At 7:00 AM, Xiaoming leaves his house and heads to school at a speed of 52 meters per minute. When he reaches the school, the hour and minute hands on his watch are positioned symmetrically around the number 7 on the clock. It's known that Xiaoming walked for less than an hour. How far is the distance between Xiaoming's house and the school, in meters?


The legs of a right triangle are 9 cm and 12 cm. Find the distance between the incenter (the point of intersection of its angle bisectors) and the centroid (the point of intersection of its medians).

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


What is the maximum number of interesting numbers among five consecutive natural numbers if we call a natural number interesting when the sum of its digits is a prime number?


In the coordinate plane, points whose x-coordinates and y-coordinates are both integers are called lattice points. For any natural number \( n \), connect the origin \( O \) with the point \( A_n(n, n+3) \). Let \( f(n) \) denote the number of lattice points on the line segment \( OA_n \) excluding the endpoints. Find the value of \( f(1) + f(2) + \cdots + f(1990) \).


On a $11 \times 11$ grid, 22 cells are marked such that there are exactly two marked cells in each row and each column. Two configurations of marked cells are considered equivalent if one can be transformed into the other by permuting rows and columns any number of times. How many distinct configurations of marked cells exist, considering these equivalence conditions?


Given the polynomial \( f(x) \) that satisfies the equation \( f(x+1) + f(x-1) = 2x^2 - 4x \) for any \( x \in \mathbb{R} \), determine the minimum value of \( f(x) \).


The side length of the base of a regular square pyramid is \( a \). A lateral face forms an angle of \( 45^{\circ} \) with the plane of the base. Find the volume of the pyramid.


Knights, who always tell the truth, and liars, who always lie, live on an island. One day, 30 inhabitants of this island sat around a round table. Each of them said one of two phrases: "My neighbor on the left is a liar" or "My neighbor on the right is a liar." What is the minimum number of knights that can be at the table?


Find the area of the figure defined in the coordinate plane by the inequality \( x^2 + y^2 \leq 2(|x| - |y|) \).

Solution: \boxed{2\pi - 4}


There was an integer number of cheese heads in the kitchen. During the night, some rats came and ate 10 cheese heads, with each rat eating an equal portion. Afterward, some of the rats got sick from overeating. The next night, the remaining seven rats finished the remaining cheese, but each rat was able to eat only half as much cheese as they did the previous night. How much cheese was originally in the kitchen?


How many numbers should there be in a lottery for the probability of getting an ambo to be $\frac{5}{473}$, when drawing five numbers?


Given the function \( f(x) = \frac{1+x}{2-x} \), and for \( n \in \mathbf{N}_{+} \), define \( f_{1}(x) = f(x) \) and \( f_{n+1}(x) = f\left[f_{n}(x)\right] \). If \( f_{13}(x) = f_{31}(x) \), what is the expression for \( f_{16}(x) \)?

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


50 balls numbered from 1 to 50 are arranged in a row and are colored as follows: (1) Color 2 balls; (2) The difference in the numbers of the two colored balls is greater than 2. If in one coloring method, the numbers of the two colored balls differ from those in another coloring method, these two coloring methods are considered "different." How many different coloring methods are there?


In triangle \(ABC\), the side lengths are \(BC = a\), \(AC = b\), and \(AB = c\), and the area is \(S\). Angle bisectors \(BM\) and \(CN\) intersect at point \(O\). Find the area of triangle \(BOC\).

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


Given two intersecting circles with radius \( r \) and centers \( O_{1} \) and \( O_{2} \), the distance between the centers is \( O_{1}O_{2} = 1 \). A ray starting from the midpoint \( O \) of the common chord and making an angle \( \alpha \) with the common chord intersects the circles at points \( A \) and \( B \). Find the distance \( AB \).

Solution: \boxed{\sin \alpha}


Calculate the area of the figure bounded by the curve \((2x + 3y + 5)^2 + (x + 2y - 3)^2 = 64\).

Solution: \boxed{S = 64 \pi}


Given a cube \( A B C D A_{1} B_{1} C_{1} D_{1} \) with an edge length of 1, let \( K \) be the midpoint of edge \( D D_{1} \). Find the distance between the lines \( C K \) and \( A_{1} D \).


Given a regular triangular prism \( S-ABC \) with height \( SO = 3 \) and base side length of 6, a perpendicular is drawn from point \( A \) to the opposite side face \( SBC \) with foot \( O' \). On \( AO' \), a point \( P \) is selected such that \( AP : PO' = 8 \). Find the area of the section passing through point \( P \) and parallel to the base.


6 different points are given on the plane, no three of which are collinear. Each pair of points is to be joined by a red line or a blue line subject to the following restriction: if the lines joining \(AB\) and \(AC\) (where \(A, B, C\) denote the given points) are both red, then the line joining \(BC\) is also red. How many different colourings of the lines are possible?


Find the coefficients \( p \) and \( q \) of the equation

\[ x^{4}-10 x^{3}+37 x^{2}+p x+q=0 \]

if among its roots there are two pairs of equal numbers.

Solution: \boxed{p = -60, q = 36}


Two digits of a number were swapped, and as a result, it increased by more than 3 times. The resulting number is 8453719. Find the original number.


The function \( f \) is such that \( f(2x - 3y) - f(x + y) = -2x + 8y \) for all \( x, y \). Find all possible values of the expression \( \frac{f(4t) - f(t)}{f(3t) - f(2t)} \).


Out of the following $2n$ numbers, how many can be selected at most such that the ratio of any two selected numbers is neither 2 nor $\frac{1}{2}$?
$3, 3 \times 2, 3 \times 2^{2}, 3 \times 2^{3}, 3 \times 2^{4}, 3 \times 2^{5}, \cdots, 3 \times 2^{2n-1}$.

Solution: \boxed{n}


Given four positive constants \( a, b, c, \) and \( d \), find the minimum value of \( c x + d y^{2} \) when the real numbers \( x \) and \( y \) satisfy \( a x^{2} + b y^{2} = 1 \).

Solution: \boxed{-\frac{c}{\sqrt{a}}}


In any permutation of the numbers \(1, 2, 3, \ldots, 18\), we can always find a set of 6 consecutive numbers whose sum is at least \(m\). Find the maximum value of the real number \(m\).


Given a natural number \( n \geq 3 \). On a board, the numbers \( 1, 2, 3, \ldots, n \) are written. In each move, two numbers are selected and replaced by their arithmetic mean. This process continues until only one number remains on the board.

Determine the smallest integer that can be achieved at the end using an appropriate sequence of moves.


Given the sequence \(\{a_n\}\) that satisfies \(a_n = n\) for \(1 \leq n \leq 5\), and \(a_{n+1} = a_1 \cdot a_2 \cdot \ldots \cdot a_n - 1\) for \(n \geq 5\), find the maximum value of \(S_m = a_1 \cdot a_2 \cdot \ldots \cdot a_m - a_1^2 - a_2^2 - \ldots - a_m^2\).


Several island inhabitants gather in a hut, with some belonging to the Ah tribe and the rest to the Uh tribe. Ah tribe members always tell the truth, while Uh tribe members always lie. One inhabitant said, "There are no more than 16 of us in the hut," and then added, "All of us are from the Uh tribe." Another said, "There are no more than 17 of us in the hut," and then noted, "Some of us are from the Ah tribe." A third person said, "There are five of us in the hut," and looking around, added, "There are at least three Uh tribe members among us." How many Ah tribe members are in the hut?


What is the number of subsets of the set $\{1, 2, 3, \ldots, n\}$ that do not contain two consecutive numbers?


In a regular hexagon \( ABCDEF \), the diagonals \( AC \) and \( CE \) are divided by interior points \( M \) and \( N \) in the following ratio: \( AM : AC = CN : CE = r \). If the points \( B, M, N \) are collinear, find the ratio \( r \).


Xiaoming formed a sequence using the digits 2, 0, 1, and 6 in that order: 2, 20, 201, 2016, 20162, 201620, 2016201, 20162016, 201620162, ..., adding these digits continuously at the end of the numbers. In this sequence, there are how many prime numbers?


In $\triangle ABC$, if $a + c = 2b$, then find the value of $\tan \frac{A}{2} \cdot \tan \frac{C}{2}$.


In quadrilateral \(ABCD\), the lengths \( |AB| = a \) and \( |AD| = b \) are given. The sides \(BC\), \(CD\), and \(AD\) are tangent to a circle whose center is at the midpoint of \(AB\). Find the length of side \( |BC| \).


With square tiles of a side length of an exact number of units, a room with a surface area of 18,144 square units has been tiled in the following manner: on the first day one tile was placed, the second day two tiles, the third day three tiles, and so on. How many tiles were necessary?


In a geometric sequence $\left\{a_{n}\right\}$ with all positive terms, there exist two terms $a_{m}$ and $a_{n}$ such that $\sqrt{a_{m} a_{n}}=8 a_{1}$, and it is known that $a_{9}=a_{8}+2 a_{7}$. Find the minimum value of $\frac{1}{m}+\frac{4}{n}$.


Determine the digits $A, B, C, D$ if

$$
\overline{A B^{2}}=\overline{A C D B}, \quad \text{and} \quad \overline{C D^{3}}=\overline{A C B D}
$$

Solution: \boxed{A=9, B=6, C=2, D=1}


Find the least positive integer \( n \) for which \( \frac{n-10}{9n+11} \) is a non-zero reducible fraction.


What is the sum of all natural numbers between 1 and 200 that are not multiples of 2 or 5?


The numbers $a, b, c, d$ belong to the interval $[-8.5,8.5]$. Find the maximum value of the expression $a + 2b + c + 2d - ab - bc - cd - da$.


Find all \( a_{0} \in \mathbb{R} \) such that the sequence \( a_{0}, a_{1}, \cdots \) determined by \( a_{n+1}=2^{n}-3a_{n}, n \in \mathbb{Z}^{+} \) is increasing.

Solution: \boxed{a_0 = \frac{1}{5}}


Two ships shuttle back and forth between the ports Mumraj and Zmatek on the same route. They spend negligible time in the ports, immediately turning around and continuing their journey. In the morning, at the same moment, the blue ship sets off from the port Mumraj and the green ship sets off from the port Zmatek. The ships first meet $20 \mathrm{~km}$ from the port Mumraj, and after some time, they meet again directly at this port. By then, the blue ship has managed to travel the route between the ports four times, while the green ship has only done so three times.

How long is the route between the ports Mumraj and Zmatek?

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


Point $A$ lies on the line $y = \frac{12}{5} x - 3$, and point $B$ lies on the parabola $y = x^2$. What is the minimum length of the segment $AB$?


Find all natural numbers \( a \) and \( b \) such that

\[ a^3 - b^3 = 633 \cdot p \]

where \( p \) is some prime number.

Solution: \boxed{a = 16, b = 13}


The elements $x_n$ of the sequence $\mathrm{Az}\left(x_{n}\right)$ are positive real numbers, and for every positive integer $n$, the following equation holds:

$$
2\left(x_{1}+x_{2}+\ldots+x_{n}\right)^{4}=\left(x_{1}^{5}+x_{2}^{5}+\ldots+x_{n}^{5}\right)+\left(x_{1}^{7}+x_{2}^{7}+\ldots+x_{n}^{7}\right)
$$

Determine the elements of the sequence.

Solution: \boxed{x_{n} = n}


There are seven red cubes, three blue cubes, and nine green cubes. Ten cubes were placed into a gift bag. In how many different ways could this have been done?


Find the number of roots of the equation

\[ z^{2} - a e^{z} = 0, \quad \text{where} \quad 0 < a < e^{-1} \]

inside the unit disk \( |z| < 1 \).


In triangle \( MNK \), \( MN = NK \). From point \( A \) on side \( MN \), a perpendicular \( AP \) is dropped to side \( NK \). It turns out that \( MA = AP \). Find the angle \( \angle PMK \).


If $2 - \sin^{2}(x + 2y - 1) = \frac{x^{2} + y^{2} - 2(x + 1)(y - 1)}{x - y + 1}$, then the minimum value of the product $xy$ is $\qquad$ .


G9.1 Find \( A \) from the sequence: \( 0,3,8, A, 24,35, \ldots \)

G9.2 The roots of the equation \( x^{2}-Ax+B=0 \) are 7 and \( C \). Find \( B \) and \( C \).

G9.3 If \( \log _{7} B=\log _{7} C+7^{X} \); find \( X \).

Solution: \boxed{X = 0}


Replace the larger of two distinct natural numbers with their difference. Continue this operation until the two numbers are the same. For example, for the pair (20, 26), the process is as follows:
\[ (20, 26) \rightarrow (20, 6) \rightarrow (14, 6) \rightarrow (8, 6) \rightarrow (2, 6) \rightarrow (2, 4) \rightarrow (2, 2) \]

1. Perform the above operation on the pair (45, 80).
2. If the operation is performed on two four-digit numbers and the final same number obtained is 17, find the maximum possible sum of these two four-digit numbers.


Find the minimum value of the expression \(\left(a^{2}+x^{2}\right) / x\), where \(a>0\) is a constant and \(x>0\) is a variable.


Triangle \( ABC \) is isosceles, and \( \angle ABC = x^\circ \). If the sum of the possible measures of \( \angle BAC \) is \( 240^\circ \), find \( x \).


For a $k$-element subset $T$ of the set $\{1,2,\cdots,242\}$, every pair of elements (which may be the same) in $T$ has a sum that is not an integer power of 3. Find the maximum value of $k$.