Consider an $n \times n$ grid consisting of $n^2$ until cells, where $n \geq 3$ is a given odd positive integer. First, Dionysus colours each cell either red or blue. It is known that a frog can hop from one cell to another if and only if these cells have the same colour and share at least one vertex. Then, Xanthias views the colouring and next places $k$ frogs on the cells so that each of the $n^2$ cells can be reached by a frog in a finite number (possible zero) of hops. Find the least value of $k$ for which this is always possible regardless of the colouring chosen by Dionysus. [i]Proposed by Tommy Walker Mackay, United Kingdom[/i]

Every planar section of a three-dimensional body $B$ is a disk. Show that B must be a ball.

Let $ABCDE$ be an inscribed pentagon such that $\angle B +\angle E = \angle C +\angle D$.Prove that $\angle CAD < \pi/3 < \angle A$. [i](Proposed by B.Frenkin)[/i]

Assume that positive numbers $a, b, c, x, y, z$ satisfy $cy + bz = a$, $az + cx = b$, and $bx + ay = c$. Find the minimum value of the function \[ f(x, y, z) = \frac{x^2}{x+1} + \frac {y^2}{y+1} + \frac{z^2}{z+1}. \]

Positive real numbers $a, b, c$ satisfy $\frac{1}{a} +\frac{1}{b} +\frac{1}{c} = 3.$ Prove the inequality \[\frac{1}{\sqrt{a^3+ b}}+\frac{1}{\sqrt{b^3 + c}}+\frac{1}{\sqrt{c^3 + a}}\leq \frac{3}{\sqrt{2}}.\]

Each spinner is divided into $3$ equal parts. The results obtained from spinning the two spinners are multiplied. What is the probability that this product is an even number? [asy] draw(circle((0,0),2)); draw(circle((5,0),2)); draw((0,0)--(sqrt(3),1)); draw((0,0)--(-sqrt(3),1)); draw((0,0)--(0,-2)); draw((5,0)--(5+sqrt(3),1)); draw((5,0)--(5-sqrt(3),1)); draw((5,0)--(5,-2)); fill((0,5/3)--(2/3,7/3)--(1/3,7/3)--(1/3,3)--(-1/3,3)--(-1/3,7/3)--(-2/3,7/3)--cycle,black); fill((5,5/3)--(17/3,7/3)--(16/3,7/3)--(16/3,3)--(14/3,3)--(14/3,7/3)--(13/3,7/3)--cycle,black); label("$1$",(0,1/2),N); label("$2$",(sqrt(3)/4,-1/4),ESE); label("$3$",(-sqrt(3)/4,-1/4),WSW); label("$4$",(5,1/2),N); label("$5$",(5+sqrt(3)/4,-1/4),ESE); label("$6$",(5-sqrt(3)/4,-1/4),WSW); [/asy] $\text{(A)}\ \frac{1}{3} \qquad \text{(B)}\ \frac{1}{2} \qquad \text{(C)}\ \frac{2}{3} \qquad \text{(D)}\ \frac{7}{9} \qquad \text{(E)}\ 1$

A square of side $1$ is divided into rectangles . We choose one of the two smaller sides of each rectangle (if the rectangle is a square, then we choose any of the four sides) . Prove that the sum of the lengths of all the chosen sides is at least $1$ . (Folklore)

Determine the number of distinct positive real solutions to $$\lfloor x \rfloor ^{\{x\}} = \frac{1}{2022}x^2$$ . Note: $\lfloor x \rfloor$ is known as the floor function, which returns the greatest integer less than or equal to $x$. Furthermore, $\{x\}$ is defined as $x - \lfloor x \rfloor$.

A nine-digit telephone number [i]abcdefghi [/i] is called [i]memorizable [/i] if the sequence of four initial digits [i]abcd [/i] is repeated in the sequence of the final five digits [i]efghi[/i]. How many [i]memorizable [/i] numbers of nine digits exist?

$(CZS 6)$ Let $d$ and $p$ be two real numbers. Find the first term of an arithmetic progression $a_1, a_2, a_3, \cdots$ with difference $d$ such that $a_1a_2a_3a_4 = p.$ Find the number of solutions in terms of $d$ and $p.$

Show that the equation $z^{n}+z+1=0$ has a solution with $|z|=1$ if and only if $n-2$ is divisble by $3$.

On a plane, $n$ straight lines are drawn. Two domains are called [i]adjacent [/i] if they border by a line segment. Prove that the domains into which the plane is divided by these lines can be painted two colors so that no two [i]adjacent [/i] domains are of the same color.

Let $a, b, r,$ and $s$ be positive integers ($a \ge 2$), where $a$ and $b$ have no common prime factor. Prove that if $a^r + b^r$ is divisible by $a^s + b^s$, then $r$ is divisible by $s$.

In a class, there are $n{}$ children of different heights. Denote by $A{}$ the number of ways to arrange them all in a row, numbered $1,2,\ldots,n$ from left to right, so that each person with an odd number is shorter than each of his neighbors. Let $B{}$ be the number of ways to organize $n-1$ badminton games between these children so that everyone plays at most two games with children shorter than himself and at most one game with children taller than himself (the order of the games is not important). Prove that $A = B$.

A sequence of integers $a_{1}, a_{2}, \cdots$ is called $good$ if: • $a_{1}=1$, and; • $a_{i+1}-a_{i}$ is either $1$ or $2$ for all $i \geq 1$. Find all positive integers n that cannot be written as a sum $n = a_{1} + a_{2} + \cdots + a_{k}$, such that the integers $a_{1} , a_{2} , \cdots , a_{k}$ forms a good sequence.

The train consists of six cars. On average, each carriage carries $18$ passengers. After one car was uncoupled, the average number of passengers in the remaining cars was reduced to $15$. How many passengers were in the uncoupled car?

Let $ a$ be a fix natural number . Prove that the set of prime divisors of $ 2^{2^{n}} \plus{} a$ for $ n \equal{} 1,2,\cdots$ is infinite

The following isosceles trapezoid consists of equal equilateral triangles with side length $1$. The side $A_1E$ has length $3$ while the larger base $A_1A_n$ has length $n-1$. Starting from the point $A_1$ we move along the segments which are oriented to the right and up(obliquely right or left). Calculate (in terms of $n$ or not) the number of all possible paths we can follow, in order to arrive at points $B,\Gamma,\Delta, E$, if $n$ is an integer greater than $3$. [color=#00CCA7][Need image][/color]

Let $A=\{P_1,P_2,…,P_{2011}\}$ be a set of points that lie in a circle $K(P_1,1)$. With $x_k$ we denote the distance between $P_k$ and the closest to it point from $A$. Prove that: $\sum_{i=1}^{2011} x_i^2 \leq \frac{9}{4}$.

$64$ people are in a single elimination rock-paper-scissors tournament, which consists of a $6$-round knockout bracket. Each person has a different rock-paper-scissors skill level, and in any game, the person with the higher skill level will always win. For how many players $P$ is it possible that $P$ wins the first four rounds that he plays? (A $6$-round knockout bracket is a tournament which works as follows: (a) In the first round, all 64 competitors are paired into $32$ groups, and the two people in each group play each other. The winners advance to the second round, and the losers are eliminated. (b) In the second round, the remaining $32$ players are paired into $16$ groups. Again, the winner of each group proceeds to the next round, while the loser is eliminated. (c) Each round proceeds in a similar way, eliminating half of the remaining players. After the sixth round, only one player will not have been eliminated. That player is declared the champion.) [i]In the game of rock-paper-scissors, two players each choose one of rock, paper, or scissors to play. Rock beats scissors, scissors beats paper, and paper beats rock. If the players play the same thing, the match is considered a draw.[/i]

Seven squares are arranged to form a rectangle as shown below. The side length of the smallest square is $3$ cm. What is the perimeter in centimeters of the rectangle formed by the $7$ squares? [asy] draw((0,0)--(57,0)--(57,63)--(0,63)--cycle); draw((12,27)--(12,39)); draw((24,27)--(24,63)); draw((27,0)--(27,30)); draw((0,27)--(27,27)); draw((24,30)--(57,30)); draw((0,39)--(24,39)); [/asy]

Let $f(x)=x^2+4x+2$. Let $r$ be the difference between the largest and smallest real solutions of the equation $f(f(f(f(x))))=0$. Then $r=a^{\frac{p}{q}}$ for some positive integers $a$, $p$, $q$ so $a$ is square-free and $p,q$ are relatively prime positive integers. Compute $a+p+q$.

Let $n$ be a positive integer. Two players $A$ and $B$ play a game in which they take turns choosing positive integers $k \le n$. The rules of the game are: (i) A player cannot choose a number that has been chosen by either player on any previous turn. (ii) A player cannot choose a number consecutive to any of those the player has already chosen on any previous turn. (iii) The game is a draw if all numbers have been chosen; otherwise the player who cannot choose a number anymore loses the game. The player $A$ takes the first turn. Determine the outcome of the game, assuming that both players use optimal strategies. [i]Proposed by Finland[/i]

Suppose that $ a, b, c, d$ are real numbers satisfying $ a \geq b \geq c \geq d \geq 0$, $ a^2 \plus{} d^2 \equal{} 1$, $ b^2 \plus{} c^2 \equal{} 1$, and $ ac \plus{} bd \equal{} 1/3$. Find the value of $ ab \minus{} cd$.

Let $BM$ be a median of right-angled nonisosceles triangle $ABC$ ($\angle B = 90$), and $H_a$, $H_c$ be the orthocenters of triangles $ABM$, $CBM$ respectively. Lines $AH_c$ and $CH_a$ meet at point $K$. Prove that $\angle MBK = 90$.