Let $n> 2$ be a positive integer. Given is a horizontal row of $n$ cells where each cell is painted blue or red. We say that a block is a sequence of consecutive boxes of the same color. Arepito the crab is initially standing at the leftmost cell. On each turn, he counts the number $m$ of cells belonging to the largest block containing the square he is on, and does one of the following: If the square he is on is blue and there are at least $m$ squares to the right of him, Arepito moves $m$ squares to the right; If the square he is in is red and there are at least $m$ squares to the left of him, Arepito moves $m$ cells to the left; In any other case, he stays on the same square and does not move any further. For each $n$, determine the smallest integer $k$ for which there is an initial coloring of the row with $k$ blue cells, for which Arepito will reach the rightmost cell.

Find the minimum possible sum of lengths of edges of a prism all of whose edges are tangent of a unit sphere. [Muller-Pfeiffer].

Let $X$ be a set of $n + 1$ elements, $n\geq 2$. Ordered $n$-tuples $(a_1,\ldots,a_n)$ and $(b_1,\ldots,b_n)$ formed from distinct elements of $X$ are called[i] disjoint [/i]if there exist distinct indices $1\leq i \neq j\leq n$ such that $a_i = b_j$. Find the maximal number of pairwise disjoint $n$-tuples.

Find the sum of the solution(s) $x$ to the equation \[x=\sqrt{2022+\sqrt{2022+x}}.\]

Let $P$ be a real polynomial with degree $n\ge1$ such that $$P(0),P(1),P(4),P(9),\ldots,P(n^2)$$ are in $\mathbb Z$. Prove that $\forall a\in\mathbb Z,P(a^2)\in\mathbb Z$. [i]Proposed by Moubinool Omarjee[/i]

1. A strip for playing "hopscotch" consists of ten squares numbered consecutively $1,2, \ldots, 10$. Clarissa and Marissa start from the center of the first square, jump 9 times to the centers of the other squares so that they visit each square once, and end up at the tenth square. (Jumps forward and backward are allowed.) Each jump of Clarissa was for the same distance as the corresponding jump of Marissa. Does this mean that they both visited the squares in the same order? Alexey Tolpygo

In this diagram the center of the circle is $O$, the radius is $a$ inches, chord $EF$ is parallel to chord $CD, O, G, H, J$ are collinear, and $G$ is the midpoint of $CD$. Let $K$ (sq. in.) represent the area of trapezoid $CDFE$ and let $R$ (sq. in.) represent the area of rectangle $ELMF$. Then, as $CD$ and $EF$ are translated upward so that $OG$ increases toward the value $a$, while $JH$ always equals $HG$, the ratio $K:R$ become arbitrarily close to: [asy] size((270)); draw((0,0)--(10,0)..(5,5)..(0,0)); draw((5,0)--(5,5)); draw((9,3)--(1,3)--(1,1)--(9,1)--cycle); draw((9.9,1)--(.1,1)); label("O", (5,0), S); label("a", (7.5,0), S); label("G", (5,1), SE); label("J", (5,5), N); label("H", (5,3), NE); label("E", (1,3), NW); label("L", (1,1), S); label("C", (.1,1), W); label("F", (9,3), NE); label("M", (9,1), S); label("D", (9.9,1), E); [/asy] $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \sqrt{2} \qquad\textbf{(D)}\ \frac{1}{\sqrt{2}}+\frac{1}{2} \qquad\textbf{(E)}\ \frac{1}{\sqrt{2}}+1$

Let $S $ be a set of three distinct integers. Show that there are $a, b \in S$ such that $a \ne b$ and $10 | a^3b - ab^3$.

Let $\mathbb{S}$ be the set of all functions $f:\mathbb{Z}\rightarrow \mathbb{R}$. Now, consider the function $g:\mathbb{S} \rightarrow \mathbb{S} ,g(f(x)) = f(x + 1)-f(x)$. Now, we call a function $f \in \mathbb{S}$ good if $g^n(f(x))=0$ for some natural $n$. Prove that if $s \not = t \in S$ are good functions then $s(m)-t(m)$ is 0 for only finitely many $m \in \mathbb{Z}$.

$(x_n)$ is sequence, such that $x_{n+2}=|x_{n+1}|-x_n$. Prove, that it is periodic.

Real numbers $a,b,c$ with $0\le a,b,c\le 1$ satisfy the condition $$a+b+c=1+\sqrt{2(1-a)(1-b)(1-c)}.$$ Prove that $$\sqrt{1-a^2}+\sqrt{1-b^2}+\sqrt{1-c^2}\le \frac{3\sqrt 3}{2}.$$ [i] (Nora Gavrea)[/i]

In an athletics tournament, five teams participate. Each athlete has a shirt numbered with a positive integer, and all athletes on the same team have different numbers. Each athlete participates in a single event and only one athlete from each team is present in each event. Emídio noticed that the sum of the athletes' jersey numbers in each event is always $20$. What is the maximum number of athletes in the tournament?

In this diagram $AB$ and $AC$ are the equal sides of an isosceles triangle $ABC$, in which is inscribed equilateral triangle $DEF$. Designate angle $BFD$ by $a$, angle $ADE$ by $b$, and angle $FEC$ by $c$. Then: [asy] size(150); defaultpen(linewidth(0.8)+fontsize(10)); pair A=(5,12),B=origin,C=(10,0),D=(5/3,4),E=(10-5*.45,12*.45),F=(6,0); draw(A--B--C--cycle^^D--E--F--cycle); draw(anglemark(E,D,A,1,45)^^anglemark(F,E,C,1,45)^^anglemark(D,F,B,1,45)); label("$b$",(D.x+.2,D.y+.25),dir(30)); label("$c$",(E.x,E.y-.4),S); label("$a$",(F.x-.4,F.y+.1),dir(150)); label("$A$",A,N); label("$B$",B,S); label("$C$",C,S); label("$D$",D,dir(150)); label("$E$",E,dir(60)); label("$F$",F,S);[/asy] $ \textbf{(A)}\ b=\frac{a+c}{2}\qquad\textbf{(B)}\ b=\frac{a-c}{2}\qquad$ $\textbf{(C)}\ a=\frac{b-c}{2} \qquad\textbf{(D)}\ a=\frac{b+c}{2}\qquad$ $\textbf{(E)}\ \text{none of these} $

Let $n$ be a positive integer. Consider sequences $a_0, a_1, ..., a_k$ and $b_0, b_1,,..,b_k$ such that $a_0 = b_0 = 1$ and $a_k = b_k = n$ and such that for all $i$ such that $1 \le i \le k $, we have that $(a_i, b_i)$ is either equal to $(1 + a_{i-1}, b_{i-1})$ or $(a_{i-1}; 1 + b_{i-1})$. Consider for $1 \le i \le k$ the number $c_i = \begin{cases} a_i \,\,\, if \,\,\, a_i = a_{i-1} \\ b_i \,\,\, if \,\,\, b_i = b_{i-1}\end{cases}$ Show that $c_1 + c_2 + ... + c_k = n^2 - 1$.

A game is played on a $n \times n$ chessboard. In the beginning Bars the cat occupies any cell according to his choice. The $d$ sparrows land on certain cells according to their choice (several sparrows may land in the same cell). Bars and the sparrows play in turns. In each turn of Bars, he moves to a cell adjacent by a side or a vertex (like a king in chess). In each turn of the sparrows, precisely one of the sparrows flies from its current cell to any other cell of his choice. The goal of Bars is to get to a cell containing a sparrow. Can Bars achieve his goal a) if $d=\lfloor \frac{3\cdot n^2}{25}\rfloor$, assuming $n$ is large enough? b) if $d=\lfloor \frac{3\cdot n^2}{19}\rfloor$, assuming $n$ is large enough? c) if $d=\lfloor \frac{3\cdot n^2}{14}\rfloor$, assuming $n$ is large enough?

For arbitrary real numbers \( x_1,x_2,\ldots,x_n \), prove that \[ \left(\max_{1\leq i \leq n}x_i \right)^2 + 4\sum_{i=1}^{n-1}\left(\max_{1\leq j \leq i}x_j\right)\left(x_{i+1}-x_i\right) \leq 4x_n^2. \]

Let $ABCD$ be a rectangle and $U, V$ two points of its circumcircle. Lines $AU,CV$ intersect at $P$ and lines $BU,DV$ intersect at $Q$, distinct from $P$. Prove that $$\frac{1}{PQ^2} \ge \frac{1}{UV^2} - \frac{1}{AC^2}$$ Michel Bataille

[list] $(a)$ A sequence $x_1,x_2,\dots$ of real numbers satisfies \[x_{n+1}=x_n \cos x_n \textrm{ for all } n\geq 1.\] Does it follows that this sequence converges for all initial values $x_1?$ (5 points) $(b)$ A sequence $y_1,y_2,\dots$ of real numbers satisfies \[y_{n+1}=y_n \sin y_n \textrm{ for all } n\geq 1.\] Does it follows that this sequence converges for all initial values $y_1?$ (5 points)[/list]

A natural number \(N\) is given. A cube with side length \(2N + 1\) is made up of \((2N + 1)^3\) unit cubes, each of which is either black or white. It turns out that among any $8$ cubes that share a common vertex and form a \(2 \times 2 \times 2\) cube, there are at most $4$ black cubes. What is the maximum number of black cubes that could have been used?

Let $x$ and $y$ be complex numbers such that $x+y=\sqrt{20}$ and $x^2+y^2=15$. Compute $|x-y|$.

Given $2009 \times 4018$ rectangular board. Frame is a rectangle $n \times n$ or $n \times(n + 2)$ for $ ( n \geq 3 )$ without all cells which don’t have any common points with boundary of rectangle. Rectangles $1\times1,1\times 2,1\times 3$ and $ 2\times 4$ are also frames. Two players by turn paint all cells of some frame that has no painted cells yet. Player that can't make such move loses. Who has a winning strategy?

Let $f:\mathbb{R}\to \mathbb{R}$ a function which transforms any closed bounded interval in a closed bounded interval and any open bounded interval in an open bounded interval. Prove that $f$ is continuous. [i]Mihai Piticari[/i]

Let $ABC$ be a scalene triangle. The inscribed circle of $ABC$ touches the sides $BC$, $CA$, and $AB$ at the points $A_1$, $B_1$, $C_1$ respectively. Let $I$ be the incenter, $O$ be the circumcenter, and lines $OI$ and $BC$ meet at point $D$. The perpendicular line from $A_1$ to $B_1 C_1$ intersects $AD$ at point $E$. Prove that $B_1 C_1$ passes through the midpoint of $EA_1$.

The curve $y=x^4+2x^3-11x^2-13x+35$ has a bitangent (a line tangent to the curve at two points). What is the equation of the bitangent?

Sixteen chairs are arranged in a row. Eight people each select a chair in which to sit so that no person sits next to two other people. Let $N$ be the number of subsets of $16$ chairs that could be selected. Find the remainder when $N$ is divided by $1000$.