Ana, Bruno and Carolina played table tennis with each other. In each game, only two of the friends played, with the third one resting. Every time one of the friends won a game, they rested during the next game. Ana played $12$ games, Bruno played $21$ games and Carolina rested for$ 8$ games. Who rested in the last game?

Let $A$ be a set of $n$ elements and $A_1, A_2, ... A_k$ subsets of $A$ such that for any $2$ distinct subsets $A_i, A_j$ either they are disjoint or one contains the other. Find the maximum value of $k$

There are $36$ cards in a deck arranged in the sequence spades, clubs, hearts, diamonds, spades, clubs, hearts, diamonds, etc. Somebody took part of this deck off the top, turned it upside down, and cut this part into the remaining part of the deck (i.e. inserted it between two consecutive cards). Then four cards were taken off the top, then another four, etc. Prove that in any of these sets of four cards, all the cards are of different suits. (A Merkov, Moscow)

Consider $ p (x) = x ^ n + a_ {n-1} x ^ {n-1} + ... + a_ {1} x + 1 $ a polynomial of positive real coefficients, degree $ n \geq 2 $ e with $ n $ real roots. Which of the following statements is always true? a) $ p (2) <2 (2 ^ {n-1} +1) $ (b) $ p (1) <3 $ c) $ p (1)> 2 ^ n $ d) $ p (3 ) <3 (2 ^ {n-1} -2) $

Three circular arcs $\gamma_1, \gamma_2,$ and $\gamma_3$ connect the points $A$ and $C.$ These arcs lie in the same half-plane defined by line $AC$ in such a way that arc $\gamma_2$ lies between the arcs $\gamma_1$ and $\gamma_3.$ Point $B$ lies on the segment $AC.$ Let $h_1, h_2$, and $h_3$ be three rays starting at $B,$ lying in the same half-plane, $h_2$ being between $h_1$ and $h_3.$ For $i, j = 1, 2, 3,$ denote by $V_{ij}$ the point of intersection of $h_i$ and $\gamma_j$ (see the Figure below). Denote by $\widehat{V_{ij}V_{kj}}\widehat{V_{kl}V_{il}}$ the curved quadrilateral, whose sides are the segments $V_{ij}V_{il},$ $V_{kj}V_{kl}$ and arcs $V_{ij}V_{kj}$ and $V_{il}V_{kl}.$ We say that this quadrilateral is $circumscribed$ if there exists a circle touching these two segments and two arcs. Prove that if the curved quadrilaterals $\widehat{V_{11}V_{21}}\widehat{V_{22}V_{12}}, \widehat{V_{12}V_{22}}\widehat{V_{23}V_{13}},\widehat{V_{21}V_{31}}\widehat{V_{32}V_{22}}$ are circumscribed, then the curved quadrilateral $\widehat{V_{22}V_{32}}\widehat{V_{33}V_{23}}$ is circumscribed, too. [i]Proposed by Géza Kós, Hungary[/i] [asy] pathpen=black; size(400); pair A=(0,0), B=(4,0), C=(10,0); draw(L(A,C,0.3)); MP("A",A); MP("B",B); MP("C",C); pair X=(5,-7); path G1=D(arc(X,C,A)); pair Y=(5,7), Z=(9,6); draw(Z--B--Y); struct T {pair C;real r;}; T f(pair X, pair B, pair Y, pair Z) { pair S=unit(Y-B)+unit(Z-B); real s=abs(sin(angle((Y-B)/(Z-B))/2)); real t=10, r=abs(X-A); pair Q; for(int k=0;k<30;++k) { Q=B+t*S; t-=(abs(X-Q)-r)/abs(S)-s*t; } T T=new T; T.C=Q; T.r=s*t*abs(S); return T; } void g(pair Q, real r) { real t=0; for(int k=0;k<30;++k) { X=(5,t); t+=(abs(X-Q)+r-abs(X-A)); } } pair Z1=(1.07,6); draw(B--Z1); T T=f(X,B,Y,Z1); draw(CR(T.C,T.r)); T T=f(X,B,Y,Z); draw(CR(T.C,T.r)); g(T.C,T.r); path G2=D(arc(X,C,A)); T T=f(X,B,Y,Z1); draw(CR(T.C,T.r)); T=f(X,B,Y,Z); draw(CR(T.C,T.r)); g(T.C,T.r); path G3=D(arc(X,C,A)); pen p=black+fontsize(8); MC("\gamma_1",G1,0.85,p); MC("\gamma_2",G2,0.85,NNW,p); MC("\gamma_3",G3,0.85,WNW,p); MC("h_1",B--Z1,0.95,E,p); MC("h_2",B--Y,0.95,E,p); MC("h_3",B--Z,0.95,E,p); path[] G={G1,G2,G3}; path[] H={B--Z1,B--Y,B--Z}; pair[][] al={{S+SSW,S+SSW,3*S},{SE,NE,NW},{2*SSE,2*SSE,2*E}}; for(int i=0;i<3;++i) for(int j=0;j<3;++j) MP("V_{"+string(i+1)+string(j+1)+"}",IP(H[i],G[j]),al[i][j],fontsize(8));[/asy]

In the tetrahedron $ABCD$ three of the faces are right-angled triangles and the other is not an obtuse triangle. Prove that: (a) the fourth wall of the tetrahedron is a right-angled triangle if and only if exactly two of the plane angles having common vertex with the some of vertices of the tetrahedron are equal. (b) its volume is equal to $\frac16$ multiplied by the multiple of two shortest edges and an edge not lying on the same wall.

Show that for every prime number $p$ and every positive integer $n\geq2$ there exists a positive integer $k$ such that the decimal representation of $p^k$ contains $n$ consecutive equal digits.

Consider the set $ S_n$ of all the $ 2^n$ numbers of the type $ 2\pm \sqrt{2 \pm \sqrt {2 \pm ...}},$ where number $ 2$ appears $ n\plus{}1$ times. $ (a)$ Show that all members of $ S_n$ are real. $ (b)$ Find the product $ P_n$ of the elements of $ S_n$.

Consider the arithmetic progression $a, a+d, a+2d,\ldots$ where $a$ and $d$ are positive integers. For any positive integer $k$, prove that the progression has either no $k$-th powers or infinitely many.

Alice has an orange $\text{3-by-3-by-3}$ cube, which is comprised of $27$ distinguishable, $\text{1-by-1-by-1}$ cubes. Each small cube was initially orange, but Alice painted $10$ of the small cubes completely black. In how many ways could she have chosen $10$ of these smaller cubes to paint black such that every one of the $27$ $\text{3-by-1-by-1}$ sub-blocks of the $\text{3-by-3-by-3}$ cube contains at least one small black cube?

Prove that there are two natural numbers $ p,q, $ satisfying $$ p<q<n\bigg|p+(p+1)+\cdots +(q-1) +q, $$ if and only if $ n $ is not a power of $ 2. $

A convex quadrilateral $ABCD$ is inscribed in a circle. Show that the line connecting the midpoints of the arcs $AB$ and $CD$ and the line connecting the midpoints of the arcs $BC$ and $DA$ are perpendicular.

A country has $2023$ cities and there are flights between these cities. Each flight connects two cities in both directions. We know that you can get from any city to any other using these flights, and from each city there are flights to at most $4$ other cities. What is the maximum possible number of cities in the country from which there is a flight to only one city?

Julia has $289$ coins stored in boxes: All the boxes contain the same number of coins (which is greater than $1$) and in each box there are coins from the same country, The coins from Bolivia are more than $6\%$ of the total, those from Chile are more than $12\%$ of the total, those of Mexico are more than $24\% $of the total and those of Peru more than $36\%$ of the total. Can Julia have any coins from Uruguay?

Let $n$ be a positive integer. On the blackboard all quadratic polynomials with positive integer coefficients, that do not exceed $n$, without real roots are written Find all $n$ for which the number of written polynomials is even [i]A. Voidelevich[/i]

Cut a cross made up of five identical squares into three polygons, equal in area and perimeter.