Let $O$ be the circumcentre of the acute triangle $ABC$. Let $c_1$ and $c_2$ be the circumcircles of triangles $ABO$ and $ACO$. Let $P$ and $Q$ be points on $c_1$ and $c_2$ respectively, such that OP is a diameter of $c_1$ and $OQ$ is a diameter of $c_2$. Let $T$ be the intesection of the tangent to $c_1$ at $P$ and the tangent to $c_2$ at $Q$. Let $D$ be the second intersection of the line $AC$ and the circle $c_1$. Prove that the points $D, O$ and $T$ are collinear

(a) Write $A = k^4 + 4$, where $k$ is a positive integer, as a product of two factors each of them is sum of two squares of integers. (b) Simplify the expression$$K=\frac{(2^4+\frac14)(4^4+\frac14)...((2n)^4+\frac14)}{(1^4+\frac14)(3^4+\frac14)...((2n-1)^4+\frac14)}$$and write it as sum of squares of two consecutive positive integers

Determine all values of $a_0$ for which the sequence of real numbers with $a_{n+1}=3a_n - 4a_n^3$ for all $n\geq 0$ is periodic from the beginning.

Find all integers $n=2k+1>1$ so that there exists a permutation $a_0, a_1,\ldots,a_{k}$ of $0, 1, \ldots, k$ such that \[a_1^2-a_0^2\equiv a_2^2-a_1^2\equiv \cdots\equiv a_{k}^2-a_{k-1}^2\pmod n.\] [i]Proposed by usjl[/i]

If $S = 6 \times10 000 +5\times 1000+ 4 \times 10+ 3 \times 1$, what is $S$? $\textbf{(A)}\ 6543 \qquad \textbf{(B)}\ 65043 \qquad \textbf{(C)}\ 65431 \qquad \textbf{(D)}\ 65403 \qquad \textbf{(E)}\ 60541$

Let $ABCD$ be a rectangle satisfying $AB = CD = 24$, and let $E$ and $G$ be points on the extension of $BA$ past $A$ and the extension of $CD$ past $D$ respectively such that $AE = 1$ and $DG = 3$. Suppose that there exists a unique pair of points $(F, H)$ on lines $BC$ and $DA$ respectively such that $H$ is the orthocenter of $\triangle EFG$. Find the sum of all possible values of $BC$.

Given the parallelogram $ABCD$. The trisection points of side $AB$ are: $H_1, H_2$, ($AH_1 = H_1H_2 =H_2B$). The trisection points of the side $DC$ are $G_1, G_2$, ($DG_1 = G_1G_2 = G_2C$), and $AD = 1, AC = 2$. Prove that triangle $AH_2G_1$ is isosceles.

Charles is playing a variant of Sudoku. To each lattice point $(x, y)$ where $1\le x,y <100$, he assigns an integer between $1$ and $100$ inclusive. These integers satisfy the property that in any row where $y=k$, the $99$ values are distinct and never equal to $k$; similarly for any column where $x=k$. Now, Charles randomly selects one of his lattice points with probability proportional to the integer value he assigned to it. Compute the expected value of $x+y$ for the chosen point $(x, y)$.

Given a polygon, a [i]triangulation[/i] is a division of the polygon into triangles whose vertices are the vertices of the polygon. Determine the values of $n$ for which the regular polygon with $n$ sides has a triangulation with all its triangles being isosceles.

Let $ T$ be a bounded linear operator on a Hilbert space $ H$, and assume that $ \|T^n \| \leq 1$ for some natural number $ n$. Prove the existence of an invertible linear operator $ A$ on $ H$ such that $ \| ATA^{\minus{}1} \| \leq 1$. [i]E. Druszt[/i]

Let $a\ge b$ and $c\ge d$ be real numbers. Prove that the equation \[(x+a)(x+d)+(x+b)(x+c)=0\] has real roots.

In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$, and $\overline{AB}$ is parallel to $\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $4:5$. What is the degree measure of angle $BCD$? [asy] unitsize(7mm); defaultpen(linewidth(.8pt)+fontsize(10pt)); dotfactor=4; real r=3; pair A=(-3cos(80),-3sin(80)); pair D=(3cos(80),3sin(80)), C=(-3cos(80),3sin(80)); pair O=(0,0), E=(-3,0), B=(3,0); path outer=Circle(O,r); draw(outer); draw(E--B); draw(E--A); draw(B--A); draw(E--D); draw(C--D); draw(B--C); pair[] ps={A,B,C,D,E,O}; dot(ps); label("$A$",A,N); label("$B$",B,NE); label("$C$",C,S); label("$D$",D,S); label("$E$",E,NW); label("$$",O,N);[/asy] $ \textbf{(A)}\ 120 \qquad \textbf{(B)}\ 125 \qquad \textbf{(C)}\ 130 \qquad \textbf{(D)}\ 135 \qquad \textbf{(E)}\ 140 $

Suppose we have a uniformly random function from $\{1, 2, 3, \ldots, 25\}$ to itself. Find the expected value of $$\sum_{x=1}^{25} (f(f(x))-x)^2.$$

On the graphic of the function $y=x^2$ were selected $1000$ pairwise distinct points, abscissas of which are integer numbers from the segment $[0; 100000]$. Prove that it is possible to choose six different selected points $A$, $B$, $C$, $A'$, $B'$, $C'$ such that areas of triangles $ABC$ and $A'B'C'$ are equals. [i]A. Tereshin[/i]

All edges of a polyhedron are painted with red or yellow. For an angle of a facet, if the edges determining it are of different colors, then the angle is called [i]excentric[/i]. The[i] excentricity [/i]of a vertex $A$, namely $S_A$, is defined as the number of excentric angles it has. Prove that there exist two vertices $B$ and $C$ such that $S_B + S_C \leq 4$.

Let $n>1$ be a positive integer. Call a rearrangement $a_1,a_2, \cdots , a_n$ of $1,2, \cdots , n$ [i]nice[/i] if for every $k = 2,3, \cdots , n$, we have that $a_1 + a_2 + \cdots + a_k$ is not divisible by $k$. (a) If $n>1$ is odd, prove that there is no nice arrangement of $1,2, \cdots , n$. (b) If $n$ is even, find a [i]nice[/i] arrangement of $1,2, \cdots , n$.

Give and Take divide $100$ coins between themselves as follows. In each step, Give chooses a handful of coins from the heap, and Take decides who gets this handful. This is repeated until all coins have been taken, or one of them has $9$ handfuls. In the latter case, the other gets all the remaining coins. What is the largest number of coins that Give can be sure of getting no matter what Take does? (A Shapovalov)

Ahmet and Betül play a game on $ n\times n$ $ \left(n\ge 7\right)$ board. Ahmet places his only piece on one of the $ n^{2}$ squares. Then Betül places her two pieces on two of the squares at the border of the board. If two squares have a common edge, we call them adjacent squares. When it is Ahmet's turn, Ahmet moves his piece either to one of the empty adjacent squares or to the out of the board if it is on one of the squares at the border of the board. When it is Betül's turn, she moves all her two pieces to the adjacent squares. If Ahmet's piece is already on one of the two squares that Betül has just moved to, Betül attacks to his piece and wins the game. If Ahmet manages to go out of the board, he wins the game. If Ahmet begins to move, he guarantees to win the game putting his piece on one of the $\dots$ squares at the beginning of the game. $\textbf{(A)}\ 0 \qquad\textbf{(B)}\ n^{2} \qquad\textbf{(C)}\ \left(n\minus{}2\right)^{2} \qquad\textbf{(D)}\ 4\left(n\minus{}1\right) \qquad\textbf{(E)}\ 2n\minus{}1$

Prove that, for any positive integer $n$, $$1^n+2^n+3^n+4^n$$ is divisible by $5$ if and only if $n$ is not divisible by $4$.

Find all sequences of positive integers $(a_n)_{n\ge 1}$ which satisfy $$a_{n+2}(a_{n+1}-1)=a_n(a_{n+1}+1)$$ for all $n\in \mathbb{Z}_{\ge 1}$. [i](Bogdan Blaga)[/i]

Find the largest positive integer $N$ such that the equation $99x + 100y + 101z = N$ has an unique solution in the positive integers $x, y, z$. [i](7 points)[/i]

Let $n\geqslant 3$ be an integer. Prove that there exists a set $S$ of $2n$ positive integers satisfying the following property: For every $m=2,3,...,n$ the set $S$ can be partitioned into two subsets with equal sums of elements, with one of subsets of cardinality $m$.

Prove that a triangle inscribed in a parallelogram has at most half the area of the parallelogram.

A website offers for $1000$ colones, the possibility of playing $4$ shifts a certain game of randomly, in each turn the user will have the same probability $p$ of winning the game and obtaining $1000$ colones (per shift). But to calculate $p$ he asks you to roll $3$ dice and add the results, with what p will be the probability of obtaining this sum. Olcoman visits the website, and upon rolling the dice, he realizes that the probability of losing his money is from $\left( \frac{103}{108}\right)^4$. a) Determine the probability $p$ that Olcoman wins a game and the possible outcomes with the dice, to get to this one. b) Which sums (with the dice) give the maximum probability of having a profit of exactly $1000$ colones? Calculate this probability and the value of $p$ for this case.

Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ denote the intersection of the internal angle bisectors of $\triangle ABC$. What is $BI$? $ \textbf{(A)}\ 15\qquad\textbf{(B)}\ 5+\sqrt{26}+3\sqrt{3}\qquad\textbf{(C)}\ 3\sqrt{26}\qquad\textbf{(D)}\ \frac{2}{3}\sqrt{546}\qquad\textbf{(E)}\ 9\sqrt{3} $