Points $A$, $B$, $C$, $D$ lie on a circle (in that order) where $AB$ and $CD$ are not parallel. The length of arc $AB$ (which contains the points $D$ and $C$) is twice the length of arc $CD$ (which does not contain the points $A$ and $B$). Let $E$ be a point where $AC=AE$ and $BD=BE$. Prove that if the perpendicular line from point $E$ to the line $AB$ passes through the center of the arc $CD$ (which does not contain the points $A$ and $B$), then $\angle ACB = 108^\circ$.

Find the smallest natural number $x> 0$ so that all following fractions are simplified $\frac{3x+9}{8},\frac{3x+10}{9},\frac{3x+11}{10},...,\frac{3x+49}{48}$ , i.e. numerators and denominators are relatively prime.

Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

(a) Prove that in the set $ S=\{2008,2009,. . .,4200\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression. (b) Bonus: Try to find a smaller integer $ n \in (2008,4200)$ such that in the set $ S'=\{2008,2009,...,n\}$ there are $ 5^3$ elements such that any three of them are not in arithmetic progression.

Let $a_1, \dots, a_n, b_1, \dots, b_n$ be $2n$ positive integers such that the $n+1$ products \[a_1 a_2 a_3 \cdots a_n, b_1 a_2 a_3 \cdots a_n, b_1 b_2 a_3 \cdots a_n, \dots, b_1 b_2 b_3 \cdots b_n\] form a strictly increasing arithmetic progression in that order. Determine the smallest possible integer that could be the common difference of such an arithmetic progression.

Find all natural numbers $n$ and nonnegative integers $x_{1},x_{2},...,x_{n}$ such that $\sum_{i=1}^{n}x_{i}^{2}=1+\frac{4}{4n+1}(\sum_{i=1}^{n}x_{i})^{2}$.

You have a 2 meter long string. You choose a point along the string uniformly at random and make a cut. You discard the shorter section. If you still have 0.5 meters or more of string, you repeat. You stop once you have less than 0.5 meters of string. On average, how many cuts will you make before stopping?

Find all functions $f :[0, +\infty) \rightarrow [0, +\infty)$ for which $f(f(x)+f(y)) = xy f (x+y)$ for every two non-negative real numbers $x$ and $y$.

Find the differentiable functions $ f:\mathbb{R}_{\ge 0 }\longrightarrow\mathbb{R} $ that verify $ f(0)=0 $ and $$ f'(x)=1/3\cdot f'\left( x/3 \right) +2/3\cdot f'\left( 2x/3 \right) , $$ for any nonnegative real number $ x. $

On a circumference, points $A$ and $B$ are on opposite arcs of diameter $CD$. Line segments $CE$ and $DF$ are perpendicular to $AB$ such that $A-E-F-B$ (i.e., $A$, $E$, $F$ and $B$ are collinear on this order). Knowing $AE=1$, find the length of $BF$.

Let $E(n)$ denote the sum of the even digits of $n$. For example, $E(5681) = 6+8 = 14$. Find $E(1) + E(2) + E(3) + \cdots + E(100)$. $\text{(A)}\ 200 \qquad \text{(B)}\ 360 \qquad \text{(C)}\ 400 \qquad \text{(D)}\ 900 \qquad \text{(E)}\ 2250$

Find the number of two-digit positive integers whose digits total $7$. $\textbf{(A)}\ 6\qquad \textbf{(B)}\ 7\qquad \textbf{(C)}\ 8\qquad \textbf{(D)}\ 9\qquad \textbf{(E)}\ 10$

The convex polygon $A_1A_2...A_n$ is given in the plane. Denote by $T_k$ $(k \le n)$ the convex $k$-gon of the largest area, with vertices at the points $A_1, A_2, ..., A_n$ and by $T_k(A+1)$ the convex k-gon of the largest area with vertices at the points $A_1, A_2, ..., A_n$ in which one of the vertices is in $A_1$. Set the relationship between the order of arrangement in the sequence $A_1, A_2, ..., A_n$ vertices: 1) $T_3$ and $T_3 (A_2)$ 2) $T_k$ and $T_k (A_1) $ 3) $T_k$ and $T_{k+1}$

Consider 2009 cards which are lying in sequence on a table. Initially, all cards have their top face white and bottom face black. The cards are enumerated from 1 to 2009. Two players, Amir and Ercole, make alternating moves, with Amir starting. Each move consists of a player choosing a card with the number $k$ such that $k < 1969$ whose top face is white, and then this player turns all cards at positions $k,k+1,\ldots,k+40.$ The last player who can make a legal move wins. (a) Does the game necessarily end? (b) Does there exist a winning strategy for the starting player? [i]Also compare shortlist 2009, combinatorics problem C1.[/i]

250 numbers are chosen among positive integers not exceeding 501. Prove that for every integer $ t$ there are four chosen numbers $ a_1$, $ a_2$, $ a_3$, $ a_4$, such that $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \minus{} t$ is divisible by 23. [i]Author: K. Kokhas[/i]

$n$ is a natural number, $f_n(x)=\frac{x^{n+1}-x^{-n-1}}{x-x^{-1}}(x\neq0,\pm1)$, let $y=x+\frac{1}{x}$. [b](a)[/b] Prove that $f_{n+1}(x)=yf_n(x)-f_{n-1}(x)$ [b](b)[/b] Prove with mathematical induction: $f_n(x)=\begin{cases} y^n-\text{C}_{n-1}^{1}y^{n-2}+\cdots+(-1)^i\text{C}_{n-i}^{i}y^{n-2i}+\cdots+(-1)^{\frac{n}{2}}(i=1,2,\cdots,\frac{n}{2},n\text{ is even})\\ y^n-\text{C}_{n-1}^{1}y^{n-2}+\cdots+(-1)^i\text{C}_{n-i}^{i}y^{n-2i}+\cdots+(-1)^{\frac{n-1}{2}}\text{C}_{\frac{n+1}{2}}^{\frac{n-1}{2}}y(i=1,2,\cdots,\frac{n-1}{2},n\text{ is odd}) \end{cases}$.

$\text{ }$ [asy] unitsize(11); for(int i=0; i<6; ++i) { if(i<5) draw( (i, 0)--(i,5) ); else draw( (i, 0)--(i,2) ); if(i < 3) draw((0,i)--(5,i)); else draw((0,i)--(4,i)); } [/asy] We are dividing the above figure into parts with shapes: [asy] unitsize(11); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,1)--(2,2)); draw((0,0)--(1,0)); draw((0,1)--(2,1)); draw((0,2)--(2,2)); [/asy][asy] unitsize(11); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,1)--(2,2)); draw((3,1)--(3,2)); draw((0,0)--(1,0)); draw((0,1)--(3,1)); draw((0,2)--(3,2)); [/asy] After that division, find the number of [asy] unitsize(11); draw((0,0)--(0,2)); draw((1,0)--(1,2)); draw((2,1)--(2,2)); draw((0,0)--(1,0)); draw((0,1)--(2,1)); draw((0,2)--(2,2)); [/asy] shaped parts.

Given a convex quadrilateral $ABCD$. Let $R_a, R_b, R_c$ and $R_d$ be the circumradii of triangles $DAB, ABC, BCD, CDA$. Prove that inequality $R_a < R_b < R_c < R_d$ is equivalent to $180^o - \angle CDB < \angle CAB < \angle CDB$ . (O.Musin)

Let $I$ be the center of the inscribed circle of $ABC$ and let $I_A$ be the center of the exscribed circle touching the side $BC$. Let $M$ be the midpoint of the side $BC$, and $N$ be the midpoint of the arc $BAC$ of the circumscribed circle of $ABC$ . The point $T$ is symmetric to the point $N$ wrt point $A$. Prove that the points $I_A,M,I,T$ lie on the same circle. (Danilo Hilko)

Consider the following operation. Given a positive integer $n$, if $n$ is a multiple of $3$, then you replace $n$ by $\dfrac{n}3$. If $n$ is not a multiple of $3$, then you replace $n$ by $n + 10$. Then continue this process. For example, beginning with $n = 4$, this procedure gives $4 \to 14 \to 24 \to 8 \to 18 \to 6 \to 2 \to 12 \to \cdots$. Suppose you start with $n = 100$. What value results if you perform this operation exactly $100$ times? $\textbf{(A) }10\qquad\textbf{(B) }20\qquad\textbf{(C) }30\qquad\textbf{(D) }40\qquad\textbf{(E) }50$

For a certain triangle all of its altitudes are integers whose sum is less than 20. If its Inradius is also an integer Find all possible values of area of the triangle.

positive reals $a_1, a_2, . . . $ satisfying (i) $a_{n+1}=a_1^2\cdot a_2^2 \cdot . . . \cdot a_n^2-3$(all positive integers $n$) (ii) $\frac{1}{2}(a_1+\sqrt{a_2-1})$ is positive integer. prove that $\frac{1}{2}(a_1 \cdot a_2 \cdot . . . \cdot a_n + \sqrt{a_{n+1}-1})$ is positive integer

On a rectangular board with $m \times n$ squares ($m, n \ge 3$) there are dominoes ($2 \times 1$ or $1\times 2$ tiles), which do not overlap and do not extend beyond the board. Every domino covers exactly two squares of the board. Assume that the dominos cover the has the property that no more dominos can be added to the board and that the four corner spaces of the board are not all empty. Prove that at least $2/3$ of the squares of the board are covered with dominos.

Find the maximal number of edges a connected graph $G$ with $n$ vertices may have, so that after deleting an arbitrary cycle, $G$ is not connected anymore.

Given a triangle $ABC$ with $AD$ is the $A-$symmedian $(D$ is on the side $BC).$ Prove that: $\dfrac{DB}{DC}=\dfrac{AB^2}{AC^2}.$