Determine all positive integers $n$ for which there exists a polynomial $P_n(x)$ of degree $n$ with integer coefficients that is equal to $n$ at $n$ different integer points and that equals zero at zero.

Find the largest integer $n$, where $2009^n$ divides $2008^{2009^{2010}} + 2010^{2009^{2008}}$ .

Right $ \triangle ABC$ has $ AB \equal{} 3$, $ BC \equal{} 4$, and $ AC \equal{} 5$. Square $ XYZW$ is inscribed in $ \triangle ABC$ with $ X$ and $ Y$ on $ \overline{AC}$, $ W$ on $ \overline{AB}$, and $ Z$ on $ \overline{BC}$. What is the side length of the square? [asy]size(200);defaultpen(fontsize(10pt)+linewidth(.8pt)); real s = (60/37); pair A = (0,0); pair C = (5,0); pair B = dir(60)*3; pair W = waypoint(B--A,(1/3)); pair X = foot(W,A,C); pair Y = (X.x + s, X.y); pair Z = (W.x + s, W.y); label("$A$",A,SW); label("$B$",B,NW); label("$C$",C,SE); label("$W$",W,NW); label("$X$",X,S); label("$Y$",Y,S); label("$Z$",Z,NE); draw(A--B--C--cycle); draw(X--W--Z--Y);[/asy] $ \textbf{(A)}\ \frac {3}{2}\qquad \textbf{(B)}\ \frac {60}{37}\qquad \textbf{(C)}\ \frac {12}{7}\qquad \textbf{(D)}\ \frac {23}{13}\qquad \textbf{(E)}\ 2$

Fine all positive integers $m,n\geq 2$, such that (1) $m+1$ is a prime number of type $4k-1$; (2) there is a (positive) prime number $p$ and nonnegative integer $a$, such that \[\frac{m^{2^n-1}-1}{m-1}=m^n+p^a.\]

We define the function $Z(A)$ where we write the digits of $A$ in base $10$ form in reverse. (For example: $Z(521)=125$). Call a number $A$ $good$ if the first and last digits of $A$ are different, none of it's digits are $0$ and the equality: $$Z(A^2)=(Z(A))^2$$ happens. Find all such good numbers greater than $10^6$.\\

The entries in a $ 3\times3$ array include all the digits from 1 through 9, arranged so that the entries in every row and column are in increasing order. How many such arrays are there? $ \textbf{(A)}\ 18\qquad\textbf{(B)}\ 24\qquad\textbf{(C)}\ 36\qquad\textbf{(D)}\ 42\qquad\textbf{(E)}\ 60$

The points $D,E,F$ belong to the sides $(AB), (BC)$ and $(CA)$ of the triangle $ABC$ respectively (but they are not vertices). Let us denote with $d_0, d_1, d_2$, and $d_3$ the maximal side length of the triangles $DEF$, $DEA$, $DBF$, $CEF$, respectively. Prove that $$d_0 \ge \frac{\sqrt3}{2} min\{d_1, d_2, d_3\}$$ When the equality takes place?

Given two sequences $x_n$ and $y_n$ defined by $x_0 = y_0 = 7$, \[x_n = 4x_{n-1}+3y_{n-1}, \text{ and}\]\[y_n = 3y_{n-1}+2x_{n-1},\] find $\lim_{n \to \infty} \frac{x_n}{y_n}$.

In a triangle $ABC$, the angle bisector of $A$ and the cevians $BD$ and $CE$ concur at a point $P$ inside the triangle. Show that the quadrilateral $ADPE$ has an incircle if and only if $AB=AC$.

What is the smallest positive integer $n$ such that $2016n$ is a perfect cube?

Let $\alpha, \beta, \gamma$ be the angles of some triangle. Prove that there is a triangle whose sides are equal to $\sin \alpha$, $\sin \beta$, $\sin \gamma$.

Let $a_1,a_2,a_3,\dots$ be a sequence of numbers such that $a_{n+2} = 2a_n$ for all integers $n.$ Suppose $a_1 = 1,$ $a_2 = 3,$ \[ \sum_{n=1}^{2021} a_{2n} = c, \quad \text{and} \quad \sum\limits_{n=1}^{2021} a_{2n-1} = b. \] Then $c - b + \tfrac{c-b}{b}$ can be written in the form $x^y,$ where $x$ and $y$ are integers such that $x$ is as small as possible. Find $x + y.$

a) Prove that if $x_1,x_2,\ldots,x_n,y_1,y_2,\ldots,y_n$ are positive real numbers satisfying the conditions [list=i] [*] $x_1y_1<x_2y_2<\ldots<x_ny_n$; [*] $x_1+x_2+\ldots+x_k \ge y_1+y_2+ \ldots +y_k,$ for $k=\overline{1,n},$[/list] then $$\frac{1}{x_1}+\frac{1}{x_2}+\ldots+\frac{1}{x_n} \le \frac{1}{y_1}+\frac{1}{y_2}+\ldots+\frac{1}{y_n}.$$ b) Let $A=\{a_1,a_2,\ldots,a_n\}$ be a set of positive integers with the property that for any distinct subsets $B$ and $C$ of $A$ we have $\sum_{x \in B} x \neq \sum_{x \in C} x.$ Prove that $$\frac{1}{a_1}+\frac{1}{a_2}+\ldots+\frac{1}{a_n}<2.$$

The product of positive numbers $x, y$ and $z$ is equal to $1$. Prove that if it holds that $$\frac1x +\frac1y + \frac1z \ge x + y + z,$$ then for any natural $k$, holds the inequality $$\frac{1}{x^k} +\frac{1}{y^k} + \frac{1}{z^k} \ge x^k + y^k + z^k.$$

Answer both (i) and (ii). Test for convergence the series (i) \[ \frac 1{\log (2!)} + \frac 1{\log (3!)} + \frac 1{\log (4!)} + \cdots + \frac 1{\log (n!)} +\cdots\] (ii) \[ \frac 13 + \frac 1{3\sqrt3} + \frac 1{3\sqrt3 \sqrt[3]3} + \cdots + \frac 1{3\sqrt3 \sqrt[3]3 \cdots \sqrt[n]3} + \cdots\]

Let $\mathbb{Z}$ be the set of integers. Determine all functions $f: \mathbb{Z} \rightarrow \mathbb{Z}$ such that, for all integers $a$ and $b$, $$f(2a)+2f(b)=f(f(a+b)).$$ [i]Proposed by Liam Baker, South Africa[/i]

A block of wood has the shape of a right circular cylinder with radius $6$ and height $8$, and its entire surface has been painted blue. Points $A$ and $B$ are chosen on the edge on one of the circular faces of the cylinder so that $\overarc{AB}$ on that face measures $120^\circ$. The block is then sliced in half along the plane that passes through point $A$, point $B$, and the center of the cylinder, revealing a flat, unpainted face on each half. The area of one of those unpainted faces is $a\cdot\pi + b\sqrt{c}$, where $a$, $b$, and $c$ are integers and $c$ is not divisible by the square of any prime. Find $a+b+c$. [asy]import three; import solids; size(8cm); currentprojection=orthographic(-1,-5,3); picture lpic, rpic; size(lpic,5cm); draw(lpic,surface(revolution((0,0,0),(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8),Z,0,120)),gray(0.7),nolight); draw(lpic,surface(revolution((0,0,0),(-3*sqrt(3),-3,8)..(-6,0,4)..(-3*sqrt(3),3,0),Z,0,90)),gray(0.7),nolight); draw(lpic,surface((3,3*sqrt(3),8)..(-6,0,8)..(3,-3*sqrt(3),8)--cycle),gray(0.7),nolight); draw(lpic,(3,-3*sqrt(3),8)..(-6,0,8)..(3,3*sqrt(3),8)); draw(lpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0),dashed); draw(lpic,(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0)--(-3,3*sqrt(3),0)..(-3*sqrt(3),3,0)..(-6,0,0),dashed); draw(lpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(-3*sqrt(3),-3,0)..(-6,0,0)); draw(lpic,(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),0)--(6*cos(atan(-1/5)+3.14159),6*sin(atan(-1/5)+3.14159),8)); size(rpic,5cm); draw(rpic,surface(revolution((0,0,0),(3,3*sqrt(3),8)..(0,6,4)..(-3,3*sqrt(3),0),Z,230,360)),gray(0.7),nolight); draw(rpic,surface((-3,3*sqrt(3),0)..(6,0,0)..(-3,-3*sqrt(3),0)--cycle),gray(0.7),nolight); draw(rpic,surface((-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(0,-6,4)..(-3,-3*sqrt(3),0)--cycle),white,nolight); draw(rpic,(-3,-3*sqrt(3),0)..(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0)); draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)..(6,0,0)..(-3,3*sqrt(3),0),dashed); draw(rpic,(3,3*sqrt(3),8)--(3,-3*sqrt(3),8)); draw(rpic,(-3,3*sqrt(3),0)..(0,6,4)..(3,3*sqrt(3),8)--(3,3*sqrt(3),8)..(3*sqrt(3),3,8)..(6,0,8)); draw(rpic,(-3,3*sqrt(3),0)--(-3,-3*sqrt(3),0)..(0,-6,4)..(3,-3*sqrt(3),8)--(3,-3*sqrt(3),8)..(3*sqrt(3),-3,8)..(6,0,8)); draw(rpic,(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),0)--(-6*cos(atan(-1/5)+3.14159),-6*sin(atan(-1/5)+3.14159),8)); label(rpic,"$A$",(-3,3*sqrt(3),0),W); label(rpic,"$B$",(-3,-3*sqrt(3),0),W); add(lpic.fit(),(0,0)); add(rpic.fit(),(1,0));[/asy]

Given an integer $a_0$, we define a sequence of real numbers $a_0, a_1, . . .$ using the relation $$a^2_i = 1 + ia^2_{i-1},$$ for $i \ge 1$. An index $j$ is called [i]good [/i] if $a_j$ can be an integer for some $a_0$. Determine the sum of the indices $j$ which lie in the interval $[0, 99]$ and which are not good.

Let $m,n,p$ be three positive integers, and let $m'=\gcd(m,np)$, $n'=\gcd(n,pm)$ and $p'=\gcd(p,mn)$. Prove that the equation $x^m+y^n=z^p$ has solutions in the set of positive integers if and only if the equation $x^{m'}+y^{n'}=z^{p'}$ has solutions in the set of positive integers. [i]Luminița Popescu[/i]

The mean of three numbers is 10 more than the least of the numbers and 15 less than the greatest. The median of the three numbers is 5. What is their sum? $ \textbf{(A)} \ 5 \qquad \textbf{(B)} \ 20 \qquad \textbf{(C)} \ 25 \qquad \textbf{(D)} \ 30 \qquad \textbf{(E)} \ 36$

The circle inscribed in the triangle $ABC$ touches the sides $AB$ and $AC$ at the points $K$ and $L$ , respectively. The angle bisectors from $B$ and $C$ intersect the altitude of the triangle from the vertex $A$ at the points $Q$ and $R$ , respectively. Prove that one of the points of intersection of the circles circumscribed around the triangles $BKQ$ and $CPL$ lies on $BC$.

Let points $M$ and $N$ lie on sides $AB$ and $BC$ of triangle $ABC$ in such a way that $MN||AC$. Points $M'$ and $N'$ are the reflections of $M$ and $N$ about $BC$ and $AB$ respectively. Let $M'A$ meet $BC$ at $X$, and let $N'C$ meet $AB$ at $Y$. Prove that $A,C,X,Y$ are concyclic.

Prove that there exists an innite sequence $<a_n>$ of positive integers such that for each $k \ge 1$ $(a_1 - 1)(a_2 - 1)(a_3 -1)...(a_k - 1)$ divides $a_1a_2a_3 ...a_k + 1$.

Three players $A, B$ and $C$ take turns taking stones from a pile of $N$ stones. They play in the order $A$, $B$, $C$, $A$, $B$, $C$, $....$, $A$ starts the game and the one who takes the last stone loses. Players $A$ and $C$ They form a team against $B$, they agree on a strategy joint. $B$ can take $1, 2, 3, 4$ or $5$ stones on each move, while that $A$ and $C$ can each draw $1, 2$ or $3$ stones in each turn. Determine for which values of $N$ have winning strategies $A$ and $C$ , and for what values the winning strategy is $B$'s.

A freight train departed from Moscow at $x$ hours and $y$ minutes and arrived at Saratov at $y$ hours and $z$ minutes. The length of its trip was $z$ hours and $x$ minutes. Find all possible values of $x$. [i]S. Tokarev[/i]