Let $A_0B_0C_0$ be a triangle. For a positive integer $n \geq 1$, we define $A_n$ on the segment $B_{n-1}C_{n-1}$ such that $B_{n-1}A_n:C_{n-1}A_n=2:1$ and $B_n, C_n$ are defined cyclically in a similar manner. Show that there exists an unique point $P$ that lies in the interior of all triangles $A_nB_nC_n$.

In a triangle $ABC$, with $AB \ne BC$, $E$ is a point on the line $AC$ such that $BE$ is perpendicular to $AC$. A circle passing through $A$ and touching the line $BE$ at a point $P \ne B$ intersects the line $AB$ for the second time at $X$. Let $Q$ be a point on the line $PB$ different from $P$ such that $BQ = BP$. Let $Y$ be the point of intersection of the lines $CP$ and $AQ$. Prove that the points $C, X, Y, A$ are concyclic if and only if $CX$ is perpendicular to $AB$.

In an obtuse triangle $ABC$ $(AB>AC)$,$O$ is the circumcentre and $D,E,F$ are the midpoints of $BC,CA,AB$ respectively.Median $AD$ intersects $OF$ and $OE$ at $M$ and $N$ respectively.$BM$ meets $CN$ at point $P$.Prove that $OP\perp AP$

Prove that the for all $n>1000$, we can arrange the number $1,2,\dots, \binom{n}{2}$ on edges of a complete graph with $n$ vertices so that the sum of the numbers assigned to edges of any length three path (possibly closed) is not less than $3n-1000log_2log_2 n$.

Prove that for any prime $p$, there exist integers $x,y,z,$ and $w$ such that $x^2+y^2+z^2-wp=0$ and $0<w<p$

A bookshelf contains $n$ volumes, labelled $1$ to $n$, in some order. The librarian wishes to put them in the correct order as follows. The librarian selects a volume that is too far to the right, say the volume with label $k$, takes it out, and inserts it in the $k$-th position. For example, if the bookshelf contains the volumes $1,3,2,4$ in that order, the librarian could take out volume $2$ and place it in the second position. The books will then be in the correct order $1,2,3,4$. (a) Show that if this process is repeated, then, however the librarian makes the selections, all the volumes will eventually be in the correct order. (b) What is the largest number of steps that this process can take?

Each of the six boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$, $B_6$ initially contains one coin. The following operations are allowed Type 1) Choose a non-empty box $B_j$, $1\leq j \leq 5$, remove one coin from $B_j$ and add two coins to $B_{j+1}$; Type 2) Choose a non-empty box $B_k$, $1\leq k \leq 4$, remove one coin from $B_k$ and swap the contents (maybe empty) of the boxes $B_{k+1}$ and $B_{k+2}$. Determine if there exists a finite sequence of operations of the allowed types, such that the five boxes $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ become empty, while box $B_6$ contains exactly $2010^{2010^{2010}}$ coins. [i]Proposed by Hans Zantema, Netherlands[/i]

Let be a real function that admits finite right-limits everywhere. Prove that the function that maps every real number to its right-limit is right-continuous everywhere. [i]Tolosi Marin[/i]

Tracy has been baking a rectangular cake whose surface is dissected by grid lines in square fields. The number of rows is $ 2^n$ and the number of columns is $ 2^{n \plus{} 1}$ where $ n \geq 1, n \in \mathbb{N}.$ Now she covers the fields with strawberries such that each row has at least $ 2n \plus{} 2$ of them. Show that there four pairwise distinct strawberries $ A,B,C$ and $ D$ which satisfy those three conditions: (a) Strawberries $ A$ and $ B$ lie in the same row and $ A$ further left than $ B.$ Similarly $ D$ lies in the same row as $ C$ but further left. (b) Strawberries $ B$ and $ C$ lie in the same column. (c) Strawberries $ A$ lies further up and further left than $ D.$

On each side of an equilateral triangle of side $n \ge 1$ consider $n - 1$ points that divide the sides into $n$ equal segments. Through these points draw parallel lines to the sides of the triangles, obtaining a net of equilateral triangles of side length $1$. On each of the vertices of the small triangles put a coin head up. A move consists in flipping over three mutually adjacent coins. Find all values of $n$ for which it is possible to turn all coins tail up after a finite number of moves. Colombia 1997

Let $c$ be a nonzero real number. Suppose that $g(x)=c_0x^r+c_1x^{r-1}+\cdots+c_{r-1}x+c_r$ is a polynomial with integer coefficients. Suppose that the roots of $g(x)$ are $b_1,\cdots,b_r$. Let $k$ be a given positive integer. Show that there is a prime $p$ such that $p>\max(k,|c|,|c_r|)$, and moreover if $t$ is a real number between $0$ and $1$, and $j$ is one of $1,\cdots,r$, then \[|(\text{ }c^r\text{ }b_j\text{}g(tb_j)\text{ })^pe^{(1-t)b}|<\dfrac{(p-1)!}{2r}.\] Furthermore, if \[f(x)=\dfrac{e^{rp-1}x^{p-1}(g(x))^p}{(p-1)!}\] then \[\left|\sum_{j=1}^r\int_0^1 e^{(1-t)b_j}f(tb_j)dt\right|\leq \dfrac{1}{2}.\]

In a triangle $ABC$, $\angle BAC = 100^{\circ}, AB = AC$. A point $D$ is chosen on the side $AC$ such that $\angle ABD = \angle CBD$. Prove that $AD + DB = BC.$

Find all pairs $(x,y)$ of positive integers such that $x(x+y)=y^2+1.$

There are two equal circles of radius $1$ placed inside the triangle $ABC$ with side $BC = 6$. The circles are tangent to each other, one is inscribed in angle $B$, the other one is inscribed in angle $C$. (a) Prove that the centroid $M$ of the triangle $ABC$ does not lie inside any of the given circles. (b) Prove that if $M$ lies on one of the circles, then the triangle $ABC$ is isosceles.

For a positive integers $n,k$ we define k-multifactorial of n as $Fk(n)$ = $(n)$ . $(n-k)$ $(n-2k)$...$(r)$, where $r$ is the reminder when $n$ is divided by $k$ that satisfy $1<=r<=k$ Determine all non-negative integers $n$ such that $F20(n)+2009$ is a perfect square.

If $ a$, $ b$ and $ c$ are positive reals prove inequality: \[ \left(1\plus{}\frac{4a}{b\plus{}c}\right)\left(1\plus{}\frac{4b}{a\plus{}c}\right)\left(1\plus{}\frac{4c}{a\plus{}b}\right) > 25.\]

In triangle $ABC$, $\angle{BAC}=60^\circ$ and the incircle of $ABC$ touches $AB$ and $AC$ at $P$ and $Q$, respectively. Lines $PC$ and $QB$ intersect at $G$. Let $R$ be the circumradius of $BGC$. Find the minimum value of $R/BC$. [i]Author: Alex Song[/i]

Let $A=(a_{ij})$ and $B=(b_{ij})$ be two real $10\times10$ matrices such that $a_{ij}=b_{ij}+1$ for all $i,j$ and $A^3=0$. Prove that $\det B=0$.

Basil has a melon in a shape of a ball, $20$ in diameter. Using a long knife, Basil makes three mutually perpendicular cuts. Each cut carves a circular segment in a plane of the cut, $h$ deep ($h$ is a height of the segment). Does it necessarily follow that the melon breaks into two or more pieces if (a) $h = 17$ ? [i](6 points)[/i] (b) $h = 18$ ? [i](6 points)[/i]

Let $a$ be some positive number. Find the number of integer solutions $x$ of inequality $100 < xa < 1000$ given that inequality $10 < xa < 100$ has exactly $5$ integer solutions. Consider all possible cases. [i](4 points)[/i]

The number $665$ is represented as a sum of $18$ natural numbers nenule $a_1, a_2, ..., a_{18}$. Determine the smallest possible value of the smallest common multiple of the numbers $a_1, a_2, ..., a_{18}$.

In a square $ABCD$ with side $k$, let $P$ and $Q$ in $BC$ and $DC$ respectively, where $PC = 3PB$ and $QD = 2QC$. Let $M$ be the point of intersection of the lines $AQ$ and $PD$, determine the area of $QMD$ in function of $k$

Let $\phi(n)$ be the number of positive integers $k<n$ which are relatively prime to $n$. For how many distinct values of $n$ is $\phi(n)$ equal to $12$? $\textbf{(A) }0\hspace{14em}\textbf{(B) }1\hspace{14em}\textbf{(C) }2$ $\textbf{(D) }3\hspace{14em}\textbf{(E) }4\hspace{14em}\textbf{(F) }5$ $\textbf{(G) }6\hspace{14em}\textbf{(H) }7\hspace{14em}\textbf{(I) }8$ $\textbf{(J) }9\hspace{14.2em}\textbf{(K) }10\hspace{13.5em}\textbf{(L) }11$ $\textbf{(M) }12\hspace{13.3em}\textbf{(N) }13$

Show that the improper integral \[ \lim_{B \rightarrow \infty} \displaystyle\int_{0}^{B} \sin (x) \sin (x^2) dx \] converges.

In an acute triangle $ABC$ the angle $\angle C$ is greater than $\angle A$. Let $E$ be such that $AE$ is a diameter of the circumscribed circle $\Gamma$ of \vartriangle ABC. Let $K$ be the intersection of $AC$ and the tangent line at $B$ to $\Gamma$. Let $L$ be the orthogonal projection of $K$ on $AE$ and let $D$ be the intersection of $KL$ and $AB$. Prove that $CE$ is the bisector of $\angle BCD$.