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 $\angle BCD$.

If $S=1!+2!+3!+ \cdots +99!$, then the units' digit in the value of $S$ is: $\textbf{(A)}\ 9 \qquad\textbf{(B)}\ 8 \qquad\textbf{(C)}\ 5 \qquad\textbf{(D)}\ 3 \qquad\textbf{(E)}\ 0$

Given a positive integer $k$, find the least integer $n_k$ for which there exist five sets $S_1, S_2, S_3, S_4, S_5$ with the following properties: \[|S_j|=k \text{ for } j=1, \cdots , 5 , \quad |\bigcup_{j=1}^{5} S_j | = n_k ;\] \[|S_i \cap S_{i+1}| = 0 = |S_5 \cap S_1|, \quad \text{for } i=1,\cdots ,4 \]

Compute the largest positive integer such that $\dfrac{2007!}{2007^n}$ is an integer.

The tangent lines from the point $A$ to the circle $C$ touches the circle at $M$ and $N$. Let $P$ a point on $[AN]$. Let $MP$ meet $C$ at $Q$. Let $MN$ meet the line through $P$ and parallel to $MA$ at $R$. If $|MA|=2$, $|MN|=\sqrt 3$, and $QR \parallel AN$, what is $|PN|$? $ \textbf{(A)}\ \dfrac 32 \qquad\textbf{(B)}\ 1 \qquad\textbf{(C)}\ \dfrac {\sqrt 3} 2 \qquad\textbf{(D)}\ \sqrt 2 \qquad\textbf{(E)}\ \sqrt 3 $

Let $n > 3$ be a positive integer. Suppose that $n$ children are arranged in a circle, and $n$ coins are distributed between them (some children may have no coins). At every step, a child with at least 2 coins may give 1 coin to each of their immediate neighbors on the right and left. Determine all initial distributions of the coins from which it is possible that, after a finite number of steps, each child has exactly one coin.

There are one-way flights between $100$ cities of a country. It is possible to fly starting from the capital city and visiting all other $99$ cities and returning again to the capital city. Let $ N$ be the smallest number of flights inorder to form such a flight combination. Among all flight combinations (satisfying previous condtions), $ N$ can be at most ? $\textbf{(A)}\ 1850 \qquad\textbf{(B)}\ 2100 \qquad\textbf{(C)}\ 2550 \qquad\textbf{(D)}\ 3060 \qquad\textbf{(E)}\ \text{None}$