Fix an integer $n \ge 2$. A fairy chess piece [i]leopard [/i] may move one cell up, or one cell to the right, or one cell diagonally down-left. A leopard is placed onto some cell of a $3n \times 3n$ chequer board. The leopard makes several moves, never visiting a cell twice, and comes back to the starting cell. Determine the largest possible number of moves the leopard could have made. Dmitry Khramtsov, Russia

Find all ordered pairs of positive integers $(m,n)$ such that : $125*2^n-3^m=271$

Let $n$ be a given positive integer. Say that a set $K$ of points with integer coordinates in the plane is connected if for every pair of points $R, S\in K$, there exists a positive integer $\ell$ and a sequence $R=T_0,T_1, T_2,\ldots ,T_{\ell}=S$ of points in $K$, where each $T_i$ is distance $1$ away from $T_{i+1}$. For such a set $K$, we define the set of vectors \[\Delta(K)=\{\overrightarrow{RS}\mid R, S\in K\}\] What is the maximum value of $|\Delta(K)|$ over all connected sets $K$ of $2n+1$ points with integer coordinates in the plane? [i]Grigory Chelnokov, Russia[/i]

In the following diagram (not to scale), $A$, $B$, $C$, $D$ are four consecutive vertices of an 18-sided regular polygon with center $O$. Let $P$ be the midpoint of $AC$ and $Q$ be the midpoint of $DO$. Find $\angle OPQ$ in degrees. [asy] pathpen = rgb(0,0,0.6)+linewidth(0.7); pointpen = black+linewidth(3); pointfontpen = fontsize(10); pen dd = rgb(0,0,0.6)+ linewidth(0.7) + linetype("4 4"); real n = 10, start = 360/n*6-15; pair O=(0,0), A=dir(start), B=dir(start+360/n), C=dir(start+2*360/n), D=dir(start+3*360/n), P=(A+C)/2, Q=(O+D)/2; D(D("O",O,NE)--D("A",A,W)--D("B",B,SW)--D("C",C,S)--D("D",D,SE)--O--D("P",P,1.6*dir(95))--D("Q",Q,NE)); D(A--C); D(A--(A+dir(start-360/n))/2, dd); D(D--(D+dir(start+4*360/n))/2, dd); [/asy]

Determine all integers $ n\geq 2$ having the following property: for any integers $a_1,a_2,\ldots, a_n$ whose sum is not divisible by $n$, there exists an index $1 \leq i \leq n$ such that none of the numbers $$a_i,a_i+a_{i+1},\ldots,a_i+a_{i+1}+\ldots+a_{i+n-1}$$ is divisible by $n$. Here, we let $a_i=a_{i-n}$ when $i >n$. [i]Proposed by Warut Suksompong, Thailand[/i]

Let $a, b, c$ be side lengths of a right triangle and $c$ be the length of the hypotenuse .Find the minimum value of $\frac{a^3+b^3+c^3}{abc}$.

[b]a)[/b] Solve the system $$ \left\{\begin{matrix} 3^y-4^x=11\\ \log_4{x} +\log_3 y =3/2\end{matrix}\right. $$ [b]b)[/b] Solve the equation $ \quad 9^{\log_5 (x-2)} -5^{\log_9 (x+2)} = 4. $

All the numbers $2, 3, 4, 5, 6, 7$ are assigned to the six faces of a cube, one number to each face. For each of the eight vertices of the cube, a product of three numbers is computed, where the three numbers are the numbers assigned to the three faces that include that vertex. What is the greatest possible value of the sum of these eight products? $\textbf{(A)}\ 312 \qquad \textbf{(B)}\ 343 \qquad \textbf{(C)}\ 625 \qquad \textbf{(D)}\ 729 \qquad \textbf{(E)}\ 1680$

Point $F$ is taken on the extension of side $AD$ of parallelogram $ABCD$. $BF$ intersects diagonal $AC$ at $E$ and side $DC$ at $G$. If $EF = 32$ and $GF = 24$, then $BE$ equals: [asy] size(7cm); pair A = (0, 0), B = (7, 0), C = (10, 5), D = (3, 5), F = (5.7, 9.5); pair G = intersectionpoints(B--F, D--C)[0]; pair E = intersectionpoints(A--C, B--F)[0]; draw(A--D--C--B--cycle); draw(A--C); draw(D--F--B); label("$A$", A, SW); label("$B$", B, SE); label("$C$", C, NE); label("$D$", D, NW); label("$F$", F, N); label("$G$", G, NE); label("$E$", E, SE); //Credit to MSTang for the asymptote [/asy] $\textbf{(A)}\ 4 \qquad \textbf{(B)}\ 8\qquad \textbf{(C)}\ 10 \qquad \textbf{(D)}\ 12 \qquad \textbf{(E)}\ 16$

Determines the largest integer $n$ that is a multiple of all positive integers less than $\sqrt{n}$.

$x$, $y$, and $z$ are positive real numbers satisfying the equation $x+y+z=\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$. Prove the following inequality: $xy + yz + zx \geq 3$.

For natural $n$, define $f_n=[2^n\sqrt{69}]+[2^n\sqrt{96}]$ Prove that there are infinite even integers and infinite odd integers that appear in number $f_1,f_2,\dots$. [i](tatari/nightmare)[/i]

What is the remainder when $(1 + x)^{2010}$ is divided by $1 + x + x^2$?

There are $6$ points on the plane such that no three of them are collinear. It's known that between every $4$ points of them, there exists a point that it's power with respect to the circle passing through the other three points is a constant value $k$.(Power of a point in the interior of a circle has a negative value.) Prove that $k=0$ and all $6$ points lie on a circle. [i]Proposed by Morteza Saghafian[/I]

Given the circumference $s$ and the straight line $l$, passing through the centre $O$ of $s$. Another circumference $s'$ passes through the point $O$ and has its centre on the $l$. Describe the set of the points $M$, where the common tangent of $s$ and $s'$ touches $s'$.

A [i]calendar[/i] is a (finite) rectangular grid. A calendar is [i]valid[/i] if it satisfies the following conditions: (i) Each square of the calendar is colored white or red, and there are exactly 10 red squares. (ii) Suppose that there are $N$ columns of squares in the calendar. Then if we fill in the numbers $1,2,\ldots$ from the top row to the bottom row, and within each row from left to right, there do not exist $N$ consecutive numbers such that the squares they are in are all white. (iii) Suppose that there are $M$ rows of squares in the calendar. Then if we fill in the numbers $1,2,\ldots$ from the left-most column to the right-most column, and within each column from bottom to top, there do not exist $M$ consecutive numbers such that the squares they are in are all white. In other words, if we rotate the calendar clockwise by $90^{\circ}$, the resulting calendar still satisfies (ii). How many different kinds of valid calendars are there? (Remark: During the actual exam, the contestants were confused about what counts as different calendars. So although this was not in the actual exam, I would like to specify that two calendars are considered different if they have different side lengths or if the $10$ red squares are at different locations.)

Show that, among all triangles whose vertices are at distances 3,5,7 respectively from a given point P, the ones with largest area have P as orthocenter. ([i]You can suppose, without demonstration, the existence of a triangle with maximal area in this question.[/i])

Evaluate the following sum: $$ \frac{1}{1} + \frac{1}{1 + 2} + \frac{1}{1 + 2 + 3} + \frac{1}{1 + 2 + 3 + 4} + \ldots + \frac{1}{1 + 2 + 3 + 4 + \dots + 2025} $$ [i]Proposed by Prudencio Guerrero Fernández[/i]

If the margin made on an article costing $ C$ dollars and selling for $ S$ dollars is $ M\equal{}\frac{1}{n}C$, then the margin is given by: $ \textbf{(A)}\ M\equal{}\frac{1}{n\minus{}1}S \qquad \textbf{(B)}\ M\equal{}\frac{1}{n}S \qquad \textbf{(C)}\ M\equal{}\frac{n}{n\plus{}1}S \\ \textbf{(D)}\ M\equal{}\frac{1}{n\plus{}1}S \qquad \textbf{(E)}\ M\equal{}\frac{n}{n\minus{}1}S$

The Bank of Oslo issues two types of coin: aluminum (denoted A) and bronze (denoted B). Marianne has $n$ aluminum coins and $n$ bronze coins arranged in a row in some arbitrary initial order. A chain is any subsequence of consecutive coins of the same type. Given a fixed positive integer $k \leq 2n$, Gilberty repeatedly performs the following operation: he identifies the longest chain containing the $k^{th}$ coin from the left and moves all coins in that chain to the left end of the row. For example, if $n=4$ and $k=4$, the process starting from the ordering $AABBBABA$ would be $AABBBABA \to BBBAAABA \to AAABBBBA \to BBBBAAAA \to ...$ Find all pairs $(n,k)$ with $1 \leq k \leq 2n$ such that for every initial ordering, at some moment during the process, the leftmost $n$ coins will all be of the same type.

In triangle $ABC$, points $M$ and $N$ are on segments $AB$ and $AC$ respectively such that $AM = MC$ and $AN = NB$. Let $P$ be the point such that $PB$ and $PC$ are tangent to the circumcircle of $ABC$. Given that the perimeters of $PMN$ and $BCNM$ are $21$ and $29$ respectively, and that $PB = 5$, compute the length of $BC$.

Let $n\ge 3$ be an integer. In a country there are $n$ airports and $n$ airlines operating two-way flights. For each airline, there is an odd integer $m\ge 3$, and $m$ distinct airports $c_1, \dots, c_m$, where the flights offered by the airline are exactly those between the following pairs of airports: $c_1$ and $c_2$; $c_2$ and $c_3$; $\dots$ ; $c_{m-1}$ and $c_m$; $c_m$ and $c_1$. Prove that there is a closed route consisting of an odd number of flights where no two flights are operated by the same airline.

Find all real numbers $x$ such that exactly one of the four numbers $x-\sqrt 2$, $x-\dfrac{1}{x}$, $x+\dfrac{1}{x}$ and $x^2+2\sqrt{2}$ is [b]not[/b] an integer.

Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?

Let $ABCD$ be a trapezoid and a tangential quadrilateral such that $AD || BC$ and $|AB|=|CD|$. The incircle touches $[CD]$ at $N$. $[AN]$ and $[BN]$ meet the incircle again at $K$ and $L$, respectively. What is $\dfrac {|AN|}{|AK|} + \dfrac {|BN|}{|BL|}$? $ \textbf{(A)}\ 8 \qquad\textbf{(B)}\ 9 \qquad\textbf{(C)}\ 10 \qquad\textbf{(D)}\ 12 \qquad\textbf{(E)}\ 16 $