Let $n > 1$ be an integer. In a circular arrangement of $n$ lamps $L_0, \ldots, L_{n-1},$ each of of which can either ON or OFF, we start with the situation where all lamps are ON, and then carry out a sequence of steps, $Step_0, Step_1, \ldots .$ If $L_{j-1}$ ($j$ is taken mod $n$) is ON then $Step_j$ changes the state of $L_j$ (it goes from ON to OFF or from OFF to ON) but does not change the state of any of the other lamps. If $L_{j-1}$ is OFF then $Step_j$ does not change anything at all. Show that: (i) There is a positive integer $M(n)$ such that after $M(n)$ steps all lamps are ON again, (ii) If $n$ has the form $2^k$ then all the lamps are ON after $n^2-1$ steps, (iii) If $n$ has the form $2^k + 1$ then all lamps are ON after $n^2 - n + 1$ steps.

Let $ABC$ be a triangle and let $P$ be a point in its interior. Denote by $D$, $E$, $F$ the feet of the perpendiculars from $P$ to the lines $BC$, $CA$, $AB$, respectively. Suppose that \[AP^2 + PD^2 = BP^2 + PE^2 = CP^2 + PF^2.\] Denote by $I_A$, $I_B$, $I_C$ the excenters of the triangle $ABC$. Prove that $P$ is the circumcenter of the triangle $I_AI_BI_C$. [i]Proposed by C.R. Pranesachar, India [/i]

Let $m$ be a positive integer. Consider a $4m\times 4m$ array of square unit cells. Two different cells are [i]related[/i] to each other if they are in either the same row or in the same column. No cell is related to itself. Some cells are colored blue, such that every cell is related to at least two blue cells. Determine the minimum number of blue cells.

Alice and Bob play the following game. They write some fractions of the form $1/n$, where $n{}$ is positive integer, onto the blackboard. The first move is made by Alice. Alice writes only one fraction in each her turn and Bob writes one fraction in his first turn, two fractions in his second turn, three fractions in his third turn and so on. Bob wants to make the sum of all the fractions on the board to be an integer number after some turn. Can Alice prevent this? [i]Andrey Arzhantsev[/i]

Determine all integer values of n for which the number $\frac{14n+25}{2n+1}$ 'is a perfect square.

Let $ a$, $ b$, $ c$ denote the sides of a triangle. Show that the quantity \[ \frac{a}{b\plus{}c}\plus{}\frac{b}{c\plus{}a}\plus{}\frac{c}{a\plus{}b}\] must lie between the limits $ 3/2$ and 2. Can equality hold at either limits?

Let $a,b,c > 0$ and $ab+bc+ca = 1$. Prove that: \[ \sqrt {a^3 + a} + \sqrt {b^3 + b} + \sqrt {c^3 + c}\geq2\sqrt {a + b + c}. \]

Find the smallest positive integer $n$ such that $2n+1$ and $3n+1$ are both squares

Let $P$ be an interior point of a triangle $ABC$ and let $BP$ and $CP$ meet $AC$ and $AB$ in $E$ and $F$ respectively. IF $S_{BPF} = 4$,$S_{BPC} = 8$ and $S_{CPE} = 13$, find $S_{AFPE}.$

Given a $ \triangle ABC $ and three points $ D, E, F $ such that $ DB = DC, $ $ EC = EA, $ $ FA = FB, $ $ \measuredangle BDC = \measuredangle CEA = \measuredangle AFB. $ Let $ \Omega_D $ be the circle with center $ D $ passing through $ B, C $ and similarly for $ \Omega_E, \Omega_F. $ Prove that the radical center of $ \Omega_D, \Omega_E, \Omega_F $ lies on the Euler line of $ \triangle DEF. $ [i]Proposed by Telv Cohl[/i]

Let $ABC$ be an acute triangle, $\Gamma$ is its circumcircle and $O$ is its circumcenter. Let $F$ be the point on $AC$ such that the $\angle COF=\angle ACB$, such that $F$ and $B$ lie in opposite sides with respect to $CO$. The line $FO$ cuts $BC$ at $G$. The line parallel to $BC$ through $A$ intersects $\Gamma$ again at $M$. The lines $CO$ and $MG$ meet at $K$. Show that the circumcircles of the triangles $BGK$ and $AOK$ meet on $AB$.

The integers $1$, $2$, $3$, $4$, and $5$ are written on a blackboard. It is allowed to wipe out two integers $a$ and $b$ and replace them with $a + b$ and $ab$. Is it possible, by repeating this procedure, to reach a situation where three of the five integers on the blackboard are $2009$?

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

Eight politicians stranded on a desert island on January 1st, 1991, decided to establish a parliament. They decided on the following rules of attendance: (a) There should always be at least one person present on each day. (b) On no two days should the same subset attend. (c) The members present on day $N$ should include for each $K<N$, $(K\ge 1)$ at least one member who was present on day $K$. For how many days can the parliament sit before one of the rules is broken?

Find the smallest positive integer whose remainder when divided by $5$ is $3$, when divided by $7$ is $5$, and when divided by $9$ is $1$.

Bob chose six of the nine digits from $1$ to $9$ and wrote the list, ordered from smallest to largest, of all three-digit numbers that can be formed using the digits you chose. At Bob's list, the number $317$ appears at position $22$. What number appears at position $60$ in the list from Bob? Find all possibilities.

There is a city with $n$ citizens. The city wants to buy [i]sceptervirus[/i] tests with which it is possible to analyze the samples of several people at the same time. The result of a test can be the following: [list] [*][i]Virus positive[/i]: there is at least one currently infected person among the people whose samples were analyzed, and none of them were healed from an earlier infection. [*][i]Antibody positive[/i]: there is at least one person who was healed from an earlier infection among the people whose samples were analyzed, and none of them are infected now. [*][i]Neutral[/i]: either all of the people whose samples were analyzed are not infected, or there is at least one currently infected person and one person who was healed from an earlier infection. (Viruses and antibodies in samples completely neutralize each other.) [/list] What is the smallest number of tests to buy if we would like to know if the sceptervirus is present now or it has been present earlier? (The samples are taken from the people at the same time. The people are either infected now, have been infected earlier, or haven't contacted the virus yet.) [i]Proposed by Csongor Beke, Cambridge[/i]

A bag contains $5$ identical blue marbles and $5$ identical green marbles. In how many ways can $5$ marbles from the bag be arranged in a row if each blue marble must be adjacent to at least $1$ green marble?

For any set $A = \{x_1, x_2, x_3, x_4, x_5\}$ of five distinct positive integers denote by $S_A$ the sum of its elements, and denote by $T_A$ the number of triples $(i, j, k)$ with $1 \le i < j < k \le 5$ for which $x_i + x_j + x_k$ divides $S_A$. Find the largest possible value of $T_A$.

Prove that $(\sqrt{2}-1)^n$ $\forall n\in \mathbb{Z}^{+}$ can be represented as $\sqrt{m}-\sqrt{m-1}$ for some $m\in \mathbb{Z}^{+}$.

A square is divided into several (greater than one) convex polygons with mutually different numbers of sides. Prove that one of these polygons is a triangle. (A.Zaslavsky)

Given positive integers $ a,b,$ find all positive integers $ x,y$ satisfying the equation: $ x^{a\plus{}b}\plus{}y\equal{}x^a y^b$.

Let $ABCD$ and $A'B'C'D'$ be two convex quadrilaterals whose corresponding sides are equal, i.e., $AB = A'B', BC = B'C'$, etc. Prove that if $\angle A > \angle A'$, then $\angle B < \angle B', \angle C > \angle C', \angle D < \angle D'$.

Thirteen children are sitting at a round table, each holding two cards. Each card has one of the numbers $1, 2, ..., 13$ written on it, and each number is written on exactly two cards. On a signal, each child gives the card with the lower number to his neighbor on the right (and at the same time receives his card with the lower number from the neighbor on the left). Prove that after a finite number of such exchanges, a situation arises when at least one of the children will have two cards with the same number.

A set of $n$ standard dice are shaken and randomly placed in a straight line. If $n < 2r$ and $r < s$, then the probability that there will be a string of at least $r$, but not more than $s$, consecutive $1$'s can be written as $\frac{P}{6^{s+2}}$. Find an explicit expression for $P$.