Find all functions $ f: N \rightarrow N $ such that: $\bullet$ $ f (m) = 1 \iff m = 1 $; $\bullet$ If $ d = \gcd (m, n) $, then $ f (mn) = \frac {f (m) f (n)} {f (d)} $; and $\bullet$ $ \forall m \in N $, we have $ f ^ {2012} (m) = m $. Clarification: $f^n (a) = f (f^{n-1} (a))$

A board $1$x$k$ is called [i]guayaco[/i] if: -Each unit square is painted with exactly one of $k$ available colors. -If $gcd(i,k)>1$, the $i$th unit square is painted with the same color as $(i-1)$th unit square. -If $gcd(i, k)=1$, the $i$th unit square is painted with the same color as $(k-i)$th unit square. Sebastian chooses a positive integer $a$ and calculates the number of boards $1$x$a$ that are guayacos. After that, David chooses a positive integer $b$ and calculates the number of boards $1$x$b$ that are guayacos. David wins if the number of boards $1$x$a$ that are guayacos is the same as the number of boards $1$x$b$ that are guayacos, otherwise, Sebastian wins. Find all the pairs $(a,b) $ such that, with those numbers, David wins.

Consider a cube $ABCDA'B'C'D'$ (where $ABCD$ is a square and $AA' \parallel BB' \parallel CC' \parallel DD'$) and a point $P$ on the line $AA'$. Construct center $S$ of a sphere which has plane $ABB'$ as a plane of symmetry, $P$ lies on the sphere and $p = AB$, $q = A'D'$ are its tangent lines. Discuss conditions of solvability with respect to different position of the point $P$ (on line $AA'$).

Let $ABCD$ be a cyclic quadrilateral and let the intersection point of lines $AB$ and $CD$ be $E$. Let the points $K$ and $L$ be arbitrary points on sides $CD$ and $AB$ respectively, which satisfy the conditions $$\angle KAD = \angle KBC \quad \text{and} \quad \angle LDA = \angle LCB.$$ Prove that $EK = EL$.

Given is an acute triangle $ABC$. $M$ is an arbitrary point at the side $AB$ and $N$ is midpoint of $AC$. The foots of the perpendiculars from $A$ to $MC$ and $MN$ are points $P$ and $Q$. Prove that center of the circumcircle of triangle $PQN$ lies on the fixed line for all points $M$ from the side $AB$.

There are $15$ lights on the ceiling of a room, numbered from $1$ to $15$. All lights are turned off. In another room, there are $15$ switches: a switch for lights $1$ and $2$, a switch for lights $2$ and $3$, a switch for lights $3$ en $4$, etcetera, including a sqitch for lights $15$ and $1$. When the switch for such a pair of lights is turned, both of the lights change their state (from on to off, or vice versa). The switches are put in a random order and all look identical. Raymond wants to find out which switch belongs which pair of lights. From the room with the switches, he cannot see the lights. He can, however, flip a number of switches, and then go to the other room to see which lights are turned on. He can do this multiple times. What is the minimum number of visits to the other room that he has to take to determine for each switch with certainty which pair of lights it corresponds to?

a) What is the least number of points that can be chosen on a circle of length $1956$, so that for each of these points there is exactly one chosen point at distance $1$, and exactly one chosen point at distance $2$ (distances are measured along the circle)? b) On a circle of length $15$ there are selected $n$ points such that for each of them there is exactly one selected point at distance $1$ from it, and exactly one is selected point at distance $2$ from it. (All distances are measured along the circle.) Prove that $n$ is divisible by $10$.

A space station consists of two living modules attached to a central hub on opposite sides of the hub by long corridors of equal length. Each living module contains $N$ astronauts of equal mass. The mass of the space station is negligible compared to the mass of the astronauts, and the size of the central hub and living modules is negligible compared to the length of the corridors. At the beginning of the day, the space station is rotating so that the astronauts feel as if they are in a gravitational field of strength $g$. Two astronauts, one from each module, climb into the central hub, and the remaining astronauts now feel a gravitational field of strength $g'$ . What is the ratio $g'/g$ in terms of $N$?[asy] import roundedpath; size(300); path a = roundedpath((0,-0.3)--(4,-0.3)--(4,-1)--(5,-1)--(5,0),0.1); draw(scale(+1,-1)*a); draw(scale(+1,+1)*a); draw(scale(-1,-1)*a); draw(scale(-1,+1)*a); filldraw(circle((0,0),1),white,black); filldraw(box((-2,-0.27),(2,0.27)),white,white); draw(arc((0,0),1.5,+35,+150),dashed,Arrow); draw(arc((0,0),1.5,-150,-35),dashed,Arrow);[/asy] $ \textbf{(A)}\ 2N/(N-1) $ $ \textbf{(B)}\ N/(N-1) $ $ \textbf{(C)}\ \sqrt{(N-1)/N} $ $ \textbf{(D)}\ \sqrt{N/(N-1)} $ $ \textbf{(E)}\ \text{none of the above} $

Given a finite set $X$, two positive integers $n,k$, and a map $f:X\to X$. Define $f^{(1)}(x)=f(x),f^{(i+1)}(x)=f^{(i)}(x)$,$i=1,2,3,\ldots$. It is known that for any $x\in X$,$f^{(n)}(x)=x$. Define $m_j$ the number of $x\in X$ satisfying $f^{(j)}(x)=x$. Prove that: (1)$\frac{1}n \sum_{j=1}^n m_j\sin {\frac{2kj\pi}{n}}=0$ (2)$\frac{1}n \sum_{j=1}^n m_j\cos {\frac{2kj\pi}{n}}$ is a non-negative integer.

Let $ABCD$ be a cyclic quadrilateral such that $AD=BD=AC$. A point $P$ moves along the circumcircle $\omega$ of triangle $ABCD$. The lined $AP$ and $DP$ meet the lines $CD$ and $AB$ at points $E$ and $F$ respectively. The lines $BE$ and $CF$ meet point $Q$. Find the locus of $Q$.

In acute triangle $ABC$ angle $B$ is greater than$C$. Let $M$ is midpoint of $BC$. $D$ and $E$ are the feet of the altitude from $C$ and $B$ respectively. $K$ and $L$ are midpoint of $ME$ and $MD$ respectively. If $KL$ intersect the line through $A$ parallel to $BC$ in $T$, prove that $TA=TM$.

Prove that if the diagonals of a certain trapezoid are perpendicular, then the sum of the lengths of the bases of this trapezoid is not greater than the sum of the lengths of the sides of this trapezoid.

You are given a balance and one copy of each of ten weights of $1, 2, 4, 8, 16, 32, 64, 128, 256$ and $512$ grams. An object weighing $M$ grams, where $M$ is a positive integer, is put on one of the pans and may be balanced in different ways by placing various combinations of the given weights on either pan of the balance. (a) Prove that no object may be balanced in more than $89$ ways. (b) Find a value of $M$ such that an object weighing $M$ grams can be balanced in $89$ ways. (A Shapovalov, A Kulakov)

Each number from the set $\{1,2,3,4,5,6,7,8\}$ is either colored red or blue, following these rules: a) The number $4$ is colored red, and there is at least one number colored blue. b) If two numbers $x, y$ have different colors and $x + y \leq 8$, then the number $x + y$ is colored blue. c) If two numbers $x, y$ have different colors and $x \cdot y \leq 8$, then the number $x \cdot y$ is colored red. Find all possible ways the numbers from this set can be colored.

Let $S = \{\frac{\pi^n}{1992^m} | m,n \in \mathbb Z \}.$ Show that every real number $x \geq 0$ is an accumulation point of $S.$

Find all positive integers $m, n$ satisfying $n!+2^{n-1}=2^m$.