On the plane are $n$ paper disks of radius $1$ whose boundaries all pass through a certain point, which lies inside the region covered by the disks. Find the perimeter of this region. (P Kozhevnikov)

Consider six points in the interior of a square of side length $3$. Prove that among the six points, there are two whose distance is less than $2$.

The sequences $a_n$ and $b_n$ members are the last digits of $[\sqrt{10}^n]$ and $[\sqrt{2}^n]$ respectively (here $[ ...]$ denotes the whole part of a number). Are those sequences periodical?

Let $\mathbb{Z}^+$ be the set of positive integers. Find all functions $f:\mathbb{Z}^+ \rightarrow\mathbb{Z}^+$ such that the following conditions both hold: (i) $f(n!)=f(n)!$ for every positive integer $n$, (ii) $m-n$ divides $f(m)-f(n)$ whenever $m$ and $n$ are different positive integers.

Let $n>1$ be an integer. For each numbers $(x_1, x_2,\dots, x_n)$ with $x_1^2+x_2^2+x_3^2+\dots +x_n^2=1$, denote $m=\min\{|x_i-x_j|, 0<i<j<n+1\}$ Find the maximum value of $m$.

The solutions to the system of equations \begin{eqnarray*} \log_{225}{x}+\log_{64}{y} &=& 4\\ \log_x{225}-\log_y{64} &=& 1 \end{eqnarray*} are $(x_1,y_1)$ and $(x_2, y_2).$ Find $\log_{30}{(x_1y_1x_2y_2)}.$

Let $n$ be a positive integer and let $a_k = \dfrac{1}{\binom{n}{k}}, b_k = 2^{k-n},\ (k=1..n)$. Show that $\sum_{k=1}^n \dfrac{a_k-b_k}{k} = 0$.

Let $ABC$ be an acute-angled triangle with $AB<AC$ and let $O$ be the centre of its circumcircle $\omega$. Let $D$ be a point on the line segment $BC$ such that $\angle BAD = \angle CAO$. Let $E$ be the second point of intersection of $\omega$ and the line $AD$. If $M$, $N$ and $P$ are the midpoints of the line segments $BE$, $OD$ and $AC$, respectively, show that the points $M$, $N$ and $P$ are collinear.

Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts ([u]not ends[/u]) with $2012$. For example, $586$ is a leader since $586^3 =201230056$. How many leaders are there in the set $\{1, 2, 3, ..., 2012\}$?

[list=a] [*]Let $a,b,c,d$ be real numbers with $0\leqslant a,b,c,d\leqslant 1$. Prove that $$ab(a-b)+bc(b-c)+cd(c-d)+da(d-a)\leqslant \frac{8}{27}.$$[/*] [*]Find all quadruples $(a,b,c,d)$ of real numbers with $0\leqslant a,b,c,d\leqslant 1$ for which equality holds in the above inequality. [/list]

Let $f(x)=(x + a)(x + b)$ where $a,b>0$. For any reals $x_1,x_2,\ldots ,x_n\geqslant 0$ satisfying $x_1+x_2+\ldots +x_n =1$, find the maximum of $F=\sum\limits_{1 \leqslant i < j \leqslant n} {\min \left\{ {f({x_i}),f({x_j})} \right\}} $.

Find all functions $f ,g,h : R\to R$ such that $f(x+y^3)+g(x^3+y) = h(xy)$ for all $x,y \in R$

Let $ABC$ be a triangle with $AB < AC$, incentre $I$, and circumcircle $\omega$. Let $D$ be the intersection of the external bisector of angle $\widehat{ BAC}$ with line $BC$. Let $E$ be the midpoint of the arc $BC$ of $\omega$ that does not contain $A$. Let $M$ be the midpoint of $DI$, and $X$ the intersection of $EM$ with $\omega$. Prove that $IX$ and $EM$ are perpendicular.

Let $\ell$ be the perimeter of an acute-angled triangle $ABC$ which is not an equilateral triangle. Let $P$ be a variable points inside the triangle $ABC$, and let $D,E,F$ be the projections of $P$ on the sides $BC,CA,AB$ respectively. Prove that \[ 2(AF+BD+CE ) = \ell \] if and only if $P$ is collinear with the incenter and the circumcenter of the triangle $ABC$.

$P_k(n) $ is the product of all positive divisors of $n$ that are divisible by $k$ (the empty product is equal to $1$). Show that $P_1(n)P_2(n)\cdots P_n(n)$ is a perfect square, for any positive integer $n$.

Let $f : (0,1) \to R$ be a function such that $f(1/n) = (-1)^n$ for all n ∈ N. Prove that there are no increasing functions $g,h : (0,1) \to R$ such that $f = g - h$.

If $78$ is divided into three parts which are proportional to $1, \frac13, \frac16$, the middle part is: $ \textbf{(A)}\ 9\frac13 \qquad\textbf{(B)}\ 13\qquad\textbf{(C)}\ 17\frac13 \qquad\textbf{(D)}\ 18\frac13\qquad\textbf{(E)}\ 26 $

Given line $AB$ and point $M$. Find all lines in space passing through $M$ at distance $d$.

The $16$ squares on a piece of paper are numbered as shown in the diagram. While lying on a table, the paper is folded in half four times in the following sequence: [list=1] [*]fold the top half over the bottom half [*]fold the bottom half over the top half [*]fold the right half over the left half [*]fold the left half over the right half.[/list] Which numbered square is on top after step $4$? [asy] unitsize(18); for(int a=0; a<5; ++a) { draw((a,0)--(a,4)); } for(int b=0; b<5; ++b) { draw((0,b)--(4,b)); } label("$1$",(0.5,3.1),N); label("$2$",(1.5,3.1),N); label("$3$",(2.5,3.1),N); label("$4$",(3.5,3.1),N); label("$5$",(0.5,2.1),N); label("$6$",(1.5,2.1),N); label("$7$",(2.5,2.1),N); label("$8$",(3.5,2.1),N); label("$9$",(0.5,1.1),N); label("$10$",(1.5,1.1),N); label("$11$",(2.5,1.1),N); label("$12$",(3.5,1.1),N); label("$13$",(0.5,0.1),N); label("$14$",(1.5,0.1),N); label("$15$",(2.5,0.1),N); label("$16$",(3.5,0.1),N); [/asy] $\text{(A)}\ 1 \qquad \text{(B)}\ 9 \qquad \text{(C)}\ 10 \qquad \text{(D)}\ 14 \qquad \text{(E)}\ 16$

A convex polyhedron $ Q$ has vertices $ V_1,V_2,\ldots,V_n$, and $ 100$ edges. The polyhedron is cut by planes $ P_1,P_2,\ldots,P_n$ in such a way that plane $ P_k$ cuts only those edges that meet at vertex $ V_k$. In addition, no two planes intersect inside or on $ Q$. The cuts produce $ n$ pyramids and a new polyhedron $ R$. How many edges does $ R$ have? $ \textbf{(A)}\ 200\qquad \textbf{(B)}\ 2n\qquad \textbf{(C)}\ 300\qquad \textbf{(D)}\ 400\qquad \textbf{(E)}\ 4n$

A group of children held a grape-eating contest. When the contest was over, the winner had eaten $ n$ grapes, and the child in $ k$th place had eaten $ n\plus{}2\minus{}2k$ grapes. The total number of grapes eaten in the contest was $ 2009$. Find the smallest possible value of $ n$.

A calculator is broken so that the only keys that still work are the $ \sin$, $ \cos$, and $ \tan$ buttons, and their inverses (the $ \arcsin$, $ \arccos$, and $ \arctan$ buttons). The display initially shows $ 0$. Given any positive rational number $ q$, show that pressing some finite sequence of buttons will yield the number $ q$ on the display. Assume that the calculator does real number calculations with infinite precision. All functions are in terms of radians.

Find the set $S$ of primes such that $p \in S$ if and only if there exists an integer $x$ such that $x^{2010} + x^{2009} + \cdots + 1 \equiv p^{2010} \pmod{p^{2011}}$. [i]Brian Hamrick.[/i]

Prove that for any prime number $p$ there are infinitely many fours $(x, y, z, t)$ pairwise distinct natural numbers such that the number $(x^2+p t^2)(y^2+p t^2)(z^2+p t^2)$ is a perfect square.

Let the circle $ {\omega}_{1}$ be internally tangent to another circle $ {\omega}_{2}$ at $ N$.Take a point $ K$ on $ {\omega}_{1}$ and draw a tangent $ AB$ which intersects $ {\omega}_{2}$ at $ A$ and $ B$. Let $M$ be the midpoint of the arc $ AB$ which is on the opposite side of $ N$. Prove that, the circumradius of the $ \triangle KBM$ doesnt depend on the choice of $ K$.