On the coordinate plane, draw a set of points $M(x;y)$, whose coordinates satisfy the equation $$\sqrt{(x - 1)^2+ y^2} +\sqrt{x^2 + (y -1)^2} = \sqrt2.$$

Let be a group and two coprime natural numbers $ m,n. $ Show that if the applications $ G\ni x\mapsto x^{m+1},x^{n+1} $ are surjective endomorphisms, then the group is commutative.

Let $ABC$ be an acute angled triangle with circumcircle $O$. Line $AO$ intersects the circumcircle of triangle $ABC$ again at point $D$. Let $P$ be a point on the side $BC$. Line passing through $P$ perpendicular to $AP$ intersects lines $DB$ and $DC$ at $E$ and $F$ respectively . Line passing through $D$ perpendicular to $BC$ intersects $EF$ at point $Q$. Prove that $EQ = FQ$ if and only if $BP = CP$.

[b]interesting function[/b] $S$ is a set with $n$ elements and $P(S)$ is the set of all subsets of $S$ and $f : P(S) \rightarrow \mathbb N$ is a function with these properties: for every subset $A$ of $S$ we have $f(A)=f(S-A)$. for every two subsets of $S$ like $A$ and $B$ we have $max(f(A),f(B))\ge f(A\cup B)$ prove that number of natural numbers like $x$ such that there exists $A\subseteq S$ and $f(A)=x$ is less than $n$. time allowed for this question was 1 hours and 30 minutes.

Let $k\ge 1$ be a positive integer. We consider $4k$ chips, $2k$ of which are red and $2k$ of which are blue. A sequence of those $4k$ chips can be transformed into another sequence by a so-called move, consisting of interchanging a number (possibly one) of consecutive red chips with an equal number of consecutive blue chips. For example, we can move from $r\underline{bb}br\underline{rr}b$ to $r\underline{rr}br\underline{bb}b$ where $r$ denotes a red chip and $b$ denotes a blue chip. Determine the smallest number $n$ (as a function of $k$) such that starting from any initial sequence of the $4k$ chips, we need at most $n$ moves to reach the state in which the first $2k$ chips are red.

Find the number of ordered pairs $(a,b)$ of positive integers less than or equal to $20$ such that \[\gcd(a,b)>1 \quad \text{and} \quad \frac{1}{\gcd(a,b)}+\frac{a+b}{\text{lcm}(a,b)} \geq 1.\] [i]Proposed by Zachary Perry[/i]

Restore the isosceles triangle $ABC$ ($AB = AC$) if the common points $I, M, H$ of bisectors, medians and altitudes respectively are given.

a) Suppose that $ RBR'B'$ is a convex quadrilateral such that vertices $ R$ and $ R'$ have red color and vertices $ B$ and $ B'$ have blue color. We put $ k$ arbitrary points of colors blue and red in the quadrilateral such that no four of these $ k\plus{}4$ point (except probably $ RBR'B'$) lie one a circle. Prove that exactly one of the following cases occur? 1. There is a path from $ R$ to $ R'$ such that distance of every point on this path from one of red points is less than its distance from all blue points. 2. There is a path from $ B$ to $ B'$ such that distance of every point on this path from one of blue points is less than its distance from all red points. We call these two paths the blue path and the red path respectively. Let $ n$ be a natural number. Two people play the following game. At each step one player puts a point in quadrilateral satisfying the above conditions. First player only puts red point and second player only puts blue points. Game finishes when every player has put $ n$ points on the plane. First player's goal is to make a red path from $ R$ to $ R'$ and the second player's goal is to make a blue path from $ B$ to $ B'$. b) Prove that if $ RBR'B'$ is rectangle then for each $ n$ the second player wins. c) Try to specify the winner for other quadrilaterals.

Four points on the plane are not concyclic, and any three of them are not collinear. Prove that there exists a point $Z$ such that the reflection of each of these four points about $Z$ lies on the circle passing through three remaining points. Proposed by:A Kuznetsov

Consider the set $M$ of all sequences of integers $s=(s_1,\ldots,s_k)$ such that $0\leqslant s_i\leqslant n,\; i=1,2,\ldots,k$ and let $M(s)=\max\{s_1,\ldots,s_k\}$. Show that \[\sum_{s\in A} M(s)=(n+1)^{k+1}-(1^k+2^k+\ldots +(n+1)^k).\] [i]Ioan Tomescu[/i]

Let $A$ be a $3$-digit positive integer and $B$ be the positive integer that comes from $A$ be replacing with each other the digits of hundreds with the digit of the units. It is also given that $B$ is a $3$-digit number. Find numbers $A$ and $B$ if it is known that $A$ divided by $B$ gives quotient $3$ and remainder equal to seven times the sum of it's digits.

Let point $T$ be on side $AB$ of $\Delta ABC$ be such that $AT-BT=AC-BC$. The perpendicular from point $T$ to $AB$ intersects $AC$ in point $E$ and the angle bisectors of $\angle B$ and $\angle C$ intersect the circumscribed circle $k$ of $ABC$ in points $M$ and $L$. If $P$ is the second intersection point of the line $ME$ with $k$, then prove that $P,T,L$ are collinear.

Let $g:(0,\infty) \rightarrow (0,\infty)$ be a differentiable function whose derivative is continuous, and such that $g(g(x)) = x$ for all $x> 0$. If $g$ is not the identity function, prove that $g$ must be strictly decreasing.

$a,b,c \geq-3/4$ and $a+b+c=1$. Show that: $\frac{a}{1+a^{2}}+\frac{b}{1+b^{2}}+\frac{c}{1+c^{2}}\leq \frac{9}{10}$

Let $P$ be a point in the interior of square $ABCD$ such that $\angle APB+\angle CPD=180^\circ$ and $\angle APB$ $ <$ $\angle CPD$. If $PC=7$ and $PD=5$, find $\tfrac{PA}{PB}$.

Someones age is equal to the sum of the digits of his year of birth. How old is he and when was he born, if it is known that he is older than $11$. P.s. the current year in the problem is $2010$.