Denote by $ S$ the set of all positive integers. Find all functions $ f: S \rightarrow S$ such that \[ f (f^2(m) \plus{} 2f^2(n)) \equal{} m^2 \plus{} 2 n^2\] for all $ m,n \in S$. [i]Bulgaria[/i]

Two circles are intersecting in points $P$ and $Q$. Construct two points $A$ and $B$ on these circles so that $P\in AB$ and the product $AP.PB$ is maximal.

We call a $6\times 6$ table consisting of zeros and ones [i]right[/i] if the sum of the numbers in each row and each column is equal to $3$. Two right tables are called [i]similar[/i] if one can get from one to the other by successive interchanges of rows and columns. Find the largest possible size of a set of pairwise similar right tables.

To be considered the following complex and distinct $a,b,c,d$. Prove that the following affirmations are equivalent: i)For every $z\in \mathbb{C}$ the inequality takes place :$\left| z-a \right|+\left| z-b \right|\ge \left| z-c \right|+\left| z-d \right|$; ii)There is $t\in \left( 0,1 \right)$ so that $c=ta+\left( 1-t \right)b$ si $d=\left( 1-t \right)a+tb$

Define a function $w:\mathbb{Z}\times\mathbb{Z}\to\mathbb{Z}$ as follows. For $|a|,|b|\le 2,$ let $w(a,b)$ be as in the table shown; otherwise, let $w(a,b)=0.$ \[\begin{array}{|lr|rrrrr|}\hline &&&&b&&\\ &w(a,b)&-2&-1&0&1&2\\ \hline &-2&-1&-2&2&-2&-1\\ &-1&-2&4&-4&4&-2\\ a&0&2&-4&12&-4&2\\ &1&-2&4&-4&4&-2\\ &2&-1&-2&2&-2&-1\\ \hline\end{array}\] For every finite subset $S$ of $\mathbb{Z}\times\mathbb{Z},$ define \[A(S)=\sum_{(\mathbf{s},\mathbf{s'})\in S\times S} w(\mathbf{s}-\mathbf{s'}).\] Prove that if $S$ is any finite nonempty subset of $\mathbb{Z}\times\mathbb{Z},$ then $A(S)>0.$ (For example, if $S=\{(0,1),(0,2),(2,0),(3,1)\},$ then the terms in $A(S)$ are $12,12,12,12,4,4,0,0,0,0,-1,-1,-2,-2,-4,-4.$)

For the real numbers $x$ and $y$, $[\sqrt{x}] = 10$ and $[\sqrt{y}] =14$. How large is $\left[\sqrt{[ \sqrt{x+y} ]}\right]$ ? (Note: the square roots are the positive values ​​and $[x]$ is the largest integer less than or equal to x.)

Four different points $A,B,C$ and $D$ lie in a plane. No three of these points lie on a single straight line. Describe the construction of a square $PQRS$ such that on each of the sides of $PQRS$, or the extensions , lies one of the points $A, B, C$ and $D$.

Let $ABC$ be a triangle where $|AB| = |AC|$. Points $P$ and $Q$ are different from the vertices of the triangle and lie on the sides $AB$ and $AC$, respectively. Prove that the circumcircle of the triangle $APQ$ passes through the circumcenter of $ABC$ if and only if $|AP| = |CQ|$.

In triangle $ABC$, arbitrary points $P,Q$ lie on side $BC$ such that $BP=CQ$ and $P$ lies between $B,Q$.The circumcircle of triangle $APQ$ intersects sides $AB,AC$ at $E,F$ respectively.The point $T$ is the intersection of $EP,FQ$.Two lines passing through the midpoint of $BC$ and parallel to $AB$ and $AC$, intersect $EP$ and $FQ$ at points $X,Y$ respectively. Prove that the circumcircle of triangle $TXY$ and triangle $APQ$ are tangent to each other. [i]Proposed by Iman Maghsoudi[/i]

Let $n \ge 2$ be a natural number and suppose that positive numbers $a_0,a_1,...,a_n$ satisfy the equality $(a_{k-1}+a_{k})(a_{k}+a_{k+1})=a_{k-1}-a_{k+1}$ for each $k =1,2,...,n -1$. Prove that $a_n< \frac{1}{n-1}$

Let $N$ be the smallest positive integer such that $N+2N+3N+\ldots +9N$ is a number all of whose digits are equal. What is the sum of digits of $N$?

A coin is tossed $n$ times, and the outcome is written in the form ($a_1,a_2,...,a_n$), where $a_i = 1$ or $2$ depending on whether the result of the $i$-th toss is the head or the tail, respectively. Set $b_j = a_1 +a_2 +...+a_j$ for $j = 1,2,...,n$, and let $p(n)$ be the probability that the sequence $b_1,b_2,...,b_n$ contains the number $n$. Express $p(n)$ in terms of $p(n-1)$ and $p(n-2)$.

Find the minimum value of $S = |x + 1| + |x + 5|+ |x + 14| + |x + 97| + |x + 1920|$.

Let $a, b, c$ be positive integers such that one of the values $$gcd(a,b) \cdot lcm(b,c), \,\,\,\, gcd(b,c)\cdot lcm(c,a), \,\,\,\, gcd(c,a)-\cdot lcm(a,b)$$ is equal to the product of the remaining two. Prove that one of the numbers $a, b, c$ is a multiple of another of them.

Prove that if for a polynomial $P(x, y)$, we have \[P(x - 1, y - 2x + 1) = P(x, y),\] then there exists a polynomial $\Phi(x)$ with $P(x, y) = \Phi(y - x^2).$

Show that any two points lying inside a regular $ n\minus{}$gon $ E$ can be joined by two circular arcs lying inside $ E$ and meeting at an angle of at least $ \left(1 \minus{} \frac{2}{n} \right) \cdot \pi.$

Find the smallest positive multiple of $77$ whose last four digits (from left to right) are $2020$.

How many ordered pairs of positive integers $(a, b)$ are there such that a right triangle with legs of length $a, b$ has an area of $p$, where $p$ is a prime number less than $100$? [i]Proposed by Joshua Hsieh[/i]

You are walking along a road of constant width with sidewalks on each side. You can only walk on the sidewalks or cross the road perpendicular to the sidewalk. Coming up on a turn, you realize that you are on the “outside” of the turn; i.e., you are taking the longer way around the turn. The turn is a circular arc. Assuming that your destination is on the same side of the road as you are currently, let $\theta$ be the smallest turn angle, in radians, that would justify crossing the road and then crossing back after the turn to take the shorter total path to your destination. What is $\lfloor 100 \cdot \theta \rfloor$ ?

Fathers childhood lasted for one sixth part of his life, and he married one $8$th after that and he immediately left to army. When one $12$th of his life passed, father returned from the army and $5$ years after he got a son. Son who lived for one half of fathers years, died $4$ years before his father. How many years lived his father, and how many years he had when his son was born?

A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle (see picture). Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is $r$. Find $100r^2$. [asy] // dragon96, replacing // [img]http://i.imgur.com/08FbQs.png[/img] size(140); defaultpen(linewidth(.7)); real alpha=10, x=-0.12, y=0.025, r=1/sqrt(3); path hex=rotate(alpha)*polygon(6); pair A = shift(x,y)*(r*dir(330+alpha)), B = shift(x,y)*(r*dir(90+alpha)), C = shift(x,y)*(r*dir(210+alpha)); pair X = (-A.x, -A.y), Y = (-B.x, -B.y), Z = (-C.x, -C.y); int i; pair[] H; for(i=0; i<6; i=i+1) { H[i] = dir(alpha+60*i);} fill(X--Y--Z--cycle, rgb(204,255,255)); fill(H[5]--Y--Z--H[0]--cycle^^H[2]--H[3]--X--cycle, rgb(203,153,255)); fill(H[1]--Z--X--H[2]--cycle^^H[4]--H[5]--Y--cycle, rgb(255,203,153)); fill(H[3]--X--Y--H[4]--cycle^^H[0]--H[1]--Z--cycle, rgb(153,203,255)); draw(hex^^X--Y--Z--cycle); draw(H[1]--B--H[2]^^H[3]--C--H[4]^^H[5]--A--H[0]^^A--B--C--cycle, linewidth(0.6)+linetype("5 5")); draw(H[0]--Z--H[1]^^H[2]--X--H[3]^^H[4]--Y--H[5]);[/asy]

A train passes an observer in $t_1$ sec. At the same speed the train crosses a bridge $\ell$ m long. It takes the train $t_2$ sec to cross the bridge from the moment the locomotive drives onto the bridge until the last car leaves it. Find the length and speed of the train.

If $n\in\mathbb N$ and $a_1,a_2,\ldots,a_n$ are arbitrary numbers in the interval $[0,1]$, find the maximum possible value of the smallest among the numbers $a_1-a_1a_2,a_2-a_2a_3,\ldots,a_n-a_na_1$.

Let $H$ and $O$ be the orthocenter and the circumcenter of triangle $ABC$. The circumcircle of triangle $AOH$ meets the perpendicular bisector of $BC$ at point $A_1 \neq O$. Points $B_1$ and $C_1$ are defined similarly. Prove that lines $AA_1$, $BB_1$ and $CC_1$ concur.

Let $ABC$ be a triangle with incircle $(I)$, tangent to $BC$, $CA$, $AB$ at $D, E, F$ respectively. On the line $DF$, take points $M, P$ such that $CM \parallel AB$, $AP \parallel BC$. On the line $DE$, take points $N$, $Q$ such that $BN \parallel AC$, $AQ \parallel BC$. Denote $X$ as intersection of $PE$, $QF$ and $K$ as the midpoint of $BC$. Prove that if $AX = IK$ then $\angle BAC \le 60^o$.