Find the area of the set of all points $ z $ in the complex plane that satisfy $ \left| z - 3i \right| + \left| z - 4 \right| \leq 5\sqrt{2} $.

Prove that an integer $n$ can be expressed as the sum of two squares if and only if $2n$ can be expressed as the sum of two squares.

A segment $AB$ is given and it's midpoint $K$. On the perpendicular line to $AB$, passing through $K$ a point $C$, different from $K$ is chosen. Let $N$ be the intersection of $AC$ and the line passing through $B$ and the midpoint of $CK$. Let $U$ be the intersection point of $AB$ and the line passing through $C$ and $L$, the midpoint of $BN$. Prove that the ratio of the areas of the triangles $CNL$ and $BUL$, is independent of the choice of the point $C$.

King Albrecht founded a family. In the family everyone has exactly $ 8$ children. The only, but really important rule is that among the grandchildren of any person at most $x$ can be named Bela. (None of Albrecht’s children is called Bela.) For which $x$ is it possible that after a certain time each newborn in the family has at least one direct ancestor in the Royal family called Bela. No two of Albrecht’s descendants (including himself) have a common child.

There are $n$ students standing in a circle, one behind the other. The students have heights $h_1<h_2<\dots <h_n$. If a student with height $h_k$ is standing directly behind a student with height $h_{k-2}$ or less, the two students are permitted to switch places. Prove that it is not possible to make more than $\binom{n}{3}$ such switches before reaching a position in which no further switches are possible.

Ankit, Box, and Clark are playing a game. First, Clark comes up with a prime number less than 100. Then he writes each digit of the prime number on a piece of paper (writing $0$ for the tens digit if he chose a single-digit prime), and gives one each to Ankit and Box, without telling them which digit is the tens digit, and which digit is the ones digit. The following exchange occurs: 1. Clark: There is only one prime number that can be made using those two digits. 2. Ankit: I don't know whether I'm the tens digit or the ones digit. 3. Box: I don't know whether I'm the tens digit or the ones digit. 4. Box: You don't know whether you're the tens digit or the ones digit. 5. Ankit: I don't know whether you're the tens digit or the ones digit. What was Clark's number?

Smallville is populated by unmarried men and women, some of which are acquainted. The two City Matchmakers know who is acquainted with whom. One day, one of the matchmakers claimed: "I can arrange it so that every red haired man will marry a woman with who he is acquainted." The other matchmaker claimed: "I can arrange it so that every blonde woman will marry a man with whom she is acquainted." An amateur mathematician overheard this conversation and said: "Then it can be arranged so that every red haired man will marry a woman with whom he is acquainted and at the same time very blonde woman will marry a man with who she is acquainted." Is the mathematician right?

Find all pairs $(m,n)$ of positive integers such that $m+n$ and $mn+1$ are both powers of $2$.

Let $A$ be set of 20 consecutive positive integers, Which sum and product of elements in $A$ not divisible by 23. Prove that product of elements in $A$ is not perfect square

Find locus of points $ M $ inside an equilateral triangle $ ABC $ such that $$ \angle MBC+\angle MCA +\angle MAB={\pi}/{2}. $$

An acute triangle $ABC$ in which $AB<AC$ is given. The bisector of $\angle BAC$ crosses $BC$ at $D$. Point $M$ is the midpoint of $BC$. Prove that the line though centers of circles escribed on triangles $ABC$ and $ADM$ is parallel to $AD$.

Let $P(x) = x^3 - px^2 + qx - r$ be a cubic polynomial with integer roots $a, b, c$. [b](a)[/b] Show that the greatest common divisor of $p, q, r$ is equal to $1$ if the greatest common divisor of $a, b, c$ is equal to $1$. [b](b)[/b] What are the roots of polynomial $Q(x) = x^3-98x^2+98sx-98t$ with $s, t$ positive integers.

It is known that $n$ is a positive integer, and the real number $x$ satisfies $$|1-|2-...|(n-1)-|n-x||...||=x.$$ Find the value of $x$.

Find all integers $k$ such that all roots of the following polynomial are also integers: $$f(x)=x^3-(k-3)x^2-11x+(4k-8).$$

Three points $X, Y,Z$ are on a striaght line such that $XY = 10$ and $XZ = 3$. What is the product of all possible values of $YZ$?

In the triangle $ABC$ it is known that$\angle A = 75^o, \angle C = 45^o$. On the ray $BC$ beyond the point $C$ the point $T$ is taken so that $BC = CT$. Let $M$ be the midpoint of the segment $AT$. Find the measure of the $\angle BMC$. (Anton Trygub)