Natural $n \geq 2$ is given. Group of people calls $n-compact$, if for any men from group, we can found $n$ people (without he), each two of there are familiar. Find maximum $N$ such that for any $n-compact$ group, consisting $N$ people contains subgroup from $n+1$ people, each of two of there are familiar.

A domino is a rectangle whose length is twice its width. Any square can be divided into seven dominoes, for example as shown in the figure below. [img]https://cdn.artofproblemsolving.com/attachments/7/6/c055d8d2f6b7c24d38ded7305446721e193203.png[/img] a) Show that you can divide a square into $n$ dominoes for all $n \ge 5$. b) Show that you cannot divide a square into three or four dominoes.

Al has three bags, each with three marbles each. Bag $1$ has two blue marbles and one red marble, Bag $2$ has one blue marble and two red marbles, and Bag $3$ has three red marbles. He chooses two distinct bags at random, then one marble at random from each of the chosen bags. What is the probability that he chooses two blue marbles?

Point $P$ lies outside a circle, and two rays are drawn from $P$ that intersect the circle as shown. One ray intersects the circle at points $A$ and $B$ while the other ray intersects the circle at $M$ and $N$. $AN$ and $MB$ intersect at $X$. Given that $\angle AXB$ measures $127^{\circ}$ and the minor arc $AM$ measures $14^{\circ}$, compute the measure of the angle at $P$. [asy] size(200); defaultpen(fontsize(10pt)); pair P=(40,10),C=(-20,10),K=(-20,-10); path CC=circle((0,0),20), PC=P--C, PK=P--K; pair A=intersectionpoints(CC,PC)[0], B=intersectionpoints(CC,PC)[1], M=intersectionpoints(CC,PK)[0], N=intersectionpoints(CC,PK)[1], X=intersectionpoint(A--N,B--M); draw(CC);draw(PC);draw(PK);draw(A--N);draw(B--M); label("$A$",A,plain.NE);label("$B$",B,plain.NW);label("$M$",M,SE); label("$P$",P,E);label("$N$",N,dir(250));label("$X$",X,plain.N);[/asy]

Let $ABCD$ ba a square and let point $E$ be the midpoint of side $AD$. Points $G$ and $F$ are located on the segment $(BE)$ such that the lines $AG$ and $CF$ are perpendicular on the line $BE$. Prove that $DF= CG$.

Two natural numbers $d$ and $d'$, where $d'>d$, are both divisors of $n$. Prove that $d'>d+\frac{d^2}{n}$.

Solve the system of equations: $\left\{ \begin{aligned} x^4+y^2-xy^3-\frac{9}{8}x = 0 \\ y^4+x^2-yx^3-\frac{9}{8}y=0 \end{aligned} \right.$

Tiham is trying to find [b]6[/b] digit positive integers$ PQRSTU$ (where $PQRSTU $are not necessarily distinct). But he only wants the numbers where the sum of the [b]3[/b] digit number$ PQR$, and the [b]3[/b] digit number $STU$ is divisible by [b]37[/b]. How many such numbers Tiham can find?

Let $ABC$ be an acute triangle. Let $\omega$ be a circle whose centre $L$ lies on the side $BC$. Suppose that $\omega$ is tangent to $AB$ at $B'$ and $AC$ at $C'$. Suppose also that the circumcentre $O$ of triangle $ABC$ lies on the shorter arc $B'C'$ of $\omega$. Prove that the circumcircle of $ABC$ and $\omega$ meet at two points. [i]Proposed by Härmel Nestra, Estonia[/i]

Let $f(x) = |x - p| + |x - 15| + |x - p - 15|$, where $0 < p < 15$. Determine the minimum value taken by $f(x)$ for $x$ in the interval $p \le x \le 15$.

In the diagram below, $BP$ bisects $\angle ABC$, $CP$ bisects $\angle BCA$, and $PQ$ is perpendicular to $BC$. If $BQ\cdot QC=2PQ^2$, prove that $AB+AC=3BC$. [img]https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvOC8zL2IwZjNjMDAxNWEwMTc1ZGNjMTkwZmZlZmJlMGRlOGRhYjk4NzczLnBuZw==&rn=VlRSTUMgMjAwNi5wbmc=[/img]

Prove that a tetrahedron with just one edge length greater than $1$ has volume at most $ \frac{1}{8}.$

How many ordered pairs of integers $(a,b)$ satisfy all of the following inequalities? \begin{eqnarray*} a^2 + b^2 &<& 16 \\ a^2 + b^2 &<& 8a \\ a^2 + b^2 &<& 8b \end{eqnarray*}

Let $ ABCD$ be a trapezoid with parallel sides $ AB > CD$. Points $ K$ and $ L$ lie on the line segments $ AB$ and $ CD$, respectively, so that $AK/KB=DL/LC$. Suppose that there are points $ P$ and $ Q$ on the line segment $ KL$ satisfying \[\angle{APB} \equal{} \angle{BCD}\qquad\text{and}\qquad \angle{CQD} \equal{} \angle{ABC}.\] Prove that the points $ P$, $ Q$, $ B$ and $ C$ are concyclic. [i]Proposed by Vyacheslev Yasinskiy, Ukraine[/i]

Let $x$ and $y$ be positive real numbers. 1. Prove: if $x^3 - y^3 \ge 4x$, then $x^2 > 2y$. 2. Prove: if $x^5 - y^3 \ge 2x$, then $x^3 \ge 2y$.

For a given natural number $ n $, find the number of solutions to the equation $ \sqrt{x} + \sqrt{y} = n $ in natural numbers $ x, y $.