IR-TST 2002

$ABCD$ is a convex quadrilateral. We draw its diagnals to divide the quadrilateral to four triabgles. $P$ is the intersection of diagnals. $I_{1},I_{2},I_{3},I_{4}$ are excenters of $PAD,PAB,PBC,PCD$ (excenters corresponding vertex $P$ ). Prove that $I_{1},I_{2},I_{3},I_{4}$ lie on a circle iff $ABCD$ is a tangential quadrilateral.

$n$ people (with names $1,2,\dots,n$ ) are around a table. Some of them are friends. At each step 2 friend can change their place. Find a necessary and sufficient condition for friendship relation between them that with these steps we can always reach to all of posiible permutations.

A "2-line" is the area between two parallel lines. Length of "2-line" is distance of two parallel lines. We have covered unit circle with some "2-lines". Prove sum of lengths of "2-lines" is at least 2.

$O$ is a point in triangle $ABC$ . We draw perpendicular from $O$ to $BC,AC,AB$ which intersect $BC,AC,AB$ at $A_{1},B_{1},C_{1}$ . Prove that $O$ is circumcenter of triangle $ABC$ iff perimeter of $ABC$ is not less than perimeter of triangles $AB_{1}C_{1},BC_{1}A_{1},CB_{1}A_{1}$ .

A school has $n$ students and $k$ classes. Every two students in the same class are friends. For each two different classes, there are two people from these classes that are not friends. Prove that we can divide students into $n-k+1$ parts taht students in each part are not friends.

Assume $x_{1},x_{2},\dots,x_{n}\in\mathbb R^{+}$ , $\sum_{i=1}^{n}x_{i}^{2}=n$ , $\sum_{i=1}^{n}x_{i}\geq s>0$ and $0\leq\lambda\leq1$ . Prove that at least $\left\lceil\frac{s^{2}(1-\lambda)^{2}}n\right\rceil$ of these numbers are larger than $\frac{\lambda s}{n}$ .

$S_{1},S_{2},S_{3}$ are three spheres in $\mathbb R^{3}$ that their centers are not collinear. $k\leq8$ is the number of planes that touch three spheres. $A_{i},B_{i},C_{i}$ is the point that $i$ -th plane touch the spheres $S_{1},S_{2},S_{3}$ . Let $O_{i}$ be circumcenter of $A_{i}B_{i}C_{i}$ . Prove that $O_{i}$ are collinear.

We call $A_{1},A_{2},A_{3}$ mangool iff there is a permutation $\pi$ that $A_{\pi(2)}\not\subset A_{\pi(1)},A_{\pi(3)}\not\subset A_{\pi(1)}\cup A_{\pi(2)}$ . A good family is a family of finite subsets of $\mathbb N$ like $X,A_{1},A_{2},\dots,A_{n}$ . To each goo family we correspond a graph with vertices $\{A_{1},A_{2},\dots,A_{n}\}$ . Connect $A_{i},A_{j}$ iff $X,A_{i},A_{j}$ are mangool sets. Find all graphs that we can find a good family corresponding to it.

$\pi(n)$ is the number of primes that are not bigger than $n$ . For $n=2,3,4,6,8,33,\dots$ we have $\pi(n)|n$ . Does exist infinitely many integers $n$ that $\pi(n)|n$ ?

Suppose from $(m+2)\times(n+2)$ rectangle we cut $4$ , $1\times1$ corners. Now on first and last row first and last columns we write $2(m+n)$ real numbers. Prove we can fill the interior $m\times n$ rectangle with real numbers that every number is average of it's $4$ neighbors.

A $10\times10\times10$ cube has $1000$ unit cubes. $500$ of them are coloured black and $500$ of them are coloured white. Show that there are at least $100$ unit squares, being the common face of a white and a black unit cube.

We call a permutation $\left(a_1, a_2, ..., a_n\right)$ of $\left(1, 2, ..., n\right)$ quadratic if there exists at least a perfect square among the numbers $a_1$ , $a_1 + a_2$ , $...$ , $a_1 + a_2 + ... + a_n$ . Find all natural numbers $n$ such that all permutations in $S_n$ are quadratic.

Remark. $S_{n}$ denotes the $n$ -th symmetric group, the group of permutations on $n$ elements.

Let $ABC$ be a triangle. The incircle of triangle $ABC$ touches the side $BC$ at $A^{\prime}$ , and the line $AA^{\prime}$ meets the incircle again at a point $P$ . Let the lines $CP$ and $BP$ meet the incircle of triangle $ABC$ again at $N$ and $M$ , respectively. Prove that the lines $AA^{\prime}$ , $BN$ and $CM$ are concurrent.