IR-TST 2007

In triangle $ABC$ , $M$ is midpoint of $AC$ , and $D$ is a point on $BC$ such that $DB=DM$ . We know that $2BC^{2}-AC^{2}=AB.AC$ . Prove that $$BD.DC=\frac{AC^{2}.AB}{2(AB+AC)}$$

Let $A$ be the largest subset of $\{1,\dots,n\}$ such that for each $x\in A$ , $x$ divides at most one other element in $A$ . Prove that $$\frac{2n}3\leq |A|\leq \left\lceil \frac{3n}4\right\rceil.$$

Find all solutions of the following functional equation: $$f(x^{2}+y+f(y))=2y+f(x)^{2}.$$

In an isosceles right-angled triangle shaped billiards table , a ball starts moving from one of the vertices adjacent to hypotenuse. When it reaches to one side then it will reflect its path. Prove that if we reach to a vertex then it is not the vertex at initial position By Sam Nariman

Find all monic polynomials $f(x)$ in $\mathbb Z[x]$ such that $f(\mathbb Z)$ is closed under multiplication. By Mohsen Jamali

Let $\omega$ be incircle of $ABC$ . $P$ and $Q$ are on $AB$ and $AC$ , such that $PQ$ is parallel to $BC$ and is tangent to $\omega$ . $AB,AC$ touch $\omega$ at $F,E$ . Prove that if $M$ is midpoint of $PQ$ , and $T$ is intersection point of $EF$ and $BC$ , then $TM$ is tangent to $\omega$ . By Ali Khezeli

Does there exist a a sequence $a_{0},a_{1},a_{2},\dots$ in $\mathbb N$ , such that for each $i\neq j, (a_{i},a_{j})=1$ , and for each $n$ , the polynomial $\sum_{i=0}^{n}a_{i}x^{i}$ is irreducible in $\mathbb Z[x]$ ? By Omid Hatami

Suppose $n$ lines in plane are such that no two are parallel and no three are concurrent. For each two lines their angle is a real number in $[0,\frac{\pi}2]$ . Find the largest value of the sum of the $\binom n2$ angles between line. By Aliakbar Daemi

Find all polynomials of degree 3, such that for each $x,y\geq 0$ : $$p(x+y)\geq p(x)+p(y)$$

Triangle $ABC$ is isosceles ( $AB=AC$ ). From $A$ , we draw a line $\ell$ parallel to $BC$ . $P,Q$ are on perpendicular bisectors of $AB,AC$ such that $PQ\perp BC$ . $M,N$ are points on $\ell$ such that angles $\angle APM$ and $\angle AQN$ are $\frac\pi2$ . Prove that $$\frac{1}{AM}+\frac1{AN}\leq\frac2{AB}$$

Let $P$ be a point in a square whose side are mirror. A ray of light comes from $P$ and with slope $\alpha$ . We know that this ray of light never arrives to a vertex. We make an infinite sequence of $0,1$ . After each contact of light ray with a horizontal side, we put $0$ , and after each contact with a vertical side, we put $1$ . For each $n\geq 1$ , let $B_{n}$ be set of all blocks of length $n$ , in this sequence. a) Prove that $B_{n}$ does not depend on location of $P$ . b) Prove that if $\frac{\alpha}{\pi}$ is irrational, then $|B_{n}|=n+1$ .