IR-TST 2009

Let $ABC$ be a triangle and $A'$ , $B'$ and $C'$ lie on $BC$ , $CA$ and $AB$ respectively such that the incenter of $A'B'C'$ and $ABC$ are coincide and the inradius of $A'B'C'$ is half of inradius of $ABC$ . Prove that $ABC$ is equilateral .

Let $a$ be a fix natural number . Prove that the set of prime divisors of $2^{2^{n}} + a$ for $n = 1,2,\cdots$ is infinite

Suppose that $a$ , $b$ , $c$ be three positive real numbers such that $a+b+c=3$ . Prove that :

$\frac{1}{2+a^{2}+b^{2}}+\frac{1}{2+b^{2}+c^{2}}+\frac{1}{2+c^{2}+a^{2}} \leq \frac{3}{4}$

Find all polynomials $f$ with integer coefficient such that, for every prime $p$ and natural numbers $u$ and $v$ with the condition: $$p \mid uv - 1$$ we always have $p \mid f(u)f(v) - 1$ .

$ABC$ is a triangle and $AA'$ , $BB'$ and $CC'$ are three altitudes of this triangle . Let $P$ be the feet of perpendicular from $C'$ to $A'B'$ , and $Q$ is a point on $A'B'$ such that $QA = QB$ . Prove that : $\angle PBQ = \angle PAQ = \angle PC'C$

We have a closed path on a vertices of a $n$ × $n$ square which pass from each vertice exactly once . prove that we have two adjacent vertices such that if we cut the path from these points then length of each pieces is not less than quarter of total path .

Suppose three direction on the plane . We draw $11$ lines in each direction . Find maximum number of the points on the plane which are on three lines .

Find all polynomials $P(x,y)$ such that for all reals $x$ and $y$ , $$P(x^{2},y^{2}) =P\left(\frac {(x + y)^{2}}{2},\frac {(x - y)^{2}}{2}\right).$$

In triangle $ABC$ , $D$ , $E$ and $F$ are the points of tangency of incircle with the center of $I$ to $BC$ , $CA$ and $AB$ respectively. Let $M$ be the foot of the perpendicular from $D$ to $EF$ . $P$ is on $DM$ such that $DP = MP$ . If $H$ is the orthocenter of $BIC$ , prove that $PH$ bisects $EF$ .

Let $ABC$ be a triangle and $AB\ne AC$ . $D$ is a point on $BC$ such that $BA = BD$ and $B$ is between $C$ and $D$ . Let $I_{c}$ be center of the circle which touches $AB$ and the extensions of $AC$ and $BC$ . $CI_{c}$ intersect the circumcircle of $ABC$ again at $T$ . If $\angle TDI_{c} = \frac {\angle B + \angle C}{4}$ then find $\angle A$

Let $n$ be a positive integer. Prove that $$3^{\dfrac{5^{2^n}-1}{2^{n+2}}} \equiv (-5)^{\dfrac{3^{2^n}-1}{2^{n+2}}} \pmod{2^{n+4}}.$$

$T$ is a subset of ${1,2,...,n}$ which has this property : for all distinct $i,j \in T$ , $2j$ is not divisible by $i$ . Prove that : $|T| \leq \frac {4}{9}n + \log_2 n + 2$