IR-TST 2015

Find all polynomials $P,Q\in \Bbb{Q}\left [ x \right ]$ such that $$P(x)^3+Q(x)^3=x^{12}+1.$$

$I_b$ is the $B$ -excenter of the triangle $ABC$ and $\omega$ is the circumcircle of this triangle. $M$ is the middle of arc $BC$ of $\omega$ which doesn't contain $A$ . $MI_b$ meets $\omega$ at $T\not =M$ . Prove that $$TB\cdot TC=TI_b^2.$$

Let $b_1<b_2<b_3<\dots$ be the sequence of all natural numbers which are sum of squares of two natural numbers. Prove that there exists infinite natural numbers like $m$ which $b_{m+1}-b_m=2015$ .

$n$ is a fixed natural number. Find the least $k$ such that for every set $A$ of $k$ natural numbers, there exists a subset of $A$ with an even number of elements which the sum of it's members is divisible by $n$ .

Let $A$ be a subset of the edges of an $n\times n$ table. Let $V(A)$ be the set of vertices from the table which are connected to at least on edge from $A$ and $j(A)$ be the number of the connected components of graph $G$ which it's vertices are the set $V(A)$ and it's edges are the set $A$ . Prove that for every natural number $l$ : $$\frac{l}{2}\leq min_{|A|\geq l}(|V(A)|-j(A)) \leq \frac{l}{2}+\sqrt{\frac{l}{2}}+1$$

$ABCD$ is a circumscribed and inscribed quadrilateral. $O$ is the circumcenter of the quadrilateral. $E,F$ and $S$ are the intersections of $AB,CD$ , $AD,BC$ and $AC,BD$ respectively. $E'$ and $F'$ are points on $AD$ and $AB$ such that $A\hat{E}E'=E'\hat{E}D$ and $A\hat{F}F'=F'\hat{F}B$ . $X$ and $Y$ are points on $OE'$ and $OF'$ such that $\frac{XA}{XD}=\frac{EA}{ED}$ and $\frac{YA}{YB}=\frac{FA}{FB}$ . $M$ is the midpoint of arc $BD$ of $(O)$ which contains $A$ . Prove that the circumcircles of triangles $OXY$ and $OAM$ are coaxal with the circle with diameter $OS$ .

$a,b,c,d$ are positive numbers such that $\sum_{cyc} \frac{1}{ab} =1$ . Prove that : $abcd+16 \geq 8 \sqrt{(a+c)(\frac{1}{a} + \frac{1}{c})}+8\sqrt{(b+d)(\frac{1}{b}+\frac{1}{d})}$

In triangle $ABC$ (with incenter $I$ ) let the line parallel to $BC$ from $A$ intersect circumcircle of $\triangle ABC$ at $A_1$ let $AI\cap BC=D$ and $E$ is tangency point of incircle with $BC$ let $EA_1\cap \odot (\triangle ADE)=T$ prove that $AI=TI$ .

Let $\triangle ABC$ be an acute triangle. Point $Z$ is on $A$ altitude and points $X$ and $Y$ are on the $B$ and $C$ altitudes out of the triangle respectively, such that: $\angle AYB=\angle BZC=\angle CXA=90$ Prove that $X$ , $Y$ and $Z$ are collinear, if and only if the length of the tangent drawn from $A$ to the nine point circle of $\triangle ABC$ is equal with the sum of the lengths of the tangents drawn from $B$ and $C$ to the nine point circle of $\triangle ABC$ .

We call a permutation $(a_1, a_2,\cdots , a_n)$ of the set $\{ 1,2,\cdots, n\}$ "good" if for any three natural numbers $i <j <k$ , $n\nmid a_i+a_k-2a_j$ find all natural numbers $n\ge 3$ such that there exist a "good" permutation of a set $\{1,2,\cdots, n\}$ .

If $a,b,c$ are positive real numbers such that $a+b+c=abc$ prove that $$\frac{abc}{3\sqrt{2}}\left ( \sum_{cyc}\frac{\sqrt{a^3+b^3}}{ab+1} \right )\geq \sum_{cyc}\frac{a}{a^2+1}$$

Assume that $a_1, a_2, a_3$ are three given positive integers consider the following sequence: $a_{n+1}=\text{lcm}[a_n, a_{n-1}]-\text{lcm}[a_{n-1}, a_{n-2}]$ for $n\ge 3$ Prove that there exist a positive integer $k$ such that $k\le a_3+4$ and $a_k\le 0$ . ( $[a, b]$ means the least positive integer such that $a\mid[a,b], b\mid[a, b]$ also because $\text{lcm}[a, b]$ takes only nonzero integers this sequence is defined until we find a zero number in the sequence)

$a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ are $2n$ positive real numbers such that $a_1,a_2,\cdots ,a_n$ aren't all equal. And assume that we can divide $a_1,a_2,\cdots ,a_n$ into two subsets with equal sums.similarly $b_1,b_2,\cdots ,b_n$ have these two conditions. Prove that there exist a simple $2n$ -gon with sides $a_1,a_2,\cdots ,a_n,b_1,b_2,\cdots ,b_n$ and parallel to coordinate axises Such that the lengths of horizontal sides are among $a_1,a_2,\cdots ,a_n$ and the lengths of vertical sides are among $b_1,b_2,\cdots ,b_n$ .(simple polygon is a polygon such that it doesn't intersect itself)

Prove that for each natural number $d$ , There is a monic and unique polynomial of degree $d$ like $P$ such that $P(1)$ ≠ $0$ and for each sequence like $a_{1}$ , $a_{2}$ , $...$ of real numbers that the recurrence relation below is true for them, there is a natural number $k$ such that $0=a_{k}=a_{k+1}= ...$ : $P(n)a_{1} + P(n-1)a_{2} + ... + P(1)a_{n}=0$ $n>1$

$AH$ is the altitude of triangle $ABC$ and $H^\prime$ is the reflection of $H$ trough the midpoint of $BC$ . If the tangent lines to the circumcircle of $ABC$ at $B$ and $C$ , intersect each other at $X$ and the perpendicular line to $XH^\prime$ at $H^\prime$ , intersects $AB$ and $AC$ at $Y$ and $Z$ respectively, prove that $\angle ZXC=\angle YXB$ .