IR-TST 2012

Find all positive integers $n \geq 2$ such that for all integers $i,j$ that $0 \leq i,j\leq n$ , $i+j$ and ${n\choose i}+ {n \choose j}$ have same parity.

Proposed by Mr.Etesami

Consider $\omega$ is circumcircle of an acute triangle $ABC$ . $D$ is midpoint of arc $BAC$ and $I$ is incenter of triangle $ABC$ . Let $DI$ intersect $BC$ in $E$ and $\omega$ for second time in $F$ . Let $P$ be a point on line $AF$ such that $PE$ is parallel to $AI$ . Prove that $PE$ is bisector of angle $BPC$ .

Proposed by Mr.Etesami

Let $n$ be a positive integer. Let $S$ be a subset of points on the plane with these conditions: $i)$ There does not exist $n$ lines in the plane such that every element of $S$ be on at least one of them. $ii)$ for all $X \in S$ there exists $n$ lines in the plane such that every element of $S - {X}$ be on at least one of them.

Find maximum of $\mid S\mid$ .

Proposed by Erfan Salavati

Consider $m+1$ horizontal and $n+1$ vertical lines ( $m,n\ge 4$ ) in the plane forming an $m\times n$ table. Cosider a closed path on the segments of this table such that it does not intersect itself and also it passes through all $(m-1)(n-1)$ interior vertices (each vertex is an intersection point of two lines) and it doesn't pass through any of outer vertices. Suppose $A$ is the number of vertices such that the path passes through them straight forward, $B$ number of the table squares that only their two opposite sides are used in the path, and $C$ number of the table squares that none of their sides is used in the path. Prove that $$A=B-C+m+n-1.$$

Proposed by Ali Khezeli

The function $f:\mathbb R^{\ge 0} \longrightarrow \mathbb R^{\ge 0}$ satisfies the following properties for all $a,b\in \mathbb R^{\ge 0}$ :a) $f(a)=0 \Leftrightarrow a=0$ b) $f(ab)=f(a)f(b)$ c) $f(a+b)\le 2 \max \{f(a),f(b)\}$ .

Prove that for all $a,b\in \mathbb R^{\ge 0}$ we have $f(a+b)\le f(a)+f(b)$ .

Proposed by Masoud Shafaei

The pentagon $ABCDE$ is inscirbed in a circle $w$ . Suppose that $w_a,w_b,w_c,w_d,w_e$ are reflections of $w$ with respect to sides $AB,BC,CD,DE,EA$ respectively. Let $A'$ be the second intersection point of $w_a,w_e$ and define $B',C',D',E'$ similarly. Prove that $$2\le \frac{S_{A'B'C'D'E'}}{S_{ABCDE}}\le 3,$$ where $S_X$ denotes the surface of figure $X$ .

Proposed by Morteza Saghafian, Ali khezeli

Is it possible to put $\binom{n}{2}$ consecutive natural numbers on the edges of a complete graph with $n$ vertices in a way that for every path (or cycle) of length $3$ where the numbers $a,b$ and $c$ are written on its edges (edge $b$ is between edges $c$ and $a$ ), $b$ is divisible by the greatest common divisor of the numbers $a$ and $c$ ?

Proposed by Morteza Saghafian

Let $g(x)$ be a polynomial of degree at least $2$ with all of its coefficients positive. Find all functions $f:\mathbb R^+ \longrightarrow \mathbb R^+$ such that $$f(f(x)+g(x)+2y)=f(x)+g(x)+2f(y) \forall x,y\in \mathbb R^+.$$

Proposed by Mohammad Jafari

Suppose $ABCD$ is a parallelogram. Consider circles $w_1$ and $w_2$ such that $w_1$ is tangent to segments $AB$ and $AD$ and $w_2$ is tangent to segments $BC$ and $CD$ . Suppose that there exists a circle which is tangent to lines $AD$ and $DC$ and externally tangent to $w_1$ and $w_2$ . Prove that there exists a circle which is tangent to lines $AB$ and $BC$ and also externally tangent to circles $w_1$ and $w_2$ .

Proposed by Ali Khezeli

For positive reals $a,b$ and $c$ with $ab+bc+ca=1$ , show that $$\sqrt{3}({\sqrt{a}+\sqrt{b}+\sqrt{c})\le \frac{a\sqrt{a}}{bc}+\frac{b\sqrt{b}}{ca}+\frac{c\sqrt{c}}{ab}.}$$

Proposed by Morteza Saghafian

Points $A$ and $B$ are on a circle $\omega$ with center $O$ such that $\tfrac{\pi}{3}< \angle AOB <\tfrac{2\pi}{3}$ . Let $C$ be the circumcenter of the triangle $AOB$ . Let $l$ be a line passing through $C$ such that the angle between $l$ and the segment $OC$ is $\tfrac{\pi}{3}$ . $l$ cuts tangents in $A$ and $B$ to $\omega$ in $M$ and $N$ respectively. Suppose circumcircles of triangles $CAM$ and $CBN$ , cut $\omega$ again in $Q$ and $R$ respectively and theirselves in $P$ (other than $C$ ). Prove that $OP\perp QR$ .

Proposed by Mehdi E'tesami Fard, Ali Khezeli

We call a subset $B$ of natural numbers loyal if there exists natural numbers $i\le j$ such that $B=\{i,i+1,\ldots,j\}$ . Let $Q$ be the set of all loyal sets. For every subset $A=\{a_1<a_2<\ldots<a_k\}$ of $\{1,2,\ldots,n\}$ we set $$f(A)=\max_{1\le i \le k-1}{a_{i+1}-a_i} \text{and} g(A)=\max_{B\subseteq A, B\in Q} |B|.$$ Furthermore, we define $$F(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} f(A) \text{and} G(n)=\sum_{A\subseteq \{1,2,\ldots,n\}} g(A).$$ Prove that there exists $m\in \mathbb N$ such that for each natural number $n>m$ we have $F(n)>G(n)$ . (By $|A|$ we mean the number of elements of $A$ , and if $|A|\le 1$ , we define $f(A)$ to be zero).

Proposed by Javad Abedi

Consider a regular $2^k$ -gon with center $O$ and label its sides clockwise by $l_1,l_2,...,l_{2^k}$ . Reflect $O$ with respect to $l_1$ , then reflect the resulting point with respect to $l_2$ and do this process until the last side. Prove that the distance between the final point and $O$ is less than the perimeter of the $2^k$ -gon.

Proposed by Hesam Rajabzade

Do there exist $2000$ real numbers (not necessarily distinct) such that all of them are not zero and if we put any group containing $1000$ of them as the roots of a monic polynomial of degree $1000$ , the coefficients of the resulting polynomial (except the coefficient of $x^{1000}$ ) be a permutation of the $1000$ remaining numbers?

Proposed by Morteza Saghafian

Find all integer numbers $x$ and $y$ such that: $$(y^3+xy-1)(x^2+x-y)=(x^3-xy+1)(y^2+x-y).$$

Proposed by Mahyar Sefidgaran

Suppose $p$ is an odd prime number. We call the polynomial $f(x)=\sum_{j=0}^n a_jx^j$ with integer coefficients $i$ -remainder if $\sum_{p-1|j,j>0}a_{j}\equiv i\pmod{p}$ . Prove that the set $\{f(0),f(1),...,f(p-1)\}$ is a complete residue system modulo $p$ if and only if polynomials $f(x), (f(x))^2,...,(f(x))^{p-2}$ are $0$ -remainder and the polynomial $(f(x))^{p-1}$ is $1$ -remainder.

Proposed by Yahya Motevassel

Let $n$ be a natural number. Suppose $A$ and $B$ are two sets, each containing $n$ points in the plane, such that no three points of a set are collinear. Let $T(A)$ be the number of broken lines, each containing $n-1$ segments, and such that it doesn't intersect itself and its vertices are points of $A$ . Define $T(B)$ similarly. If the points of $B$ are vertices of a convex $n$ -gon (are in convex position), but the points of $A$ are not, prove that $T(B)<T(A)$ .

Proposed by Ali Khezeli

Let $O$ be the circumcenter of the acute triangle $ABC$ . Suppose points $A',B'$ and $C'$ are on sides $BC,CA$ and $AB$ such that circumcircles of triangles $AB'C',BC'A'$ and $CA'B'$ pass through $O$ . Let $\ell_a$ be the radical axis of the circle with center $B'$ and radius $B'C$ and circle with center $C'$ and radius $C'B$ . Define $\ell_b$ and $\ell_c$ similarly. Prove that lines $\ell_a,\ell_b$ and $\ell_c$ form a triangle such that it's orthocenter coincides with orthocenter of triangle $ABC$ .

Proposed by Mehdi E'tesami Fard