IR-TST 2022

Morteza Has $100$ sets. at each step Mahdi can choose two distinct sets of them and Morteza tells him the intersection and union of those two sets. Find the least steps that Mahdi can find all of the sets.

Proposed by Morteza Saghafian

For a positive integer $n$ , let $\tau(n)$ and $\sigma(n)$ be the number of positive divisors of $n$ and the sum of positive divisors of $n$ , respectively. let $a$ and $b$ be positive integers such that $\sigma(a^n)$ divides $\sigma(b^n)$ for all $n\in \mathbb{N}$ . Prove that each prime factor of $\tau(a)$ divides $\tau(b)$ .

Proposed by MohammadAmin Sharifi

Incircle $\omega$ of triangle $ABC$ is tangent to sides $CB$ and $CA$ at $D$ and $E$ , respectively. Point $X$ is the reflection of $D$ with respect to $B$ . Suppose that the line $DE$ is tangent to the $A$ -excircle at $Z$ . Let the circumcircle of triangle $XZE$ intersect $\omega$ for the second time at $K$ . Prove that the intersection of $BK$ and $AZ$ lies on $\omega$ .

Proposed by Mahdi Etesamifard and Alireza Dadgarnia

Cyclic quadrilateral $ABCD$ with circumcenter $O$ is given. Point $P$ is the intersection of diagonals $AC$ and $BD$ . Let $M$ and $N$ be the midpoint of the sides $AD$ and $BC$ , respectively. Suppose that $\omega_1$ , $\omega_2$ and $\omega_3$ be the circumcircle of triangles $ADP$ , $BCP$ and $OMN$ , respectively. The intersection point of $\omega_1$ and $\omega_3$ , which is not on the arc $APD$ of $\omega_1$ , is $E$ and the intersection point of $\omega_2$ and $\omega_3$ , which is not on the arc $BPC$ of $\omega_2$ , is $F$ . Prove that $OF=OE$ .

Proposed by Seyed Amirparsa Hosseini Nayeri

Find all $C\in \mathbb{R}$ such that every sequence of integers $\{a_n\}_{n=1}^{\infty}$ which is bounded from below and for all $n\geq 2$ satisfy $$0\leq a_{n-1}+Ca_n+a_{n+1}<1$$ is periodic.

Proposed by Navid Safaei

Let $m,n$ and $a_1,a_2,\dots,a_m$ be arbitrary positive integers. Ali and Mohammad Play the following game. At each step, Ali chooses $b_1,b_2,\dots,b_m \in \mathbb{N}$ and then Mohammad chosses a positive integers $s$ and obtains a new sequence $\{c_i=a_i+b_{i+s}\}_{i=1}^m$ , where $$b_{m+1}=b_1,\ b_{m+2}=b_2, \dots,\ b_{m+s}=b_s$$ The goal of Ali is to make all the numbers divisible by $n$ in a finite number of steps. FInd all positive integers $m$ and $n$ such that Ali has a winning strategy, no matter how the initial values $a_1, a_2,\dots,a_m$ are.

Proposed by Shayan Gholami

Suppose that $n$ is a positive integer number. Consider a regular polygon with $2n$ sides such that one of its largest diagonals is parallel to the $x$ -axis. Find the smallest integer $d$ such that there is a polynomial $P$ of degree $d$ whose graph intersects all sides of the polygon on points other than vertices.

Proposed by Mohammad Ahmadi

In triangle $ABC$ , with $AB<AC$ , $I$ is the incenter, $E$ is the intersection of $A$ -excircle and $BC$ . Point $F$ lies on the external angle bisector of $BAC$ such that $E$ and $F$ lieas on the same side of the line $AI$ and $\angle AIF=\angle AEB$ . Point $Q$ lies on $BC$ such that $\angle AIQ=90$ . Circle $\omega_b$ is tangent to $FQ$ and $AB$ at $B$ , circle $\omega_c$ is tangent to $FQ$ and $AC$ at $C$ and both circles pass through the inside of triangle $ABC$ . if $M$ is the Midpoint od the arc $BC$ , which does not contain $A$ , prove that $M$ lies on the radical axis of $\omega_b$ and $\omega_c$ .

Proposed by Amirmahdi Mohseni

consider $n\geq 6$ points $x_1,x_2,\dots,x_n$ on the plane such that no three of them are colinear. We call graph with vertices $x_1,x_2,\dots,x_n$ a "road network" if it is connected, each edge is a line segment, and no two edges intersect each other at points other than the vertices. Prove that there are three road networks $G_1,G_2,G_3$ such that $G_i$ and $G_j$ don't have a common edge for $1\leq i,j\leq 3$ .

Proposed by Morteza Saghafian

We call an infinite set $S\subseteq\mathbb{N}$ good if for all parwise different integers $a,b,c\in S$ , all positive divisors of $\frac{a^c-b^c}{a-b}$ are in $S$ . for all positive integers $n>1$ , prove that there exists a good set $S$ such that $n \not \in S$ .

Proposed by Seyed Reza Hosseini Dolatabadi

Consider a table with $n$ rows and $2n$ columns. we put some blocks in some of the cells. After putting blocks in the table we put a robot on a cell and it starts moving in one of the directions right, left, down or up. It can change the direction only when it reaches a block or border. Find the smallest number $m$ such that we can put $m$ blocks on the table and choose a starting point for the robot so it can visit all of the unblocked cells. (the robot can't enter the blocked cells.)

Proposed by Seyed Mohammad Seyedjavadi and Alireza Tavakoli

suppose that $A$ is the set of all Closed intervals $[a,b] \subset \mathbb{R}$ . Find all functions $f:\mathbb{R} \rightarrow A$ such that $\bullet$ $x \in f(y) \Leftrightarrow y \in f(x)$ $\bullet$ $|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing$ $\bullet$ For all real numbers $0\leq r\leq 1$ , $f(r)=[r^2-1,r^2+1]$ Proposed by Matin Yousefi