IR-TST 2021

In acute scalene triangle $ABC$ the external angle bisector of $\angle BAC$ meet $BC$ at point $X$ .Lines $l_b$ and $l_c$ which tangents of $B$ and $C$ with respect to $(ABC)$ .The line pass through $X$ intersects $l_b$ and $l_c$ at points $Y$ and $Z$ respectively. Suppose $(AYB)\cap(AZC)=N$ and $l_b\cap l_c=D$ . Show that $ND$ is angle bisector of $\angle YNZ$ . Proposed by Alireza Haghi

In the simple and connected graph $G$ let $x_i$ be the number of vertices with degree $i$ . Let $d>3$ be the biggest degree in the graph $G$ . Prove that if : $$x_d \ge x_{d-1} + 2x_{d-2}+... +(d-1)x_1$$ Then there exists a vertex with degree $d$ such that after removing that vertex the graph $G$ is still connected.

Proposed by Ali Mirzaie

There exist $4$ positive integers $a,b,c,d$ such that $abcd \neq 1$ and each pair of them have a GCD of $1$ . Two functions $f,g : \mathbb{N} \rightarrow \{0,1\}$ are multiplicative functions such that for each positive integer $n$ we have : $$f(an+b)=g(cn+d)$$ Prove that at least one of the followings hold. $i)$ for each positive integer $n$ we have $f(an+b)=g(cn+d)=0$ $ii)$ There exists a positive integer $k$ such that for all $n$ where $(n,k)=1$ we have $g(n)=f(n)=1$ (Function $f$ is multiplicative if for any natural numbers $a,b$ we have $f(ab)=f(a)f(b)$ )

Proposed by Navid Safaii

Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have : $$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$ Who has the winning strategy.

Proposed by Alireza Haghi

Call a triple of numbers Nice if one of them is the average of the other two. Assume that we have $2k+1$ distinct real numbers with $k^2$ Nice triples. Prove that these numbers can be devided into two arithmetic progressions with equal ratios

Proposed by Morteza Saghafian

Point $D$ is chosen on the Euler line of triangle $ABC$ and it is inside of the triangle. Points $E,F$ are were the line $BD,CD$ intersect with $AC,AB$ respectively. Point $X$ is on the line $AD$ such that $\angle EXF =180 - \angle A$ , also $A,X$ are on the same side of $EF$ . If $P$ is the second intersection of circumcircles of $CXF,BXE$ then prove the lines $XP,EF$ meet on the altitude of $A$

Proposed by Alireza Danaie

Natural numbers are placed in an infinite grid. Such that the number in each cell is equal to the number of its adjacent cells having the same number. Find the most distinct numbers this infinite grid can have.

(Two cells of the grid are adjacent if they have a common vertex)

Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that for any two positive integers $m,n$ we have : $$f(n)+1400m^2|n^2+f(f(m))$$

Prove there exist two relatively prime polynomials $P(x),Q(x)$ having integer coefficients and a real number $u>0$ such that if for positive integers $a,b,c,d$ we have: $$|\frac{a}{c}-1|^{2021} \le \frac{u}{|d||c|^{1010}}$$ $$| (\frac{a}{c})^{2020}-\frac{b}{d}| \le \frac{u}{|d||c|^{1010}}$$ Then we have : $$bP(\frac{a}{c})=dQ(\frac{a}{c})$$ (Two polynomials are relatively prime if they don't have a common root)

Proposed by Navid Safaii and Alireza Haghi

Find all functions $f : \mathbb{N} \rightarrow \mathbb{R}$ such that for all triples $a,b,c$ of positive integers the following holds : $$f(ac)+f(bc)-f(c)f(ab) \ge 1$$ Proposed by Mojtaba Zare

Point $X$ is chosen inside the non trapezoid quadrilateral $ABCD$ such that $\angle AXD +\angle BXC=180$ . Suppose the angle bisector of $\angle ABX$ meets the $D$ -altitude of triangle $ADX$ in $K$ , and the angle bisector of $\angle DCX$ meets the $A$ -altitude of triangle $ADX$ in $L$ .We know $BK \perp CX$ and $CL \perp BX$ . If the circumcenter of $ADX$ is on the line $KL$ prove that $KL \perp AD$ .

Proposed by Alireza Dadgarnia

Prove that we can color every subset with $n$ element of a set with $3n$ elements with $8$ colors . In such a way that there are no $3$ subsets $A,B,C$ with the same color where : $$|A \cap B| \le 1,|A \cap C| \le 1,|B \cap C| \le 1$$ Proposed by Morteza Saghafian and Amir Jafari