TH-TMO 2022

Let $BC$ be a chord of a circle $\Gamma$ and $A$ be a point inside $\Gamma$ such that $\angle BAC$ is acute. Outside $\triangle ABC$ , construct two isosceles triangles $\triangle ACP$ and $\triangle ABR$ such that $\angle ACP$ and $\angle ABR$ are right angles. Let lines $BA$ and $CA$ meet $\Gamma$ again at points $E$ and $F$ , respectively. Let lines $EP$ and $FR$ meet $\Gamma$ again at points $X$ and $Y$ , respectively. Prove that $BX=CY$ .

Define a function $f:\mathbb{N}\times \mathbb{N}\to\{-1,1\}$ such that $$f(m,n)=\begin{cases} 1 &\text{if }m,n\text{ have the same parity, and} -1 &\text{if }m,n\text{ have different parity}\end{cases}$$ for every positive integers $m,n$ . Determine the minimum possible value of $$\sum_{1\leq i<j\leq 2565} ijf(x_i,x_j)$$ across all permutations $x_1,x_2,x_3,\dots,x_{2565}$ of $1,2,\dots,2565$ .

Let $\Omega$ be a circle in a plane. $2022$ pink points and $2565$ blue points are placed inside $\Omega$ such that no point has two colors and no two points are collinear with the center of $\Omega$ . Prove that there exists a sector of $\Omega$ such that the angle at the center is acute and the number of blue points inside the sector is greater than the number of pink points by exactly $100$ . (Note: such sector may contain no pink points.)

Find all positive integers $n$ such that there exists a monic polynomial $P(x)$ of degree $n$ with integers coefficients satisfying $$P(a)P(b)\neq P(c)$$ for all integers $a,b,c$ .

Determine all functions $f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ that satisfies the equation $$f\left(\frac{x+y+z}{3},\frac{a+b+c}{3}\right)=f(x,a)f(y,b)f(z,c)$$ for any real numbers $x,y,z,a,b,c$ such that $az+bx+cy\neq ay+bz+cx$ .

In an examination, there are $3600$ students sitting in a $60 \times 60$ grid, where everyone is facing toward the top of the grid. After the exam, it is discovered that there are $901$ students who got infected by COVID-19. Each infected student has a spreading region, which consists of students to the left, to the right, or in the front of them. Student in spreading region of at least two students are considered a close contact. Given that no infected student sat in the spreading region of other infected student, prove that there is at least one close contact.

Let $d \geq 2$ be a positive integer. Define the sequence $a_1,a_2,\dots$ by $$a_1=1 \ \text{and} \ a_{n+1}=a_n^d+1 \ \text{for all }n\geq 1.$$ Determine all pairs of positive integers $(p, q)$ such that $a_p$ divides $a_q$ .

Determine all possible values of $a_1$ for which there exists a sequence $a_1, a_2, \dots$ of rational numbers satisfying $$a_{n+1}^2-a_{n+1}=a_n$$ for all positive integers $n$ .

Let $PQRS$ be a quadrilateral that has an incircle and $PQ\neq QR$ . Its incircle touches sides $PQ,QR,RS,$ and $SP$ at $A,B,C,$ and $D$ , respectively. Line $RP$ intersects lines $BA$ and $BC$ at $T$ and $M$ , respectively. Place point $N$ on line $TB$ such that $NM$ bisects $\angle TMB$ . Lines $CN$ and $TM$ intersect at $K$ , and lines $BK$ and $CD$ intersect at $H$ . Prove that $\angle NMH=90^{\circ}$ .

For each positive integers $u$ and $n$ , say that $u$ is a friend of $n$ if and only if there exists a positive integer $N$ that is a multiple of $n$ and the sum of digits of $N$ (in base 10) is equal to $u$ . Determine all positive integers $n$ that only finitely many positive integers are not a friend of $n$ .