OSN 2020

Since this is already 3 PM (GMT +7) in Jakarta, might as well post the problem here.

Problem 1. Given an acute triangle $ABC$ and the point $D$ on segment $BC$ . Circle $c_1$ passes through $A, D$ and its centre lies on $AC$ . Whereas circle $c_2$ passes through $A, D$ and its centre lies on $AB$ . Let $P \neq A$ be the intersection of $c_1$ with $AB$ and $Q \neq A$ be the intersection of $c_2$ with $AC$ . Prove that $AD$ bisects $\angle{PDQ}$ .

Let $P(x) = ax^2 + bx + c$ where $a, b, c$ are real numbers. If $$P(a) = bc, \hspace{0.5cm} P(b) = ac, \hspace{0.5cm} P(c) = ab$$ then prove that $$(a - b)(b - c)(c - a)(a + b + c) = 0.$$

The wording is just ever so slightly different, however the problem is identical.

Problem 3. Determine all functions $f: \mathbb{N} \to \mathbb{N}$ such that $n^2 + f(n)f(m)$ is a multiple of $f(n) + m$ for all natural numbers $m, n$ .

A chessboard with $2n \times 2n$ tiles is coloured such that every tile is coloured with one out of $n$ colours. Prove that there exists 2 tiles in either the same column or row such that if the colours of both tiles are swapped, then there exists a rectangle where all its four corner tiles have the same colour.

A set $A$ contains exactly $n$ integers, each of which is greater than $1$ and every of their prime factors is less than $10$ . Determine the smallest $n$ such that $A$ must contain at least two distinct elements $a$ and $b$ such that $ab$ is the square of an integer.

Given a cyclic quadrilateral $ABCD$ . Let $X$ be a point on segment $BC$ ( $X \not= BC$ ) such that line $AX$ is perpendicular to the angle bisector of $\angle CBD$ , and $Y$ be a point on segment $AD$ ( $Y \not= D)$ such that $BY$ is perpendicular to the angle bisector of $\angle CAD$ . Prove that $XY$ is parallel to $CD$ .

Determine all real-coefficient polynomials $P(x)$ such that $$P(\lfloor x \rfloor) = \lfloor P(x) \rfloor$$for every real numbers $x$ .

Determine the smallest natural number $n > 2$ , or show that no such natural numbers $n$ exists, that satisfy the following condition: There exists natural numbers $a_1, a_2, \dots, a_n$ such that $$\gcd(a_1, a_2, \dots, a_n) = \sum_{k = 1}^{n - 1} \underbrace{\left( \frac{1}{\gcd(a_k, a_{k + 1})} + \frac{1}{\gcd(a_k, a_{k + 2})} + \dots + \frac{1}{\gcd(a_k, a_n)} \right)}_{n - k \ \text{terms}}$$