TH-TMO 2023

Let $A$ be set of 20 consecutive positive integers, Which sum and product of elements in $A$ not divisible by 23. Prove that product of elements in $A$ is not perfect square

Let $\triangle ABC$ which $\angle ABC$ are right angle, Let $D$ be point on $AB$ ( $D \neq A , B$ ), Let $E$ be point on line $AB$ which $B$ is the midpoint of $DE$ , Let $I$ be incenter of $\triangle ACE$ and $J$ be $A$ -excenter of $\triangle ACD$ . Prove that perpendicular bisector of $BC$ bisects $IJ$

Defined all $f : \mathbb{R} \to \mathbb{R}$ that satisfied equation $$f(x)f(y)f(x-y)=x^2f(y)-y^2f(x)$$ for all $x,y \in \mathbb{R}$

A table tennis tournament has $101$ contestants, where each pair of contestants will play each other exactly once. In each match, the player who gets $11$ points first is the winner, and the other the loser. At the end of the tournament, it turns out that there exist matches with scores $11$ to $0$ and $11$ to $10$ . Show that there exists 3 contestants $A,B,C$ such that the score of the losers in the matches between $A,B$ and $A,C$ are equal, but different from the score of the loser in the match between $B,C$ .

Let $\ell$ be a line in the plane and let $90^\circ<\theta<180^\circ$ . Consider any distinct points $P,Q,R$ that satisfy the following:

(i) $P$ lies on $\ell$ and $PQ$ is perpendicular to $\ell$ (ii) $R$ lies on the same side of $\ell$ as $Q$ , and $R$ doesn't lie on $\ell$ (iii) for any points $A,B$ on $\ell$ , if $\angle ARB=\theta$ then $\angle AQB \geq \theta$ .

Find the minimum value of $\angle PQR$ .

Let $a,b,c,x,y$ be positive real numbers such that $abc=1$ . Prove that $$\frac{a^5}{xc+yb}+\frac{b^5}{xa+yc}+\frac{c^5}{xb+ya}\geq \frac{9}{(x+y)(a^2+b^2+c^2)}.$$

Let $n$ be positive integer and $S$ = { $0,1,…,n$ }, Define set of point in the plane. $$A = \{(x,y) \in S \times S \mid -1 \leq x-y \leq 1 \}$$ , We want to place a electricity post on a point in $A$ such that each electricity post can shine in radius 1.01 unit. Define minimum number of electricity post such that every point in $A$ is in shine area

Let $ABC$ be an acute triangle. The tangent at $A,B$ of the circumcircle of $ABC$ intersect at $T$ . Line $CT$ meets side $AB$ at $D$ . Denote by $\Gamma_1,\Gamma_2$ the circumcircle of triangle $CAD$ , and the circumcircle of triangle $CBD$ , respectively. Let line $TA$ meet $\Gamma_1$ again at $E$ and line $TB$ meet $\Gamma_2$ again at $F$ . Line $EF$ intersects sides $AC,BC$ at $P,Q$ , respectively. Prove that $EF=PQ+AB$ .

Prove that there exists an infinite sequence of positive integers $a_1,a_2,a_3,\dots$ such that for any positive integer $k$ , $a_k^2+a_k+2023$ has at least $k$ distinct positive divisors.

To celebrate the 20th Thailand Mathematical Olympiad (TMO), Ratchasima Witthayalai School put up flags around the Thao Suranari Monument so that [list=i] [] Each flag is painted in exactly one color, and at least $2$ distinct colors are used. [] The number of flags are odd. [] Every flags are on a regular polygon such that each vertex has one flag. [] Every flags with the same color are on a regular polygon. [/list] Prove that there are at least $3$ colors with the same amount of flags.