Mock 22nd Thailand TMO P10

by korncrazy, Apr 13, 2025, 6:57 PM

Prove that there exists infinitely many triples of positive integers $(a,b,c)$ such that $a>b>c,\,\gcd(a,b,c)=1$ and $a^2-b^2,a^2-c^2,b^2-c^2$ are all perfect square.

number theory

Mock 22nd Thailand TMO P9

by korncrazy, Apr 13, 2025, 6:57 PM

Let $H_A,H_B,H_C$ be the feet of the altitudes of the triangle $ABC$ from $A,B,C$ , respectively. $P$ is the point on the circumcircle of the triangle $ABC$ , $H$ is the orthocenter of the triangle $ABC$ , and the incircle of triangle $H_AH_BH_C$ has radius $r$ . Let $T_A$ be the point such that $T_A$ and $H$ are on the opposite side of $H_BH_C$ , line $T_AP$ is perpendicular to the line $H_BH_C$ , and the distance from $T_A$ to line $H_BH_C$ is $r$ . Define $T_B$ and $T_C$ similarly. Prove that $T_A,T_B,T_C$ are collinear.

geometry

Mock 22nd Thailand TMO P8

by korncrazy, Apr 13, 2025, 6:56 PM

Let $S$ be the set of positive integers with at least two elements. Suppose that there exist a positive integer $a$ such that $\{x+y|\,x,y\in S,\,x<y\}=\bigg\{n\bigg|\,a\leq n\leq a+\dfrac{|S|(|S|-1)}{2}-1\bigg\}.$ Find all possible values of $|S|$ .

algebra

Mock 22nd Thailand TMO P7

by korncrazy, Apr 13, 2025, 6:55 PM

Let $\{a_i\}$ be the sequence of positive integers such that $a_1=3$ and $a_{k+1}=a_k^2-2$ for all positive integers $k$ . Prove that there exists a positive integer $N$ such that the greatest prime divisor of $a_n$ is more than $10^{2025}$ for all $n>N$ .

number theory

Mock 22nd Thailand TMO P6

by korncrazy, Apr 13, 2025, 6:55 PM

Let $S$ be a subset of $\{1,2,\dots,2025\}$ such that $|S|=35$ . Prove that there exists two distinct subsets $X,Y$ of $S$ such that $|X|=|Y|=3$ and the sum of all elements in $X$ is equal to the sum of all elements in $Y$ .

combinatorics

Mock 22nd Thailand TMO P5

by korncrazy, Apr 13, 2025, 6:54 PM

Find all functions $f:\mathbb{R}^+\rightarrow\mathbb{R}$ such that $f(1)=0$ and $f(x)f(y)f(z)=f(x^2)+f(y^2)+f(z^2)$ for all positive real numbers $x,y,z$ such that $xyz=1$ .

functional equation

Abusing surjectivity

by Sadigly, Apr 13, 2025, 6:54 PM

Find all functions $f:\mathbb{Q}\rightarrow\mathbb{Q}$ and $g:\mathbb{Q}\rightarrow\mathbb{Q}$ such that

$f(f(x)+yg(x))=(x+1)g(y)+f(y)$
for any $x;y\in\mathbb{Q}$

This post has been edited 1 time. Last edited by Sadigly, 14 minutes ago

function

Cauchy functional equation

algebra

functional equation

Mock 22nd Thailand TMO P4

by korncrazy, Apr 13, 2025, 6:53 PM

Let $n$ be a positive integer. In an $n\times n$ table, an upright path is a sequence of adjacent cells starting from the southwest corner to the northeast corner such that the next cell is either on the top or on the right of the previous cell. Find the smallest number of grids one needs to color in an $n\times n$ table such that there exists only one possible upright path not containing any colored cells.

This post has been edited 1 time. Last edited by korncrazy, 22 minutes ago

combinatorics

one cyclic formed by two cyclic

by CrazyInMath, Apr 13, 2025, 12:38 PM

Let $ABC$ be an acute triangle. Points $B, D, E$ , and $C$ lie on a line in this order and satisfy $BD = DE = EC$ . Let $M$ and $N$ be the midpoints of $AD$ and $AE$ , respectively. Suppose triangle $ADE$ is acute, and let $H$ be its orthocentre. Points $P$ and $Q$ lie on lines $BM$ and $CN$ , respectively, such that $D, H, M,$ and $P$ are concyclic and pairwise different, and $E, H, N,$ and $Q$ are concyclic and pairwise different. Prove that $P, Q, N,$ and $M$ are concyclic.

geometry

EGMO 2025

pairwise coprime sum gcd

by InterLoop, Apr 13, 2025, 12:34 PM

For a positive integer $N$ , let $c_1 < c_2 < \dots < c_m$ be all the positive integers smaller than $N$ that are coprime to $N$ . Find all $N \ge 3$ such that
$\gcd(N, c_i + c_{i+1}) \neq 1$ for all $1 \le i \le m - 1$ .