2023 RMO

Problem 1

Let $\mathbb{N}$ be the set of all positive integers and $S = {(a,b,c,d) \in \mathbb{N}^{4} : a^{2} + b^{2} + c^{2} = d^{2}}$ . Find the largest positive integer $m$ such that $m$ divides $abcd$ for all $(a,b,c,d) \in S$ .

Problem 2

Let $\omega$ be a semicircle with $AB$ as the bounding diameter and let $CD$ be a variable chord of the semicircle of constant length such that $C,D$ lie in the interior of the arc $AB$ . Let $E$ be a point on the diameter $AB$ such that $CE$ and $DE$ are equally inclined to the line $AB$ . Prove that

(a) the measure of $\angle CED$ is a constant;

(b) the circumcircle of triangle $CED$ passes through a fixed point.

Problem 3

Problem 4

For any natural number $n$ , expressed in base $10$ , let $s(n)$ denote the sum of all its digits. Find all natural numbers $m$ and $n$ such that $m < n$ and $(s(n))^{2} = m$ and $(s(m))^{2} = n$ .

Problem 5

Problem 6

Consider a set of $16$ points arranged in a $4\times4$ square grid formation. Prove that if any $7$ of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.

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