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

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 4

Let $\Omega_1,\Omega_2$ be two intersecting circles with centres $O_1,O_2$ respectively. Let $l$ be a line that intersects $\Omega_1$ at points $A,C$ and $\Omega_2$ at points $B,D$ such that $A, B, C, D$ are collinear in that order. Let the perpendicular bisector of segment $AB$ intersect $\Omega_1$ at points $P,Q$ ; and the perpendicular bisector of segment $CD$ intersect $\Omega_1$ at points $R,S$ such that $P,R$ are on the same side of $l$ . Prove that the midpoints of $PR, QS$ and $\Omega_{1} \Omega_{2}$ are collinear.

Problem 5

Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, ..., a_n$ which satisfy $\sum_{i=1}^{n}$ $\sqrt {\frac {ka_{i}^{k}}{k-1a_{i}^{k}+1}}$ $=\sum_{i=1}^{n}$ $a_i$ $=n$ .

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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