RMO Previous Paper Oct 29 2023 Paper 1

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

Let $$ω$$ 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.

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 $$(s(m))^{2} = n$$.

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

Let $$n > k > 1$$ be positive integers. Determine all positive real numbers $$a_{1},a_{2},....,a_{n}$$ which satisfy $$\sum_{i=1}^{n} \sqrt{\frac{ka_{i}^{k}}{(k-1)a_{i}^{k} + 1}} = \sum_{i=1}^{n}a_{i} = n$$.

Consider a set of 16 points arranged in a $$4 × 4$$ 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.