Difference between revisions of "2023 RMO"

Revision as of 09:00, 2 November 2024

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.

@@ Line 3: / Line 3: @@
 ==Problem 2==
-Let <math>\varomega</math> be a semicircle with <math>AB</math> as the bounding diameter and let <math>CD</math> be a variable chord of the semicircle of constant length such that <math>C,D</math> lie in the interior of the arc <math>AB</math>. Let <math>E</math> be a point on the diameter <math>AB</math> such that <math>CE</math> and <math>DE</math> are equally inclined to the line <math>AB</math>. Prove that
+Let <math>\omega</math> be a semicircle with <math>AB</math> as the bounding diameter and let <math>CD</math> be a variable chord of the semicircle of constant length such that <math>C,D</math> lie in the interior of the arc <math>AB</math>. Let <math>E</math> be a point on the diameter <math>AB</math> such that <math>CE</math> and <math>DE</math> are equally inclined to the line <math>AB</math>. Prove that
-(a) the measure of <math>\angleCED</math> is a constant;
+(a) the measure of <math>\angle CED</math> is a constant;
 (b) the circumcircle of triangle <math>CED</math> passes through a fixed point.

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