Difference between revisions of "2023 RMO"

Latest revision as of 13:30, 9 December 2024

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 ==Problem 1==
-Let  <math>\mathbb{N}</math> be the set of all positive integers and <math>S = {(a,b,c,d)  \in  \mathbb{N}^{4} : a^{2} + b^{2} + c^{2} = d^{2}}</math>. Find the largest positive integer <math>m</math> such that <math>m</math> divides <math>abcd</math> for all <math>(a,b,c,d)  \in S</math>.
 ==Problem 2==
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 ==Problem 6==
-Consider a set of <math>16</math> points arranged in a <math>4\times4</math> square grid formation. Prove that if any <math>7</math> of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.