Difference between revisions of "2023 RMO"
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==Problem 3== | ==Problem 3== | ||
+ | For any natural number <math>n</math>, expressed in base <math>10</math>, let <math>s(n)</math> denote the sum of all its digits. Find all natural numbers <math>m</math> and <math>n</math> such that <math>m < n</math> and | ||
+ | |||
+ | <math>(s(n))^{2} = m</math> and <math>(s(m))^{2} = n</math>. | ||
==Problem 4== | ==Problem 4== |
Revision as of 09:01, 2 November 2024
Problem 1
Let be the set of all positive integers and . Find the largest positive integer such that divides for all .
Problem 2
Let be a semicircle with as the bounding diameter and let be a variable chord of the semicircle of constant length such that lie in the interior of the arc . Let be a point on the diameter such that and are equally inclined to the line . Prove that
(a) the measure of is a constant;
(b) the circumcircle of triangle passes through a fixed point.
Problem 3
For any natural number , expressed in base , let denote the sum of all its digits. Find all natural numbers and such that and
and .
Problem 4
Problem 5
Problem 6
Consider a set of points arranged in a square grid formation. Prove that if any of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.