Difference between revisions of "2023 RMO"
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==Problem 2== | ==Problem 2== | ||
− | Let <math>\ | + | 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>\ | + | |
+ | (a) the measure of <math>\angle CED</math> is a constant; | ||
+ | |||
(b) the circumcircle of triangle <math>CED</math> passes through a fixed point. | (b) the circumcircle of triangle <math>CED</math> passes through a fixed point. | ||
Revision as of 09:00, 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
Problem 4
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 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.