Difference between revisions of "2023 RMO"
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==Problem 2== | ==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 | ||
+ | (a) the measure of <math>\angleCED</math> is a constant; | ||
+ | (b) the circumcircle of triangle <math>CED</math> passes through a fixed point. | ||
==Problem 3== | ==Problem 3== |
Revision as of 08:59, 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 $\varomega$ (Error compiling LaTeX. Unknown error_msg) 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 $\angleCED$ (Error compiling LaTeX. Unknown error_msg) 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.