Difference between revisions of "2023 RMO"
(→Problem 3) |
(→Problem 4) |
||
Line 15: | Line 15: | ||
==Problem 4== | ==Problem 4== | ||
+ | Let <math>\Omega_1,\Omega_2</math> be two intersecting circles with centres <math>O_1,O_2</math> respectively. Let <math>l</math> be a line that intersects <math>\Omega_1</math> at points <math>A,C</math> and <math>\Omega_2</math> at points <math>B,D</math> such that <math>A, B, C, D</math> are collinear in that order. Let the perpendicular bisector of segment <math>AB</math> intersect <math>\Omega_1</math> at points <math>P,Q</math>; and the perpendicular bisector of segment <math>CD</math> intersect <math>\Omega_1</math> at points <math>R,S</math> such that <math>P,R</math> are on the same side of <math>l</math>. Prove that the midpoints of <math>PR, QS</math> and <math>\Omega_{1} \Omega_{2}</math> are collinear. | ||
==Problem 5== | ==Problem 5== |
Revision as of 09:06, 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
Let be two intersecting circles with centres respectively. Let be a line that intersects at points and at points such that are collinear in that order. Let the perpendicular bisector of segment intersect at points ; and the perpendicular bisector of segment intersect at points such that are on the same side of . Prove that the midpoints of and are collinear.
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.