Difference between revisions of "2023 RMO"
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==Problem 2== | ==Problem 2== | ||
+ | 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>\angle CED</math> is a constant; | ||
+ | |||
+ | (b) the circumcircle of triangle <math>CED</math> passes through a fixed point. | ||
==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== | ||
+ | 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== | ||
+ | Let <math>n>k>1</math> be positive integers. Determine all positive real numbers <math>a_1, a_2, ..., a_n</math> which satisfy <math>\sum_{i=1}^{n}</math> <math>\sqrt {\frac {ka_{i}^{k}}{k-1a_{i}^{k}+1}}</math> <math>=\sum_{i=1}^{n}</math> <math>a_i</math> <math>=n</math>. | ||
==Problem 6== | ==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. | 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. |
Latest revision as of 12:46, 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
Let be positive integers. Determine all positive real numbers which satisfy .
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.