Difference between revisions of "2023 RMO"

Revision as of 12:30, 9 December 2024

Problem 1

Problem 2

Problem 3

Problem 4

Let $\Omega_1,\Omega_2$ be two intersecting circles with centres $O_1,O_2$ respectively. Let $l$ be a line that intersects $\Omega_1$ at points $A,C$ and $\Omega_2$ at points $B,D$ such that $A, B, C, D$ are collinear in that order. Let the perpendicular bisector of segment $AB$ intersect $\Omega_1$ at points $P,Q$ ; and the perpendicular bisector of segment $CD$ intersect $\Omega_1$ at points $R,S$ such that $P,R$ are on the same side of $l$ . Prove that the midpoints of $PR, QS$ and $\Omega_{1} \Omega_{2}$ are collinear.

Problem 5

Let $n>k>1$ be positive integers. Determine all positive real numbers $a_1, a_2, ..., a_n$ which satisfy $\sum_{i=1}^{n}$ $\sqrt {\frac {ka_{i}^{k}}{k-1a_{i}^{k}+1}}$ $=\sum_{i=1}^{n}$ $a_i$ $=n$ .

Problem 6

Consider a set of $16$ points arranged in a $4\times4$ square grid formation. Prove that if any $7$ of these points are coloured blue, then there exists an isosceles right-angled triangle whose vertices are all blue.

@@ Line 4: / Line 4: @@
 ==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==

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