Difference between revisions of "2023 RMO"

Revision as of 13:29, 9 December 2024

Problem 1

Problem 2

Problem 3

For any natural number $n$ , expressed in base $10$ , let $s(n)$ denote the sum of all its digits. Find all natural numbers $m$ and $n$ such that $m < n$ and

$(s(n))^{2} = m$ and $(s(m))^{2} = n$ .

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 2: / Line 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==

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