# Blog

Viewing posts by Marcello Mamino

### Thinking Out Loud – EGMO 2013 Problem 1

EGMO 2013, problem 1. The side $$\textrm{BC}$$ of the triangle $$\textrm{ABC}$$ is extended beyond $$\textrm{C}$$ to $$\textrm{D}$$ so that $$\textrm{CD} = \textrm{BC}$$. The side $$\textrm{CA}$$ is extended beyond $$\textrm{A}$$ to $$\textrm{E}$$ so that $$\textrm{AE} = 2\textrm{CA}$$. Prove that if $$\textrm{AD} = \textrm{BE}$$, then the triangle $$\textrm{ABC}$$ is right-angled.

### Thinking Out Loud – EGMO 2013 Problem 2

EGMO 2013, problem 2. Determine all integers $$m$$ for which the $$m\times m$$ square can be dissected into five rectangles, the side lengths of which are the integers $$1, 2, 3, \dotsc, 10$$ in some order.

### Thinking Out Loud – EGMO 2013 Problem 3

EGMO 2013, problem 3. Let $$n$$ be a positive integer.

1. Prove that there exists a set $$S$$ of $$6n$$ pairwise different positive integers, such that the least common multiple of any two elements of $$S$$ is no larger than $$32n^2$$.
2. Prove that every set $$T$$ of $$6n$$ pairwise different positive integers contains two elements the least common multiple of which is larger than $$9n^2$$.