# Blog

Viewing posts for the category Olimpiadi

### 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 $$9n^2$$.

### Thinking Out Loud – EGMO 2013 Problem 4

EGMO 2013, problem 4. Find all positive integers $$a$$ and $$b$$ for which there are three consecutive integers at which the polynomial $P(n)=\frac{n^5+a}{b}$ takes integer values.