# Blog

Viewing posts for the category EGMO

### Thinking Out Loud – EGMO 2014 Problem 2

EGMO 2014, problem 2. Let $$D$$ and $$E$$ be points in the interiors of sides $$AB$$ and $$AC$$, respectively, of a triangle $$ABC$$, such that $$DB=BC=CE$$. Let the lines $$CD$$ and $$BE$$ meet at $$F$$. Prove that the incentre $$I$$ of triangle $$ABC$$, the orthocentre $$H$$ of triangle $$DEF$$ and the midpoint $$M$$ of the arc $$BAC$$ of the circumcircle of triangle $$ABC$$ are collinear.

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

### 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.