# Blog

Viewing posts for the category Olimpiadi

### Thinking Out Loud – EGMO 2017 Problem 1

EGMO 2017, Problem 1. Let $$ABCD$$ be a convex quadrilateral with $$\widehat{DAB} = \widehat{BCD} = 90^\circ$$ and $$\widehat{ABC} > \widehat{CDA}$$. Let $$Q$$ and $$R$$ be points on segments $$BC$$ and $$CD$$, respectively, such that line $$QR$$ intersects lines $$AB$$ and $$AD$$ at points $$P$$ and $$S$$, respectively. It is given that $$PQ = RS$$. Let the midpoint of $$BD$$ be $$M$$ and the midpoint of $$QR$$ be $$N$$. Prove that the points $$M, N, A$$ and $$C$$ lie on a circle.

### Thinking Out Loud – EGMO 2014 Problem 3

EGMO 2014, Problem 3. We denote the number of positive divisors of a positive integer $$m$$ by $$d(m)$$ and the number of distinct prime divisors of $$m$$ by $$\omega(m)$$. Let $$k$$ be a positive integer. Prove that there exist infinitely many positive integers $$n$$ such that $$\omega(n) = k$$ and $$d(n)$$ does not divide $$d(a^2+b^2)$$ for any positive integers $$a, b$$ satisfying $$a + b = n$$.

### 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.