2010 Portugal MO

Part of Portugal MO

Subcontests

(3)

2

Portuguese food - Portuguese MO, Problem 3, 2010

On each day, more than half of the inhabitants of Évora eats sericaia as dessert. Show that there is a group of 10 inhabitants of Évora such that, on each of the last 2010 days, at least one of the inhabitants ate sericaia as dessert.

Primes and unit squares - Portuguese MO, Problem 6, 2010

Consider a square

(p-1)\times(p-1)

p

is a prime number, which is divided by squares

1\times 1

whose sides are parallel to the initial square's sides. Show that it is possible to select

p

vertices such that there are no three collinear vertices.

2

Easy line segment lenght - Portuguese MO, Problem 2, 2010

On a circumference, points

A

B

are on opposite arcs of diameter

CD

. Line segments

CE

DF

are perpendicular to

AB

A-E-F-B

A

E

F

B

are collinear on this order). Knowing

AE=1

, find the length of

BF

Triangle sides quotient - Portuguese MO, Problem 5, 2010

Show that any triangle has two sides whose lengths

a

b

\frac{\sqrt{5}-1}{2}<\frac{a}{b}<\frac{\sqrt{5}+1}{2}

2

Bones and Candles - Portuguese MO, Problem 1, 2010

There are several candles of the same size on the Chapel of Bones. On the first day a candle is lit for a hour. On the second day two candles are lit for a hour, on the third day three candles are lit for a hour, and successively, until the last day, when all the candles are lit for a hour. On the end of that day, all the candles were completely consumed. Find all the possibilities for the number of candles.

Numbers on a pentagon - Portuguese MO, Problem 4, 2010

Giraldo wrote five distinct natural numbers on the vertices of a pentagon. And next he wrote on each side of the pentagon the least common multiple of the numbers written of the two vertices who were on that side and noticed that the five numbers written on the sides were equal. What is the smallest number Giraldo could have written on the sides?