3

Part of 2008 Argentina Iberoamerican TST

Problems(2)

Sequence of squares $a^2+b^2=2c^2$

Source: Argentina TST for iberoamerican 2008 Problem 3

Show that exists a sequence of

100

terms such that: 1)Every term is a perfect square 2) every term is greater than the one before it ( it is strictly increasing) 3)Every two terms of the sequence are relative prime 4) The average between two consecutive terms is also a perfect square Daniel

number theory proposednumber theory

Number of regions at most $\frac{n(n+1)}{3}$

Source: Argentina TST Iberoamerican 2008 problem 6

The plane is divided into regions by

n \ge 3

lines, no two of which are parallel, and no three of which are concurrent. Some regions are coloured , in such a way that no two coloured regions share a common segment or half-line of their borders. Prove that the number of coloured regions is at most \frac{n(n\plus{}1)}{3}

inequalitiesinductiongraph theorycombinatorics unsolvedcombinatorics