6

Part of 2011 NZMOC Camp Selection Problems

Problems(2)

3 x 2^m + 1 = n^2

Source: New Zealand NZMOC Camp Selection Problems 2011 Juniors 6

Find all pairs of non-negative integers

m

n

3 \cdot 2^m + 1 = n^2.

Diophantine equationdiophantinenumber theory

2011^2 lattice points (x, y), subset with >= 4x2011x\sqrt{2011} points

Source: New Zealand NZMOC Camp Selection Problems 2011 Seniors 6

Consider the set

G

2011^2

(x, y)

in the plane where

x

y

are both integers between

1

2011

A

be any subset of

G

containing at least

4\times 2011\times \sqrt{2011}

points. Show that there are at least

2011^2

parallelograms whose vertices lie in

A

and all of whose diagonals meet at a single point.

combinatoricscombinatorial geometry