1

Part of 2010 Pan African

Problems(2)

Pairing off so that sum is a prime

Source: Pan African Olympiad 2010

a) Show that it is possible to pair off the numbers

1,2,3,\ldots ,10

so that the sums of each of the five pairs are five different prime numbers. b) Is it possible to pair off the numbers

1,2,3,\ldots ,20

so that the sums of each of the ten pairs are ten different prime numbers?

number theoryprime numbersnumber theory proposed

At least one of the 7 arcs created is at most 1/24

Source: Pan African Olympiad 2010

Seven distinct points are marked on a circle of circumference

c

. Three of the points form an equilateral triangle and the other four form a square. Prove that at least one of the seven arcs into which the seven points divide the circle has length less than or equal

\frac{c}{24}

pigeonhole principlecombinatorics proposedcombinatorics