Problem Solving Set 21

24 July 2012

1. (a) A sequence x_{1}, x_{2}, … of real numbers satises

x_{n+1} = x_{n} cos x_{n}

for all n ≥ 1. Does it follow that this sequence converges for all initial values x_{1}?

(b) A sequence y_{1}, y_{2}, … of real numbers satises

y_{n+1} = y_{n} sin y_{n}

for all n ≥ 1. Does it follow that this sequence converges for all initial values y_{1}?

2. Show that in any collection of 52 distinct positive integers there are two distinct numbers whose sum or difference is divisible by 100.