1

Part of 1995 IMO Shortlist

Problems(2)

Infinitely many perfect squares of the form n*2^k-7

Source: IMO Shortlist 1995, N1

k

be a positive integer. Show that there are infinitely many perfect squares of the form n \cdot 2^k \minus{} 7 where

n

is a positive integer.

algebramodular arithmeticnumber theoryIMO Shortlist

F(F(n^163)) = F(F(n)) + F(F(361))

Source: IMO Shortlist 1995, S1

Does there exist a sequence

F(1), F(2), F(3), \ldots

of non-negative integers that simultaneously satisfies the following three conditions? (a) Each of the integers

0, 1, 2, \ldots

occurs in the sequence. (b) Each positive integer occurs in the sequence infinitely often. (c) For any

n \geq 2,

F(F(n^{163})) \equal{} F(F(n)) \plus{} F(F(361)).

algebraIMO Shortlistfunctional equationIteration