2

Part of 1991 Cono Sur Olympiad

Problems(2)

Find all numbers

Source: Cono Sur 1991-problem 2

A

B

, play the following game:

A

start choosing a positive integrer number and then, each player in it's turn, say a number due to the following rule: If the last number said was odd, the player add

7

to this number; If the last number said was even, the player divide it by

2

. The winner is the player that repeats the first number said. Find all numbers that

A

can choose in order to win. Justify your answer.

inequalitiesnumber theory unsolvednumber theory

Source: Cono Sur 1991-problem 5

ABCD

1

, and a square inside

ABCD

x

, find (in terms of

x

r

of the circle tangent to two sides of

ABCD

and touches the square with side

x

. (See picture).

geometry unsolvedgeometry