4

Part of 1999 Akdeniz University MO

Problems(2)

Source:

In a sequence ,first term is

2

2.

term all terms is equal to sum of the previous number's digits'

5.

power. (Like this

2.

2^5=32

3.

3^5+2^5=243+32=275\dotsm

) Prove that, this infinite sequence has at least

2

two numbers are equal.

geometric combinatorics

Source:

n \in {\mathbb N}

circles with radius

1

unit

inside a square with side

100

unit

such that, whichever line segment with lenght

10

unit

intersect at least one circle. Prove that

n \geq 416

geometrycombinatoricscombinatorial geometry