4

Part of 2009 Junior Balkan Team Selection Test

Problems(2)

Divisibility by 11 with digits 5 and 8

Source: Serbian JBTST 4, Day 1

In the decimal expression of a

2009

-digit natural number there are only the digits

5

8

. Prove that we can get a

2008

-digit number divisible by

11

if we remove just one digit from the number.

inductionnumber theory proposednumber theory

Strange condition, easy inequality

Source: Serbian JBTST 4, Day 2

For positive real numbers

x,y,z

the inequality \frac1{x^2\plus{}1}\plus{}\frac1{y^2\plus{}1}\plus{}\frac1{z^2\plus{}1}\equal{}\frac12 holds. Prove the inequality \frac1{x^3\plus{}2}\plus{}\frac1{y^3\plus{}2}\plus{}\frac1{z^3\plus{}2}<\frac13.

inequalitiesinequalities proposed