4

Part of 2011 Belarus Team Selection Test

Problems(2)

beetles creeping on neighbouring cells on a n x n square

Source: 2011 Belarus TST 1.4

n \times n

square table. Exactly one beetle sits in each cell of the table. At

12.00

all beetles creeps to some neighbouring cell (two cells are neighbouring if they have the common side). Find the greatest number of cells which can become empty (i.e. without beetles) if a)

n=8

n=9

Problem Committee of BMO 2011

(1/a+1/b+1/c)(a+b+c-2) if a+b+c=a^2+b^2+c^2=a^3+b^3+c^3

Source: 2011 Belarus TST 2.4

Given nonzero real numbers a,b,c with

a+b+c=a^2+b^2+c^2=a^3+b^3+c^3

*

\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{b}\right)(a+b+c-2)

b) Do there exist pairwise different nonzero

a,b,c

*

algebrasystem of equationsBelarus