2008 Kyiv Mathematical Festival

Part of Kyiv Mathematical Festival

Subcontests

(5)

1

marking the squares on the chessboard-1

m

squares on the chessboard are marked. If among four squares at the intersection of some two rows and two columns three squares are marked then it is allowed to mark the fourth square. Find the smallest

m

for which it is possible to mark all squares after several such operations.

1

points on two circles

K,L,M

N

be the midpoints of sides

AB,

BC,

CD

AD

of the convex quadrangle

ABCD.

Is it possible that points

A,B,L,M,D

lie on the same circle and points

K,B,C,D,N

lie on the same circle?

1

7 integers

Prove that among any 7 integers there exist three numbers

a,b,c

such that a^2\plus{}b^2\plus{}c^2\minus{}ab\minus{}bc\minus{}ac is divisible by 7.

1

Aladdin's set of coins

Aladdin has a set of coins with weights

1, 2, \ldots, 20

grams. He can ask Genie about any two coins from the set which one is heavier, but he should pay Genie some other coin from the set before. (So, with every question the set of coins becomes smaller.) Can Aladdin find two coins from the set with total weight at least

28

1

quite easy equation

Find all positive integers

k

for which equation n^m\minus{}m^n\equal{}k has solution in positive integers.