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Part of 2010 Bulgaria National Olympiad

Problems(2)

Winning Strategy For The Game - [Bulgaria NMO 2010]

Source:

2 \times 2010

is divided to unit cells. Ivan and Peter are playing the following game. Ivan starts, and puts horizontal

2 \times 1

domino that covers exactly two unit table cells. Then Peter puts vertical

1 \times 2

domino that covers exactly two unit table cells. Then Ivan puts horizontal domino. Then Peter puts vertical domino, etc. The person who cannot put his domino will lose the game. Find who have winning strategy.

geometryrectanglecombinatorics unsolvedcombinatorics

Does there exist such n? - [Bulgaria NMO 2010]

Source:

Does there exist a number

n=\overline{a_1a_2a_3a_4a_5a_6}

\overline{a_1a_2a_3}+4 = \overline{a_4a_5a_6}

10

n=a^k

for some positive integers

a,k

k \geq 3 \ ?

modular arithmeticnumber theory proposednumber theory