MathDB

6

Part of 1994 IMO Shortlist

Problems(2)

Nice sequence

Source: IMO Shortlist 1994, N6

3/29/2005
Define the sequence a1,a2,a3,... a_1, a_2, a_3, ... as follows. a1 a_1 and a2 a_2 are coprime positive integers and a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1. Show that for every m>1 m > 1 there is an n>m n > m such that amm a_m^m divides ann a_n^n. Is it true that a1 a_1 must divide ann a_n^n for some n>1 n > 1?
modular arithmeticnumber theoryInteger sequencerecurrence relationDivisibilityIMO ShortlistKazakhstan NMO 2022 p6
Preventing first player from getting 11 Xs in a row

Source: IMO Shortlist 1994, C6

10/22/2005
Two players play alternatively on an infinite square grid. The first player puts an XX in an empty cell and the second player puts an OO in an empty cell. The first player wins if he gets 1111 adjacent XX's in a line - horizontally, vertically or diagonally. Show that the second player can always prevent the first player from winning.
combinatorics unsolvedcombinatoricsIMO ShortlistCombinatorial gamesgame strategygame