2
Part of 2009 Argentina Iberoamerican TST
Problems(2)
Prove the lead wins the game
Source: 2009 Korean MO #5
3/29/2009
There are m \plus{} 1 horizontal lines and vertical lines on the plane so that m(m \plus{} 1) intersections are made.
A mark is placed at one of the points of the lowest horizontal line.
2 players play the game of the following rules on this lines and points.
1. Each player moves a mark from a point to a point along the lines in turns.
2. The segment is erased after a mark moved along it.
3. When a player cannot make a move, then he loses.
Prove that the lead always wins the game.
PS I haven't found a student who solved it. There can be no one.
analytic geometrycombinatorics proposedcombinatorics
Sequence and product of digits
Source: Argentina TST Iberoamerican 2009 Problem 5
8/25/2009
Let and be positive integers. Let be the sequence defined by
a_1 \equal{} a and
a_{n \plus{} 1} \equal{} a_n \plus{} k\pi(a_n)
where
is the product of the digits of (written in base ten)
Prove that we can choose and such that the infinite sequence contains exactly distinct terms
number theory unsolvednumber theory