1

Part of 2013 CentroAmerican

Problems(2)

Drawing Squares on an Infinite Grid

Source: CentroAmerican 2013 Problem 4

Ana and Beatriz take turns in a game that starts with a square of side

1

drawn on an infinite grid. Each turn consists of drawing a square that does not overlap with the rectangle already drawn, in such a way that one of its sides is a (complete) side of the figure already drawn. A player wins if she completes a rectangle whose area is a multiple of

5

. If Ana goes first, does either player have a winning strategy?

geometryrectanglecombinatorics unsolvedcombinatorics

Writing (n, 3^n) On a Board

Source: CentroAmerican 2013 Problem 1

Juan writes the list of pairs

(n, 3^n)

n=1, 2, 3,...

on a chalkboard. As he writes the list, he underlines the pairs

(n, 3^n)

n

3^n

have the same units digit. What is the

2013^{th}

underlined pair?

modular arithmeticnumber theory unsolvednumber theoryLast digit