MathDB

4

Part of 1999 Irish Math Olympiad

Problems(2)

divisors

Source: Ireland 1999

7/4/2009
Find all positive integers m m with the property that the fourth power of the number of (positive) divisors of m m equals m m.
number theory
tiling

Source: Ireland 1999

7/4/2009
A 100×100 100 \times 100 square floor consisting of 10000 10000 squares is to be tiled by rectangular 1×3 1 \times 3 tiles, fitting exactly over three squares of the floor. (a) (a) If a 2×2 2 \times 2 square is removed from the center of the floor, prove that the rest of the floor can be tiled with the available tiles. (b) (b) If, instead, a 2×2 2 \times 2 square is removed from the corner, prove that such a tiling is not possble.
geometryrectanglecombinatorics proposedcombinatorics