MathDB

3

Part of 1998 Irish Math Olympiad

Problems(2)

base b

Source: Ireland 1998

7/4/2009
Show that no integer of the form xyxy xyxy in base 10 10 can be a perfect cube. Find the smallest base b>1 b>1 for which there is a perfect cube of the form xyxy xyxy in base b b.
geometry3D geometrynumber theory proposednumber theory
sets

Source: Ireland 1998

7/4/2009
(a) (a) Prove that N \mathbb{N} can be partitioned into three (mutually disjoint) sets such that, if m,n∈N m,n \in \mathbb{N} and |m\minus{}n| is 2 2 or 5 5, then m m and n n are in different sets. (b) (b) Prove that N \mathbb{N} can be partitioned into four sets such that, if m,n∈N m,n \in \mathbb{N} and |m\minus{}n| is 2,3, 2,3, or 5 5, then m m and n n are in different sets. Show, however, that N \mathbb{N} cannot be partitioned into three sets with this property.
modular arithmeticcombinatorics proposedcombinatorics