MathDB

4

Part of 2002 Tuymaada Olympiad

Problems(2)

1x2 tiling of a 2001x2002 table.

Source: Tuymaada 2002, day 1, problem 4. - Author : S. Volchenkov.

5/3/2007
A rectangular table with 2001 rows and 2002 columns is partitioned into 1×21\times 2 rectangles. It is known that any other partition of the table into 1×21\times 2 rectangles contains a rectangle belonging to the original partition. Prove that the original partition contains two successive columns covered by 2001 horizontal rectangles.
Proposed by S. Volchenkov
geometryrectanglecombinatorics proposedcombinatorics
Tuymaada 2002

Source:

1/22/2007
A real number aa is given. The sequence n1<n2<n3<...n_{1}< n_{2}< n_{3}< ... consists of all the positive integral nn such that {na}<110\{na\}< \frac{1}{10}. Prove that there are at most three different numbers among the numbers n2−n1n_{2}-n_{1}, n3−n2n_{3}-n_{2}, n4−n3n_{4}-n_{3}, …\ldots.
A corollary of a theorem from ergodic theory
calculusintegrationfloor functionpigeonhole principlenumber theory unsolvednumber theory