MathDB

6

Part of 2000 All-Russian Olympiad

Problems(3)

2n x 2n board with white/black markers

Source: All-Russian MO 2000

12/30/2012
On some cells of a 2n×2n2n \times 2n board are placed white and black markers (at most one marker on every cell). We first remove all black markers which are in the same column with a white marker, then remove all white markers which are in the same row with a black one. Prove that either the number of remaining white markers or that of remaining black markers does not exceed n2n^2.
combinatorics unsolvedcombinatorics
v_p(n) != 1 for perfect n>6

Source: All-Russian MO 2000

12/30/2012
A perfect number, greater than 66, is divisible by 33. Prove that it is also divisible by 99.
number theory unsolvednumber theoryeasy problem
v_7(n) != 1 for perfect n > 28

Source: All-Russian MO 2000

12/30/2012
A perfect number, greater than 2828 is divisible by 77. Prove that it is also divisible by 4949.
Eulernumber theory unsolvednumber theory