MathDB

1

Part of 2001 Putnam

Problems(2)

Putnam 2001 A1

Source:

2/26/2012
Consider a set SS and a binary operation ∗*, i.e. for each a,b∈Sa,b\in S, a∗b∈Sa*b\in S. Assume (a∗b)∗a=b(a*b)*a=b for all a,b∈Sa,b\in S. Prove that a∗(b∗a)=ba*(b*a)=b for all a,b∈Sa,b \in S.
Putnamcollege contests
Putnam 2001 B1

Source:

2/27/2012
Let nn be an even positive integer. Write the numbers 1,2,⋯ ,n21, 2, \cdots, n^2 in the squares of an n×nn \times n grid so that the kkth row, from left to right, is (k−1)n+1, (k−1)n+2, ⋯ , (k−1)n+n. (k-1)n + 1, \ (k-1)n + 2, \ \cdots, \ (k-1)n + n. Color the squares of the grid so that half of the squares in each row and in each column are red and the other half are black (a checkerboard coloring is one possibility). Prove that for each coloring, the sum of the numbers on the red squares is equal to the sum of the numbers on the black squares.
Putnamcollege contests