Here are 2 interesting problems from chapter 2 of Arthur Engel's book:

1. The positive integers are colored black and white. The sum of two differently colored

numbers is black, and their product is white. What is the product of two white

numbers? Find all such colorings.

numbers is black, and their product is white. What is the product of two white

numbers? Find all such colorings.

2. Each element of a 25 × 25 matrix is either +1 or −1. Let ai be the product of all

elements of the i-th row and bj be the product of all elements of the j-th column.

Prove that a

elements of the i-th row and bj be the product of all elements of the j-th column.

Prove that a

_{1}+ b_{1}+ · · · + a_{25}+ b_{25}≠ 0.
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