By: Tao Steven Zheng (郑涛)

【Problem】

Let  be four consecutive integers.

Part 1: Prove that  is a perfect square.

Part 2: Prove that the square root of  is equal to the average of  and .

【Solution】

Part 1

Let  be represented as , where are integers. Thus,

A smart way to factor the right hand side of the above equation is to rearrange the terms as follows:

Then multiply as follows:

Let , then

Since

then

Therefore,

Since is an integer, we prove the result is a perfect square.

Part 2

The average of  and is

which is the square root of . This should be obvious from the above factoring method.