By: Tao Steven Zheng (鄭濤)

【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.