Prove that there is a positive integer that 
can be written as the sum of squares of 
positive integers in two different ways.

There are lots of such positive integers.
65 and 85 are the two smallest examples:

65 = 1 + 64 = 1² + 8² 
65 = 16 + 49 = 4² + 7²

85 = 4 + 81 = 2² + 9²
85 = 36 + 49 = 5² + 7² 

Here's one that can be written as the 
sum of squares in three different ways:

325 = 1 + 324 = 1² + 18²
325 = 36 + 289 = 6² + 17²
325 = 100 + 225 = 10² + 15²

Here's one that can be written as the 
sum of squares in four different ways:

1105 = 16 + 1089 = 4² + 33² 
1105 = 81 + 1024 = 9² + 32²
1105 = 144 + 961 = 12² + 31²
1105 = 529 + 576 = 23² + 24²

Here's one that can be written as the 
sum of squares in five different ways:

5525 = 49 + 5476 = 7² + 74²
5525 = 196 + 5329 = 14² + 73²
5525 = 484 + 5041 = 22² + 71²
5525 = 625 + 4900 = 25² + 70²
5525 = 1681 + 3844 = 41² + 62²

Edwin