Question 1093005
<br>The sum of ANY two perfect square unit fractions is a rational number:<br>
{{{1/a^2 + 1/b^2 = (a^2+b^2)/(a^2b^2)}}}<br>
The denominator of that combined fraction is a perfect square, so the whole fraction is a perfect square rational number if the numerator is a perfect square.  So the sum of two perfect square unit fractions is a perfect square rational number if a^2+b^2 is a perfect square -- i.e., if<br>
{{{a^2+b^2 = c^2}}}
You should recognize that as the Pythagorean Theorem.  So we are looking for integers a, b, and c that can be the sides of a right triangle.<br>
In the problem, we are looking for the LARGEST perfect square unit fractions for which the sum of the two fractions is again a perfect square fraction.  If we want the largest fractions, then we want the fractions with the SMALLEST denominators.<br>
Since (3,4,5) is the smallest set of three integers for which a^2+b^2 = c^2, we want the denominators of our two perfect square unit fractions to be 3^2 and 4^2.<br>
So the answer to the problem is that the two largest perfect square unit fractions for which the sum of them is a perfect square rational number are 1/9 and 1/16 -- answer C.