A square is partitioned into some rectangles. For each rectangle with side lengths $a,b$, its score is $a^2+b^2$. It turns out that the sum of scores of all rectangles is equal to the score of the big square. Is it true that all the rectangles in the partition are squares?
The converse is true: if all the rectangles in the partition are squares, then the score is simply two times the area of the square. Since the areas of the little squares add up to the area of the big square, the scores also do.