Can a Square be completely filled by smaller squares when none of the smaller squares have same area?

Question

Can a Square $S$ be completely filled by smaller squares $S_i$ when area of $S_i \neq S_j$ whenever $i \neq j$?

PS:The image is only meant to clarify the complete filling of squares otherwise it includes two squares of same area and a rectangle

Answer 1

Yes, it is possible. The Mathworld article has more details.

Regrettably, there is no way to pack squares with sides $1,2,3,\ldots, 24$ into a square of size $70$.

Answer 2

I'm not sure but there is some ways with Extremal or Infinite Descent idea in Mathematical Olympiad.

Source: Problem-Solving Strategies Part 14_2 Problem 2: Prove that we can not fill a cube, with smaller different cubes. (No two cubes are equal.)

This can be rejected if we prove that we can not square a square into smaller different squares. Check my post out on https://artofproblemsolving.com/community/c6h1522640_strategy_needed