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Singapore Mathematical Olympiad (SMO)

Geometry Difficulty 4.8 AIME Prove it Singapore

A square is cut into several rectangles, none of which is a square, so that the sides of each rectangle are parallel to the sides of the square. For each rectangle with sides $a$ , $b$ , $a < b$ , compute the ratio $a/b$ . Prove that sum of these ratios is at least $1$ .

Solution

Without loss of generality, we may assume the square has area $1$ . Let the sides of the rectangles be $a_i$ , $b_i$ , $a_i \leq b_i$ and $S_i = a_i b_i$ . We have $\sum S_i = 1$ and
$\sum \frac{a_i}{b_i} = \sum \frac{S_i}{b_i^2} \geq \sum S_i = 1.$

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