Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 5.0 AIME Find the answer

14. The lengths of the sides of a rectangle are positive integers, and the numerical value of its area is equal to twice the numerical value of its perimeter. There are $\qquad$ such rectangles.

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Note: The blank space represented by $\qquad$ is kept as is in the translation.

Solution

14.3.

Let the sides of the rectangle be $a$ and $b$ ( $a \geqslant b$ ). Then $ab = 2(2a + 2b)$ , which simplifies to $(a-4)(b-4) = 16$ .
Since $a$ and $b$ are positive integers, we have $\left\{\begin{array}{l}a-4=16,8,4, \\ b-4=1,2,4 .\end{array}\right.$ Solving this, we get $\left\{\begin{array}{l}a=20,12,8, \\ b=5,6,8 .\end{array}\right.$ There are 3 solutions.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.