Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.9 AIME Prove it North Macedonia

If one of the sides of a square is increased two times and the other one is increased by $22\text{ mm}$ , then the new rectangle has a perimeter that is $2000\text{ mm}$ greater than the perimeter of the square. What is the side length of the square?

Solution

If we denote the side of the square by $a$ , then the side lengths of the rectangle are $2a$ and $a+22$ . The sum of the two side lengths of the rectangle with length $2a$ is equal to the perimeter of the square. Hence the sum of the other two side lengths, with length $a+22$ , is $2000\text{ mm}$ . Now we obtain $2a+44=2000$ , $2a=1956$ , $a=1956:2=978\text{ mm}$ .

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