Competition problem — Number theory

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Number theory Difficulty 4.7 AIME Find the answer

5. ( a ) Find two natural numbers $x$ and $y$ such that $x y + x$ and $x y + y$ are squares of two different natural numbers.
(b) Can such a pair of $x$ and $y$ be found between 988 and 1991?

Your answer

A number or a short expression. Fractions can be typed as 3/2, and spacing doesn't matter.

Solution

5. (a) For example, we can take $x=1, y=8$ .
( b) For definiteness, let $y>x$ . Then, we have $x^{2}3 x+1$ , which is impossible in the given interval.

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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.