Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.7 AIME Prove it Soviet Union

Problem:

The quadratic $x^{2} + a x + b + 1$ has roots which are positive integers. Show that $(a^{2} + b^{2})$ is composite.

Solution

Solution:

Let the roots be $c$ , $d$ , so $c + d = -a$ , $cd = b+1$ . Hence $a^{2} + b^{2} = (c^{2} + 1)(d^{2} + 1)$ .

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