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Perfect square Problem | AIME I, 1999 | Question 3

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 2011 based on Rectangles and sides.

Try this beautiful problem from the American Invitational Mathematics Examination I, AIME I, 1999 based on Perfect square and Integers.

Perfect square Problem – AIME I, 1999

Find the sum of all positive integers n for which $n^{2}-19n+99$ is a perfect square.

• is 107
• is 38
• is 840
• cannot be determined from the given information

Key Concepts

Perfect Square

Integers

Inequalities

AIME I, 1999, Question 3

Elementary Number Theory by David Burton

Try with Hints

$(n-10)^{2}$ $\lt$ $n^{2}-19n+99$ $\lt$ $(n-8)^{2}$ and $n^{2}-19n+99$ is perfect square then $n^{2}-19n+99$=$(n-9)^{2}$ that is n=18

and $n^{2}-19n+99$ also perfect square for n=1,9,10