HMMT February

Algebra Difficulty 4.9 AIME Find the answer

Find the smallest integer $n$ such that $\sqrt{n+99}-\sqrt{n}<1$ .

Your answer

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

Solution

This is equivalent to $\begin{aligned} \sqrt{n+99} & <\sqrt{n}+1 \\ n+99 & <n+1+2 \sqrt{n} \\ 49 & <\sqrt{n} \end{aligned}$ So the smallest integer $n$ with this property is $49^{2}+1=2402$ .

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