Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 4.0 AIME Find the answer South Africa

For how many positive values of $n$ are both $\frac{1}{2}n$ and $2n$ two-digit integers?

Your answer

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

Solution

The smallest value of $n$ is 20 (for which $\frac{1}{2}n$ is 10) and the largest value of $n$ is 49 (for which $2n$ is 98). From 20 to 49 inclusive represents 30 integers, half of which are even (so that $\frac{1}{2}n$ is an integer).

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