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A triangle whose side lengths are whole numbers has one side which measures $25$ inches and a perimeter of $80$ inches. What is the fewest number of inches that can be the length of one of the remaining sides?

 Feb 12, 2021
 #1
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By the triangle inequality, the minimum of the remaining side is 80 - 25 = 55.

 Feb 12, 2021
 #2
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I'm sorry guest but thet is wrong

Let $s$ and $\ell$ be the shorter and longer of the remaining sides of the triangle. We are told that $25+s+\ell=80$. By the Triangle Inequality,$$s+25>\ell.$$Adding $\ell$ to both sides of the inequality gives \begin{align*} s+25+\ell &> 2\ell \\ 80 &> 2\ell, \end{align*}so the largest value of $\ell$ is $39$. In that case, $s=80-(25+\ell)=80-64=\boxed{16}$.

 Feb 12, 2021

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