# 2008 UNCO Math Contest II Problems/Problem 8

## Problem

Triangle $ABC$ has integer side lengths. One side is twice the length of a second side.

$[asy] draw((0,0)--(185/16,sqrt(225-(185/16)^2))--(40,0)--cycle,black); MP("A",(0,0),W);MP("C",(185/16,sqrt(225-(185/16)^2)),N);MP("B",(40,0),E); [/asy]$

(a) If the third side has length $40$ what is the greatest possible perimeter?

(b) If the third side has length $n$ what is the greatest possible perimeter?

(c) Now suppose one side is three times the length of a second side and the third side has length of $40$. What is the maximum perimeter?

(d) Generalize

## Solution

a) 157 b) 4n - 3 c) 116 d) Given that triangle ABC has integer side lengths and that one side is $a$ times as long as the second, the maximum perimeter given third side, n, is $$(a + 1)(\lfloor\frac{n}{a-1}\rfloor - 1) + n$$

## See Also

 2008 UNCO Math Contest II (Problems • Answer Key • Resources) Preceded byProblem 7 Followed byProblem 9 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 All UNCO Math Contest Problems and Solutions