Difference between revisions of "2008 UNCO Math Contest II Problems/Problem 8"
(Created page with "== Problem == Triangle <math>ABC</math> 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...") |
(→Solution) |
||
| Line 22: | Line 22: | ||
== Solution == | == Solution == | ||
| − | + | a) 157 | |
| + | b) 4n - 3 | ||
| + | c) 116 | ||
| + | d) Given that triangle ABC has integer side lengths and that one side is <math>a</math> times as long as the second, the maximum perimeter given third side, n, is <cmath>(a + 1)(\lfloor\frac{n}{a-1}\rfloor - 1) + n</cmath> | ||
== See Also == | == See Also == | ||
Latest revision as of 17:37, 23 November 2016
Problem
Triangle 
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]](http://latex.artofproblemsolving.com/d/6/e/d6e0cbadcbdd84a96434e27c99b7e125940f4053.png)
(a) If the third side has length 
what is the greatest possible perimeter?
(b) If the third side has length 
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 . 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 
times as long as the second, the maximum perimeter given third side, n, is ![\[(a + 1)(\lfloor\frac{n}{a-1}\rfloor - 1) + n\]](http://latex.artofproblemsolving.com/2/d/f/2dfcccc9e9c86552921528d0b6a1f938b4af87b3.png)
See Also
| 2008 UNCO Math Contest II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 7 |
Followed by Problem 9 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 | ||
| All UNCO Math Contest Problems and Solutions | ||
