Difference between revisions of "2017 AIME II Problems/Problem 13"
m (→Solution) |
|||
| Line 3: | Line 3: | ||
==Solution== | ==Solution== | ||
| − | <math>\boxed{245}</math> | + | Considering <math>n \pmod{6}</math>, we have the following formulas: |
| + | |||
| + | Even and a multiple of 3: <math>\frac{n(n-4)}{2} + \frac{n}{3}</math> | ||
| + | Even and not a multiple of 3: <math>\frac{n(n-2)}{2}</math> | ||
| + | Odd and a multiple of 3: <math>\frac{n(n-3)}{2} + \frac{n}{3}</math> | ||
| + | Odd and not a multiple of 3: <math>\frac{n(n-1)}{2}</math> | ||
| + | |||
| + | Considering the six possibilities <math>n \equiv 0,1,2,3,4,5 \pmod{6}</math> and solving, we find that the only valid solutions are <math>n = 36, 52, 157</math>, from which the answer is <math>36 + 52 + 157 = \boxed{245}</math>. | ||
=See Also= | =See Also= | ||
{{AIME box|year=2017|n=II|num-b=12|num-a=14}} | {{AIME box|year=2017|n=II|num-b=12|num-a=14}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 17:47, 23 March 2017
Problem
For each integer 
, let 
be the number of 
-element subsets of the vertices of the regular 
-gon that are the vertices of an isosceles triangle (including equilateral triangles). Find the sum of all values of such that 
.
Solution
Considering 
, we have the following formulas:
Even and a multiple of 3: 
Even and not a multiple of 3: 
Odd and a multiple of 3: 
Odd and not a multiple of 3: 
Considering the six possibilities 
and solving, we find that the only valid solutions are 
, from which the answer is 
.
See Also
| 2017 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
