Difference between revisions of "2017 AIME II Problems/Problem 13"
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==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 18:47, 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.