2022 AIME II Problems/Problem 5
Problem
Twenty distinct points are marked on a circle and labeled through
in clockwise order. A line segment is drawn between every pair of points whose labels differ by a prime number. Find the number of triangles formed whose vertices are among the original
points.
Solution
Let ,
, and
be the vertex of a triangle that satisfies this problem, where
.
. Because
is the sum of two primes,
and
,
or
must be
. Let
, then
. There are only
primes less than
:
. Only
plus
equals another prime.
.
Once is determined,
and
. There are
values of
where
, and
values of
. Therefore the answer is
Note: I believe that you also need , so you cannot just simply do
. In addition, it is possible for
to occur. You would need to do casework to make sure
does not go above 20.
See Also
2022 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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.