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==Problem 11== | ==Problem 11== |
Revision as of 17:06, 12 March 2021
2021 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
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Problem 11
Let be a cyclic quadrilateral with
and
. Let
and
be the feet of the perpendiculars from
and
, respectively, to line
and let
and
be the feet of the perpendiculars from
and
respectively, to line
. The perimeter of
is
, where
and
are relatively prime positive integers. Find
.
Problem 12
Let be a dodecagon (12-gon). Three frogs initially sit at
and
. At the end of each minute, simultaneously, each of the three frogs jumps to one of the two vertices adjacent to its current position, chosen randomly and independently with both choices being equally likely. All three frogs stop jumping as soon as two frogs arrive at the same vertex at the same time. The expected number of minutes until the frogs stop jumping is
, where
and
are relatively prime positive integers. Find
.
Problem 13
Circles and
with radii
and
, respectively, intersect at distinct points
and
. A third circle
is externally tangent to both
and
. Suppose line
intersects
at two points
and
such that the measure of minor arc
is
. Find the distance between the centers of
and
.
Problem 14
For any positive integer
denotes the sum of the positive integer divisors of
. Let
be the least positive integer such that
is divisible by
for all positive integers
. Find the sum of the prime factors in the prime factorization of
.
Problem 15
Let be the set of positive integers
such that the two parabolas
intersect in four distinct points, and these four points lie on a circle with radius at most
. Find the sum of the least element of
and the greatest element of
.
See also
2021 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2020 AIME II |
Followed by 2021 AIME II | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
- American Invitational Mathematics Examination
- AIME Problems and Solutions
- Mathematics competition resources
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.