2016 AIME I Problems
2016 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
Instructions
| ||
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 |
Contents
Problem 1
Problem 2
Problem 3
Problem 4
Problem 5
Problem 6
In let
be the center of the inscribed circle, and let the bisector of
intersect
at
. The line through
and
intersects the circumscribed circle of
at the two points
and
. If
and
, then
, where
and
are relatively prime positive integers. Find
.
Problem 7
Problem 8
For a permutation of the digits
, let
denote the sum of the three
-digit numbers
,
, and
. Let
be the minimum value of
subject to the condition that the units digit of
is
. Let
denote the number of permutations
with
. Find
.
Solution
Problem 9
Problem 10
A strictly increasing sequence of positive integers ,
,
,
has the property that for every positive integer
, the subsequence
,
,
is geometric and the subsequence
,
,
is arithmetic. Suppose that
. Find
.
Problem 11
Let be a nonzero polynomial such that
for every real
, and
. Then
, where
and
are relatively prime positive integers. Find
.
Problem 12
Problem 13
Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line . A fence is located at the horizontal line
. On each jump Freddy randomly chooses a direction parallel to one of the coordinate axes and moves one unit in that direction. When he is at a point where
, with equal likelihoods he chooses one of three directions where he either jumps parallel to the fence or jumps away from the fence, but he never chooses the direction that would have him cross over the fence to where
. Freddy starts his search at the point
and will stop once he reaches a point on the river. Find the expected number of jumps it will take Freddy to reach the river.
Solution
Problem 14
Problem 15
2016 AIME I (Problems • Answer Key • Resources) | ||
Preceded by 2015 AIME II |
Followed by 2016 AIME II | |
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.