Difference between revisions of "2016 AIME I Problems"
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==Problem 13== | ==Problem 13== | ||
[[2016 AIME I Problems/Problem 13 | Solution]] | [[2016 AIME I Problems/Problem 13 | Solution]] | ||
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| + | Freddy the frog is jumping around the coordinate plane searching for a river, which lies on the horizontal line <math>y = 24</math>. A fence is located at the horizontal line <math>y = 0</math>. 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 <math>y=0</math>, 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 <math>y < 0</math>. Freddy starts his search at the point <math>(0, 21)</math> 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. | ||
==Problem 14== | ==Problem 14== | ||
Revision as of 14:20, 4 March 2016
| 2016 AIME I (Answer Key) | AoPS Contest Collections • PDF | ||
|
Instructions
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| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
to 
, inclusive. Your score will be the number of correct answers; i.e., there is neither partial credit nor a penalty for wrong answers.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
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.
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. 
