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==Problem 2== | ==Problem 2== | ||
| − | Lily pads <math>1,2,3,\ldots</math> lie in a row on a pond. A frog makes a sequence of jumps starting on pad <math>1</math>. From any pad <math>k</math> the frog jumps to either pad <math>k+1</math> or pad <math>k+2</math> chosen randomly with probability <math>\tfrac{1}{2}</math> and independently of other jumps. The probability that the frog visits pad <math>7</math> is <math>\ | + | Lily pads <math>1,2,3,\ldots</math> lie in a row on a pond. A frog makes a sequence of jumps starting on pad <math>1</math>. From any pad <math>k</math> the frog jumps to either pad <math>k+1</math> or pad <math>k+2</math> chosen randomly with probability <math>\tfrac{1}{2}</math> and independently of other jumps. The probability that the frog visits pad <math>7</math> is <math>\tfrac{p}{q}</math>, where <math>p</math> and <math>q</math> are relatively prime positive integers. Find <math>p+q</math>. |
[[2019 AIME II Problems/Problem 2 | Solution]] | [[2019 AIME II Problems/Problem 2 | Solution]] | ||
Revision as of 15:24, 22 March 2019
| 2019 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
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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
Two different points, 
and 
, lie on the same side of line 
so that 
and 
are congruent with 
, and 
. The intersection of these two triangular regions has area 
, where 
and 
are relatively prime positive integers. Find 
.
Problem 2
Lily pads 
lie in a row on a pond. A frog makes a sequence of jumps starting on pad 
. From any pad 
the frog jumps to either pad 
or pad 
chosen randomly with probability 
and independently of other jumps. The probability that the frog visits pad 
is 
, where 
and 
are relatively prime positive integers. Find 
.
Problem 3
Find the number of -tuples of positive integers 
that satisfy the following system of equations:
![\[abc=70\]](http://latex.artofproblemsolving.com/5/5/0/5500cb940a2bbc21aadaf9b1689b2a607908d76d.png)
![\[cde=71\]](http://latex.artofproblemsolving.com/1/f/4/1f434b64e58553de1c3f6b5b17d11e1ab4ed2550.png)
![\[efg=72.\]](http://latex.artofproblemsolving.com/4/6/4/464cc931f514d80bf2aacddebd43e8f477f2bbee.png)
Problem 4
A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is , where and are relatively prime positive integers. Find .
Problem 5
Problem 6
Problem 7
Problem 8
Problem 9
Problem 10
Problem 11
Problem 12
Problem 13
Problem 14
Problem 15
| 2019 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by 2019 AIME I |
Followed by 2020 AIME I | |
| 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. 
