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==Problem 6== | ==Problem 6== | ||
| − | + | In a Martian civilization, all logarithms whose bases are not specified as assumed to be base <math>b</math>, for some fixed <math>b\ge2</math>. A Martian student writes down | |
| + | <cmath>3\log(\sqrt{x}\log x)=56</cmath> | ||
| + | <cmath>\log_{\log x}(x)=54</cmath> | ||
| + | and finds that this system of equations has a single real number solution <math>x>1</math>. Find <math>b</math>. | ||
[[2019 AIME II Problems/Problem 6 | Solution]] | [[2019 AIME II Problems/Problem 6 | Solution]] | ||
Revision as of 15:29, 22 March 2019
| 2019 AIME II (Answer Key) | AoPS Contest Collections • PDF | ||
|
Instructions
| ||
| 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
Four ambassadors and one advisor for each of then are to be seated at a round table with 
chairs numbered in order to . Each ambassador must sit in an even-numbered chair. Each advisor must sit in a chair adjacent to his or her ambassador. There are 
ways for the 
people to be seated at the table under these conditions. Find the remainder when is divided by 
.
Problem 6
In a Martian civilization, all logarithms whose bases are not specified as assumed to be base 
, for some fixed 
. A Martian student writes down
![\[3\log(\sqrt{x}\log x)=56\]](http://latex.artofproblemsolving.com/e/d/d/edd14880fb8a07eb5e444758008df636655d25fa.png)
![\[\log_{\log x}(x)=54\]](http://latex.artofproblemsolving.com/a/4/f/a4f7d9de7f66eef8b9752b307cdda8a029dbc98b.png)
and finds that this system of equations has a single real number solution 
. Find .
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. 
