Difference between revisions of "2019 AIME II Problems/Problem 3"
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Revision as of 15:20, 22 March 2019
Problem 3
Find the number of 
-tuples of positive integers 
that satisfy the following systems of equations:
Solution
As 71 is prime, 
, 
, and 
must be 1, 1, and 71 (up to ordering). However, since and are divisors of 70 and 72 respectively, the only possibility is 
. Now we are left with finding the number of solutions 
satisfying 
and 
, which separates easily into two subproblems. The number of positive integer solutions to simply equals the number of divisors of 70 (as we can choose a divisor for 
, which uniquely determines 
). As 
, we have 
solutions. Similarly, 
, so 
.
Then the answer is simply 
.
-scrabbler94
See Also
| 2019 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 2 |
Followed by Problem 4 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
