2021 AIME II Problems/Problem 7
Problem
Let 
and 
be real numbers that satisfy the system of equations
![\[a + b = -3\]](http://latex.artofproblemsolving.com/4/4/e/44e5b38260b7960662814ee711e8b85687684e4f.png)
![\[ab + bc + ca = -4\]](http://latex.artofproblemsolving.com/8/4/1/841e750802a3225f6fc501109e008cb0bc9ba970.png)
![\[abc + bcd + cda + dab = 14\]](http://latex.artofproblemsolving.com/5/0/4/5048cd3f5676696b0e72c258ecd3ffa2796794b8.png)
![\[abcd = 30.\]](http://latex.artofproblemsolving.com/f/a/6/fa6bc62da686109e2d14ee62a247b590baa24219.png)
There exist relatively prime positive integers 
and 
such that
![\[a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.\]](http://latex.artofproblemsolving.com/c/2/0/c20fc78cacf6996c3bfc20ed66cd4b0ac20f7bb3.png)
Find 
.
Solution
We can't have a solution without a problem.
See also
| 2021 AIME II (Problems • Answer Key • Resources) | ||
| Preceded by Problem 6 |
Followed by Problem 8 | |
| 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. 
