Difference between revisions of "2021 AIME II Problems/Problem 7"
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==Problem== | ==Problem== | ||
| − | + | Let <math>a, b, c,</math> and <math>d</math> be real numbers that satisfy the system of equations | |
| + | <cmath>a + b = -3</cmath><cmath>ab + bc + ca = -4</cmath><cmath>abc + bcd + cda + dab = 14</cmath><cmath>abcd = 30.</cmath>There exist relatively prime positive integers <math>m</math> and <math>n</math> such that | ||
| + | <cmath>a^2 + b^2 + c^2 + d^2 = \frac{m}{n}.</cmath>Find <math>m + n</math>. | ||
| + | |||
==Solution== | ==Solution== | ||
We can't have a solution without a problem. | We can't have a solution without a problem. | ||
Revision as of 14:33, 22 March 2021
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 | ||
These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. 
