Difference between revisions of "2023 AIME II Problems/Problem 4"
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| + | == Problem == | ||
| + | Let <math>x,y,</math> and <math>z</math> be real numbers satisfying the system of equations | ||
| + | <cmath>\begin{align*} | ||
| + | xy + 4z &= 60 \\ | ||
| + | yz + 4x &= 60 \\ | ||
| + | zx + 4y &= 60. | ||
| + | \end{align*}</cmath> | ||
| + | Let <math>S</math> be the set of possible values of <math>x.</math> Find the sum of the squares of the elements of <math>S.</math> | ||
| + | |||
==Solution 1== | ==Solution 1== | ||
Revision as of 15:36, 16 February 2023
Problem
Let 
and 
be real numbers satisfying the system of equations
Let 
be the set of possible values of 
Find the sum of the squares of the elements of 
Solution 1
We first subtract the 2nd equation from the first, noting that they both equal 
.
![\[xy+4z-yz-4x=0\]](http://latex.artofproblemsolving.com/4/1/d/41de42a673c31f5a1ca95cb600947d8bde42b2fc.png)
![\[4(z-x)-y(z-x)=0\]](http://latex.artofproblemsolving.com/2/b/f/2bfeba47cf80c7f2ad7fa7cfebf03a46bae04eed.png)
![\[(z-x)(4-y)=0\]](http://latex.artofproblemsolving.com/f/1/2/f12a0d84e3439024da8bdbb59fc8151543b456d2.png)
Case 1: Let 
The first and third equations simplify to:
![\[x+z=15\]](http://latex.artofproblemsolving.com/c/f/4/cf4c2df2a5fedc96c127939686fe5e7eaa7fe3bf.png)
![\[xz=44\]](http://latex.artofproblemsolving.com/e/e/7/ee7532bfa95b5e9a79b354d631e4dda2f3c56f5b.png)
From which it is apparent that 
and 
are solutions.
Case 2: Let 
The first and third equations simplify to:
![\[xy+4x=60\]](http://latex.artofproblemsolving.com/1/c/7/1c77cbdb81faf12f89911633f6e154e4fc47c466.png)
![\[x^2+4y=60\]](http://latex.artofproblemsolving.com/3/1/0/31086e95fff556528b95ed2012be9f1aaf9a4285.png)
We subtract the following equations, yielding:
![\[x^2+4y-xy-4x=0\]](http://latex.artofproblemsolving.com/7/d/a/7dac40e4608060ab7dbd2f7d0a56d93b7113ffcc.png)
![\[x(x-4)-y(x-4)=0\]](http://latex.artofproblemsolving.com/6/c/5/6c53fb219271b714b3f891be37b4b79409cb6b40.png)
![\[(x-4)(x-y)=0\]](http://latex.artofproblemsolving.com/2/9/e/29e733edf39cffea61237b64b3cd816c24402646.png)
We thus have and 
, substituting in 
and solving yields 
and 
Then, we just add the squares of the solutions (make sure not to double count the 4), and get: 
~SAHANWIJETUNGA
