2023 AIME II Problems/Problem 4
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 .
Case 1: Let
The first and third equations simplify to:
From which it is apparent that and
are solutions.
Case 2: Let
The first and third equations simplify to:
We subtract the following equations, yielding:
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
Solution 2
We index these equations as (1), (2), and (3), respectively.
Taking , we get
Denote ,
,
.
Thus, the above equation can be equivalently written as
Similarly, by taking , we get
By taking , we get
From , we have the following two cases.
Case 1: .
Plugging this into and
, we get
.
Thus,
or
.
Because we only need to compute all possible values of , without loss of generality, we only need to analyze one case that
.
Plugging and
into (1), we get a feasible solution
,
,
.
Case 2: and
.
Plugging this into and
, we get
.
Case 2.1: .
Thus, . Plugging
and
into (1), we get a feasible solution
,
,
.
Case 2.2: and
.
Thus, . Plugging these into (1), we get
or
.
Putting all cases together, .
Therefore, the sum of the squares of the elements of
is
Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)