Difference between revisions of "2023 AIME II Problems/Problem 4"
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Revision as of 16:11, 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 .
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)