Difference between revisions of "2023 AIME II Problems/Problem 4"
(→Solution 2) |
MRENTHUSIASM (talk | contribs) m (→Solution 2) |
||
Line 107: | Line 107: | ||
~ Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ~ Steven Chen (Professor Chen Education Palace, www.professorchenedu.com) | ||
+ | |||
+ | == See also == | ||
+ | {{AIME box|year=2023|num-b=4|num-a=5|n=II}} | ||
+ | {{MAA Notice}} |
Revision as of 16:14, 16 February 2023
Contents
[hide]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)
See also
2023 AIME II (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 5 | |
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.