2021 AIME II Problems/Problem 7
Contents
[hide]Problem
Let and
be real numbers that satisfy the system of equations
There exist relatively prime positive integers
and
such that
Find
.
Solution 1
From the fourth equation we get substitute this into the third equation and you get
. Hence
. Solving we get
or
. From the first and second equation we get
, if
, substituting we get
. If you try solving this you see that this does not have real solutions in
, so
must be
. So
. Since
,
or
. If
, then the system
and
does not give you real solutions. So
. From here you already know
and
, so you can solve for
and
pretty easily and see that
. So the answer is
.
~ math31415926535
Solution 2
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 |
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