2007 iTest Problems/Problem 46
Problem
Let be an ordered triplet of real numbers that satisfies the following system of equations:
If
is the minimum possible value of
, find the modulo
residue of
.
Solution
Rearrange the terms to get
Since the left hand side of all three equations is greater than or equal to 0,
. Also, note that the equations have symmetry, so WLOG, let
. By substitution, we have
Note that and
. That means
. Since
,
Since
, then
. Because
and
are nonpositive,
.
Using substitution in the original system,
To find the real solutions, we use casework and the Zero Product Property.
Case 1:
If , then since
and
are nonpositive, then
. Substitution results in
That means
or
. For the first equation,
. For the second equation, note that
, and since
,
, where
is a real number. Since
and
, the root of
is less than
but more than
, so
Case 2:
Because ,
. From one of the original equations,
Using the Rational Root Theorem,
Note that if
, then
, so that won’t work. Let
(where
since
), so
If
, then
Thus, there are no solutions in this case.
From the two cases, the smallest possible value of is
, so the modulo
residue of
is
.
See Also
2007 iTest (Problems, Answer Key) | ||
Preceded by: Problem 45 |
Followed by: Problem 47 | |
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