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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