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) | ||
Preceded by: Problem 45 |
Followed by: Problem 47 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • TB1 • TB2 • TB3 • TB4 |