2007 iTest Problems/Problem 46

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Problem

Let $(x,y,z)$ be an ordered triplet of real numbers that satisfies the following system of equations: \begin{align*}x+y^2+z^4&=0,\\y+z^2+x^4&=0,\\z+x^2+y^4&=0.\end{align*} If $m$ is the minimum possible value of $\lfloor x^3+y^3+z^3\rfloor$, find the modulo $2007$ residue of $m$.

Solution

Rearrange the terms to get \[y^2 + z^4 = -x\] \[z^2 + x^4 = -y\] \[x^2 + y^4 = -z\] Since the left hand side of all three equations is greater than or equal to 0, $x,y,z \le 0$. Also, note that the equations have symmetry, so WLOG, let $0 \ge x \ge y \ge z$. By substitution, we have

\[y^2 + z^4 \le z^2 + x^4 \le x^2 + y^4\]

Note that $0 \le x^2 \le y^2 \le z^2$ and $0 \le x^4 \le y^4 \le z^4$. That means $x^2 + y^4 \le x^2 + z^4$. Since $y^2 + z^4 \le x^2 + y^4$, \[y^2 + z^4 \le x^2 + z^4\] \[y^2 \le x^2\] Since $x^2 \le y^2$, then $x^2 = y^2$. Because $x$ and $y$ are nonpositive, $x = y$.


Using substitution in the original system, \[x^2 + z^4 = z^2 + x^4\] \[(z^2 + x^2)(z^2 - x^2) - (z^2 - x^2) = 0\] \[(z^2 - x^2)(x^2 + z^2 - 1) = 0\] To find the real solutions, we use casework and the Zero Product Property.


Case 1: $z^2 = x^2$

If $z^2 = x^2$, then since $z$ and $x$ are nonpositive, then $z = x$. Substitution results in \[x+x^2+x^4 = 0\] \[x(1+x+x^3) = 0\] That means $x = 0$ or $x^3 + x + 1 = 0$. For the first equation, $m = 0$. For the second equation, note that $x^3 = -x-1$, and since $x = y = z$, $m = \lfloor -3x-3 \rfloor$, where $x$ is a real number. Since $-\tfrac{1}{3}^3 - \tfrac13 + 1 = \tfrac{16}{27}$ and $-\tfrac{2}{3}^3 - \tfrac23 + 1 = \tfrac{1}{27}$, the root of $x$ is less than $-\tfrac23$ but more than $-1$, so \[0 > -3x-3 > -1\] \[m = \lfloor -3x-3 \rfloor = -1\]


Case 2: $x^2 + z^2 = 1$

Because $x^2 \le z^2$, $x^2 \le \frac12$. From one of the original equations, \[z^2 + x^4 = -y\] \[1 - x^2 + x^4 + x = 0\] Using the Rational Root Theorem, \[(x+1)(x^3 - x^2 + 1) = 0\] Note that if $x = -1$, then $x^2 \ge \tfrac12$, so that won’t work. Let $x = -a$ (where $a \ge 0$ since $x \le 0$), so \[a^3 + a^2 = 1\] If $a \le \tfrac{\sqrt2}{2}$, then \[a^3 + a^2 \le \frac{\sqrt2}{4} + \frac12\] \[a^3 + a^2 \le \frac{\sqrt2 + 2}{4} < 1\] Thus, there are no solutions in this case.


From the two cases, the smallest possible value of $m$ is $-1$, so the modulo $2007$ residue of $m$ is $\boxed{2006}$.

See Also

2007 iTest (Problems)
Preceded by:
Problem 45
Followed by:
Problem 47
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