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# Systems of Equations

Vieta root jumping is a descent method that occurs when you have to solve a Diophantine equation whose solutions have a recursive structure. Popular in advanced math olympiad number theory problems.

# System of Equations: Level 4 Challenges

$\dfrac{x}{1-x^2}+\dfrac{y}{1-y^2}+\dfrac{z}{1-z^2} \\ =\dfrac{kxyz}{(1-x^2)(1-y^2)(1-z^2)}$

If $$x,y,$$ and $$z$$ are real numbers satisfying $$xy+yz+xz=1$$, find the value of $$k$$ for which the equation above holds true.

If $$a, b$$ and $$c$$ are non-zero real numbers, that satisfy the equations $$a^2 + b^2 + c^2 = 1$$ and $$a\left(\frac {1}{b} + \frac {1}{c}\right) + b\left(\frac {1}{c} + \frac {1}{a}\right) + c\left( \frac {1}{a} + \frac {1}{b}\right) = -3$$, how many possible values are there for $$a + b + c$$?

How many unordered triplets $$(x,y,z)$$ are in the range $$[0,2]$$ and satisfy the system of equations: $\left\{\begin{array}{l}2x^2-4x+2=y\\ 2y^2-4y+2=z\\ 2z^2-4z+2=x\end{array}\right.$

Details and Assumptions:

An unordered triplet means that $$(1,2,3)$$ is the same as $$(3,2,1)$$ or $$(1,3,2)$$.

The positive reals $$x,y$$ and $$z$$ satisfy the given equations:

$x^{2}+xy+\frac{y^{2}}{3}=25$

$z^{2}+\frac{y^{2}}{3}=9$

$z^{2}+zx+x^{2}=16$

Find the value of $$(xy+2yz+3xz)^{2}$$.

$$x$$ satisfies the equation $\sqrt{x+\sqrt{x}}-\sqrt{x-\sqrt{x}}=\dfrac{199}{100}\sqrt{\dfrac{x}{x+\sqrt{x}}}$

Also, its value can be expressed in the form $$\frac{m}{n}$$, where $$m$$ and $$n$$ are coprime positive integers, find $$m+n$$.

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