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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.

Level 4

         

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

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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