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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 5

\[\begin{cases} 4\dfrac{\sqrt{x^{2} + 1}}{x} = 5\dfrac{\sqrt{y^{2} + 1}}{y}=6\dfrac{\sqrt{z^{2} + 1}}{z} \\ xyz = x + y + z\end{cases}\]

If \(x = \dfrac{\sqrt{a}}{b} , y = c\dfrac{\sqrt{d}}{e} , z = f\sqrt{g}\) satisfy the above system of equations, then find \( (a + b + \ldots + g )\).

  • The expression is in simplest terms, meaning that all of the variables are positive integers, \( a, d, g \) are square free and \(c, e \) are coprime.

Let \(a\) and \(b\) be positive integers such that \(a\ge\ b\) and\[\frac{a+1}{b}+\frac{b+1}{a}\]is also a positive integer. Find sum of all possible values of \(a\) which are less than 1000.

The system of equations \[ 4a^4+36a^2b^2+9b^4=20a^2+30b^2-1\] \[2a^3b+3ab^3=5ab\] has \(n\) nonnegative (meaning both \(a\) and \(b\) are nonnegative) real solutions. Let these solutions be \((a_1,b_1),(a_2,b_2),\cdots (a_n,b_n)\). The value of the sum \[\sum_{i=1}^n a_i+b_i\] can be expressed in the form \(\dfrac{a+b\sqrt{c}}{d}\) where \(\gcd (a,b,d)=1\) and \(c\) is squarefree. Find \(a+b+c+d\).

A positive real number is given. In each move, we can either add \(3\) to it, subtract \(3\) to it, multiply it by \(3\) and divide it by \(3\). The sum of all numbers such that after exactly \(3\) moves, the original number comes back is \(\frac{a}{b}\), where \(a\) and \(b\) are relatively prime positive integers. Find \(a + b\).

Find the largest possible value of \(x^3+y^3+z^3\) for real \(x,\ y,\ z,\) such that \[ \begin{cases} xyz^2=-64y-128x\\ x^2yz=-32y-32z\\ 3xy^2z=128x-64z \end{cases} \]

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