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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.

There are two containers named \(A\) and \(B\). Container \(A\) has milk in it, container \(B\) has water (same amount as of milk in container \(A\)) in it. Some amount of milk is transferred from container \(A\) to container \(B\) and then , the same amount is transferred from the container \(B\) to the container \(A\).

Let container \(A\) has \(x\) amount of water in it and container \(B\) has \(y\) amount of milk in it.

Which one option is correct?

**Note :** Finally the amount in both the containers is same as it was initially.

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Find the number of real solutions \((x,y,z)\) of

\[ \begin{cases} (x+y)^{3}=z \\ (y+z)^{3}=x \\ (z+x)^{3}=y \end{cases} \]

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What is the largest integer \( n \leq 1000 \), such that there exist 2 non-negative integers \((a, b)\) satisfying

\[ n = \frac{ a^2 + b^2 } { ab - 1 } ? \]

**Hint**:

\( (a,b) = (0,0) \) gives us \( \frac{ 0^2 + 0^2 } { 0 \times 0 - 1 } = 0\), so the answer is at least \( 0 .\)

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\[\begin{cases} (1+x)(1+x^2)(1+x^4) = 1+y^7 \\ (1+y)(1+y^2)(1+y^4) = 1+x^7 \end{cases} \]

How many ordered pairs of real numbers \((x, y)\) satisfy the above equations?

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