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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 3 Challenges

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.

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

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

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 $$0 .$$

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

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

$\left\{ \begin{array}{l} \frac{{bcdef}}{a} = 4\\ \frac{{acdef}}{b} = 9\\ \frac{{abdef}}{c} = 16\\ \frac{{abcef}}{d} = \frac{1}{4}\\ \frac{{abcdf}}{e} = \frac{1}{9}\\ \frac{{abcde}}{f} = \frac{1}{{16}} \end{array} \right.$ Positive numbers $a,b,c,d,e,f$ satisfy the formulas above. Find the value of $(a+ c + e) - (b + d + f)$

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