Algebra
# Systems of Equations

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?

×

Problem Loading...

Note Loading...

Set Loading...