1. If \(\frac{1}{x^3} + \frac{1}{y^3} \propto \frac{1}{x^3 + y^3}\), then show that \(x \propto y\).

2. If \(\cfrac{1}{x}+\cfrac{1}{y}\propto \cfrac{1}{x+y}\), then show that \(x\propto y\).

3. If \((u^2 + v^2) \propto (x^2 + y^2)\) and\(uv \propto xy\), then show that\((u + v) \propto (x + y)\) when \(\frac{u}{v} + \frac{v}{u} = \frac{x}{y} + \frac{y}{x}\)."

4. If \(\left(\frac{1}{x} - \frac{1}{y}\right) \propto \frac{1}{x - y}\), then show that \((x^2 + y^2) \propto xy\).

5. If \(\left(x^3 - \frac{1}{y^3}\right) \propto \left(x^3 + \frac{1}{y^3}\right)\), then show that \(x \propto \frac{1}{y}\).

6. If \(x + y \propto x - y\), then show that \(x^3 + y^3 \propto x^3 - y^3\).

7. If \(x^3 + y^3 \propto x^3 - y^3\), then prove that \(x + y \propto x - y\).

8. If \(x \propto y\) and \(y \propto z\), then show that \(x + y \propto z\).

9. If \(3x - 4y \propto \sqrt{xy}\), then prove that \(x^2 + y^2 \propto xy\).

10. If \(x \propto y\), then show that \((x+y) \propto (x-y)\).

11. If \(\cfrac{x}{y} \propto (x+y)\) and \(\cfrac{y}{x} \propto (x-y)\), then show that \(x^2 - y^2\) is a constant.

12. If \(\cfrac{x}{y} \propto x + y\) and \(\cfrac{y}{x} \propto x - y\), then show that \(x^2 - y^2 =\) constant.

13. If \(\cfrac{x}{y} \propto x + y\) and \(\cfrac{y}{x} \propto x - y\), then show that \(x^2 - y^2\) is constant.

14. If \(x \propto y\) and \(y \propto z\), then show that \[ \frac{x}{yz} + \frac{y}{zx} + \frac{z}{xy} \propto \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \]

15. If \(\cfrac{x}{y} \propto x + y\) and \(\cfrac{y}{x} \propto x - y\), then show that \(x^2 - y^2\) is a constant.

16. "If \(x^3 + y^3\propto \x^3 - y^3\), then prove that \(x + y\) is proportional to \(x - y\)."

17. If \(x \propto y\) and \(y \propto z\), then show that \[ x^2 + y^2 + z^2 \propto xy + yz + zx \]

18. If \(x^2 + y^2 - z^2 \propto x^2 - y^2 + z^2\), then show that \(x^2 \propto (y^2 - z^2)\)

19. If \(\cfrac{x}{y} \propto (x + y)\) and \(\cfrac{y}{x} \propto (x - y)\), then show that \((x^2 - y^2)\) is constant.

20. If \(\frac{x}{y} \propto x + y\) and \(\frac{y}{x} \propto x - y\), then show that \((x^2 - y^2)\) is a constant quantity.

21. If \(a \propto b\) and \(b \propto c\), then show that \[ a^3b^3 + b^3c^3 + c^3a^3 \propto abc(a^3 + b^3 + c^3) \]

22. If \((a + b) \propto \sqrt{ab}\), then show that \((\sqrt{a} + \sqrt{b}) \propto (\sqrt{a} - \sqrt{b})\).

23. If \(x + y \propto x - y\), then prove that \(x^3 + y^3 \propto x^3 - y^3\).

24. If \(x+y \propto x-y\), prove that \(ax+by \propto px+qy\).

25. If \(x+y \propto x-y\), then prove that \(x^3+y^3\propto xy(x+y)\).

26. If \(a+b \propto \sqrt{ab}\), then show that \(\sqrt{a}+\sqrt{b}\propto \sqrt{a}-\sqrt{b}\).

27. If \(x \propto \cfrac{1}{y}\), then the value of \(x+y\) will be the smallest when \(x=y\).

28. If \(x^3 + y^3 \propto x^3 - y^3\), then prove that \(x^2 + xy + y^2 \propto x^2 - xy + y^2\).

29. If \(x + y \propto x - y\), then prove that \(x^4 + y^4 \propto x^2 y^2\).

30. If \(\cfrac{p}{q} \propto p+q\) and \(\cfrac{q}{p} \propto p-q\), then show that \(p^2 - q^2\) is constant.