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

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

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

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

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

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

7. If \(x+y ∝ x−y\), show that \(x^3+y^3 ∝ x^3−y^3\).

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

9. If \(2x^2 + 3y^2 \propto xy\), then show that \(x \propto y\).

10. If \(x \propto y\) and \(y \propto z\), prove that \(x^3 + y^3 + z^3 \propto xyz\).

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

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

15. If \(\left(x^3 - \frac{1}{y^3}\right) \propto \left(x^3 + \frac{1}{y^3}\right)\), then prove that \(xy\) is a constant.

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

17. If \(x : a = y : b = z : c\), then show that \(\frac{x^3}{a^3} + \frac{y^3}{b^3} + \frac{z^3}{c^3} = \frac{3xyz}{abc}\).

18. 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.

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

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

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

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

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

24. If \(x : a = y : b = z : c\), then show that \(\cfrac{x^3}{a^3} + \cfrac{y^3}{b^3} + \cfrac{z^3}{c^3} = \cfrac{3xyz}{abc}\).

25. If \(x : a = y : b = z : c\), then show that \[ \frac{x^3 + y^3 + z^3}{a^3 + b^3 + c^3} = \frac{xyz}{abc} \]

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

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

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

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

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