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

2. If \(y - z \propto \cfrac{1}{x}\), \(z - x \propto \cfrac{1}{y}\), and \(x - y \propto \cfrac{1}{z}\), then the sum of the three proportional constants is \(0\).

3. If \(x \propto \cfrac{1}{y}\), then \(y \propto \cfrac{1}{x}\).

4. If \( \cfrac{1}{y} - \cfrac{1}{x} \propto \cfrac{1}{x - y} \), prove that \( x \propto y \).

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

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

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

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

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

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

11. 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.

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

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

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

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

16. If \(x \propto \cfrac{1}{y}\) and \(z \propto \cfrac{1}{y}\), then \(x \propto y\).

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

18. If \(y \propto \cfrac{1}{x}\), then \(\cfrac{y}{x}\) is a nonzero constant.

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

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

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

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

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

24. If \(\frac{x}{y} \propto (x + y)\) and \(\frac{y}{x} \propto (x - y)\), then find the value of \(x^2 - y^2\).

(a) "Proportional to \(x\)" (b) "Proportional to \(y\)" (c) "Proportional to \(xy\)" (d) constant

25. If \( x \propto yz \) and \( y \propto zx \), then show that \( z \ne 0 \) is a constant.

26. If \(\cfrac{y+z-x}{b+c-a} = \cfrac{z+x-y}{c+a-b} = \cfrac{x+y-z}{a+b-c}\), then show that \(\cfrac{x}{a} = \cfrac{y}{b} = \cfrac{z}{c}\).

27. If \(x \propto y\) and \(y \propto z\), then \((x + y + z) \propto (\sqrt{yz} + \sqrt{zx} + \sqrt{xy})\).

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

29. If \(\frac{x}{y} \propto z\), \(\frac{y}{z} \propto x\), and \(\frac{z}{x} \propto y\), then \(xyz = 1\).

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