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

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

3. 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 \(0\).

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

19. If \( x \propto y \) and \( y \propto z \), prove that \( x^2 + y^2 + z^2 \propto xy + yz + zx \).

20. If \( x \propto y \) and \( y \propto z \), then prove that \( x^2+y^2+z^2 \propto xy+yz+zx \).

21. If \(x\propto y, y\propto z\), then prove that \(x^2+y^2+z^2\propto xy+yz+zx\).

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

23. 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

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

25. If \(\frac{ay - bx}{c} = \frac{cx - az}{b} = \frac{bz - cy}{a}\), then prove that \(\frac{x}{a} = \frac{y}{b} = \frac{z}{c}\).

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

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

28. 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.

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

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