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

2. Given: \(a \propto b\) and \(b \propto c\) Show that: \(a^3 + b^3 + c^3 \propto abc\)

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

4. If \(a \propto b\) and \(b \propto c\), prove that \(a^3 + b^3 + c^3 \propto 3abc\). Let me know if you'd like this proven step by step. I'm here for it.

5. If \(a \propto b\) and \(b \propto c\), prove that \(a^3 + b^3 + c^3 \propto 3abc\).

6. If \(a \propto b\) and \(b \propto c\), then prove that \(a^3 + b^3 + c^3 \propto 3abc\)

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

8. If \(a \propto b\) and \(b \propto c\), then prove that \(a^3 + b^3 + c^3 \propto 3abc\).

9. If \(a \propto b\) and \(b \propto c\), then prove that \(a + b + c \propto 5abc\).

10. If \(a\propto b\) and \(b\propto c\), then show that \((pa+qb+rc) \propto (l\sqrt{bc}+m\sqrt{ca}+n\sqrt{ab})\).

11. If \(a ∝ b\) and \(b ∝ c\), then prove that \(a^3 + b^3 + c^3 ∝ 3abc\).

12. If \(a\propto b\) and \(b\propto c\), prove that: \[ a^2+b^2+c^2 \propto ab+bc+ca \]

13. If \(a \propto b+c\), \(b \propto c+a\), and \(c \propto a+b\), and \(l, m, n\) are three distinct constants, prove that \(\cfrac{l}{l+1}+\cfrac{m}{m+1}+\cfrac{n}{n+1}=1\).

14. Given: \[ a \propto b \quad \text{and} \quad b \propto c \] Show that: \[ a^3b^3 + b^3c^3 + c^3a^3 \propto abc(a^3 + b^3 + c^3) \]

15. If the roots of \((a^2-bc)x^2+2(b^2-ca)x+(c^2-ab)=0\) are equal, show that \(b=0\) or \(a^3+b^3+c^3=3abc\).

16. If \( a \propto b \) and \( b \propto c \), then prove that \( a^3 + b^3 + c^3 \propto 3abc \).

17. If \( a \propto b \) and \( b \propto c \), then prove that \( a^3 + b^3 + c^3 \propto 3abc \).

18. If \(a^2 + b^2 \propto 5ab\), then **show that** \(a + b \propto a - b\).

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

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

21. 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.

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

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

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

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

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

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

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

29. If \(x = \cfrac{4ab}{a + b}\), show that \[ \cfrac{x + 2a}{x - 2a} + \cfrac{x + 2b}{x - 2b} = 2 \] [Given: \(a \ne 0\), \(b \ne 0\), and \(a \ne b\)]

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