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

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

3. If \(x \alpha y\) and \(y \alpha z\), then prove that: \[ x^2 + y^2 + z^2 \alpha xy + yz + zx \]

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

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

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

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

8. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} = \frac{z^2}{1 - c^2} \]

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

10. If \(\cfrac{x^2 - yz}{a} = \cfrac{y^2 - zx}{b} = \cfrac{z^2 - xy}{c}\), then prove that > \((a + b + c)(x + y + z) = ax + by + cz\)

11. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

12. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(xy = 1\), prove that \[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{7\sqrt{3}}{12} \]

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

14. If \(x = \cfrac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then show that \[ \cfrac{x^2 + xy + y^2}{x^2 - xy + y^2} = \cfrac{12}{11} \]

15. If \[ \frac{x^2 - yz}{a} = \frac{y^2 - zx}{b} = \frac{z^2 - xy}{c} \] then prove that \[ (a + b + c)(x + y + z) = ax + by + cz \]

16. If \[ \cfrac{x^2 - yz}{a} = \cfrac{y^2 - zx}{b} = \cfrac{z^2 - xy}{c}, \] then prove that \[ (a + b + c)(x + y + z) = ax + by + cz. \]

17. If \[ \cfrac{x^2 - yz}{a} = \cfrac{y^2 - zx}{b} = \cfrac{z^2 - xy}{c}, \] then prove that \[ (a + b + c)(x + y + z) = ax + by + cz. \]

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

19. If \(x + y ∝ x - y\), then show that \(x^2 + y^2 ∝ xy\).

20. If \(x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}}\) and \(xy = 1\), then show that \[ \frac{x^2 + xy + y^2}{x^2 - xy + y^2} = \frac{12}{11} \]

21. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then show that: \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

22. If \(x, y, z\) are positive and unequal, can \(\cos\theta = \frac{x^2 + y^2 + z^2}{xy + yz + zx}\) be valid? Justify your answer.

23. If \((x + y) ∝ (x - y)\), then \((x^2 + y^2) ∝ xy\).

24. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(y = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\), then show that \[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{7\sqrt{3}}{12} \]

25. If \(\frac{x}{y} = \frac{a + 2}{a - 2}\), then prove that \[ \frac{x^2 - y^2}{x^2 + y^2} = \frac{4a}{a^2 + 4} \]

26. If \( \cot\theta = \frac{x}{y} \), then prove that \[ \frac{x\cos\theta - y\sin\theta}{x\cos\theta + y\sin\theta} = \frac{x^2 - y^2}{x^2 + y^2} \]

27. If \(a^2 = by + cz\), \(b^2 = cz + ax\), and \(c^2 = ax + by\), then prove that \[ \frac{x}{a + x} + \frac{y}{b + y} + \frac{z}{c + z} = 1 \]

28. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(xy = 1\), prove that the simplified value of \(\frac{x^2}{y} + \frac{y^2}{x}\) is 52.

29. If \(x=\cfrac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}\) and \(xy=1\), prove that \(\cfrac{x^2+xy+y^2}{x^2-xy+y^2}=\cfrac{12}{11}\).

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