1. If \(\cfrac{a}{b + c} = \cfrac{b}{c + a} = \cfrac{c}{a + b}\) and \(a + b + c \ne 0\), then prove that \(a = b = c\).

2. Assume, \(\cfrac{a}{b} = \cfrac{b}{c} = \cfrac{c}{d} = k\) (where \(k \ne 0\)) So, \(a = bk,\; b = ck,\; c = dk\) \(\therefore b = dk \cdot k = dk^2\) and \(a = dk^2 \cdot k = dk^3\)

Left-hand side \[ = (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = \{(dk^3)^2 + (dk^2)^2 + (dk)^2\} \cdot \{(dk^2)^2 + (dk)^2 + d^2\} \] \[ = \{d^2k^6 + d^2k^4 + d^2k^2\} \cdot \{d^2k^4 + d^2k^2 + d^2\} = d^2k^2(k^4 + k^2 + 1) \cdot d^2(k^4 + k^2 + 1) \] \[ = d^4k^2(k^4 + k^2 + 1)^2 \] Right-hand side \[ = (ab + bc + cd)^2 = (dk^3 \cdot dk^2 + dk^2 \cdot dk + dk \cdot d)^2 = (d^2k^5 + d^2k^3 + d^2k)^2 = \{d^2k(k^4 + k^2 + 1)\}^2 = d^4k^2(k^4 + k^2 + 1)^2 \] ∴ Left-hand side = Right-hand side (Hence Proved)

3. Given that \[ \frac{a + b - c}{a + b} = \frac{b + c - a}{b + c} = \frac{c + a - b}{c + a} \] and \[ a + b + c \ne 0 \] prove that \(a = b = c\).

4. If \(\cfrac{b}{a + b} = \cfrac{a + c - b}{b + c - a} = \cfrac{a + b + c}{2a + b + 2c}\), where \(a + b + c \ne 0\), then show that \(\cfrac{a}{2} = \cfrac{b}{3} = \cfrac{c}{4}\).

5. If \( \cfrac{b+c}{a^2}=\cfrac{c+a}{b^2}=\cfrac{a+b}{c^2}=1 \), then show that \( \cfrac{1}{1+a}+\cfrac{1}{1+b}+\cfrac{1}{1+c}=1 \).

6. If \(\cfrac{b}{a+b} = \cfrac{a+c-b}{b+c-a} = \cfrac{a+b+c}{2a+b+2c}\), (where \(a + b + c \ne 0\)), then show that \(\cfrac{a}{2} = \cfrac{b}{3} = \cfrac{c}{4}\).

7. If \[ \frac{a}{b + c} = \frac{b}{c + a} = \frac{c}{a + b} \] then prove that each of these ratios is either \(\frac{1}{2}\) or \(-1\).

8. If \(a + b + c = 0\), then prove that \[ \frac{bc}{2a^2 + bc} + \frac{ca}{2b^2 + ca} + \frac{ab}{2c^2 + ab} = 1 \]

9. If \(\cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} + \cfrac{c^2}{a+b} = 1\), then prove that \[ \cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c} = 1 \]

10. If the roots of the quadratic equation \((a^2 + b^2)x^2 + 2(ac + bd)x + (c^2 + d^2) = 0\) are equal, then prove that \(\cfrac{a}{b} = \cfrac{c}{d}\).

11. If \(a + b + c = 0\), then prove that: \[ \cfrac{1}{2a^2 + bc} + \cfrac{1}{2b^2 + ca} + \cfrac{1}{2c^2 + ab} = 0 \]

12. If \(a + c = 2b\) and \(q^2 = pr\), then prove that \(p^{b-c} \cdot q^{c-a} \cdot r^{a-b} = 1\).

13. If \(\cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} = \cfrac{c^2}{a+b} = 1\), then prove that \(\cfrac{1}{a+1} + \cfrac{1}{b+1} + \cfrac{1}{c+1} = 1\).

14. If \(\cfrac{a}{1-a}+\cfrac{b}{1-b}+\cfrac{c}{1-c}=1\), then prove that \(\cfrac{1}{1-a}+\cfrac{1}{1-b}+\cfrac{c}{1-c}=4\).

15. If \(\cfrac{a}{3} = \cfrac{b}{4} = \cfrac{c}{7}\), then prove that \[ \cfrac{a + b + c}{c} = 2 \]

16. If \[ \cfrac{a + b}{b + c} = \cfrac{c + d}{d + a} \] then prove that either \(c = a\) or \(a + b + c + d = 0\)

17. If \(a + \frac{1}{b} = 1\) and \(b + \frac{1}{c} = 1\), then prove that \(c + \frac{1}{a} = 1\).

18. **"If the roots of the quadratic equation \((a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + d^2) = 0\) are equal, then prove that \(\cfrac{a}{b} = \cfrac{c}{d}\)."**

19. “If \(\cfrac{a}{b} = \cfrac{3}{2}\) and \(\cfrac{b}{c} = \cfrac{3}{2}\), then what is \((a + b) : (b + c)\)?”

20. If \(a(\tan\theta + \cot\theta) = 1\) and \(\sin\theta + \cos\theta = b\), then prove that \(2a = b^2 - 1\), where \(0^\circ < \theta < 90^\circ\).

21. If \(a + \frac{1}{b} = 1\) and \(b + \frac{1}{c} = 1\), then show that \(c + \frac{1}{a} = 1\).

22. If a rectangular parallelepiped has volume \(v\), and its length, width, and height are \(a, b, c\) respectively, with a total surface area \(s\), prove that: \(\cfrac{1}{v}=\cfrac{2}{3}\left(\cfrac{1}{a}+\cfrac{1}{b}+\cfrac{1}{c}\right)\).

23. If \(a = \cfrac{\sqrt7+\sqrt3}{\sqrt7-\sqrt3}\) and \(ab = 1\), then prove that \(\cfrac{a^2+ab+b^2}{a^2-ab+b^2}=\cfrac{12}{11}\).

24. If \(\cfrac{bz+cy}{a} = \cfrac{cx+az}{b} = \cfrac{ay+bx}{c}\), then prove that \(\cfrac{x}{a(b^2+c^2-a^2)} = \cfrac{y}{b(c^2+a^2-b^2)} = \cfrac{z}{c(a^2+b^2-c^2)}\).

25. If \(\alpha\) and \(\beta\) are the two roots of the equation \(ax^2 + bx + c = 0\), then \(\cfrac{\alpha}{\beta}\) and \(\cfrac{\beta}{\alpha}\) are also roots of a quadratic equation. Determine that quadratic equation.

26. If the roots of the quadratic equation \((a^2 + b^2)x^2 - 2(ac + bd)x + (c^2 + a^2) = 0\) are equal, prove that \(\cfrac{a}{b} = \cfrac{c}{d}\).

27. If \(\cfrac{b}{a+b}=\cfrac{a+c-b}{b+c-a}=\cfrac{a+b+c}{2a+b+2c}\), then show that \(\cfrac{a}{2}=\cfrac{b}{3}=\cfrac{c}{4}\).

28. If \[ \cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} = \cfrac{c^2}{a+b} = 1 \] then prove that \[ \cfrac{1}{a+1} + \cfrac{1}{b+1} + \cfrac{1}{c+1} = 1. \]

29. If \(\cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} + \cfrac{c^2}{a+b} = 1\), prove that \(\cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c} = 1\).

30. If \(\cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} + \cfrac{c^2}{a+b} = 1\), then show that \(\cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c} = 1\).