1. If \[ \frac{x}{y + z} = \frac{y}{z + x} = \frac{z}{x + y} \] then prove that the value of each ratio is either \(\frac{1}{2}\) or \(-1\).

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

3. If \(\frac{a}{b} = \frac{c}{d} = \frac{e}{f}\), then prove that each ratio equals \[ \left(\frac{la^n + mc^n + pe^n}{lb^n + md^n + pf^n}\right)^{\frac{1}{n}} \]

4. If \(\cfrac{x}{y+z}=\cfrac{y}{z+x}=\cfrac{z}{x+y}\), then prove that the value of each ratio is equal to either \(\cfrac{1}{2}\) or \(-1\).

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

6. If \[ \cfrac{ax + by}{a} = \cfrac{bx - ay}{b} \] then prove that each ratio is equal to \(x\).

7. If \[ \cfrac{x}{xa + yb + zc} = \cfrac{y}{ya + zb + xc} = \cfrac{z}{za + xb + yc} \] and \(x + y + z \ne 0\), then show that each of the above ratios is equal to \(\cfrac{1}{a + b + c}\).

8. If \( \tan 9^\circ = \frac{a}{b} \), then prove that \[ \frac{\sec^2 81^\circ}{1 + \cot^2 81^\circ} = \frac{b^2}{a^2} \]

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

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

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

12. If \[ \cfrac{a+b}{b+c}=\cfrac{c+d}{d+a} \] then prove that \[ c=a \quad \text{or} \quad a+b+c+d=0. \]

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

14. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then prove that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

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