1. If \(a : b = b : c\), then prove that \[ a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3 \]

2. If \(a:b = b:c\), then prove that \[ a^2b^2c^2\left(\cfrac{1}{a^3}+\cfrac{1}{b^3}+\cfrac{1}{c^3}\right) = a^3 + b^3 + c^3 \]

3. If \(a : b = b : c\), then prove that \[ \left(\cfrac{a + b}{b + c}\right)^2 = \cfrac{a^2 + b^2}{b^2 + c^2} \]

4. If \(a : b = b : c\), then prove that \[ a^2b^2c^2\left(\cfrac{1}{a^3} + \cfrac{1}{b^3} + \cfrac{1}{c^3}\right) = a^3 + b^3 + c^3 \]

5. If \(a : b = b : c\), then prove that \[ \cfrac{abc(a + b + c)^3}{(ab + bc + ca)^3} = 1 \]

6. If \(a : b = b : c\), then show that \[abc (a + b + c)^3 = (ab + bc + ca)^3\]

7. If \(a:b = b:c\), prove that: \[ \left(\cfrac{a+b}{b+c}\right)^2 = \cfrac{a^2+b^2}{b^2+c^2} \]

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

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

10. If \(A\) and \(B\) are complementary angles, prove that: \[ \frac{\sin A + \sin B}{\cos B - \cos A} = \frac{\cos A + \cos B}{\sin A - \sin B} \]

11. Translate of your statement in English: If the roots of the equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, then prove that: \(a + c = 2b\).

12. If \(a : b = c : d\), then prove that \((a^2 + c^2)(b^2 + d^2) = (ab + cd)^2\).

13. If \(a : b = b : c\), then prove that \((a + b + c)(a - b + c) = a^2 + b^2 + c^2\).

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

15. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : s\), then show that \[ \frac{(s + 1)^2}{s} = \frac{b^2}{ac} \]

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

17. If \(a:b=c:d\), then prove that \((a^2+c^2)(b^2+d^2)=(ab+cd)^2\).

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

19. If \(a : b = b : c\), then prove that \(\cfrac{abc(a+b+c)^3}{(ab+bc+ca)^3} = 1\).

20. If \(x : a = y : b = z : c\), then prove that \(\cfrac{x^3}{a^2} + \cfrac{y^3}{b^2} + \cfrac{z^3}{c^2} = \cfrac{(x + y + z)^3}{(a + b + c)^2}\)

21. If \(a : b = c : d = e : f\), then prove that \((a^2 + c^2 + e^2)(b^2 + d^2 + f^2) = (ab + cd + ef)^2\)

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

23. 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 \(a \ne b\)]

24. If \[ (a + b + c + d):(a + b - c - d) = (a - b + c - d):(a - b - c + d) \] then prove that \[ a : b = c : d \]

25. If \[ x^2 : (by + cz) = y^2 : (cz + ax) = z^2 : (ax + by) = 1 \] then show that \[ \frac{a}{a + x} + \frac{b}{b + y} + \frac{c}{c + z} = 1 \]

26. If \(a : b = c : d\), then prove: \[ \frac{a + b}{a - b} = \frac{\sqrt{ac + ad + bc + bd}}{\sqrt{ac - ad - bc + bd}} \]

27. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : r\), then prove that \(b^2r = ac(r + 1)^2\).

28. If ABCD is a cyclic quadrilateral inscribed in a circle with center O, then prove that: \[ AB + CD = BC + DA \]

29. In triangle △ABC, ∠A is a right angle and BP and CQ are medians. Prove that: \[ 5BC^2 = 4(BP^2 + CQ^2) \]

30. In triangle △ABC, ∠C = 90°, and if BC = \(m\) and AC = \(n\), then prove that: \[ m \sin A + n \sin B = \sqrt{m^2 + n^2} \]