1. If \(\sqrt a\), \(\sqrt b\), and \(\sqrt c\) are in continuous proportion, prove that \((a+c)^2 - b^2 = a^2 + b^2 + c^2\).

2. If \(a, b, c\) are in continued geometric progression (i.e., in successive proportion), then prove that \[ \cfrac{1}{b^2} = \cfrac{1}{b^2 - a^2} + \cfrac{1}{b^2 - c^2} \]

3. If \(a, b, c\) are in continued geometric progression, prove that \[ a^2b^2c^2\left(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3}\right) = a^3 + b^3 + c^3 \]

4. If \(a, b, c\), and \(d\) are in continued proportion, then prove that \((b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2\)

5. If \(a, b, c, d\) are in continued proportion, then prove that \[ (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2 \]

6. If \(a, b, c\) are in continued geometric progression, then prove that \[ a^2 b^2 c^2 \left(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3 \]

7. If \(a, b, c, d\) are in continued geometric progression, prove that \[(a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2\]

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

9. If \(a, b, c,\) and \(d\) are in continued geometric proportion, then prove that \[ (b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2 \]

10. If \(a, b, c,\) and \(d\) are in continued proportion, then prove that \((b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2\).

11. If \(a, b, c, d\) are in continued geometric progression, then prove that \[ (b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2 \]

12. If \(a, b, c\) are in continued geometric progression, prove that \(\cfrac{1}{b} = \cfrac{1}{b - a} + \cfrac{1}{b - c}\).

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

14. If \( a, b, c \) are in continuous proportion, prove that \( a^2b^2c^2\left(\cfrac{1}{a^3}+\cfrac{1}{b^3}+\cfrac{1}{c^3}\right)=a^3+b^3+c^3 \).

15. If \(a, b, c,\) and \(d\) are in continuous proportion, then prove that \((b-c)^2+(c-a)^2+(b-d)^2=(a-d)^2\).

16. If \(a, b, c,\) and \(d\) are in continued geometric progression, then prove that \((b - c)^2 + (c - a)^2 + (b - d)^2 = (a - d)^2\)

17. 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 \(a^3+b^3+c^3=3abc\).

18. If \(a, b, c, d\) are in continued proportion, then what is the value of \(\frac{abc(a + b + c)}{ab + bc + ca}\)?

(a) \(\cfrac{1}{a^2}\) (b) \(\cfrac{1}{c^2}\) (c) \(b^2\) (d) \(a^2\)

19. If \(a, b, c, d\) are in continuous proportion, then what is the value of \(\cfrac{a^2 + b^2 + c^2}{b^2 + c^2 + d^2}\)?

(a) \(\cfrac{a}{c}\) (b) \(\cfrac{b}{c}\) (c) \(\cfrac{c}{d}\) (d) \(\cfrac{b}{d}\)

20. 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 \(\cfrac{a}{b} = \cfrac{c}{d}\).

21. 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: \(\cfrac{1}{v}=\cfrac{2}{3}\left(\cfrac{1}{a}+\cfrac{1}{b}+\cfrac{1}{c}\right)\).

22. 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 \(\cfrac{a}{b} = \cfrac{c}{d}\).

23. If the roots of the quadratic equation \(ax^2+bx+c=0\) are in the ratio \(1:2\), then prove that \(2b^2=9ac\).

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

25. If \(x : a = y : b = z : c\), then prove that \((a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = (ax + by + cz)^2\)

26. **"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 \(\cfrac{a}{b} = \cfrac{c}{d}\)."**

27. 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: \(a + c = 2b\).

28. Write whether the equation \(x - 1 + \frac{1}{x} = 6,\ (x ≠ 0)\) can be expressed in the form \(ax^2 + bx + c = 0\), where \(a, b, c\) are real numbers and \(a ≠ 0\).

29. If \(x : a = y : b = z : c\), then show that \((a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = (ax + by + cz)^2\)

30. ABCD is a rectangle and O is any point inside the rectangle. Prove that: \[ OA^2 + OC^2 = OB^2 + OD^2 \]