RMO 1992 original question paper

1. Determine the set of integers n for which n2 + 19n + 92 is a square of an integer.

2. If \frac{1}{a} + \frac{1}{b} = \frac{1}{c} , where a, b, c are positive integers with no common factor, prove that (a + b) is the square of an integer.

3. Determine the largest 3-digit prime factor of the integer 2000C1000.

4. ABCD is a cyclic quadrilateral with AC⊥BD; AC meets BD at E. Prove that EA2 + EB2 + EC2 + ED2 = 4R2,
where R is the radius of the circumscribing circle.

5. ABCD is a cyclic quadrilateral; x, y, z are the distances of A from the lines BD, BC, CD respectively. Prove that \frac{BD}{x} = \frac{BC}{y} + \frac{CD}{z}

6. ABCD is a quadrilateral and P, Q are mid-points of CD, AB respectively. Let AP, DQ meet at X, and BP, CQ meet at Y . Prove that area of ADX + area of BCY = area of quadrilateral PXQY .

7. Prove that
1 < \frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + …. + \frac{1}{3001} < 1\frac{1}{3}

8. Solve the system
(x + y)(x + y + z) = 18
(y + z)(x + y + z) = 30
(z + x)(x + y + z) = 2A
in terms of the parameter A.

9. The cyclic octagon ABCDEFGH has sides a, a, a, a, b, b, b, b respectively. Find the radius of the circle that circumscribes ABCDEFGH in terms of a and b.
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