RMO–1992
1. Determine the set of integers n for which n² + 19n + 92 is a square of an integer.

2. If 1/a+1/b=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 perpendicular to 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 BD/x= BC/y+ 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 <1/1001+1/1002+1/1003+ . . . +1/3001 < (4/3)

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.