RMO–2000
1. Let AC be a line segment in the plane and B a point between A and C. Construct
isoscelestriangles PAB and QBC on one side of the segment AC such that angleAPB =
angleBQC = 120^o and an isosceles triangle RAC on the other side of AC such that
angleARC = 120^o . Show that PQR is an equilateral triangle.

2. Solve the equation y³ = x³ + 8x² − 6x + 8, for positive integers x and y.

3. Suppose (x1, x2, · · · , xn, · · ·) is a sequence of positive real numbers such that
x_1>= x_2>= x₃ . . . x_n . . ., and for all n
x₁/1+ x₄/2+ x₉/3+ . . . + x_n²/n<=1.
Show that for all k the following inequality is satisfied:
x₁/1+ x₂/2+ x₃/3+ . . . + x_k/k<=3.

4. All the 7-digit numbers containing each of the digits 1, 2, 3, 4, 5, 6, 7 exactly once,
and not divisible by 5, are arranged in increasing order. Find the 2000-th number in this list.

5. The internal bisector of angle A in a triangle ABC with AC > AB, meets the circumcircle
􀀀 of the triangle in D. Join D to the centre O of the circle 􀀀 and suppose DO meets AC in E,
possibly when extended. Given that BE is perpendicular to AD, show that AO is parallel to BD.

6. (i) Consider two positive integers a and b which are such that a^ab^b is divisible by 2000. What
is the least possible value of the product ab?
(ii) Consider two positive integers a and b which are such that a^bb^a is divisible by 2000. What
is the least possible value of the product ab?

7. Find all real values of a for which the equation x⁴−2ax²+x+a² -a = 0 has all its roots real.