THE 1997 ASIAN PACIFIC MATHEMATICAL OLYMPIAD

Time allowed: 4 hours NO calculators are to be used. Each question is worth seven points.

Question 1

Given

S = 1 + \frac{1}{1 + \frac{1}{3}} + \frac{1}{1 + \frac{1}{3} + \frac{1}{6}} + ⋯ + \frac{1}{1 + \frac{1}{3} + \frac{1}{6} + ⋯ + \frac{1}{1993006}} ,

where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e. k = n(n + 1)/2 for n = 1, 2, …, 1996). Prove that S > 1001.

Question 2

Find an integer n, where 100 ≤ n ≤ 1997, such that

\frac{2^{n} + 2}{n}

is also an integer.

Question 3

Let ABC be a triangle inscribed in a circle and let

l_{a} = \frac{m_{a}}{M_{a}} , l_{b} = \frac{m_{b}}{M_{b}} , l_{c} = \frac{m_{c}}{M_{c}} ,

where m_{a}, m_{b}, m_{c} are the lengths of the angle bisectors (internal to the triangle) and M_{a}, M_{b}, M_{c} are the lengths of the angle bisectors extended until they meet the circle. Prove that

\frac{l_{a}}{sin^{2} A} + \frac{l_{b}}{sin^{2} B} + \frac{l_{c}}{sin^{2} C} ≥ 3,

and that equality holds iff ABC is an equilateral triangle.

Question 4

Triangle A_{1}A_{2}A_{3} has a right angle at A_{3}. A sequence of points is now defined by the following iterative process, where n is a positive integer. From A_{n} (n ≥ 3), a perpendicular line is drawn to meet A_{n−2}A_{n−1} at A_{n+1}.

(a) Prove that if this process is continued indefinitely, then one and only one point P is interior to every triangle A_{n−2}A_{n−1}A_{n}, n ≥ 3.

(b) Let A_{1} and A_{3} be fixed points. By considering all possible locations of A_{2} on the plane, find the locus of P.

Question 5

Suppose that n people A_{1}, A_{2}, …, A_{n}, (n ≥ 3) are seated in a circle and that A_{i} has a_{i} objects such that

a_{1} + a_{2} + ⋯ + a_{n} = nN,

where N is a positive integer. In order that each person has the same number of objects, each person A_{i} is to give or to receive a certain number of objects to or from its two neighbours A_{i−1} and A_{i+1}. (Here A_{n+1} means A_{1} and A_{n} means A_{0}.) How should this redistribution be performed so that the total number of objects transferred is minimum?