1997 APMO

Part of APMO

Subcontests

(5)

1

N people passing objects

n

A_1

A_2

\ldots

A_n

n \geq 3

) are seated in a circle and that

A_i

a_i

objects such that

a_1 + a_2 + \cdots + a_n = nN

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}

A_{i+1}

A_{n+1}

A_1

A_n

A_0

.) How should this redistribution be performed so that the total number of objects transferred is minimum?

1

Locus of a point

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 \geq 3

), a perpendicular line is drawn to meet

A_{n-2}A_{n-1}

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 \geq 3

A_1

A_3

be fixed points. By considering all possible locations of

A_2

on the plane, find the locus of

P

1

Sum

S = 1 + \frac{1}{1 + \frac{1}{3}} + \frac{1}{1 + \frac{1}{3} + \frac{1} {6}} + \cdots + \frac{1}{1 + \frac{1}{3} + \frac{1}{6} + \cdots + \frac{1} {1993006}}

where the denominators contain partial sums of the sequence of reciprocals of triangular numbers (i.e.

k=\frac{n(n+1)}{2}

n = 1

2

\ldots

1996

S>1001

1

(2^n+2)/n being an integer

Find an integer

n

100 \leq n \leq 1997

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

is also an integer.

1

I am finding another solution! new ideas!!

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} \ ,

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} \geq 3

and that equality holds iff

ABC

is an equilateral triangle.