1996 Nordic

Part of Nordic

Subcontests

(4)

1

f(a) = f(1995),..., f(n + a) = (f(n) - 1)/(f(n) + 1)

The real-valued function

f

is defined for positive integers, and the positive integer

a

f(a) = f(1995), f(a+1) = f(1996), f(a+2) = f(1997), f(n + a) = \frac{f(n) - 1}{f(n) + 1}

for all positive integers

n

. (i) Show that

f(n+ 4a) = f(n)

for all positive integers

n

. (ii) Determine the smallest possible

a

1

circle with diameter the altitude

The circle whose diameter is the altitude dropped from the vertex

A

of the triangle

ABC

intersects the sides

AB

AC

D

E

(A\ne D, A \ne E)

. Show that the circumcenter of

ABC

lies on the altitude drawn from the vertex

A

of the triangle

ADE

, or on its extension.

1

x^n+x^{-n} is an integer for all integers n

Determine all real numbers

x

x^n+x^{-n}

is an integer for all integers

n

1

1996 divides x when sum of decimal digits of x is 1996

Show that there exists an integer divisible by

1996

such that the sum of the its decimal digits is

1996