1999 Nordic

Part of Nordic

Subcontests

(4)

1

inequality with sums of inverses of positive reals

a_1, a_2, . . . , a_n

be positive real numbers and

n \ge 1

n (\frac{1}{a_1}+...+\frac{1}{a_n}) \ge (\frac{1}{1+a_1}+...+\frac{1}{1+a_n})(n+\frac{1}{a_1}+...+\frac{1}{a_n})

When does equality hold?

1

moving in the infinite integer plane

The infinite integer plane

Z\times Z = Z^2

consists of all number pairs

(x, y)

x

y

are integers. Let

a

b

be non-negative integers. We call any move from a point

(x, y)

to any of the points

(x\pm a, y \pm b)

(x \pm b, y \pm a)

(a, b)

-knight move. Determine all numbers

a

b

, for which it is possible to reach all points of the integer plane from an arbitrary starting point using only

(a, b)

1

maximal number of 120^o angles in specific 7-gons

7

-gons inscribed in a circle such that all sides of the

7

-gon are of different length. Determine the maximal number of

120^\circ

angles in this kind of a

7

1

f(n) = f(f(n + 11)) if n<=1999, f(n) = n - 5 if n > 1999

f

is defined for non-negative integers and satisfies the condition

f(n) = f(f(n + 11))

n \le 1999

f(n) = n - 5

n > 1999

. Find all solutions of the equation

f(n) = 1999