2000 Nordic

Part of Nordic

Subcontests

(4)

1

1/3 <= f(1/3) <= 4/7

The real-valued function

f

0 \le x \le 1, f(0) = 0, f(1) = 1

\frac{1}{2} \le \frac{ f(z) - f(y)}{f(y) - f(x)} \le 2

0 \le x < y < z \le 1

z - y = y -x

\frac{1}{7} \le f (\frac{1}{3} ) \le \frac{4}{7}

1

isosceles or <A = 60^o, because of angle bisectors

In the triangle

ABC

, the bisector of angle

\angle B

AC

D

and the bisector of angle

\angle C

AB

E

. The bisectors meet each other at

O

OD = OE

. Prove that either

ABC

is isosceles or

\angle BAC = 60^\circ

1

persons exchanging coins around a table

P_1, P_2, . . . , P_{n-1}, P_n

sit around a table, in this order, and each one of them has a number of coins. In the start,

P_1

has one coin more than

P_2, P_2

has one coin more than

P_3

P_{n-1}

who has one coin more than

P_n

P_1

gives one coin to

P_2

, who in turn gives two coins to

P_3

Pn

who gives n coins to

P_1

. Now the process continues in the same way:

P_1

n+ 1

P_2

P_2

n+2

P_3

; in this way the transactions go on until someone has not enough coins, i.e. a person no more can give away one coin more than he just received. At the moment when the process comes to an end in this manner, it turns out that there are two neighbours at the table such that one of them has exactly five times as many coins as the other. Determine the number of persons and the number of coins circulating around the table.

1

ways to write 2000 as a sum of 3 positive integers

In how many ways can the number

2000

be written as a sum of three positive, not necessarily different integers? (Sums like

1 + 2 + 3

3 + 1 + 2

etc. are the same.)