9.4

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

n points marked on a circle (V Soros Olympiad 1998-99 Round 1 9.4)

Source:

There are n points marked on the circle. It is known that among all possible distances between two marked points there are no more than

100

different ones. What is the largest possible value for

n

combinatoricscombinatorial geometry

simplify fraction (V Soros Olympiad 1998-99 Round 2 9.4)

Source:

Simplify the fraction

\frac{123456788...87654321}{1234567899...987654321}

’ if the digit

8

in the numerator occurs

2000

times, and the digit

9

in the denominator

1999

occurs times (as a result you need to get an irreducible fraction).

algebranumber theory

max <AKM, 2 midpoints (V Soros Olympiad 1998-99 Round 3 9.4)

Source:

ABC

be a triangle without obtuse angles,

M

the midpoint of

BC

K

the midpoint of

BM

. What is the largest value of the angle

\angle KAM