1996 South africa National Olympiad

Part of South Africa National Olympiad

Subcontests

(6)

1

f(f(x))=3^x

f

is increasing and convex (i.e. every straight line between two points on the graph of

f

lies above the graph) and satisfies

f(f(x))=3^x

x\in\mathbb{R}

f(0)=0.5

f(0.75)

with an error of at most

0.025

. The following are corrent to the number of digits given:
3^{0.25}=1.31607, 3^{0.50}=1.73205, 3^{0.75}=2.27951.

1

Inscribed equilateral triangle

ABC

is a triangle with sides

1

2

\sqrt3

. Determine the smallest possible area of an equilateral triangle with a vertex on each side of triangle

ABC

1

Red Rockets and Blue Boeings

In the Rainbow Nation there are two airways: Red Rockets and Blue Boeings. For any two cities in the Rainbow Nation it is possible to travel from the one to the other using either or both of the airways. It is known, however, that it is impossible to travel from Beanville to Mieliestad using only Red Rockets - not directly nor by travelling via other cities. Show that, using only Blue Boeings, one can travel from any city to any other city by stopping at at most one city along the way.

1

Triangle with integral sides

The sides of triangle

ABC

has integer lengths. Given that

AC=6

\angle BAC=120^\circ

, determine the lengths of the other two sides.

1

3^x+4^x=5^x

Find all real numbers for which

3^x+4^x=5^x

1

Highest power of 2 dividing 1996!

Find the highest power of

2

that divides exactly into

1996!=1\times2\times\cdots\times1996