10.1

Part of V Soros Olympiad 1998 - 99 (Russia)

Problems(3)

perfect powers (V Soros Olympiad 1998-99 Round 1 10.1)

Source:

Find some natural number

a

2a

is a perfect square,

3a

is a perfect cube,

5a

is the fifth power of some natural number.

number theoryPerfect power

average speed from 3 different speeds (V Soros Olympiad 1998-99 Round 2 10.1)

Source:

A car drove from one city to another. She drove the first third of the journey at a speed of

50

km/h, the second third at

60

km/h, and the last third at

70

km/h. What is the average speed of the car along the entire journey?

y=(a-6x)/{2+x) summteric to y=1/x(V Soros Olympiad 1998-99 Round 3 10.1)

Source:

It is known that the graph of the function

y =\frac{a-6x}{2+x}

is centrally summetric to the graph of the function

y = \frac{1}{x}

with respect to some point. Find the value of the parameter

a

and the coordinates of the center of symmetry.

analytic geometrygeometrysymmetrySymmetric