2006 Flanders Math Olympiad

Part of Flanders Math Olympiad

Subcontests

(4)

1

To be the projection, or the be projected, that's the q!

\triangle ABC

be an equilateral triangle and let

P

\left[AB\right]

Q

is the point on

BC

PQ

is perpendicular to

AB

R

is the point on

AC

QR

is perpendicular to

BC

S

is the point on

AB

RS

is perpendicular to

AC

Q'

is the point on

BC

PQ'

is perpendicular to

BC

R'

is the point on

AC

Q'R'

is perpendicular to

AC

S'

is the point on

AB

R'S'

is perpendicular to

AB

\frac{|PB|}{|AB|}

S=S'

1

Elfs and trolls

Elfs and trolls are seated at a round table, 60 creatures in total. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake. Everybody claims to sit between an elf and a troll, but exactly two elfs made a mistake! How many trolls are there at this table?

1

Trigonometry: 3 roots are given

\theta\in\mathbb{R}

\cos(4\theta) = \cos(3\theta)

\cos\left(\frac{2\pi}{7}\right)

\cos\left(\frac{4\pi}{7}\right)

\cos\left(\frac{6\pi}{7}\right)

are the roots of an equation of the form

ax^3+bx^2+cx+d = 0

a, b, c, d

are integers. Determine

a, b, c

d

1

Functional equation

Find all functions

f: \mathbb{R}\backslash\{0,1\} \rightarrow \mathbb{R}

f(x)+f\left(\frac{1}{1-x}\right) = 1+\frac{1}{x(1-x)}.