2006 German National Olympiad

Part of German National Olympiad

Subcontests

(6)

1

A point far, far away...

D

be a point inside a triangle

ABC

|AC| -|AD| \geq 1

|BC|- |BD| \geq 1.

Prove that for any point

E

AB

|EC| -|ED| \geq 1.

1

A Geometry problem

Let a circle through

B

C

ABC

AB

AC

Y

Z

, respectively. Let

P

be the intersection of

BZ

CY

X

be the intersection of

AP

BC

M

be the point that is distinct from

X

and on the intersection of the circumcircle of the triangle

XYZ

BC

M

is the midpoint of

BC

1

Max value for min distance

Five points are on the surface of of a sphere of radius

1

a_{\text{min}}

denote the smallest distance (measured along a straight line in space) between any two of these points. What is the maximum value for

a_{\text{min}}

, taken over all arrangements of the five points?

1

Not so standard inequality involving absolute values, roots

x \neq 0

be a real number satisfying

ax^2+bx+c=0

a,b,c \in \mathbb{Z}

|a|+|b|+|c| > 1

|x| \geq \frac{1}{|a|+|b|+|c|-1}.

1

Germany National Olympiad

For which positive integer n can you color the numbers 1,2...2n with n colors, such that every color is used twice and the numbers 1,2,3...n occur as difference of two numbers of the same color exatly once.

1

101...101

n\in \mathbb Z^+

z_n = \underbrace{ 101\dots101}_{2n+1 \text{ digits} }