2003 Flanders Math Olympiad

Part of Flanders Math Olympiad

Subcontests

(4)

1

[move to other probs plz] Points

Consider all points with integer coordinates in the carthesian plane. If one draws a circle with M(0,0) and a well-chose radius r, the circles goes through some of those points. (like circle with

r=2\sqrt2

goes through 4 points) Prove that

\forall n\in \mathbb{N}, \exists r

so that the circle with midpoint 0,0 and radius

r

goes through at least

n

1

[to be solved] - Challenged Again (easy one - solved by all)

Playing soccer with 3 goes as follows: 2 field players try to make a goal past the goalkeeper, the one who makes the goal stands goalman for next game, etc. Arne, Bart and Cauchy played this game. Later, they tell their math teacher that A stood 12 times on the field, B 21 times on the field, C 8 times in the goal. Their teacher knows who made the 6th goal. Who made it?

1

Tangents to intersecting circles...

C_1

C_2

S

. The tangent in

S

C_1

C_2

A

S

. The tangent in

S

C_2

C_1

B

S

. Another circle

C_3

A, B, S

. The tangent in

S

C_3

C_1

P

S

C_2

Q

S

. Prove that the distance

PS

is equal to the distance

QS

1

Easy sum of digits problem

A number consists of 3 different digits. The sum of the 5 other numbers formed with those digits is 2003. Find the number.