MathDB

2

Prove that triangle ABC is equilateral

ABC

be a triangle. Points

A',B',C'

are on the sides

BC, CA, AB,

respectively such that we have \overline{A'B'} \equal{} \overline{B'C'} \equal{} \overline{C'A'} and \overline{AB'} \equal{} \overline{BC'} \equal{} \overline{CA'}. Prove that triangle

ABC

is equilateral.

Quadrilateral BECD and triangle ABC have the same area

Let

ABC

an acute triangle with circumcircle center

O.

The line

(BO)

intersects the circumcircle again in

D,

and the extension of the altitude from

A

intersects the circle in

E.

Prove that the quadrilateral

BECD

and the triangle

ABC

have the same area.

2

x^2 + p_n * x + q_n is the square of a natural number

For each

n \in \mathbb{N}

we have two numbers

p_n, q_n

with the following property: For exactly

n

distinct integer numbers

x

the number x^2 \plus{} p_n \cdot x \plus{} q_n is the square of a natural number. (Note the definition of natural numbers includes the zero here.)

Find explicit function for recursively defined function

For a sequence

a_i \in \mathbb{R}, i \in \{0, 1, 2, \ldots\}

we have a_0 \equal{} 1 and a_{n\plus{}1} \equal{} a_n \plus{} \sqrt{a_{n\plus{}1} \plus{} a_n} \forall n \in \mathbb{N}. Prove that this sequence is unique and find an explicit formula for this recursively defined sequence.

1

100-gon

10 vertices of a regular 100-gon are coloured red and ten other (distinct) vertices are coloured blue. Prove that there is at least one connection edge (segment) of two red which is as long as the connection edge of two blue points. [hide="Hint"]Possible approaches are pigeon hole principle, proof by contradiction, consider turns (bijective congruent mappings) which maps red in blue points.

4

2

Segment AB has no common point with the square R

A square

R

of sidelength

250

lies inside a square

Q

of sidelength

500

. Prove that: One can always find two points

A

and

B

on the perimeter of

Q

such that the segment

AB

has no common point with the square

R

, and the length of this segment

AB

is greater than

521

.

Nb. of divisors whose dec. representations ends with 1 or 9

Prove: For each positive integer is the number of divisors whose decimal representations ends with a 1 or 9 not less than the number of divisors whose decimal representations ends with 3 or 7.

2001 Bundeswettbewerb Mathematik

Subcontests

Prove that triangle ABC is equilateral

Quadrilateral BECD and triangle ABC have the same area

x^2 + p_n * x + q_n is the square of a natural number

Find explicit function for recursively defined function

100-gon

Segment AB has no common point with the square R

Nb. of divisors whose dec. representations ends with 1 or 9