MathDB

2

3 isosceles given on a line with an angle 120, equilateral wanted

AC

and a point

B

on the segment. Let us draw the positively oriented isosceles triangles

ABS_1, BCS_2

, and

CAS_3

with the angles at

S_1,S_2,S_3

equal to

120^o

. Prove that the triangle

S_1S_2S_3

is equilateral.

\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}, parallelograms inscribed a convex ABCD

Let

P

be a point inside a convex quadrilateral

ABCD

. Points

K,L,M,N

are given on the sides

AB,BC,CD,DA

respectively such that

PKBL

and

PMDN

are parallelograms. Let

S,S_1

, and

S_2

be the areas of

ABCD, PNAK

, and

PLCM

. Prove that

\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}

.

2

a_1 = b_1 = 1 and a_{n+1} = a_n +b_n, b_{n+1} = a_nb_n, coprime terms

The sequences

(a_n)

and

(b_n)

are defined by

a_1 = b_1 = 1

and

a_{n+1} = a_n +b_n, b_{n+1} = a_nb_n

for

n = 1,2,...

Show that every two distinct terms of the sequence

(a_n)

are coprime

Q(3^k) \ge Q(3^{k+1} for infinite n, whereas Q(n) is sum of decimal digits of n

For every natural number

n

, let

Q(n)

denote the sum of the decimal digits of

n

. Prove that there are infinitely many positive integers

k

with

Q(3^k) \ge Q(3^{k+1})

.

4

2

min no of triangular faces of polyhedron with faces more numbered than vertices

It is known that there are polyhedrons whose faces are more numbered than the vertices. Find the smallest number of triangular faces that such a polyhedron can have.

bright number is a sum of a perfect square and a perfect cube

A natural number is called bright if it is the sum of a perfect square and a perfect cube. Prove that if

r

and

s

are any two positive integers, then (a) there exist infinitely many positive integers

n

such that both

r+n

and

s+n

are bright, (b) there exist infinitely many positive integers

m

such that both rm and sm are bright.

1

2

Monkeys and Coconuts

Exactly 1600 Coconuts are distributed on exactly 100 monkeys, where some monkeys also can have 0 coconuts. Prove that, no matter how you distribute the coconuts, at least 4 monkeys will always have the same amount of coconuts.
(The original problem is written in German. So, I apologize when I've changed the original problem or something has become unclear while translating.)

vertices of a 2n-gon are labelled with the numbers 1,2,...,2n

The vertices of a regular

2n

-gon (with

n > 2

an integer) are labelled with the numbers

1,2,...,2n

in some order. Assume that the sum of the labels at any two adjacent vertices equals the sum of the labels at the two diametrically opposite vertices. Prove that this is possible if and only if

n

is odd.

1999 Bundeswettbewerb Mathematik

Subcontests

3 isosceles given on a line with an angle 120, equilateral wanted

\sqrt{S}\ge \sqrt{S_1} +\sqrt{S_2}, parallelograms inscribed a convex ABCD

a_1 = b_1 = 1 and a_{n+1} = a_n +b_n, b_{n+1} = a_nb_n, coprime terms

Q(3^k) \ge Q(3^{k+1} for infinite n, whereas Q(n) is sum of decimal digits of n

min no of triangular faces of polyhedron with faces more numbered than vertices

bright number is a sum of a perfect square and a perfect cube

Monkeys and Coconuts

vertices of a 2n-gon are labelled with the numbers 1,2,...,2n