MathDB

2

q^2 + r = 2011 when ab = q (a + b) + r , 0 \le r <a + b

a

and

b

be positive integers. As is known, the division of of

a \cdot b

with

a + b

determines integers

q

and

r

uniquely such that

a \cdot b = q (a + b) + r

and

0 \le r <a + b

. Find all pairs

(a, b)

for which

q^2 + r = 2011

.

mininal sum of distances from a point to vertices of a tetrahedron

Let

ABCD

be a tetrahedron that is not degenerate and not necessarily regular, where sides

AD

and

BC

have the same length

a

, sides

BD

and

AC

have the same length

b

, side

AB

has length

c_1

and the side

CD

has length

c_2

. There is a point

P

for which the sum of the distances to the vertices of the tetrahedron is minimal. Determine this sum depending on the quantities

a, b, c_1

and

c_2

.

3

2

convex pentagon with each diagonal an angle trisector, is it regular?

The diagonals of a convex pentagon divide each of its interior angles into three equal parts. Does it follow that the pentagon is regular?

prove that all teams play the same number of games.

When preparing for a competition with more than two participating teams two of them play against each other at most once. When looking at the game plan it turns out: (1) If two teams play against each other, there are no more team playing against both of them. (2) If two teams do not play against each other, then there is always exactly two other teams playing against them both. Prove that all teams play the same number of games.

2

16 children are sitting at a round table

16

children are sitting at a round table. After the break, they sit down again on table. They find that each child is either sitting on its original [lace or in one of the two neighboring places. How many seating arrangements are possible in this way after the break?

3n + 1 and 10n + 1 are perfect squares , then 29n+11 not prime

Proove that if for a positive integer

n

, both

3n + 1

and

10n + 1

are perfect squares , then

29n + 11

is not a prime number.

1

2

split a square into finitely many hexagons with inner angles all < 180^o

Prove that you can't split a square into finitely many hexagons, whose inner angles are all less than

180^o

.

10 bowls with marbles in a circles

Ten bowls are in a circle. They will go clockwise - starting somewhere filled with

1, 2, 3, ..., 9

or

10

marbles. You can have two choices in every move . Add a marble to neighboring shells or from two neighboring shells - if both of them are not empty - remove one marble each. Can you achieve that after finally many moves in each bowl exactly

2011

marbles lying?

2011 Bundeswettbewerb Mathematik

Subcontests

q^2 + r = 2011 when ab = q (a + b) + r , 0 \le r <a + b

mininal sum of distances from a point to vertices of a tetrahedron

convex pentagon with each diagonal an angle trisector, is it regular?

prove that all teams play the same number of games.

16 children are sitting at a round table

3n + 1 and 10n + 1 are perfect squares , then 29n+11 not prime

split a square into finitely many hexagons with inner angles all < 180^o

10 bowls with marbles in a circles

2011 Bundeswettbewerb Mathematik

Subcontests

q^2 + r = 2011 when ab = q (a + b) + r , 0 \le r &lt;a + b

mininal sum of distances from a point to vertices of a tetrahedron

convex pentagon with each diagonal an angle trisector, is it regular?

prove that all teams play the same number of games.

16 children are sitting at a round table

3n + 1 and 10n + 1 are perfect squares , then 29n+11 not prime

split a square into finitely many hexagons with inner angles all &lt; 180^o

10 bowls with marbles in a circles

q^2 + r = 2011 when ab = q (a + b) + r , 0 \le r <a + b

split a square into finitely many hexagons with inner angles all < 180^o