MathDB

1

f (f (...( f (y)) ...)) = a where f (x) = x - 1/x

X = Q-\{-1,0,1\}

. We consider the function

f: X\to X

given by

f (x) = x -\frac{1}{x} .

Is there an

a \in X

such that for every natural number n there is a

y \in X

with

f (f (...( f (y)) ...)) = a

where

f

occurs exactly

n

times on the left side?

10

1

n coins trick, a magician and his assistant

The great Zagier and his assistant performed a magic trick, which was a natural one number

n

involved. To do this, Zagier distributes

n

identical coins (with heads and tails on the sides) and then leave the hall. The audience now arranges the coins in a row, you can choose the top side of each coin as you like, and then tell the assistant one natural number

k

from

1

to

n

. This then turns over exactly one coin, whereupon Zagier is brought in and placed in front of said coins. To the surprise of the non-mathematical, he can name the number

k

to the public. For which natural numbers

n

can this trick be carried out, if the two are allowed to coordinate only before the show, but during the show they do not use any magic tricks?

8

1

coloring sides and diagonals of a regular polygon

Find for which natural numbers

n

one can color the sides and diagonals of a regular

n

-gon with

n

colors in such a way that for each triplet in pairs of different colors, a triangle can be found, the sides of which are sides or diagonals of

n

-gon and which is colored with exactly these three colors.

7

1

cookies on a 7x7 chessboard

Cookies should be placed on a

7 \times 7

chess board, so that never four cookies can form a rectangle whose sides are parallel to those of the chessboard. Find the maximum number of biscuits that can be positioned in this way.
Note: If you do the same job for a

13 \times 13

chessboard, you get a biscuit. If you solve it for an infinite number of squares of chessboards, you get two biscuits. If you solve them for all sidelengths, , you even get two three biscuits (We cannot distribute more cookies, otherwise we run the risk of them to form a rectangle).

6

1

n noble men with different no of posts meet k spammers at Council of War

An ancient noble family has

n

members, each holding a different number of posts . As every year in December, they gather at a very specific place for a Council of War to be held, where also k, from the point of view of the high nobility, unimportant spammers speak up, which, due to their irrelevance, should and cannot be further differentiated. The Council is held as follows: those present speak one after the other, each one carefully put forward his request once. In addition, for reasons of respect, a nobleman never speaks right after a nobleman who holds more posts, while the common people disregarde such rules. Find the number of possible sequences of the Council of war.

5

1

f(x) is completely reducible if f (x^{2011}) is also

A polynomial

f (x)

with real coefficients is called completely reducible if it is a product of at least two non-constant polynomials whose coefficientsare all nonnegative real numbers. Show: If

f (x^{2011})

is completely reducible, then

f(x)

is also.

2

1

G(n)/U(n) = (2^n + 1)(2^n - 1) for even, odd sum of products of 0s and 1s

Let

n

be a positive integer. Let

G (n)

be the number of

x_1,..., x_n, y_1,...,y_n \in \{0,1\}

, for which the number

x_1y_1 + x_2y_2 +...+ x_ny_n

is even, and similarly let

U (n)

be the number for which this sum is odd. Prove that

\frac{G(n)}{U(n)}= \frac{2^n + 1}{2^n - 1}.

4

1

n members from n identical and 2n+1 different

In year

2525

the QED has

3n + 1

members, of which

n

are identical robots and

2n + 1

(uncloned and therefore distinguishable) people. For the

263^{th}

board election in Wurzburg there will be exactly

n

members. Find out how many distinguishable compositions are conceivable for this.

3

1

a = b if (a^2 + ab + 1) a multiple of (b^2 + ab + 1)

Let

a, b

be positive integers such that

a^2 + ab + 1

a multiple of

b^2 + ab + 1

. Prove that

a = b

.

1

xf (y) + yf (x) = (x + y) f (xy)

Find all functions

f: R\to R

with the property that

xf (y) + yf (x) = (x + y) f (xy)

for all

x, y \in R

.

2011 QEDMO 10th

Subcontests

f (f (...( f (y)) ...)) = a where f (x) = x - 1/x

n coins trick, a magician and his assistant

coloring sides and diagonals of a regular polygon

cookies on a 7x7 chessboard

n noble men with different no of posts meet k spammers at Council of War

f(x) is completely reducible if f (x^{2011}) is also

G(n)/U(n) = (2^n + 1)(2^n - 1) for even, odd sum of products of 0s and 1s

n members from n identical and 2n+1 different

a = b if (a^2 + ab + 1) a multiple of (b^2 + ab + 1)

xf (y) + yf (x) = (x + y) f (xy)