MathDB

1

perpendicular from interior point of a convex lies on a side

P

be a convex polygon, so have all interior angles smaller than

180^o

, and let

X

be a point in the interior of

P

. Prove that

P

has a side

[AB]

such that the perpendicular from

X

to the line

AB

lies on the side

[AB]

.

9

1

open - closed numbered doors in corridor

In a very long corridor there is an infinite number of cabinets, which start with

1,2,3,...

numbered and initially all are closed. There is also a horde of QEDlers, whose number lies in set

A \subseteq \{1, 2,3,...\}

. In ascending order, the QED people now cause chaos: the person with number

a \in A

visits the cabinet with the numbers

a,2a,3a,...

opening all of the closed ones and closes all open. Show that in the end the cabinet has never exactly the same numbers from

A

open.

10

1

cities and streets in kingdom of Pinguinia

The kingdom of Pinguinia has various cities and streets, the latter being all one-way streets always run between exactly two cities, so there are no intermediate stops. Every city has exactly two streets that lead out of it and exactly two that lead into it. Prove that the streets can be divided into black and white streets so that exactly one exit of each city is black and one is white and exactly one white and one black entrance in each city.

8

1

min m to find heaviest and light lamp in a set of 256, comparing 2 each time

There are

256

lumps of metal that have different weights in pairs. With the help of a beam balance , one may now compare every two lumps. Find the smallest number

m

such that you can be sure to find the heaviest as well as the lightest lump with the weighing process.

7

1

f(xy + x + y) + f(xy-x-y)=2f (x) + 2f (y)

Find all functions

f: R\to R

, such that

f(xy + x + y) + f(xy-x-y)=2f (x) + 2f (y)

for all

x, y \in R

.

6

1

m^m- 1 is divisible by n and n^n- 1 is divisible by m

Show that there are infinitely many pairs

(m, n)

of natural numbers

m, n \ge 2

, for

m^m- 1

is divisible by

n

and

n^n- 1

is divisible by

m

.

3

1

money transfers between two banks, no of ways is a power of 2

A numerist has

n

eurodollars and distributes them to two bank accounts

A, B

in Germany and Switzerland, whereby the Eurodollars cannot be split into smaller monetary units due to the lack of a suitable name. In order to hide all money from the tax authorities if necessary, he would like to be able to move all of his money to account

B

. Due to the immense bureaucracy, money is only allowed to be moved between two accounts if the deposited amount in one account is double. Of course, he can carry out several such transfers in a row. Show that the number of ways to initially distribute the money appropriately is a power of two.

4

1

(n!)! is a multiple of (n!)^{(n-1)!}

Prove that

(n!)!

is a multiple of

(n!)^{(n-1)!}

2

1

a^4+b^5+c^6-c if a,b,c roots of x^3-x-1 = 0

Let

a,b,c

be the three different solutions of

x^3-x-1 = 0

. Compute

a^4+b^5+c^6-c

.

1

4n + 1 and 9n + 1 are perfect squares

Find all integers

n

for which both

4n + 1

and

9n + 1

are perfect squares.

2011 QEDMO 9th

Subcontests

perpendicular from interior point of a convex lies on a side

open - closed numbered doors in corridor

cities and streets in kingdom of Pinguinia

min m to find heaviest and light lamp in a set of 256, comparing 2 each time

f(xy + x + y) + f(xy-x-y)=2f (x) + 2f (y)

m^m- 1 is divisible by n and n^n- 1 is divisible by m

money transfers between two banks, no of ways is a power of 2

(n!)! is a multiple of (n!)^{(n-1)!}

a^4+b^5+c^6-c if a,b,c roots of x^3-x-1 = 0

4n + 1 and 9n + 1 are perfect squares