MathDB

1

sum of n from 1 to n^2 in a n x n board under conditions

n \times n

board are labelled with the numbers

1

through

n^2

in the usual way. Let

n

of these cells be selected, no two of which are in the same row or column. Find all possible values of the sum of their labels.

1

2

set of points in space is allowed to mirror a point from the set , image inside

For a given set of points in space it is allowed to mirror a point from the set with respect to another point from the set, and to include the image in the set. Starting with a set of seven vertices of a cube, is it possible to include the eight vertex in the set after finitely many such steps?

Can a square of side length 5 be covered by three squares of side length 4?

Can a square of side length

5

be covered by three squares of side length

4

?

4

2

n2^{n-1} +1 is a perfect square

Find all natural numbers

n

for which

n2^{n-1} +1

is a perfect square.

a prime

Let

p

be an odd prime. Determine the positive integers

x

and

y

with

x\leq y

for which the number

\sqrt{2p}-\sqrt{x}-\sqrt{y}

is non-negative and as small as possible.

3

2

set of 4 lines such one is parallel to a medina determined by other 3

Four lines are given in a plane so that any three of them determine a triangle. One of these lines is parallel to a median in the triangle determined by the other three lines. Prove that each of the other three lines also has this property.

Rectangles erected on sides of a triangle; a concurrence

Let

ABC

be a triangle, and erect three rectangles

ABB_1A_2

,

BCC_1B_2

,

CAA_1C_2

externally on its sides

AB

,

BC

,

CA

, respectively. Prove that the perpendicular bisectors of the segments

A_1A_2

,

B_1B_2

,

C_1C_2

are concurrent.

1996 Bundeswettbewerb Mathematik

Subcontests

sum of n from 1 to n^2 in a n x n board under conditions

set of points in space is allowed to mirror a point from the set , image inside

Can a square of side length 5 be covered by three squares of side length 4?

n2^{n-1} +1 is a perfect square

a prime

set of 4 lines such one is parallel to a medina determined by other 3

Rectangles erected on sides of a triangle; a concurrence