MathDB

2

10000 trees in a park, arranged in a 100x100 square grid

10000

trees in a park, arranged in a square grid with

100

rows and

100

columns. Find the largest number of trees that can be cut down, so that sitting on any of the tree stumps one cannot see any other tree stump.

3n+1 , 4n+1 are perfect squares then n is divisible by 56

Prove that if

n

is a natural number such that both

3n+1

and

4n+1

are squares, then

n

is divisible by

56

.

3

2

ST \ge 2r(\sqrt{2}-1) , inscribed circles in a semicircle

A semicircle with diameter

AB = 2r

is divided into two sectors by an arbitrary radius. To each of the sectors a circle is inscribed. These two circles touch A

B

at

S

and

T

. Show that

ST \ge 2r(\sqrt{2}-1)

.

congruent squares inscribed in a triangle

A square

S_a

is inscribed in an acute-angled triangle

ABC

with two vertices on side

BC

and one on each of sides

AC

and

AB

. Squares

S_b

and

S_c

are analogously inscribed in the triangle. For which triangles are the squares

S_a,S_b

, and

S_c

congruent?

1

2

15 integers from 100 with difference divisible by 7

Given

100

integers, is it always possible to choose

15

of them such that the difference of any two of the chosen numbers is divisible by

7

? What is the answer if

15

is replaced by

16

?

3 faced of regular tetrahedron are painted in white and one in black

Three faces of a regular tetrahedron are painted in white and the remaining one in black. Initially, the tetrahedron is positioned on a plane with the black face down. It is then tilted several times over its edges. After a while it returns to its original position. Can it now have a white face down?

2

Prime Subject

Find a prime number

p

such that

\frac{p+1}{2}

and

\frac{p^2+1}{2}

are perfect square

y =\sqrt{x^2 +a} has infinite solutions with rational coordinates

Show that for any rational number

a

the equation

y =\sqrt{x^2 +a}

has infinitely many solutions in rational numbers

x

and

y

.

1997 Bundeswettbewerb Mathematik

Subcontests

10000 trees in a park, arranged in a 100x100 square grid

3n+1 , 4n+1 are perfect squares then n is divisible by 56

ST \ge 2r(\sqrt{2}-1) , inscribed circles in a semicircle

congruent squares inscribed in a triangle

15 integers from 100 with difference divisible by 7

3 faced of regular tetrahedron are painted in white and one in black

Prime Subject

y =\sqrt{x^2 +a} has infinite solutions with rational coordinates