MathDB

1

f(2x) = 2f(x), f(4x+1) = 4f(x) + 3, f(4x-1) = 2f(2x -1)-1

\{0, 1, 2,...\}

of non-negative integers, satisfying

f(2x) = 2f(x), f(4x+1) = 4f(x) + 3

, and

f(4x-1) = 2f(2x - 1) -1

. Show that

f

is an injection, i.e. if

f(x) = f(y)

, then

x = y

.

3

1

4 points, 6 fixed equal distances, ratio of distances ?

Let

A, B, C

, and

D

be four different points in the plane. Three of the line segments

AB, AC, AD, BC, BD

, and

CD

have length

a

. The other three have length

b

, where

b > a

. Determine all possible values of the quotient

\frac{b}{a}

. .

2

1

4 equal areas from an interior P of a convex quadrilateral

Let

ABCD

be a convex quadrilateral. We assume that there exists a point

P

inside the quadrilateral such that the areas of the triangles

ABP, BCP, CDP

, and

DAP

are equal. Show that at least one of the diagonals of the quadrilateral bisects the other diagonal.

1

maximal number (x, y, z) of 7 elements with x < y & x + y=z

Let

A

be a set of seven positive numbers. Determine the maximal number of triples

(x, y, z)

of elements of

A

satisfying

x < y

and

x + y = z

.

1997 Nordic

Subcontests

f(2x) = 2f(x), f(4x+1) = 4f(x) + 3, f(4x-1) = 2f(2x -1)-1

4 points, 6 fixed equal distances, ratio of distances ?

4 equal areas from an interior P of a convex quadrilateral

maximal number (x, y, z) of 7 elements with x < y & x + y=z

1997 Nordic

Subcontests

f(2x) = 2f(x), f(4x+1) = 4f(x) + 3, f(4x-1) = 2f(2x -1)-1

4 points, 6 fixed equal distances, ratio of distances ?

4 equal areas from an interior P of a convex quadrilateral

maximal number (x, y, z) of 7 elements with x &lt; y &amp; x + y=z

maximal number (x, y, z) of 7 elements with x < y & x + y=z