MathDB

1

essentially different Fibonacci sequences

A sequence of real numbers is called a Fibonacci sequence if

t_{n+2} = t_{n+1} + t_n

for

n= 1,2,3,. .

. Two Fibonacci sequences are said to be essentially different if the terms of one sequence cannot be obtained by multiplying the terms of the other by a constant. For example, the Fibonacci sequences

1,2,3,5,8,...

and

1,3,4,7,11,...

are essentially different, but the sequences

1,2,3,5,8,...

and

2,4,6,10,16,...

are not.
(a) Prove that there exist real numbers

p

and

q

such that the sequences

1,p,p^2,p^3,...

and

1,q,q^2,q^3,...

are essentially different Fibonacci sequences.
(b) Let

a_1,a_2,a_3,...

and

b_1,b_2,b_3,...

be essentially different Fibonacci sequences. Prove that for every Fibonacci sequence

t_1,t_2,t_3,...

, there exists exactly one number

\alpha

and exactly one number

\beta

, such that:

t_n = \alpha a_n + \beta b_n

for

n = 1,2,3,...

(c)

t_1,t_2,t_3,...

, is the Fibonacci sequence with

t_1 = 1

and

t_2= 2

. Express

t_n

in terms of

n

.

4

1

S(k)=a(1) + a(2) + ... + a(2^k) when n = 2^{a(n)} (2b(n)+1),

For every positive integer

n

there exist unambiguously determined non-negative integers

a(n)

and

b(n)

such that

n = 2^{a(n)}(2b(n)+1),

For positive integer

k

we define

S(k)

by:

a(1) + a(2) + ... + a(2^k) = S(k)

Express

S(k)

in terms of

k

.

5

1

lines on paper and intersection points

Someone draws at least three lines on paper. Each cuts the other lines two by two. No three lines pass through one point. He chooses a line and counts the intersection points on either side of the line. The numbers of intersections turn out to be the same. He chooses another line. Now the intersections number on one side appears to be six times as large as that on the other side. What is the minimum number of lines where this is possible?
[hide=original wording of second sentence]De lijnen snijden elkaar twee aan twee.

1

PB//DQ if AQ//CP and AB// CD (trapezoid ABCD)

Given a trapezoid

ABCD

, where sides

AB

and

CD

are parallel; the points

P

on

AD

and

Q

on

BC

lie such that the lines

AQ

and

CP

are parallel. Prove that lines

PB

and

DQ

are parallel.

3

1

diophantine x^2 + x +1 = y^2

Prove that

(0,1)

,

(0, -1)

,

( -1,1)

and

(-1,-1)

are the only integer solutions of

x^2 + x +1 = y^2.

1971 Dutch Mathematical Olympiad

Subcontests

essentially different Fibonacci sequences

S(k)=a(1) + a(2) + ... + a(2^k) when n = 2^{a(n)} (2b(n)+1),

lines on paper and intersection points

PB//DQ if AQ//CP and AB// CD (trapezoid ABCD)

diophantine x^2 + x +1 = y^2