2005 Hungary-Israel Binational

Part of Hungary-Israel Binational

Subcontests

(3)

1

functions defined by Fibonacci numbers

F_{n}

n-

th Fibonacci number (where

F_{1}= F_{2}= 1

). Consider the functions

f_{n}(x)=\parallel . . . \parallel |x|-F_{n}|-F_{n-1}|-...-F_{2}|-F_{1}|, g_{n}(x)=| . . . \parallel x-1|-1|-...-1|

F_{1}+...+F_{n}

one’s). Show that

f_{n}(x) = g_{n}(x)

for every real number

x.

2

equation $\frac{1}{2}=\sum_{i=1}^n\frac{1}{x_i^2}$

Find all sequences

x_{1},x_{2},...,x_{n}

of distinct positive integers such that

\frac{1}{2}=\sum_{i=1}^{n}\frac{1}{x_{i}^{2}}

7 rods erected at the vertices of a regular heptagonal area

There are seven rods erected at the vertices of a regular heptagonal area. The top of each rod is connected to the top of its second neighbor by a straight piece of wire so that, looking from above, one sees each wire crossing exactly two others. Is it possible to set the respective heights of the rods in such a way that no four tops of the rods are coplanar and each wire passes one of the crossings from above and the other one from below?

2

right triangle and squares externally drawn

ABB_{1}A_{2}

BCC_{1}B_{2}

are externally drawn on the hypotenuse

AB

BC

of a right triangle

ABC

. Show that the lines

CA_{2}

AB_{2}

meet on the perimeter of a square with the vertices on the perimeter of triangle

ABC .

consecutive positive integers contains exactly 25 primes

Does there exist a sequence of

2005

consecutive positive integers that contains exactly

25