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Problem A. 378. (September 2005)

A. 378. Does there exist a function f:\mathbb{Q}\to\{-1,1\} such that f(x)=-f(y) whenever x and y are different rationals and xy=1 or x+y\in{0,1}?

(5 pont)

Deadline expired on 17 October 2005.


Solution. There exists such a function.

Consider the continued fraction form of the rational numbers. As is well-known, each rational x can be uniquely written as

x=a_0+\frac1{a_1+\frac1{a_2+\dots+\frac1{a_n}}}

where a0 is an integer, a1,...,an are positive integers and an>1.

For an arbitrary rational x, lett \ellx be the number of divisions in the continued fraction form of x (the number n above). Then \ell(x)=0, for inegers and \ell(x)=\ell(1/x)+1 for all 0<x<1.

Now define function f. Set

f(0)=-1;    f(x)=(-1)^{\ell(x)} for all x>0;    f(x)=-(-1)^{\ell(|x|)}, for all x<0.

By the definition of \ell(x) we have that for all positive rational x and positive integer k, \ell(x+k)=\ell(x) and f(x+k)=f(x).

Now prove f(1/x)=-f(x), if x\ne0,\pm1. Since f(-x)=-f(x) and f(-1/x)=-f(1/x), it can assumed that x>0. Moreover, the roles of x and 1/x are symmetric so we can assume 0<x<1. Then \ell(x)=\ell(1/x)+1, therefore f(1/x)=-f(x).

From the definition of f we know that f(x)=-f(y) for all x+y=0, except x=y=0. The final property which must be verified is that f(1-x)=-f(x) if x\ne\frac12. By the symmetry assume x>\frac12.

If x>1 then f(x)=f(x-1)=-f(1-x), done.

If x=1 then f(x)=f(1)=1 and f(1-x)=f(0)=-1, done.

The only remaining case is when \frac12<x<1. then

f(x)=-f
\left(\frac1x\right)=
-f\left(\frac1x-1\right)=
-f\left(\frac{1-x}x\right)=

=f\left(\frac{x}{1-x}\right)=
f\left(\frac{x}{1-x}+1\right)=
f\left(\frac1{1-x}\right)=
-f(1-x).

Therefore, function f satisfies all the required properties.

Remark. It is easy to see that there exist exactly two solutions: f and -f.


Statistics:

23 students sent a solution.
5 points:Erdélyi Márton, Gyenizse Gergő, Hujter Bálint, Jankó Zsuzsanna, Kisfaludi-Bak Sándor, Kónya 495 Gábor, Molnár 999 András, Nagy 224 Csaba, Paulin Roland, Sümegi Károly, Szabó 108 Tamás, Ureczky Bálint.
4 points:Korándi Dániel, Radnai András, Szilágyi Dániel.
3 points:2 students.
2 points:3 students.
0 point:3 students.

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