Let $ x,y\in \mathbb{R} $. Show that if the set $ A_{x,y}=\{ \cos {(n\pi x)}+\cos {(n\pi y)} \mid n\in \mathbb{N}\} $ is finite then $ x,y \in \mathbb{Q} $. Vasile Pop
Problem
Source: Romanian IMO Team Selection Test TST 1996, problem 3
Tags: trigonometry, algebra unsolved, algebra
31.08.2005 16:21
Does finite sequence mean that $ A_{x,y} $ take on only a finite no. of values???
31.08.2005 17:16
From the equality \[\left( \cos n\pi x -\cos n\pi y \right) ^2=2+\cos2n\pi x +\cos2n\pi y -\left( \cos n\pi x +\cos n\pi y \right)^2 \]we deduce that the set \[B_{x,y}=\{\cos n\pi x -\cos n\pi y |n\in \mathbb{N}\} \]is also finite. Since \[\cos n\pi x =\frac{(\cos n\pi x +\cos n\pi y )+(\cos n\pi x -\cos n\pi y )}{2} \]we obtain that the set \[A=\{\cos n\pi x |n \in \mathbb{N}\} \]is finite as well. It follows that there exist $n,m \in \mathbb{N},n \ne m$ such that $\cos n\pi x=\cos m\pi x$, hence $x=\frac {2k}{n\pm m}$, for some integer $k$. Similarly, $y \in \mathbb{Q}$.
31.08.2005 23:56
enescu where did you get that $\displaystyle B_{x,y}=\{\cos n\pi x -\cos n\pi y |n\in \mathbb{N}\}$ is finite? i really want to learn from this solution can u explain
01.09.2005 00:08
Look at the right hand side of the equality \[\left( \cos n\pi x -\cos n\pi y \right) ^2=2+\cos2n\pi x +\cos2n\pi y -\left( \cos n\pi x +\cos n\pi y \right)^2 \]and observe that it only takes finitely many values, according to the hypothesis (because $\cos2n\pi x +\cos2n\pi y \in A_{x,y}$ as well)
01.09.2005 00:30
thanks a lot really nice solution...
03.09.2005 23:30
ehsan2004 wrote: Let $x,y\in \mathbb{R}$. Show that if $A_{x,y}={\{\cos n\pi x+\cos n\pi y \mid n\in \mathbb{N}\}}$ be the finite sequence then $x,y \in \mathbb{Q}$. I heard Vasile Pop is the author of this problem ? Is it true ? Are there some others articles related to this problem ?
03.09.2005 23:46
Moubinool wrote: I heard Vasile Pop is the author of this problem ? Is it true ? Yes. Moubinool wrote: Are there some others articles related to this problem ? None to my knowledge.
26.10.2005 13:32
Well, Vasile Pop wrote an article in Gazeta Matematica (I think in 1999 or later) about some of his problems given in national olympiads and among them this one was present too. He even generalized these problems. Unfortunately, I'm in Paris now and I don't have Gazeta Matematica with me.