Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem A. 689. (February 2017)

A. 689. Let $\displaystyle f_1,f_2,\ldots$ be an infinite sequence of continuous $\displaystyle \mathbb{R}\to\mathbb{R}$ functions such that for arbitrary positive integer $\displaystyle k$ and arbitrary real numbers $\displaystyle r>0$ and $\displaystyle c$ there exists a number $\displaystyle x\in(-r,r)$ with $\displaystyle f_k(x)\ne cx$. Show that there exists a sequence $\displaystyle a_1,a_2,\ldots$ of real numbers such that $\displaystyle \sum_{n=1}^\infty a_n$ is convergent, but $\displaystyle \sum_{n=1}^\infty f_k(a_n)$ is divergent for every positive integer $\displaystyle k$.

(5 pont)

Deadline expired on March 10, 2017.

### Statistics:

 8 students sent a solution. 5 points: Bukva Balázs, Gáspár Attila, Kovács 246 Benedek, Lajkó Kálmán, Williams Kada. 4 points: Matolcsi Dávid. 3 points: 1 student. 0 point: 1 student.

Problems in Mathematics of KöMaL, February 2017