Found problems: 38
For a given positive integer $n$, let $f(x) =x^{n}$. Is it possible for the decimal number
$$0.f(1)f(2)f(3)\ldots$$
to be rational? (Example: for $n=2$, we are considering $0.1491625\ldots$)
A function $f$ satisfies the conditions $f (x) \ne 0$ and $f (x+2) = f (x-1) f (x+5)$ for all real x. Show that $f (x+18) = f (x)$ for any real $x$.
For each positive integer $n$, let $S(n)$ be the sum of the digits of $n^2 +1$. A sequence $\{a_n\}$ is defined, with $a_0$ an arbitrary positive integer and $a_{n+1} = S(a_n)$. Prove that the sequence $\{a_n\}$ is eventually periodic with period three.
The sequence of integers $a_1, a_2, ,,$ is defined as follows:
$$a_n=\begin{cases} 0\,\,\,\, if\,\,\,\, n\,\,\,\, has\,\,\,\, an\,\,\,\, even\,\,\,\, number\,\,\,\, of\,\,\,\, divisors\,\,\,\, greater\,\,\,\, than\,\,\,\, 2014 \\ 1 \,\,\,\, if \,\,\,\, n \,\,\,\, has \,\,\,\, an \,\,\,\, odd \,\,\,\, number \,\,\,\, of \,\,\,\, divisors \,\,\,\, greater \,\,\,\, than \,\,\,\, 2014\end{cases}$$
Show that the sequence $a_n$ never becomes periodic.
Denote by $a_n$ the last digit of the number $n^{(n^n)}$ (let $n\ne 0$ be a natural number ). Prove that the numbers $a_n$ form a periodic sequence and state this period!
Find the smallest positive period of the function $f(x)=\sin (1998x)+ \sin (2000x)$
Define a sequence $ a_n = n^n + (n - 1)^{n+1}$ when $n$ is a positive integer.
Define all those positive integer $m$ , for which this sequence of numbers is eventually periodic modulo $m$, e.g. there are such positive integers $K$ and $s$ such that $a_k \equiv a_{k+s}$ ($mod \,m$), where $k$ is an integer with $k \ge K$.
The function $f (x)$ satisfies the relation $f(x+\pi)=\frac{f(x)}{3f(x) -1}$ for any real number $x$. Prove that the function $f (x)$ is periodic.
Let $a> 0$ and $f: R\to R$ a function such that $f (x) + f (x + 2a) + f (x + 3a) + f (x + 5a) = 1$ for all $x\in R$ . Show that $f$ is periodic, that is, that there is some $b> 0$ for which $f (x) = f (x + b)$ for every $x \in R$ holds. Find the smallest such $b$, which works for all these functions .
Function $f: R\to R$ is said periodic , if $f$ is not a constant function and there is a number real positive $p$ with the property of $f (x) = f (x + p)$ for every $x \in R$. The smallest positive real number p which satisfies the condition $f (x) = f (x + p)$ for each $x \in R$ is named period of $f$. Given $a$ and $b$ real positive numbers, show that there are periodic functions $f_1$ and $f_2$, with periods $a$ and $b$ respectively, so that $f_1 (x)\cdot f_2 (x)$ is also a periodic function.
The periods of two periodic sequences are $7$ and $13$. What is the maximal length of initial sections of the two sequences which can coincide? (The period $p$ of a sequence $a_1$,$a_2$, $...$ is the minimal $p$ such that $a_n = a_{n+p}$ for all $n$.)
(AY Belov)
Let $m$ and $n$ be two natural numbers and let $d = gcd (m, n)$ their greatest common divisor.
Let $a_1, a_2,...$ and $b_1, b_2, ...$ be two sequences of integers which are periodic with periods $m$ and $n$ respectively (this means that $a_{i + m} = a_i$ and $b_{i + n} = b_i$ for all natural numbers $i \ge 1$, note that there could be smaller periods).
Prove that if the two sequences on the first $m + n - d$ terms match (i.e. $a_i = b_i$ for all $i \in \{1, 2, ..., m + n - d\}$), then they are the same (so $a_i = b_i$ for all natural $i \ge 1$).
The numerical sequence $\{x_n\}$ satisfies the condition $$x_{n+1}=|x_n|-x_{n-1}$$ for all $n > 1$. Prove that the sequence is periodic with period $9$, i.e. for any $n > 1$ we have $x_n = x_{n+9}$.
(M Kontsevich, Moscow)