Found problems: 38
VI Soros Olympiad 1999 - 2000 (Russia), 10.8
Find the smallest positive period of the function $f(x)=\sin (1998x)+ \sin (2000x)$
1976 Chisinau City MO, 129
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.
1998 Belarus Team Selection Test, 1
Do there exist functions $f : R \to R$ and $g : R \to R$, $g$ being periodic, such that $$x^3= f(\lfloor x \rfloor ) + g(x)$$
for all real $x$ ?
2019 Finnish National High School Mathematics Comp, 4
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 $k \ge K$.
1996 All-Russian Olympiad Regional Round, 11.8
Is there an infinite periodic sequence consisting of the letters $a$ and$ b$, such that if all letters are replaced simultaneously $a$ to $aba$ and letters $b$ to $bba$ does it transform into itself (possibly with a shift)? (A sequence is called periodic if there is such natural number $n$, which for every $i = 1, 2, . . . i$-th member of this sequence is equal to the ($i + n$)- th.)
1990 Tournament Of Towns, (271) 5
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)
2016 NZMOC Camp Selection Problems, 9
An $n$-tuple $(a_1, a_2 . . . , a_n)$ is [i]occasionally periodic[/i] if there exist a non-negative integer $i$ and a positive integer $p$ satisfying $i + 2p \le n$ and $a_{i+j} = a_{i+j+p}$ for every $j = 1, 2, . . . , p$. Let $k$ be a positive integer. Find the least positive integer $n$ for which there exists an $n$-tuple $(a_1, a_2 . . . , a_n)$ with elements from the set $\{1, 2, . . . , k\}$, which is not occasionally periodic but whose arbitrary extension $(a_1, a_2, . . . , a_n, a_{n+1})$ is occasionally periodic for any $a_{n+1} \in \{1, 2, . . . , k\}$.
1998 Estonia National Olympiad, 3
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 $x$.
2013 QEDMO 13th or 12th, 4
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 .
2023 Canadian Mathematical Olympiad Qualification, 4
Let $a_1$, $a_2$, $ ...$ be a sequence of numbers, each either $1$ or $-1$. Show that if
$$\frac{a_1}{3}+\frac{a_2}{3^2} + ... =\frac{p}{q}$$ for integers $p$ and $q$ such that $3$ does not divide $q$, then the sequence $a_1$, $a_2$, $ ...$ is periodic; that is, there is some positive integer $n$ such that $a_i = a_{n+i}$ for $i = 1$, $2$,$...$.
2006 Miklós Schweitzer, 7
Suppose that the function $f: Z \to Z$ can be written in the form $f = g_1+...+g_k$ , where $g_1,. . . , g_k: Z \to R$ are real-valued periodic functions, with period $a_1,...,a_k$. Does it follow that f can be written in the form $f = h_1 +. . + h_k$ , where $h_1,. . . , h_k: Z \to Z$ are periodic functions with integer values, also with period $a_1,...,a_k$?
2006 Argentina National Olympiad, 1
Let $A$ be the set of positive real numbers less than $1$ that have a periodic decimal expansion with a period of ten different digits. Find a positive integer $n$ greater than $1$ and less than $10^{10}$ such that $na-a$ is a positive integer for all $a$. of set $A$.
1990 Czech and Slovak Olympiad III A, 1
Let $(a_n)_{n\ge1}$ be a sequence given by
\begin{align*}
a_1 &= 1, \\
a_{2^k+j} &= -a_j\text{ for any } k\ge0,1\le j\le 2^k.
\end{align*}
Show that the sequence is not periodic.
1998 Tuymaada Olympiad, 6
Prove that the sequence of the first digits of the numbers in the form $2^n+3^n$ is nonperiodic.
1994 Tournament Of Towns, (414) 2
Consider a sequence of numbers between $0$ and $1$ in which the next number after $x$ is $1 - |1 - 2x|$. ($|x| = x$ if$ x \ge 0$, $|x| = -x$ if $x < 0$.) Prove that
(a) if the first number of the sequence is rational, then the sequence will be periodic (i.e. the terms repeat with a certain cycle length after a certain term in the sequence);
(b) if the sequence is periodic, then the first number is rational.
(G Shabat)
1975 Chisinau City MO, 88
Prove that the fraction $0.123456789101112...$ is not periodic.
1998 Switzerland Team Selection Test, 10
5. Let $f : R \to R$ be a function that satisfies for all $x \in R$
(i) $| f(x)| \le 1$, and
(ii) $f\left(x+\frac{13}{42}\right)+ f(x) = f\left(x+\frac{1}{6}\right)+f\left(x+\frac{1}{7}\right)$
Prove that $f$ is a periodic function
2010 QEDMO 7th, 11
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 $i \ge 1$).
2001 All-Russian Olympiad Regional Round, 11.5
Given a sequence $\{x_k\}$ such that $x_1 = 1$, $x_{n+1} = n \sin x_n+ 1$. Prove that the sequence is non-periodic.
2021 Saudi Arabia Training Tests, 26
Given an infinite sequence of numbers $a_1, a_2, a_3, ...$ such that for each positive integer $k$, there exists positive integer $t$ for which $a_k = a_{k+t} = a_{k+2t} = ....$ Does this sequences must be periodic?
1987 Greece National Olympiad, 2
If for function $f$ holds that $$f(x)+f(x+1)+f(x+2)+...+f(x+1986)=0$$ for any $\in\mathbb{R}$, prove that $f$ is periodic and find one period of her.
1994 Tournament Of Towns, (428) 5
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)
2015 Indonesia MO Shortlist, A1
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.
2014 Switzerland - Final Round, 9
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.
1997 ITAMO, 2
Let a real function $f$ defined on the real numbers satisfy the following conditions:
(i) $f(10+x) = f(10- x)$
(ii) $f(20+x) = - f(20- x)$
for all $x$. Prove that f is odd and periodic.