Olympiad problems

1989 Greece National Olympiad, 1

Consider two functions $f , \,g \,:\mathbb{R} \to \mathbb{R}$ such that from some $a>0$ holds $g(x)=f(x+a)$ for any $x \in \mathbb{R}$. If $f$ is even and $g$ is odd, prove that both functions are periodic.

1994 Bulgaria National Olympiad, 5

Tags: periodic , remainder , number theory

Let $k$ be a positive integer and $r_n$ be the remainder when ${2 n} \choose {n}$ is divided by $k$. Find all $k$ for which the sequence $(r_n)_{n=1}^{\infty}$ is eventually periodic.

1989 Bundeswettbewerb Mathematik, 1

Tags: rational , decimal representation , periodic , number theory

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 $0.1491625\ldots$)

1987 Greece National Olympiad, 2

Tags: periodic , periodic function , function

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.

1997 ITAMO, 2

Tags: periodic function , periodic , odd , algebra , function

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.

2025 Bulgarian Spring Mathematical Competition, 12.2

Tags: periodic , algebra

Determine all values of $a_0$ for which the sequence of real numbers with $a_{n+1}=3a_n - 4a_n^3$ for all $n\geq 0$ is periodic from the beginning.

1998 Estonia National Olympiad, 3

Tags: functional equation , periodic , algebra , functional , function

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 $x$.

1998 Switzerland Team Selection Test, 1

Tags: algebra , periodic , functional , function

A function $f : R -\{0\} \to R$ has the following properties: (i) $f(x)- f(y) = f(x)f\left(\frac{1}{y}\right)- f(y)f\left(\frac{1}{x}\right)$ for all $x,y \ne 0$, (ii) $f$ takes the value $\frac12$ at least once. Determine $f(-1)$. Prove that $f$ is a periodic function

2008 District Olympiad, 2

Tags: analysis , primitive , integral , periodic , function , real analysis

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a countinuous and periodic function, of period $ T. $ If $ F $ is a primitive of $ f, $ show that: [b]a)[/b] the function $ G:\mathbb{R}\longrightarrow\mathbb{R}, G(x)=F(x)-\frac{x}{T}\int_0^T f(t)dt $ is periodic. [b]b)[/b] $ \lim_{n\to\infty}\sum_{i=1}^n\frac{F(i)}{n^2+i^2} =\frac{\ln 2}{2T}\int_0^T f(x)dx. $

1964 Putnam, A4

Tags: recurrence relation , periodic

Let $p_n$ be a bounded sequence of integers which satisfies the recursion $$p_n =\frac{p_{n-1} +p_{n-2} + p_{n-3}p _{n-4}}{p_{n-1} p_{n-2}+ p_{n-3} +p_{n-4}}.$$ Show that the sequence eventually becomes periodic.

2006 Miklós Schweitzer, 7

Tags: periodic , real analysis , algebra

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 $a_1,...,a_k$?

1994 Tournament Of Towns, (439) 5

Tags: algebra , periodic

The periods of two periodic sequences are coprime (i.e. relatively prime) numbers $m$ and $n$.. 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)

2021 Saudi Arabia Training Tests, 26

Tags: algebra , sequence , periodic

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?

1994 Tournament Of Towns, (428) 5

Tags: algebra , sequence , periodic

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)

1964 German National Olympiad, 4

Tags: number theory , last digit , periodic , digit

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!

VI Soros Olympiad 1999 - 2000 (Russia), 10.8

Tags: periodic , trigonometry , algebra

Find the smallest positive period of the function $f(x)=\sin (1998x)+ \sin (2000x)$

2013 QEDMO 13th or 12th, 4

Tags: functional , algebra , periodic , functional equation

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 .

2007 Cuba MO, 8

Tags: periodic sequence , periodic , number theory , sum of digits

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.

1996 All-Russian Olympiad Regional Round, 11.8

Tags: periodic , combinatorics

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 Czech and Slovak Olympiad III A, 1

Tags: periodic , recurrence relation , algebra , sequence

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.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.8

Tags: polynomial , algebra , periodic

A polynomial with rational coefficients is called [i]integer[/i], if it takes integer values for all integer values of the variable. For an integer polynomial $P$, consider the sequence $(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...$ a) Prove that this sequence is periodic, the period of which is some power of two (i.e. for some integer $k$ and for all natural $i$, the $i$-th and ($i+2^k$)th members of the sequence are equal). b) Prove that for any periodic sequence consisting of $(- 1)$ and $ 1$ and with a period of some power of two, there exists a integer, polynomial P for which this sequence is $(-1)^{P(1)},(-1)^{P(2)},(-1)^{P(3)},...$

2010 QEDMO 7th, 11

Tags: periodic , algebra

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 $i \ge 1$).

2019 Finnish National High School Mathematics Comp, 4

Tags: periodic sequence , sequence , periodic , bir tinga qimmat masala

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 $k \ge K$.

1976 Chisinau City MO, 129

Tags: algebra , function , functional , periodic

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.

2000 Chile National Olympiad, 3

Tags: divide , periodic , divisible , number theory

A number $N_k$ is defined as [i]periodic[/i] if it is composed in number base $N$ of a repeated $k$ times . Prove that $7$ divides to infinite periodic numbers of the set $N_1, N_2, N_3,...$