Found problems: 20
Let $f : \mathbb{R} \to \mathbb{R}$ a periodic and continue function with period $T$ and $F : \mathbb{R} \to \mathbb{R}$ antiderivative of $f$. Then $$\int \limits_0^T \left[F(nx)-F(x)-f(x)\frac{(n-1)T}{2}\right]dx=0$$
Denote by S the set of all primes such the decimal representation of $\frac{1}{p}$ has the fundamental period divisible by 3. For every $p \in S$ such that $\frac{1}{p}$ has the fundamental period $3r$ one may write
\[\frac{1}{p}=0,a_{1}a_{2}\ldots a_{3r}a_{1}a_{2} \ldots a_{3r} \ldots , \]
where $r=r(p)$; for every $p \in S$ and every integer $k \geq 1$ define $f(k,p)$ by \[ f(k,p)= a_{k}+a_{k+r(p)}+a_{k+2.r(p)}\]
a) Prove that $S$ is infinite.
b) Find the highest value of $f(k,p)$ for $k \geq 1$ and $p \in S$
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties:
(i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$;
(ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite.
Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic.
[i]Proposed by Ang Jie Jun, Singapore[/i]
Let $f(x) = \sin \frac{x}{3}+ \cos \frac{3x}{10}$ for all real $x$.
Find the least natural number $n$ such that $f(n\pi + x)= f(x)$ for all real $x$.
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.
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties:
(i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$;
(ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite.
Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic.
[i]Proposed by Ang Jie Jun, Singapore[/i]
Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$.
Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.
Let be a continuous and periodic function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ of period $ 1. $ Show:
[b]a)[/b] $ a\in\mathbb{R}\implies\int_a^{a+1} f(x)dx =\int_0^1 f(x) dx . $
[b]b)[/b] $ \lim_{n\to\infty} \int_0^1 f(x)f(nx) dx=\left( \int_0^1 f(x) dx \right)^2 . $
[i]C. Mortici[/i]
Prove that for any $a$ the function $y (x) = \cos (\cos x) + a \cdot \sin (\sin x)$ is periodic.
Find its smallest period in terms of $a$.
Let $f, g$ be functions $\mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y$, $$f(g(x) + y) = g(x + y)$$
Prove that either $f$ is the identity function or $g$ is periodic.
[i]Proposed by Pranjal Srivastava[/i]
Let ${f}$ be a bounded real function defined for all real numbers and satisfying for all real numbers ${x}$ the condition ${ f \Big(x+\frac{1}{3}\Big) + f \Big(x+\frac{1}{2}\Big)=f(x)+ f \Big(x+\frac{5}{6}\Big)}$ . Show that ${f}$ is periodic.
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.
Let $f$ be a real-valued function defined for all real numbers, such that for some $a>0$ we have \[ f(x+a)={1\over2}+\sqrt{f(x)-f(x)^2} \] for all $x$.
Prove that $f$ is periodic, and give an example of such a non-constant $f$ for $a=1$.
Let $\mathbb{Z}_{>0}$ denote the set of positive integers. Consider a function $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$. For any $m, n \in \mathbb{Z}_{>0}$ we write $f^n(m) = \underbrace{f(f(\ldots f}_{n}(m)\ldots))$. Suppose that $f$ has the following two properties:
(i) if $m, n \in \mathbb{Z}_{>0}$, then $\frac{f^n(m) - m}{n} \in \mathbb{Z}_{>0}$;
(ii) The set $\mathbb{Z}_{>0} \setminus \{f(n) \mid n\in \mathbb{Z}_{>0}\}$ is finite.
Prove that the sequence $f(1) - 1, f(2) - 2, f(3) - 3, \ldots$ is periodic.
[i]Proposed by Ang Jie Jun, Singapore[/i]
Let $ f$ be a function from the set of real numbers $ \mathbb{R}$ into itself such for all $ x \in \mathbb{R},$ we have $ |f(x)| \leq 1$ and
\[ f \left( x \plus{} \frac{13}{42} \right) \plus{} f(x) \equal{} f \left( x \plus{} \frac{1}{6} \right) \plus{} f \left( x \plus{} \frac{1}{7} \right).\]
Prove that $ f$ is a periodic function (that is, there exists a non-zero real number $ c$ such $ f(x\plus{}c) \equal{} f(x)$ for all $ x \in \mathbb{R}$).
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.
If $f$ and $g$ are continuous and periodic functions with period $1$ on the real line, then
$$\lim_{n\to \infty} \int_{0}^{1} f(x)g (nx)\; dx =\left( \int_{0}^{1} f(x)\; dx\right)\left( \int_{0}^{1} g(x)\; dx\right).$$
Let $f, g$ be functions $\mathbb{R} \rightarrow \mathbb{R}$ such that for all reals $x,y$, $$f(g(x) + y) = g(x + y)$$
Prove that either $f$ is the identity function or $g$ is periodic.
[i]Proposed by Pranjal Srivastava[/i]
Let be a continuous function $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having a positive period $ T. $ Prove that:
$$ \lim_{n\to\infty } e^{-nT}\int_0^{nT} e^tf(t)dt=\frac{1}{e^T-1}\int_0^T e^tf(t)dt $$
Let $(a_{n})_{n \ge 1}$ be a sequence of integers satisfying the inequality \[ 0\le a_{n-1}+\frac{1-\sqrt{5}}{2}a_{n}+a_{n+1} <1 \] for all $n \ge 2$. Prove that the sequence $(a_{n})$ is periodic.
Any Hints or Sols for this hard problem?? :help: