Found problems: 4776
$f(x)$ is a continuous function defined in $x>0$. For all $a,\ b\ (a>0,\ b>0)$, if $\int_a^b f(x)\ dx$ is determined by only $\frac{b}{a}$, then prove that $f(x)=\frac{c}{x}\ (c: constant).$
The sequence of functions $f_0,f_1,f_2,...$ is given by the conditions:
$f_0(x) = |x|$ for all $x \in R$
$f_{n+1}(x) = |f_n(x)-2|$ for $n=0,1,2,...$ and all $x \in R$.
For each positive integer $n$, solve the equation $f_n(x)=1$.
Does there exist a function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that
\[f(f(n-1)=f(n+1)-f(n)\quad\text{for all}\ n\ge 2\text{?} \]
How many functions $f: \left\{1,2,3\right\} \to \left\{1,2,3 \right\}$ satisfy $f(f(x))=f(f(f(x)))$ for every $ x $?
Let $|a|<\frac{\pi}{2}.$ Evaluate the following definite integral.
\[\int_{0}^{\frac{\pi}{2}}\frac{dx}{\{\sin (a+x)+\cos x\}^{2}}\]
Prove that there exist infinitely many natural numbers $n$, for which there $\exists \, f:\{0,1…n-1\}\rightarrow \{0,1…n-1\}$, satisfying the following conditions:
1) $f(x)\neq x$;
2) $f(f(x))=x$;
3) $f(f(f(x+1)+1)+1)=x$ for $\forall x\in \{0,1…n-1\}$.
Given a real number $\alpha$ and consider function $\varphi(x)=x^2e^{\alpha x}$ for $x\in\mathbb R$. Find all function $f:\mathbb R\to\mathbb R$ that satisfy: $$f(\varphi(x)+f(y))=y+\varphi(f(x))$$ forall $x,y\in\mathbb R$
Find all continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that satisfy:
$ \text{(i)}\text{id}+f $ is nondecreasing
$ \text{(ii)} $ There is a natural number $ m $ such that $ \text{id}+f+f^2\cdots +f^m $ is nonincreasing.
Here, $ \text{id} $ represents the identity function, and ^ denotes functional power.
Let $ n$ be positive integer, $ A,B\subseteq[0,n]$ are sets of integers satisfying $ \mid A\mid \plus{} \mid B\mid\ge n \plus{} 2.$ Prove that there exist $ a\in A, b\in B$ such that $ a \plus{} b$ is a power of $ 2.$
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.
[i]Proposed by Juhan Aru, Estonia[/i]
Let $A = \left\{ 1,2,3,4,5,6,7 \right\}$ and let $N$ be the number of functions $f$ from set $A$ to set $A$ such that $f(f(x))$ is a constant function. Find the remainder when $N$ is divided by $1000$.
Let $\mathbb{Q^+}$ denote the set of positive rational numbers. Determine all functions $f: \mathbb{Q^+} \to \mathbb{Q^+}$ that satisfy the conditions
\[ f \left( \frac{x}{x+1}\right) = \frac{f(x)}{x+1} \qquad \text{and} \qquad f \left(\frac{1}{x}\right)=\frac{f(x)}{x^3}\]
for all $x \in \mathbb{Q^+}.$
Let $A$ be a set with $|A|=225$, meaning that $A$ has 225 elements. Suppose further that there are eleven subsets $A_1, \ldots, A_{11}$ of $A$ such that $|A_i|=45$ for $1\leq i\leq11$ and $|A_i\cap A_j|=9$ for $1\leq i<j\leq11$. Prove that $|A_1\cup A_2\cup\ldots\cup A_{11}|\geq 165$, and give an example for which equality holds.
For any quadratic functions $ f(x)$ such that $ f'(2)\equal{}1$, evaluate $ \int_{2\minus{}\pi}^{2\plus{}\pi}f(x)\sin\left(\frac{x}{2}\minus{}1\right) dx$.
For every positive integer $n$, let $\operatorname{mod_5}(n)$ be the remainder obtained when $n$ is divided by $5$. Define a function $f : \{0, 1, 2, 3, \dots\} \times \{0, 1, 2, 3, 4\} \to \{0, 1, 2, 3, 4\}$ recursively as follows:
\[f(i, j) = \begin{cases}
\operatorname{mod_5}(j+1) & \text{if }i=0\text{ and }0\leq j\leq 4 \\
f(i-1, 1) & \text{if }i\geq 1\text{ and }j=0 \text{, and}\\
f(i-1, f(i, j-1)) & \text{if }i\geq 1\text{ and }1\leq j\leq 4
\end{cases}\]
What is $f(2015, 2)$?
$\textbf{(A) }0 \qquad\textbf{(B) }1 \qquad\textbf{(C) }2 \qquad\textbf{(D) }3 \qquad\textbf{(E) }4$
Find all the functions $f:[0,1]\rightarrow \mathbb{R}$ for which we have:
\[|x-y|^2\le |f(x)-f(y)|\le |x-y|,\]
for all $x,y\in [0,1]$.
Find all functions $f:\mathbb{R} \rightarrow \mathbb{R}$ which satisfies the followings. (Note that $\mathbb{R}$ stands for the set of all real numbers)
(1) For each real numbers $x$, $y$, the equality $f(x+f(x)+xy) = 2f(x)+xf(y)$ holds.
(2) For every real number $z$, there exists $x$ such that $f(x) = z$.
Let $f$ be the function such that
\[f(x)=\begin{cases}2x & \text{if }x\leq \frac{1}{2}\\2-2x & \text{if }x>\frac{1}{2}\end{cases}\]
What is the total length of the graph of $\underbrace{f(f(\ldots f}_{2012\text{ }f's}(x)\ldots))$ from $x=0$ to $x=1?$
Determine all functions $f:\mathbb R\rightarrow\mathbb R$ satisfying the condition
\[f(y^2+2xf(y)+f(x)^2)=(y+f(x))(x+f(y))\]
for all real numbers $x$ and $y$.
Prove that there's no function $f: \mathbb{R}^+\rightarrow\mathbb{R}^+$ such that $f(x)^2\ge f(x+y)\left(f(x)+y\right)$ for all $x,y>0$.
Find all functions $f:\mathbb{Z}^+\rightarrow \mathbb{Z}^+$ satisfying $$m!!+n!!\mid f(m)!!+f(n)!!$$for each $m,n\in \mathbb{Z}^+$, where $n!!=(n!)!$ for all $n\in \mathbb{Z}^+$.
[i]Proposed by DVDthe1st[/i]
Let $\mathbb Q_{>0}$ be the set of all positive rational numbers. Let $f:\mathbb Q_{>0}\to\mathbb R$ be a function satisfying the following three conditions:
(i) for all $x,y\in\mathbb Q_{>0}$, we have $f(x)f(y)\geq f(xy)$;
(ii) for all $x,y\in\mathbb Q_{>0}$, we have $f(x+y)\geq f(x)+f(y)$;
(iii) there exists a rational number $a>1$ such that $f(a)=a$.
Prove that $f(x)=x$ for all $x\in\mathbb Q_{>0}$.
[i]Proposed by Bulgaria[/i]
Find the strictly monotone functions $ f:\{ 0\}\cup\mathbb{N}\longrightarrow\{ 0\}\cup\mathbb{N} $ that satisfy the following two properties:
$ \text{(i)} f(2n)=n+f(n), $ for any nonnegative integers $ n. $
$ \text{(ii)} f(n) $ is a perfect square if and only if $ n $ is a perfect square.
Given any two positive real numbers $x$ and $y$, then $x\Diamond y$ is a positive real number defined in terms of $x$ and $y$ by some fixed rule. Suppose the operation $x\Diamond y$ satisfies the equations $(x\cdot y)\Diamond y=x(y\Diamond y)$ and $(x\Diamond 1)\Diamond x=x\Diamond 1$ for all $x,y>0$. Given that $1\Diamond 1=1$, find $19\Diamond 98$.
Find all functions $f:\mathbb{R} \to \mathbb{R}$ satisfying the equation \[
f(x^2+y^2+2f(xy)) = (f(x+y))^2.
\] for all $x,y \in \mathbb{R}$.