Found problems: 2349
Determine all polynomials $P(x)$ with real coefficients such that
$P(x)^2 + P\left(\frac{1}{x}\right)^2= P(x^2)P\left(\frac{1}{x^2}\right)$ for all $x$.
a) The real number $a$ and the continuous function $f : [a, \infty) \rightarrow [a, \infty)$ are such that $|f(x)-f(y)| < |x–y|$ for every two different $x, y \in [a, \infty)$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, \infty)$?
b) The real numbers $a$ and $b$ and the continuous function $f : [a, b] \rightarrow [a, b]$ are such that $|f(x)-f(y)| < |x–y|$, for every two different $x, y \in [a, b]$. Is it always true that the equation $f(x)=x$ has only one solution in the interval $[a, b]$?
Find all functions $f : R \to R$ such that $$2f(x) =f(x+y)+f(x+2y)$$, for all $x \in R$ and for all $y \ge 0$.
Determine all bounded functions $f: R \to R$, such that $f (f (x) + y) = f (x) + f (y)$, for all real $x, y$.
Find all functions $f:\mathbb{R^+} \to \mathbb{R^+}$ such that for all $x,y,z\in \mathbb{R^+}$:
$$\biggl\{\frac{f(x)}{f(y)}\biggl\}+\biggl\{\frac{f(y)}{f(z)}\biggl\}+ \biggl\{\frac{f(z)}{f(x)}\biggl\}= \biggl\{\frac{x}{y}\biggl\} +\biggl\{\frac{y}{z}\biggl\}+ \biggl\{\frac{z}{x}\biggl\}$$
Note: For any real number $x$, let $\{x\}$ denote the fractional part of $x$, defined as For example, $\{2,7\}=0,7$ .
Let $ f(x)$ be a function with the two properties:
[list=a]
[*] for any two real numbers $ x$ and $ y$, $ f(x \plus{} y) \equal{} x \plus{} f(y)$, and
[*] $ f(0) \equal{} 2$
[/list]
What is the value of $ f(1998)$?
$ \textbf{(A)}\ 0\qquad
\textbf{(B)}\ 2\qquad
\textbf{(C)}\ 1996\qquad
\textbf{(D)}\ 1998\qquad
\textbf{(E)}\ 2000$
Find all functions $f:\mathbb Z_{>0}\to \mathbb Z_{>0}$ such that $a+f(b)$ divides $a^2+bf(a)$ for all positive integers $a$ and $b$ with $a+b>2019$.
Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying
\[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\]
for any two positive integers $ m$ and $ n$.
[i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers:
$ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$.
By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$).
[i]Proposed by Mohsen Jamali, Iran[/i]
Let $a\in\mathbb{R}-\{0\}$. Find all functions $f: \mathbb{R}\to\mathbb{R}$ such that $f(a+x) = f(x) - x$ for all $x\in\mathbb{R}$.
[i]Dan Schwartz[/i]
Find all functions $f : R \to R$ such that $f(2xy) + f(f(x + y)) = xf(y) + yf(x) + f(x + y)$ for all real numbers $x$ and $y$.
Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y \in \mathbb{Q}$: \[f(x+y)+f(x-y)=2(f(x)+f(y)).\]
Determine all functions $f : \mathbb R \to \mathbb R$ satisfying the following two conditions:
(a) $f(x + y) + f(x - y) = 2f(x)f(y)$ for all $x, y \in \mathbb R$, and
(b) $\lim_{x\to \infty} f(x) = 0$.
Find all functions $ f: \mathbb{R} \to \mathbb{R}$ satisfying
\[ f\left(\frac {x \plus{} y}{x \minus{} y}\right) \equal{} \frac {f\left(x\right) \plus{} f\left(y\right)}{f\left(x\right) \minus{} f\left(y\right)}
\]
for all $ x \neq y$.
Find all functions $f : \mathbb{N}\rightarrow{\mathbb{N}}$ such that for all positive integers $m$ and $n$ the number $f(m)+n-m$ is divisible by $f(n)$.
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$
Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.
Let p be an odd prime, A be a non-empty subset of residue classes modulo p, $f:A\to\mathbb R$. Suppose that f is not constant and satisfies $f(x) \leq \frac{f(x + h) + f(x-h)}{2}$ whenever $x,x+h,x-h\in A$. Prove that $|A| \leq \frac{p + 1}{2}$.
Let $\mathbb{Z}$ be the set of integers. Determine, with proof, all primes $p$ for which there exists a function $f:\mathbb{Z}\to\mathbb{Z}$ such that for any integer $x,$
$\quad \bullet \ f(x+p)=f(x)\text{ and}$
$\quad \bullet \ p \text{ divides } f(x+f(x))-x.$
[color=darkred]Determine all functions $ f : [0; \plus{} \infty) \rightarrow [0; \plus{} \infty)$, such that
\[ f(x \plus{} y \minus{} z) \plus{} f(2\sqrt {xz}) \plus{} f(2\sqrt {yz}) \equal{} f(x \plus{} y \plus{} z)\]
for all $ x,y,z \in [0; \plus{} \infty)$, for which $ x \plus{} y\ge z$.[/color]
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N} $ satisfying
\[ f(mn) = \operatorname{lcm} (m,n) \cdot \gcd( f(m), f(n) ) \]
for all positive integer $m,n$.
Given a function $f$ such that $f(x)\le x\forall x\in\mathbb{R}$ and $f(x+y)\le f(x)+f(y)\forall \{x,y\}\in\mathbb{R}$, prove that $f(x)=x\forall x\in\mathbb{R}$.
Let $ f(x) \equal{} \frac{x^2\plus{}1}{2x}$ for $ x \neq 0.$ Define $ f^{(0)}(x) \equal{} x$ and $ f^{(n)}(x) \equal{} f(f^{(n\minus{}1)}(x))$ for all positive integers $ n$ and $ x \neq 0.$ Prove that for all non-negative integers $ n$ and $ x \neq \{\minus{}1,0,1\}$
\[ \frac{f^{(n)}(x)}{f^{(n\plus{}1)}(x)} \equal{} 1 \plus{} \frac{1}{f \left( \left( \frac{x\plus{}1}{x\minus{}1} \right)^{2n} \right)}.\]
Let $f : \mathbb R \to \mathbb R$ be a real-valued function defined on the set of real numbers that satisfies
\[f(x + y) \leq yf(x) + f(f(x))\]
for all real numbers $x$ and $y$. Prove that $f(x) = 0$ for all $x \leq 0$.
[i]Proposed by Igor Voronovich, Belarus[/i]
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m,n\in \mathbb{Z}$: \[f(m+f(n))=f(m)-n.\]
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$.
[i]Proposed by Dorlir Ahmeti, Albania[/i]