Found problems: 1513
Find all $f:R^+ \rightarrow R^+$ such that
$f(f(x) + y)f(x) = f(xy + 1)\ \ \forall x, y \in R^+$
@below: [url]https://artofproblemsolving.com/community/c6h2254883_2020_imoc_problems[/url]
[quote]Feel free to start individual threads for the problems as usual[/quote]
Let function pair $f,g : \mathbb{R^+} \rightarrow \mathbb{R^+}$ satisfies
\[
f(g(x)y + f(x)) = (y+2015)f(x)
\]
for every $x,y \in \mathbb{R^+} $
a. Prove that $f(x) = 2015g(x)$ for every $x \in \mathbb{R^+}$
b. Give an example of function pair $(f,g)$ that satisfies the statement above and $f(x), g(x) \geq 1$ for every $x \in \mathbb{R^+}$
Determine, whether there exist a function $f$ defined on the set of all positive real numbers and taking positive values such that $f(x+y)\geqslant yf(x)+f(f(x))$ for all positive x and y?
Functions $f(x)$ and $g(x)$ are defined on the set of real numbers and take real values. It is known that $g(x)$ takes all real values, $g(0)=0$, and for all $x,y \in \mathbb{R}$ the following equality holds
$$f(x+f(y))=f(x)+g(y)$$
Prove that $g(x+y)=g(x)+g(y)$ for all $x,y \in \mathbb{R}$.
Find all functions $ f: \mathbb{R}\to\mathbb{R}$ such that $ f(x+y)+f(x)f(y)=f(xy)+2xy+1$ for all real numbers $ x$ and $ y$.
[i]Proposed by B.J. Venkatachala, India[/i]
Find all polynomials $p$ with real coefficients such that for all real $x$
$$(x-8)p(2x)=8(x-1)p(x).$$
Find all functions $f : \mathbb{Z} \to \mathbb{Z}$ that satisfy the conditions:
$i) f(f(x)) = xf(x) - x^2 + 2,\forall x\in\mathbb{Z}$
$ii) f$ takes all integer values
Determine is there a function $a: \mathbb{N} \rightarrow \mathbb{N}$ such that:
$i)$ $a(0)=0$
$ii)$ $a(n)=n-a(a(n))$, $\forall n \in$ $ \mathbb{N}$.
If exists prove:
$a)$ $a(k)\geq a(k-1)$
$b)$ Does not exist positive integer $k$ such that $a(k-1)=a(k)=a(k+1)$.
Find all functions \( f: \mathbb{R} \rightarrow \mathbb{R} \) such that for all \( x, y \in \mathbb{R} \),
\[
f(x + f(2y)) + f(x^2 - y) = f(f(x)) f(x + 1) + 2y - f(y).
\]
Let $n\in\mathbb{Z}$, $n\geq 2$. Find all functions $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ such that $$f(x_1+\dots +x_n)^2=\sum_{i=1}^nf(x_i) ^2+ 2\sum_{i<j}f(x_ix_j),$$ for all $x_1,\dots ,x_n\in\mathbb{R}_{>0}$.
[i]Proposed by Andrei Vila[/i]
Determine all functions $f: R \longrightarrow R$ such that
$f(x+y)-f(x-y)=f(x)*f(y)$ for $x,y \in R$
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ for which these two conditions hold simultaneously
(i) For all $m,n \in \mathbb{N}$ we have:
$$ \frac{f(mn)}{\gcd(m,n)} = \frac{f(m)f(n)}{f(\gcd(m,n))};$$
(ii) For all prime numbers $p$, there exists a prime number $q$ such that $f(p^{2025})=q^{2025}$.
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}$).
Find the continuous functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ having the following property:
$$ f\left( x+\frac{1}{n}\right) \le f(x) +\frac{1}{n},\quad\forall n\in\mathbb{Z}^* ,\quad\forall x\in\mathbb{R} . $$
Find all $g:\mathbb{R}\to\mathbb{R}$ so that there exists a unique $f:\mathbb{R}\to\mathbb{R}$ satisfying $f(0)=g(0)$ and
\[f(x+g(y))+f(-x-g(-y))=g(x+f(y))+g(-x-f(-y))\]
for all $x,y\in\mathbb{R}$.
[i]
Proposed by usjl[/i]
The function $f(x,y)$ satisfies: $f(0,y)=y+1, f(x+1,0) = f(x,1), f(x+1,y+1)=f(x,f(x+1,y))$ for all non-negative integers $x,y$. Find $f(4,1981)$.
Suppose $f:\mathbb{N}\rightarrow \mathbb{N}$ is a function such that $$f^n(n) = 2n$$ for all $n\in \mathbb{N}$. Must $f(n) = n+1$ for all $n$?
Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that
\[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0.
\]
Find all $n$ such that there is a real polynomial $f(x)$ of degree $n$ such that $f(x) \ge f'(x)$ for all real $x$.
Find all functions $f : \mathbb{Q}[\sqrt{2}] \to \mathbb{Q}[\sqrt{2}]$ such that for all $x, y \in \mathbb{Q}[\sqrt{2}]$,
$$
f(xy) = f(x)f(y) \quad \text{and} \quad f(x + y) = f(x) + f(y),
$$
where $\mathbb{Q}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Q} \}$.
[I]Proposed by Stijn Cambie, Belgium[/i]
Find all functions $f : R \to R$ such that $x f(x)-y f(y) = (x-y)f(x+y)$ for all $x,y \in R$.
$\mathbb{Z}[x]$ represents the set of all polynomials with integer coefficients. Find all functions $f:\mathbb{Z}[x]\rightarrow \mathbb{Z}[x]$ such that for any 2 polynomials $P,Q$ with integer coefficients and integer $r$, the following statement is true. \[P(r)\mid Q(r) \iff f(P)(r)\mid f(Q)(r).\]
(We define $a|b$ if and only if $b=za$ for some integer $z$. In particular, $0|0$.)
[i]Proposed by the4seasons.[/i]
Find all non constant polynomials $P(x),Q(x)$ with real coefficients such that: $P((Q(x))^3)=xP(x)(Q(x))^3$
Find all functions $f:\mathbb R_{\ge0}\times\mathbb R_{\ge0}\to\mathbb R_{\ge0}$ such that
$1$. $f(x,0)=f(0,x)=x$ for all $x\in\mathbb R_{\ge0}$,
$2$. $f(f(x,y),z)=f(x,f(y,z))$ for all $x,y,z\in\mathbb R_{\ge0}$ and
$3$. there exists a real $k$ such that $f(x+y,x+z)=kx+f(y,z)$ for all $x,y,z\in\mathbb R_{\ge0}$.
Does there exist a function $f: \mathbb R \to \mathbb R $ satisfying the following conditions:
(i) for each real $y$ there is a real $x$ such that $f(x)=y$ , and
(ii) $f(f(x)) = (x - 1)f(x) + 2$ for all real $x$ ?
[i]Proposed by Igor I. Voronovich, Belarus[/i]