Found problems: 2349
The function $f$, with domain on the set of non-negative integers, is defined by the following :
$\bullet$ $f (0) = 2$
$\bullet$ $(f (n + 1) -1)^2 + (f (n)-1) ^2 = 2f (n) f (n + 1) + 4$, taking $f (n)$ the largest possible value.
Determine $f (n)$.
Find all functions $f : \mathbb{Q}[x] \to \mathbb{Q}[x]$ such that two following conditions holds :
$$\forall P , Q \in \mathbb{Q}[x] : f(P+Q)=f(P)+f(Q)$$
$$\forall P \in \mathbb{Q}[x] : gcd(P , f(P))=1 \iff$$
$P$ is square-free.
Which a square-free polynomial with rational coefficients is a polynomial such that there doesn't exist square of a non-constant polynomial with rational coefficients that divides it.
[i]Proposed by Sina Azizedin[/i]
Let $\mathbb{R}_{>0}$ be the set of the positive real numbers. Find all functions $f:\mathbb{R}_{>0} \rightarrow \mathbb{R}_{>0}$ such that $$f(xy+f(x))=f(f(x)f(y))+x$$ for all positive real numbers $x$ and $y$.
Find all functions $f :R \to R$ such that for all real numbers $x$ and $y$
$$f(x^3+y^3)=f(x^3)+3x^3f(x)f(y)+3f(x)(f(y))^2+y^6f(y)$$
Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.
Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy \[f^{g(n)+1}(n) + g^{f(n)}(n) = f(n+1) - g(n+1) + 1\] for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\ldots f)}_{k}(n) \ldots ))$.
[i]Proposed by Bojan Bašić, Serbia[/i]
Find all functions $f : \mathbb{R} \rightarrow \mathbb{R}$ such that
$$(y + 1)f(x) + f(xf(y) + f(x + y))= y$$
for all $x, y \in \mathbb{R}$.
Let $f(x)=x^2-2x+1.$ For some constant $k, f(x+k) = x^2+2x+1$ for all real numbers $x.$ Determine the value of $k.$
Do there exist numbers $a,b \in R$ and surjective function $f: R \to R$ such that $f(f(x)) = bx f(x) +a$ for all real $x$?
I.Voronovich
Let $a$ be a real number and let $f : \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(0)=\frac{1}{2}$ and
\[f(x+y)=f(x)f(a-y)+f(y)f(a-x), \quad \forall x,y \in \mathbb{R}.\]
Prove that $f$ is constant.
Determine all functionf $f : \mathbb{R} \mapsto \mathbb{R}$ such that
\[ f(x+y) = f(x)f(y) - c \sin{x} \sin{y} \] for all reals $x,y$ where $c> 1$ is a given constant.
$ k$ is a given natural number. Find all functions $ f: \mathbb{N}\rightarrow\mathbb{N}$ such that for each $ m,n\in\mathbb{N}$ the following holds: \[ f(m)\plus{}f(n)\mid (m\plus{}n)^k\]
Determine all functions $f:(0;\infty)\to \mathbb{R}$ that satisfy
$$f(x+y)=f(x^2+y^2)\quad \forall x,y\in (0;\infty)$$
Let $f:\mathbb{Z}^2\rightarrow\mathbb{Z}$ be a function satisfying
\[f(x+1,y)+f(x,y+1)+1=f(x,y)+f(x+1,y+1)\]
for all integers $x$ and $y$. Can it happen that $|f(x,y)|\leq 2024$ for all $x,y\in\mathbb{Z}$?
Find all real numbers $a$ for which there exist functions $f,g: \mathbb{R} \to \mathbb{R}$, where $g$ is strictly increasing, such that $f(1) = 1$, $f(2) = a$, and
\[
f(x) - f(y) \leq (x-y)(g(x) - g(y))
\]
for all real numbers $x$ and $y$.
Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a function satisfying $f(x)f(y)=f(x-y)$. Find all possible values of $f(2017)$.
Find all the functions $f:R \to R$ satisfying
$$(x + y)(f (x)-f (y)) = f (x^2) - f (y^2),\, \forall x, y \in R$$
Find all functions $f (x) : Z \to Z$ defined on integers, take integer values, and for all $x,y \in Z$ satisfy $$f(x+y)+f(xy)=f(x)f(y)+1$$
Let $n\geq 3$ be an integer. Find the number of functions $f:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ such that
\[ f(f(k)) = f^3(k) - 6f^2(k) + 12f(k) - 6 , \ \textrm{ for all } k \geq 1 . \]
Find all injective functions $f : N\to N$ such that holds for all natural numbers $n$:
$$f(f(n)) \le \frac{f(n) + n}{2}$$
Source: 2017 Canadian Open Math Challenge, Problem C2
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A function $f(x)$ is periodic with period $T > 0$ if $f(x + T) = f(x)$ for all $x$. The smallest such number $T$ is called the least period. For example, the functions $\sin(x)$ and $\cos(x)$ are periodic with least period $2\pi$.
$\qquad$(a) Let a function $g(x)$ be periodic with the least period $T = \pi$. Determine the least period of $g(x/3)$.
$\qquad$(b) Determine the least period of $H(x) = sin(8x) + cos(4x)$
$\qquad$(c) Determine the least periods of each of $G(x) = sin(cos(x))$ and $F(x) = cos(sin(x))$.
Define $f(x)=2x+3$ and suppose that
$g(x+2)=f(f(x-1)\cdot f(x+1)+f(x))$. Find $g(6)$.
Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all integers $x$ and $y$, the number $$f(x)^2+2xf(y)+y^2$$ is a perfect square.
[i]Proposed by Barış Koyuncu[/i]
Find all functions $f: \mathbb N \to \mathbb N$ Such that:
1.for all $x,y\in N$:$x+y|f(x)+f(y)$
2.for all $x\geq 1395$:$x^3\geq 2f(x)$
Let $F(x)$ be a real valued function defined for all real $x$ except for $x=0$ and $x=1$ and satisfying the functional equation $F(x)+F\{(x-1)/x\}=1+x.$ Find all functions $F(x)$ satisfying these conditions.