Found problems: 2349
Let $2\mathbb{Z} + 1$ denote the set of odd integers. Find all functions $f:\mathbb{Z} \mapsto 2\mathbb{Z} + 1$ satisfying \[ f(x + f(x) + y) + f(x - f(x) - y) = f(x+y) + f(x-y) \] for every $x, y \in \mathbb{Z}$.
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $f(0)\neq 0$ and for all $x,y\in\mathbb{R}$,
\[ f(x+y)^2 = 2f(x)f(y) + \max \left\{ f(x^2+y^2), f(x^2)+f(y^2) \right\}. \]
Find all functions $f:\mathbb R\to\mathbb R$ such that for reals $x,y$,
$$f(xf(y)+y)=yf(x)+f(y).$$
Determine all functions $f : \mathbb{R}^2 \to\mathbb {R}$ for which \[f(A)+f(B)+f(C)+f(D)=0,\]whenever $A,B,C,D$ are the vertices of a square with side-length one.
[i]Ilir Snopce[/i]
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all $x,y\in \mathbb{R}$:
$$f\left(f(x)^2-y^2\right)^2+f(2xy)^2=f\left(x^2+y^2\right)^2$$
[i]Proposed by Ali Behrouz - Mojtaba Zare Bidaki[/i]
A function $f$ from the positive integers to the positive integers is called [i]Canadian[/i] if it satisfies $$\gcd\left(f(f(x)), f(x+y)\right)=\gcd(x, y)$$ for all pairs of positive integers $x$ and $y$.
Find all positive integers $m$ such that $f(m)=m$ for all Canadian functions $f$.
Find all monotonic positive functions $f(x)$ defined on the positive reals such that $f(xy) f\left( \frac{f(y)}{x}\right) = 1$ for all $x, y$.
Determine all functions $f:\mathbb{Z}\rightarrow\mathbb{Z}$ with the property that \[f(x-f(y))=f(f(x))-f(y)-1\] holds for all $x,y\in\mathbb{Z}$.
Find all strictly increasing functions $f : Z \to R$ such that for any $m, n \in Z$ there exists a $k \in Z$ such that $f(k) = f(m) - f(n)$.
Nguyễn Duy Thái Sơn
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 $f(4,1981)$.
A function $f$ has its domain equal to the set of integers $0$, $1$, $\ldots$, $11$, and $f(n)\geq 0$ for all such $n$, and $f$ satisfies
[list]
[*]$f(0)=0$
[*]$f(6)=1$
[*]If $x\geq 0$, $y\geq 0$, and $x+y\leq 11$, then $f(x+y)=\tfrac{f(x)+f(y)}{1-f(x)f(y)}$.[/list]
Find $f(2)^2+f(10)^2$.
Find all functions $f:\mathbb R\to \mathbb R$ such that $$xy(f(x+y)-f(x)-f(y))=2f(xy)$$ for all $x,y\in \mathbb R.$
[i]Proposed by USJL[/i]
Let $S_n = \{1, 2,\cdots, n\}$ and $f_n : S_n \to S_n$ be defined inductively as follows: $f_1(1) = 1, f_n(2j) = j \ (j = 1, 2, \cdots , [n/2])$ and
[list]
[*][b][i](i)[/i][/b] if $n = 2k \ (k \geq 1)$, then $f_n(2j - 1) = f_k(j) + k \ (j = 1, 2, \cdots, k);$
[*][b][i](ii)[/i][/b] if $n = 2k + 1 \ (k \geq 1)$, then $f_n(2k + 1) = k + f_{k+1}(1), f_n(2j - 1) = k + f_{k+1}(j + 1) \ (j = 1, 2,\cdots , k).$[/list]
Prove that $f_n(x) = x$ if and only if $x$ is an integer of the form
\[\frac{(2n + 1)(2^d - 1)}{2^{d+1} - 1}\]
for some positive integer $d.$
Let $\mathbb R_{>0}$ be the set of positive real numbers. Determine all functions $f \colon \mathbb R_{>0} \to \mathbb R_{>0}$ such that \[x \big(f(x) + f(y)\big) \geqslant \big(f(f(x)) + y\big) f(y)\] for every $x, y \in \mathbb R_{>0}$.
For a positive integer $n$, let $A(n)$ be the equal to the remainder when $n$ is divided by $11$ and let $T(n)=A(1)+A(2)+A(3)+ \dots + A(n)$. Find the value of $$A(T(2021))$$
Let $f(n)$ be the sum of the first $n$ terms of the sequence $0,1,1,2,2,3,3,4, \dots,$ where the $n$th term is given by $$a_n= \begin{cases} n/2 & \text{if } n \text{ is even,} \\ (n-1)/2 & \text{if } n \text{ is odd.} \end{cases}$$ Show that if $x$ and $y$ are positive integers and $x>y$ then $xy=f(x+y)-f(x-y)$.
Let $\mathbb{N}$ be the set of positive integers. Let $f$ be a function defined on $\mathbb{N}$, which satisfies the inequality $f(n + 1) > f(f(n))$ for all $n \in \mathbb{N}$. Prove that for any $n$ we have $f(n) = n.$
Find all functions $f : \mathbb{R}\to\mathbb{R}$ such that for all real numbers $x$ and $y$,
$$f(x+f(y))+xy=f(x)f(y)+f(x)+y.$$
[i]Andrew Carratu[/i]
Is there a function $f(x)$, which satisfies both of the following conditions:
a) if $x \ne y$, then $f(x)\ne f(y)$
b) for all real $x$, holds the inequality $f(x^2-1998x)-f^2(2x-1999)\ge \frac14$?
Given a polynomial $P(x,y)$ with real coefficients, suppose that some real function $f:\mathbb R \to \mathbb R$ satisfies
$$P(x,y) = f(x+y)-f(x)-f(y)$$for all $x,y\in\mathbb R$. Show that some polynomial $q$ satisfies
$$P(x,y) = q(x+y)-q(x)-q(y)$$
Let $P_1(x, y)$ and $P_2(x, y)$ be two relatively prime polynomials with complex coefficients. Let $Q(x, y)$ and $R(x, y)$ be polynomials with complex coefficients and each of degree not exceeding $d$. Prove that there exist two integers $A_1, A_2$ not simultaneously zero with $|A_i| \leq d + 1 \ (i = 1, 2)$ and such that the polynomial $A_1P_1(x, y) + A_2P_2(x, y)$ is coprime to $Q(x, y)$ and $R(x, y).$
Find all functions $f: R \to R$ that satisfy
$$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$
Find all functions $f : \mathbb{R} \to\mathbb{R}$ such that $f(0)\neq 0$ and
\[f(f(x)) + f(f(y)) = f(x + y)f(xy),\]
for all $x, y \in\mathbb{R}$.
Let $f: \Bbb{R} \rightarrow \Bbb{R}$ be a function such that
(i) For all $x,y \in \Bbb{R}$, \[ f(x)+f(y)+1 \geq f(x+y) \geq f(x)+f(y) \]
(ii) For all $x \in [0,1)$, $f(0) \geq f(x)$,
(iii) $-f(-1) = f(1) = 1$.
Find all such functions $f$.
Find all polynomials $P$ such that
$P(x) + \binom{2018}{2}P(x+2)+...+\binom{2018}{2106}P(x+2016)+P(x+2018)=$
$=\binom{2018}{1}P(x+1)+\binom{2018}{3}P(x+3)+...+\binom{2018}{2105}P(x+2015)+\binom{2018}{2107}P(x+2017)$
for all real numbers $x$.