Found problems: 1513
Find all functions $f \colon \mathbb{R} \to \mathbb{R}$ that satisfy the inequality
\[ f(y) - \left(\frac{z-y}{z-x} f(x) + \frac{y-x}{z-x}f(z)\right) \leq f\left(\frac{x+z}{2}\right) - \frac{f(x)+f(z)}{2} \]
for all real numbers $x < y < z$.
[i]Proposed by Gabriel Carroll[/i]
For a given natural number $k > 1$, find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for all $x, y \in \mathbb{R}$, $f[x^k + f(y)] = y +[f(x)]^k$.
Find all functions $f$ defined on the integers greater than or equal to $1$ that satisfy:
(a) for all $n,f(n)$ is a positive integer.
(b) $f(n + m) =f(n)f(m)$ for all $m$ and $n$.
(c) There exists $n_0$ such that $f(f(n_0)) = [f(n_0)]^2$ .
Prove that there is a function $f$ from the set of all natural numbers into itself such that $f(f(n))=n^2$ for all $n \in \mathbb{N}$.
Find all functions $f: Q \to R$ satisfying $f (x + y) = f (x)f (y) - f(xy) + 1$ for all $x,y \in Q$
If $f$ is a function from the set of positive integers to itself such that $f(x) \leq x^2$ for all natural $x$, and $f\left( f(f(x)) f(f(y))\right) = xy$ for all naturals $x$ and $y$. Find the number of possible values of $f(30)$.
[i]Author: Alex Zhu[/i]
Find all functions $f : \mathbb R \to \mathbb R$ for which
\[f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),\]
for all real numbers $a, b, c$ and $d$. Note that there is only one occurrence of $f$ on the right hand side!
Find all pairs of positive integer $\alpha$ and function $f : N \to N_0$ that satisfies
(i) $f(mn^2) = f(mn) + \alpha f(n)$ for all positive integers $m, n$.
(ii) If $n$ is a positive integer and $p$ is a prime number with $p|n$, then $f(p) \ne 0$ and $f(p)|f(n)$.
Find all functions $f,g: \mathbb{Q}\to \mathbb{Q}$ such that the following two conditions hold:
$$f(g(x)-g(y))=f(g(x))-y \ \ (1)$$
$$g(f(x)-f(y))=g(f(x))-y\ \ (2)$$
for all $x,y \in \mathbb{Q}$.
Let $\mathbb{Z} _{>0}$ be the set of positive integers. Find all functions $f: \mathbb{Z} _{>0}\rightarrow \mathbb{Z} _{>0}$ such that
\[ m^2 + f(n) \mid mf(m) +n \]
for all positive integers $m$ and $n$.
Let ${R^* = R-\{0\}}$ be the set of non-zero real numbers. Find all functions ${f : R^* \rightarrow R^*}$ satisfying ${f(x) + f(y) = f(xy f(x + y))}$, for ${x, y \in R^*}$ and ${ x + y\ne 0 }$.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that
$f(f(x)f(1-x))=f(x)$ and $f(f(x))=1-f(x)$,
for all real $x$.
Find all functions $f:\mathbb{Q}\to\mathbb{Q}$ such that the equation
\[f(xf(x)+y) = f(y) + x^2\]holds for all rational numbers $x$ and $y$.
Here, $\mathbb{Q}$ denotes the set of rational numbers.
Find all functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the following conditions: $f(x+1)=f(x)+1$ and $f(x^2)=f(x)^2.$
[i]Based on a problem of Romanian Masters of Mathematics[/i]
Find the function $f(x)$ such that :
\[f(x)=\cos x+\int_0^{2\pi} f(y)\sin (x-y)\ dy\]
Find all functions $f:\mathbb{R} \to \mathbb{R}$, such that for any $x, y \in \mathbb{R}$ holds the following:
$$f(x)f(yf(x)) + yf(xy) = xf(xy) + y^2f(x)$$
[i]Proposed by Mykhailo Shtandenko[/i]
Find all functions $f:\mathbb{Z}_{\ge 1}\to \mathbb{R}_{>0}$ such that for all positive integers $n$ the following relation holds: $$\sum_{d|n} f(d)^3=\left (\sum_{d|n} f(d) \right )^2,$$
where both sums are taken over the positive divisors of $n$.
[i] (Vlad Robu) [/i]
Let $f: \mathbb{R}^{2} \to \mathbb{R}^{+}$such that for every rectangle $A B C D$ one has
$$
f(A)+f(C)=f(B)+f(D).
$$
Let $K L M N$ be a quadrangle in the plane such that $f(K)+f(M)=f(L)+f(N)$, for each such function. Prove that $K L M N$ is a rectangle.
[i]Proposed by Navid.[/i]
Determine all functions $f : \mathbb R \to \mathbb R$ such that for all real numbers $x$ and $y$,
$$f(x^2 + xy) = f(x)f(y) + yf(x) + xf(x+y).$$
[i]Proposed by Walther Janous, Austria[/i]
Find all functions $f\colon \mathbb{R} \rightarrow \mathbb{R}$ such that for all $x,y \in \mathbb{R}$,
\[xf(x+f(y))=(y-x)f(f(x)).\]
[i]Proposed by Nikola Velov, Macedonia[/i]
Find all function $f:\mathbb{N}\rightarrow\mathbb{N}$ such that for all $a,b\in\mathbb{N}$ ,
$(f(a)+b) f(a+f(b))=(a+f(b))^2$
Find all functions $f,g: \mathbb{R} \to \mathbb{R}$ such that satisfies
$$f(x^2-g(y))=g(x)^2-y$$
for all $x,y \in \mathbb{R}$
Find all functions $f\colon \mathbb{Z}^2 \to [0, 1]$ such that for any integers $x$ and $y$,
\[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\]
[i]Proposed by Yang Liu and Michael Kural[/i]
Let $f\colon\mathbb{N}\rightarrow\mathbb{N}$ be a function such that for all positive integers $x$ and $y$, the number $f(x)+y$ is a perfect square if and only if $x+f(y)$ is a perfect square. Prove that $f$ is injective.
[i]Remark.[/i] A function $f\colon\mathbb{N}\rightarrow\mathbb{N}$ is injective if for all pairs $(x,y)$ of distinct positive integers, $f(x)\neq f(y)$ holds.
[i]Ivan Novak[/i]
Suppose that a function $f : R^+ \to R^+$ satisfies $$f(f(x))+x = f(2x).$$
Prove that $f(x) \ge x$ for all $x >0$