Found problems: 1513
2021 Science ON all problems, 4
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]
2022 Turkey Team Selection Test, 2
Find all functions $f: \mathbb{Q^+} \rightarrow \mathbb{Q}$ satisfying $f(x)+f(y)= \left(f(x+y)+\frac{1}{x+y} \right) (1-xy+f(xy))$ for all $x, y \in \mathbb{Q^+}$.
2016 Thailand Mathematical Olympiad, 9
A real number $a \ne 0$ is given. Determine all functions $f : R \to R$ satisfying $f(x)f(y) + f(x + y) = axy$ for all real numbers $x, y$.
2015 Switzerland Team Selection Test, 7
Find all finite and non-empty sets $A$ of functions $f: \mathbb{R} \mapsto \mathbb{R}$ such that for all $f_1, f_2 \in A$, there exists $g \in A$ such that for all $x, y \in \mathbb{R}$
$$f_1 \left(f_2 (y)-x\right)+2x=g(x+y)$$
2009 IMO Shortlist, 3
Determine all functions $ f$ from the set of positive integers to the set of positive integers such that, for all positive integers $ a$ and $ b$, there exists a non-degenerate triangle with sides of lengths
\[ a, f(b) \text{ and } f(b \plus{} f(a) \minus{} 1).\]
(A triangle is non-degenerate if its vertices are not collinear.)
[i]Proposed by Bruno Le Floch, France[/i]
1986 Traian Lălescu, 2.1
Let be a nonnegative integer $ n. $ Find all continuous functions $ f:\mathbb{R}_{\ge 0}\longrightarrow\mathbb{R} $ for which the following equation holds:
$$ (1+n)\int_0^x f(t) dt =nxf(x) ,\quad\forall x>0. $$
1973 Bulgaria National Olympiad, Problem 4
Find all functions $f(x)$ defined in the range $\left(-\frac\pi2,\frac\pi2\right)$ that are differentiable at $0$ and satisfy
$$f(x)=\frac12\left(1+\frac1{\cos x}\right)f\left(\frac x2\right)$$
for every $x$ in the range $\left(-\frac\pi2,\frac\pi2\right)$.
[i]L. Davidov[/i]
2014 Contests, 1
Determine all functions $f:\mathbb{R} \to \mathbb{R}$ such that
\[ xf(y) + f(xf(y)) - xf(f(y)) - f(xy) = 2x + f(y) - f(x+y)\]
holds for all $x,y \in \mathbb{R}$.
1997 Nordic, 4
Let f be a function defined in the set $\{0, 1, 2,...\}$ of non-negative integers, satisfying $f(2x) = 2f(x), f(4x+1) =
4f(x) + 3$, and $f(4x-1) = 2f(2x - 1) -1$.
Show that $f $ is an injection, i.e. if $f(x) = f(y)$, then $x = y$.
2020 Serbian Mathematical Olympiad, Problem 5
For a natural number $n$, with $v_2(n)$ we denote the largest integer $k\geq0$ such that $2^k|n$. Let us assume that the function $f\colon\mathbb{N}\to\mathbb{N}$ meets the conditions:
$(i)$ $f(x)\leq3x$ for all natural numbers $x\in\mathbb{N}$.
$(ii)$ $v_2(f(x)+f(y))=v_2(x+y)$ for all natural numbers $x,y\in\mathbb{N}$.
Prove that for every natural number $a$ there exists exactly one natural number $x$ such that $f(x)=3a$.
2005 Taiwan TST Round 3, 1
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]
1990 IMO Longlists, 6
Let function $f : \mathbb Z_{\geq 0}^0 \to \mathbb N$ satisfy the following conditions:
(i) $ f(0, 0, 0) = 1;$
(ii) $f(x, y, z) = f(x - 1, y, z) + f(x, y - 1, z) + f(x, y, z - 1);$
(iii) when applying above relation iteratively, if any of $x', y', z$' is negative, then $f(x', y', z') = 0.$
Prove that if $x, y, z$ are the side lengths of a triangle, then $\frac{\left(f(x,y,z) \right) ^k}{ f(mx ,my, mz)}$ is not an integer for any integers $k, m > 1.$
2010 Belarus Team Selection Test, 2.4
Find all functions $f, g : Q \to Q$ satisfying the following equality $f(x + g(y)) = g(x) + 2 y + f(y)$ for all $x, y \in Q$.
(I. Voronovich)