Found problems: 1513
1967 Putnam, A4
Show that if $\lambda > \frac{1}{2}$ there does not exist a real-valued function $u(x)$ such that for all $x$ in the closed interval $[0,1]$ the following holds:
$$u(x)= 1+ \lambda \int_{x}^{1} u(y) u(y-x) \; dy.$$
2018-IMOC, N1
Find all functions $f:\mathbb N\to\mathbb N$ satisfying
$$x+f^{f(x)}(y)\mid2(x+y)$$for all $x,y\in\mathbb N$.
PEN K Problems, 30
Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(f(n)))+f(f(n))+f(n)=3n.\]
2019 Dutch IMO TST, 4
Find all functions $f : Z \to Z$ satisfying
$\bullet$ $ f(p) > 0$ for all prime numbers $p$,
$\bullet$ $p| (f(x) + f(p))^{f(p)}- x$ for all $x \in Z$ and all prime numbers $p$.
1985 IMO Shortlist, 12
A sequence of polynomials $P_m(x, y, z), m = 0, 1, 2, \cdots$, in $x, y$, and $z$ is defined by $P_0(x, y, z) = 1$ and by
\[P_m(x, y, z) = (x + z)(y + z)P_{m-1}(x, y, z + 1) - z^2P_{m-1}(x, y, z)\]
for $m > 0$. Prove that each $P_m(x, y, z)$ is symmetric, in other words, is unaltered by any permutation of $x, y, z.$
2017 Abels Math Contest (Norwegian MO) Final, 1b
Find all functions $f : R \to R$ which satisfy $f(x)f(y) = f(x + y) + xy$ for all $x, y \in R$.
2020 Peru IMO TST, 4
Find all functions $\,f: {\mathbb{N}}\rightarrow {\mathbb{N}}\,$ such that\[f(a)^{bf(b^2)}\le a^{f(b)^3}\hspace{0.2in}\text{for all}\,a,b\in \mathbb{N}. \]
2015 Canada National Olympiad, 1
Let $\mathbb{N} = \{1, 2, 3, \ldots\}$ be the set of positive integers. Find all functions $f$, defined on $\mathbb{N}$ and taking values in $\mathbb{N}$, such that $(n-1)^2< f(n)f(f(n)) < n^2+n$ for every positive integer $n$.
2009 IMO Shortlist, 7
Find all functions $f$ from the set of real numbers into the set of real numbers which satisfy for all $x$, $y$ the identity \[ f\left(xf(x+y)\right) = f\left(yf(x)\right) +x^2\]
[i]Proposed by Japan[/i]
2020 IMO Shortlist, N5
Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions:
[list]
[*] $(i)$ $f(n) \neq 0$ for at least one $n$;
[*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$;
[*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$.
[/list]
2021 Middle European Mathematical Olympiad, 1
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that the inequality
\[ f(x^2)-f(y^2) \le (f(x)+y)(x-f(y)) \]
holds for all real numbers $x$ and $y$.
2007 Thailand Mathematical Olympiad, 9
Let $f : R \to R$ be a function satisfying the equation $f(x^2 + x + 3) + 2f(x^2 - 3x + 5) =6x^2 - 10x + 17$ for all real numbers $x$. What is the value of $f(85)$?
2015 India National Olympiad, 3
Find all real functions $f: \mathbb{R} \to \mathbb{R}$ such that $f(x^2+yf(x))=xf(x+y)$.
2010 Contests, 1
Find all function $f:\mathbb{R}\rightarrow\mathbb{R}$ such that for all $x,y\in\mathbb{R}$ the following equality holds \[
f(\left\lfloor x\right\rfloor y)=f(x)\left\lfloor f(y)\right\rfloor \] where $\left\lfloor a\right\rfloor $ is greatest integer not greater than $a.$
[i]Proposed by Pierre Bornsztein, France[/i]
2014 Grand Duchy of Lithuania, 1
Determine all functions $f : R \to R$ such that $f(xy + f(x)) = xf(y) + f(x)$ holds for any $x, y \in R$.
2019 Korea National Olympiad, 5
Find all functions $f$ such that $f:\mathbb{R}\rightarrow \mathbb{R}$ and $f(f(x)-x+y^2)=yf(y)$
2021 Pan-African, 5
Find all functions $f$ $:$ $\mathbb{R} \rightarrow \mathbb{R}$ such that $\forall x,y \in \mathbb{R}$ :
$$(f(x)+y)(f(y)+x)=f(x^2)+f(y^2)+2f(xy)$$
2019 ELMO Shortlist, A2
Find all functions $f:\mathbb Z\to \mathbb Z$ such that for all surjective functions $g:\mathbb Z\to \mathbb Z$, $f+g$ is also surjective. (A function $g$ is surjective over $\mathbb Z$ if for all integers $y$, there exists an integer $x$ such that $g(x)=y$.)
[i]Proposed by Sean Li[/i]
1992 IMO Longlists, 71
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 $R(x, y).$
1984 AIME Problems, 7
The function $f$ is defined on the set of integers and satisfies \[ f(n)=\begin{cases} n-3 & \text{if } n\ge 1000 \\ f(f(n+5)) & \text{if } n<1000\end{cases} \] Find $f(84)$.
2018 Estonia Team Selection Test, 4
Find all functions $f : R \to R$ that satisfy $f (xy + f(xy)) = 2x f(y)$ for all $x, y \in R$
2017-IMOC, A5
Find all functions $f:\mathbb Z\to\mathbb Z$ such that
$$f(mf(n+1))=f(m+1)f(n)+f(f(n))+1$$for all integer pairs $(m,n)$.
2004 Cuba MO, 8
Determine all functions $f : R_+ \to R_+$ such that:
a) $f(xf(y))f(y) = f(x + y)$ for $x, y \ge 0$
b) $f(2) = 0$
c) $f(x) \ne 0$ for $0 \le x < 2$.
1992 IMO Longlists, 8
Given two positive real numbers $a$ and $b$, suppose that a mapping $f : \mathbb R^+ \to \mathbb R^+$ satisfies the functional equation
\[f(f(x)) + af(x) = b(a + b)x.\]
Prove that there exists a unique solution of this equation.
2020 Grand Duchy of Lithuania, 1
Find all functions $f: R \to R$, such that equality $f (xf (y) - yf (x)) = f (xy) - xy$ holds for all $x, y \in R$.