Found problems: 1513
1995 Grosman Memorial Mathematical Olympiad, 6
(a) Prove that there is a unique function $f : Q \to Q$ satisfying:
(i) $f(q)= 1 + f\left(\frac{q}{1-2q}\right)$ for $0<q< \frac12$
(ii) $f(q)= 1 + f(q-1)$ for $1<q\le 2$
(iii) $f(q)f\left(\frac{1}{q}\right)=1$ for all $q\in Q^+$
(b) For this function $f$ , find all $r\in Q^+$ such that $f(r) = r$
2008 Grigore Moisil Intercounty, 3
Let be two nonzero real numbers $ a,b, $ and a function $ f:\mathbb{R}\longrightarrow [0,\infty ) $ satisfying the functional equation
$$ f(x+a+b)+f(x)=f(x+a)+f(x+b) . $$
[b]1)[/b] Prove that $ f $ is periodic if $ a/b $ is rational.
[b]2)[/b] If $ a/b $ is not rational, could $ f $ be nonperiodic?
2015 Indonesia MO Shortlist, A6
Let functions $f, g: \mathbb{R}^+ \to \mathbb{R}^+$ satisfy the following:
\[ f(g(x)y + f(x)) = (y+2015)f(x) \]
for every $x,y \in \mathbb{R}^+$.
(a) Prove that $g(x) = \frac{f(x)}{2015}$ for every $x \in \mathbb{R}^+. $
(b) State an example of function that satisfy the equation above and $f(x), g(x) \ge 1$ for every $x \in \mathbb{R}^+$.
2024 India IMOTC, 19
Denote by $\mathbb{S}$ the set of all proper subsets of $\mathbb{Z}_{>0}$. Find all functions $f : \mathbb{S} \mapsto \mathbb{Z}_{>0}$ that satisfy the following:\\
[color=#FFFFFF]___[/color]1. For all sets $A, B \in \mathbb{S}$ we have \[f(A \cap B) = \text{min}(f(A), f(B)).\]
[color=#FFFFFF]___[/color]2. For all positive integers $n$ we have \[\sum \limits_{X \subseteq [1, n]}
f(X) = 2^{n+1}-1.\]
(Here, by a proper subset $X$ of $\mathbb{Z}_{>0}$ we mean $X \subset \mathbb{Z}_{>0}$ with $X \ne \mathbb{Z}_{>0}$. It is allowed for $X$ to have infinite size.) \\
[i]Proposed by MV Adhitya, Kanav Talwar, Siddharth Choppara, Archit Manas[/i]
2024 CAPS Match, 5
Let $\alpha\neq0$ be a real number. Determine all functions $f:\mathbb R\to\mathbb R$ such that \[f\left(x^2+y^2\right)=f(x-y)f(x+y)+\alpha yf(y)\] holds for all $x, y\in\mathbb R.$
2022 Vietnam TST, 1
Given a real number $\alpha$ and consider function $\varphi(x)=x^2e^{\alpha x}$ for $x\in\mathbb R$. Find all function $f:\mathbb R\to\mathbb R$ that satisfy: $$f(\varphi(x)+f(y))=y+\varphi(f(x))$$ forall $x,y\in\mathbb R$
2016 Peru IMO TST, 6
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 $x,y\in\mathbb{Z}$.
2013 Iran MO (3rd Round), 4
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that $f(0) \in \mathbb Q$ and
\[f(x+f(y)^2 ) = {f(x+y)}^2.\]
(25 points)
Mexican Quarantine Mathematical Olympiad, #5
Let $\mathbb{N} = \{1, 2, 3, \dots \}$ be the set of positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$, such that for all positive integers $n$ and prime numbers $p$:
$$p \mid f(n)f(p-1)!+n^{f(p)}.$$
[i]Proposed by Dorlir Ahmeti[/i]
2020 Hong Kong TST, 4
Find all real-valued functions $f$ defined on the set of real numbers such that $$f(f(x)+y)+f(x+f(y))=2f(xf(y))$$ for any real numbers $x$ and $y$.
1993 Brazil National Olympiad, 5
Find at least one function $f: \mathbb R \rightarrow \mathbb R$ such that $f(0)=0$ and $f(2x+1) = 3f(x) + 5$ for any real $x$.
2019 Philippine TST, 2
Find all functions $f : \mathbb{R} \to \mathbb{R}$ that satisfy the equation $$f(x^{2019} + y^{2019}) = x(f(x))^{2018} + y(f(y))^{2018}$$ for all real numbers $x$ and $y$.
1977 Germany Team Selection Test, 2
Determine the polynomials P of two variables so that:
[b]a.)[/b] for any real numbers $t,x,y$ we have $P(tx,ty) = t^n P(x,y)$ where $n$ is a positive integer, the same for all $t,x,y;$
[b]b.)[/b] for any real numbers $a,b,c$ we have $P(a + b,c) + P(b + c,a) + P(c + a,b) = 0;$
[b]c.)[/b] $P(1,0) =1.$
2013 Dutch BxMO/EGMO TST, 4
Determine all functions $f:\mathbb{R}\to\mathbb{R}$ satisfying
\[f(x+yf(x))=f(xf(y))-x+f(y+f(x))\]
2018 SIMO, Q3
Suppose $f:\mathbb{N}\rightarrow \mathbb{N}$ is a function such that $$f^n(n) = 2n$$ for all $n\in \mathbb{N}$. Must $f(n) = n+1$ for all $n$?
2013 NIMO Problems, 2
Let $f$ be a non-constant polynomial such that \[ f(x-1) + f(x) + f(x+1) = \frac {f(x)^2}{2013x} \] for all nonzero real numbers $x$. Find the sum of all possible values of $f(1)$.
[i]Proposed by Ahaan S. Rungta[/i]
1990 Vietnam National Olympiad, 2
Suppose $ f(x)\equal{}a_0x^n\plus{}a_1x^{n\minus{}1}\plus{}\ldots\plus{}a_{n\minus{}1}x\plus{}a_n$ ($ a_0\neq 0$) is a polynomial with real coefficients satisfying $ f(x)f(2x^2) \equal{} f(2x^3 \plus{} x)$ for all $ x \in\mathbb{R}$. Prove that $ f(x)$ has no real roots.
2021 Science ON all problems, 2
Let $X$ be a set with $n\ge 2$ elements. Define $\mathcal{P}(X)$ to be the set of all subsets of $X$. Find the number of functions $f:\mathcal{P}(X)\mapsto \mathcal{P}(X)$ such that
$$|f(A)\cap f(B)|=|A\cap B|$$
whenever $A$ and $B$ are two distinct subsets of $X$.
[i] (Sergiu Novac)[/i]
2021 Azerbaijan IMO TST, 3
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]
2009 Ukraine Team Selection Test, 4
Let $n$ be some positive integer. Find all functions $f:{{R}^{+}}\to R$ (i.e., functions defined by the set of all positive real numbers with real values) for which equality holds $f\left( {{x}^{n+1}}+ {{y}^{n+1}} \right)={{x}^{n}}f\left( x \right)+{{y}^{n}}f\left( y \right)$ for any positive real numbers $x, y$
2009 Belarus Team Selection Test, 3
Find all real numbers $a$ for which there exists a function $f: R \to R$ asuch that $x + f(y) =a(y + f(x))$ for all real numbers $x,y\in R$.
I.Voronovich
2016 Thailand Mathematical Olympiad, 3
Determine all functions $f : R \to R$ satisfying $f (f(x)f(y) + f(y)f(z) + f(z)f(x))= f(x) + f(y) + f(z)$ for all real numbers $x, y, z$.
2023 ELMO Shortlist, A2
Let \(\mathbb R_{>0}\) denote the set of positive real numbers. Find all functions \(f:\mathbb R_{>0}\to\mathbb R_{>0}\) such that for all positive real numbers \(x\) and \(y\), \[f(xy+1)=f(x)f\left(\frac1x+f\left(\frac1y\right)\right).\]
[i]Proposed by Luke Robitaille[/i]
Kvant 2021, M2661
An infinite table whose rows and columns are numbered with positive integers, is given. For a sequence of functions
$f_1(x), f_2(x), \ldots $ let us place the number $f_i(j)$ into the cell $(i,j)$ of the table (for all $i, j\in \mathbb{N}$).
A sequence $f_1(x), f_2(x), \ldots $ is said to be {\it nice}, if all the numbers in the table are positive integers, and each positive integer appears exactly once. Determine if there exists a nice sequence of functions $f_1(x), f_2(x), \ldots $, such that each $f_i(x)$ is a polynomial of degree 101 with integer coefficients and its leading coefficient equals to 1.
2010 Contests, 2
Find all the continuous functions $f : \mathbb{R} \mapsto\mathbb{R}$ such that $\forall x,y \in \mathbb{R}$,
$(1+f(x)f(y))f(x+y)=f(x)+f(y)$.