This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

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Found problems: 1513

2020 Indonesia MO, 7
Tags: algebra , polynomial , floor function , functional equation
Determine all real-coefficient polynomials $P(x)$ such that \[ P(\lfloor x \rfloor) = \lfloor P(x) \rfloor \]for every real numbers $x$.

2017 Azerbaijan BMO TST, 3
Tags: algebra , functional equation
Find all funtions $f:\mathbb R\to\mathbb R$ such that: $$f(xy-1)+f(x)f(y)=2xy-1$$ for all $x,y\in \mathbb{R}$.

1988 IMO Shortlist, 26
Tags: algebra , functional equation , binary representation , function
A function $ f$ defined on the positive integers (and taking positive integers values) is given by: $ \begin{matrix} f(1) \equal{} 1, f(3) \equal{} 3 \\ f(2 \cdot n) \equal{} f(n) \\ f(4 \cdot n \plus{} 1) \equal{} 2 \cdot f(2 \cdot n \plus{} 1) \minus{} f(n) \\ f(4 \cdot n \plus{} 3) \equal{} 3 \cdot f(2 \cdot n \plus{} 1) \minus{} 2 \cdot f(n), \end{matrix}$ for all positive integers $ n.$ Determine with proof the number of positive integers $ \leq 1988$ for which $ f(n) \equal{} n.$

2019 Latvia Baltic Way TST, 2
Tags: function , functional equation , algebra
Let $\mathbb R$ be set of real numbers. Determine all functions $f:\mathbb R\to \mathbb R$ such that $$f(y^2 - f(x)) = yf(x)^2+f(x^2y+y)$$ holds for all real numbers $x; y$

2019 IFYM, Sozopol, 5
Tags: algebra , functional equation
The non-decreasing functions $f,g: \mathbb{R}\rightarrow \mathbb{R}$ are such that $f(r)\leq g(r)$ for $\forall$ rational numbers $r$. Is it true that $f(x)\leq g(x)$ for $\forall$ real numbers $x$?

2020 Moldova Team Selection Test, 11
Tags: algebra , functional equation , function , trigonometry
Find all functions $f:[-1,1] \rightarrow \mathbb{R},$ which satisfy $$f(\sin{x})+f(\cos{x})=2020$$ for any real number $x.$

2024 Thailand TST, 3
Tags: integer , functional equation
Determine all functions $f\colon\mathbb{Z}_{>0}\to\mathbb{Z}_{>0}$ such that, for all positive integers $a$ and $b$, \[ f^{bf(a)}(a+1)=(a+1)f(b). \]

2022 European Mathematical Cup, 3
Tags: function , functional equation , substitution , cauchy functional equation , algebra
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that $$ f(x^3) + f(y)^3 + f(z)^3 = 3xyz $$ for all real numbers $x$, $y$ and $z$ with $x+y+z=0$.

2000 IMO Shortlist, 4
Tags: function , algebra , functional equation
The function $ F$ is defined on the set of nonnegative integers and takes nonnegative integer values satisfying the following conditions: for every $ n \geq 0,$ (i) $ F(4n) \equal{} F(2n) \plus{} F(n),$ (ii) $ F(4n \plus{} 2) \equal{} F(4n) \plus{} 1,$ (iii) $ F(2n \plus{} 1) \equal{} F(2n) \plus{} 1.$ Prove that for each positive integer $ m,$ the number of integers $ n$ with $ 0 \leq n < 2^m$ and $ F(4n) \equal{} F(3n)$ is $ F(2^{m \plus{} 1}).$

2012 Harvard-MIT Mathematics Tournament, 7
Tags: hmmt , algebra , functional equation , logarithm
Let $\otimes$ be a binary operation that takes two positive real numbers and returns a positive real number. Suppose further that $\otimes$ is continuous, commutative $(a\otimes b=b\otimes a)$, distributive across multiplication $(a\otimes(bc)=(a\otimes b)(a\otimes c))$, and that $2\otimes 2=4$. Solve the equation $x\otimes y=x$ for $y$ in terms of $x$ for $x>1$.

2023 Abelkonkurransen Finale, 4b
Tags: functional equation , algebra
Find all functions $f: \mathbb R^{+} \to \mathbb R^{+}$ satisfying \begin{align*} f(f(x)+y) = f(y) + x, \qquad \text{for all } x,y \in \mathbb R^{+}. \end{align*} Note that $\mathbb R^{+}$ is the set of all positive real numbers.

2021 Regional Olympiad of Mexico Center Zone, 6
Tags: algebra , number theory , functional equation , sequence
The sequence $a_1,a_2,\dots$ of positive integers obeys the following two conditions: [list] [*] For all positive integers $m,n$, it happens that $a_m\cdot a_n=a_{mn}$ [*] There exist infinite positive integers $n$ such that $(a_1,a_2,\dots,a_n)$ is a permutation of $(1,2,\dots,n)$ [/list] Prove that $a_n=n$ for all positive integers $n$. [i]Proposed by José Alejandro Reyes González[/i]

1983 Canada National Olympiad, 2
Tags: function , functional equation , algebra , logarithm
For each $r\in\mathbb{R}$ let $T_r$ be the transformation of the plane that takes the point $(x, y)$ into the point $(2^r x; r2^r x+2^r y)$. Let $F$ be the family of all such transformations (i.e. $F = \{T_r : r\in\mathbb{R}\}$). Find all curves $y = f(x)$ whose graphs remain unchanged by every transformation in $F$.

2018 IMO Shortlist, A1
Tags: algebra , functional equation
Let $\mathbb{Q}_{>0}$ denote the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}_{>0}\to \mathbb{Q}_{>0}$ satisfying $$f(x^2f(y)^2)=f(x)^2f(y)$$ for all $x,y\in\mathbb{Q}_{>0}$

2021 Thailand Online MO, P8
Tags: functional equation , number theory , greatest common divisor
Let $\mathbb N$ be the set of positive integers. Determine all functions $f:\mathbb N\times\mathbb N\to\mathbb N$ that satisfy both of the following conditions: [list] [*]$f(\gcd (a,b),c) = \gcd (a,f(c,b))$ for all $a,b,c \in \mathbb{N}$. [*]$f(a,a) \geq a$ for all $a \in \mathbb{N}$. [/list]

1992 IMO Shortlist, 9
Tags: algebra , polynomial , functional equation , iteration
Let $ f(x)$ be a polynomial with rational coefficients and $ \alpha$ be a real number such that \[ \alpha^3 \minus{} \alpha \equal{} [f(\alpha)]^3 \minus{} f(\alpha) \equal{} 33^{1992}.\] Prove that for each $ n \geq 1,$ \[ \left [ f^{n}(\alpha) \right]^3 \minus{} f^{n}(\alpha) \equal{} 33^{1992},\] where $ f^{n}(x) \equal{} f(f(\cdots f(x))),$ and $ n$ is a positive integer.

2022 Balkan MO Shortlist, A4
Tags: functional equation , algebra
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 $x, y \in\mathbb{R}$.

1986 Tournament Of Towns, (116) 4
Tags: algebra , continuous , analysis , function , functional equation , functional
The function $F$ , defined on the entire real line, satisfies the following relation (for all $x$ ) : $F(x +1 )F(x) + F(x + 1 ) + 1 = 0$ . Prove that $F$ is not continuous. (A.I. Plotkin, Leningrad)

1987 IMO Longlists, 6
Tags: function , algebra , functional equation
Let f be a function that satisfies the following conditions: $(i)$ If $x > y$ and $f(y) - y \geq v \geq f(x) - x$, then $f(z) = v + z$, for some number $z$ between $x$ and $y$. $(ii)$ The equation $f(x) = 0$ has at least one solution, and among the solutions of this equation, there is one that is not smaller than all the other solutions; $(iii)$ $f(0) = 1$. $(iv)$ $f(1987) \leq 1988$. $(v)$ $f(x)f(y) = f(xf(y) + yf(x) - xy)$. Find $f(1987)$. [i]Proposed by Australia.[/i]

1980 IMO Shortlist, 7
Tags: function , algebra , functional equation
The function $f$ is defined on the set $\mathbb{Q}$ of all rational numbers and has values in $\mathbb{Q}$. It satisfies the conditions $f(1) = 2$ and $f(xy) = f(x)f(y) - f(x+y) + 1$ for all $x,y \in \mathbb{Q}$. Determine $f$.

2017 Estonia Team Selection Test, 6
Tags: algebra , functional equation
Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

BIMO 2022, 1
Tags: algebra , functional equation
Find all functions $f:\mathbb{R}\rightarrow \mathbb{R}$ such that for all real numbers $x,y$, we have $$f(xf(x)+2y)=f(x)^2+x+2f(y)$$

2020 Peru Iberoamerican Team Selection Test, P6
Tags: number theory , functional equation
Find all functions $f : \mathbb{Z}\to \mathbb{Z}$ that satisfy: $i) f(f(x))=x, \forall x\in\mathbb{Z}$ $ii)$ For any integer $x$ and $y$ such that $x + y$ is odd, it holds that $f(x) + f(y) \ge x + y.$

2010 Thailand Mathematical Olympiad, 6
Tags: algebra , functional equation , functional
Let $f : R \to R$ be a function satisfying the functional equation $f(3x + y) + f(3x-y) = f(x + y) + f(x - y) + 16f(x)$ for all reals $x, y$. Show that $f$ is even, that is, $f(-x) = f(x)$ for all reals $x$

2019 Canadian Mathematical Olympiad Qualification, 1
Tags: algebra , function , injective function , injective , functional equation
A function $f$ is called injective if when $f(n) = f(m)$, then $n = m$. Suppose that $f$ is injective and $\frac{1}{f(n)}+\frac{1}{f(m)}=\frac{4}{f(n) + f(m)}$. Prove $m = n$