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

Tags were heavily modified to better represent problems.

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Found problems: 66

2000 Moldova National Olympiad, Problem 5
Tags: functional equation , fe , polynomial
Prove that there is no polynomial $P(x)$ with real coefficients that satisfies $$P'(x)P''(x)>P(x)P'''(x)\qquad\text{for all }x\in\mathbb R.$$Is this statement true for all of the thrice differentiable real functions?

2023 APMO, 4
Tags: functional equation , fe
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]

2014 Middle European Mathematical Olympiad, 1
Tags: function , algebra , functional equation , fe , real
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 $x,y \in \mathbb{R}$.

2025 Vietnam Team Selection Test, 1
Tags: function , algebra , fe , functional equation
Find all functions $f: \mathbb{Q}^+ \to \mathbb{Q}^+$ such that $$\dfrac{f(x)f(y)}{f(xy)} = \dfrac{\left( \sqrt{f(x)} + \sqrt{f(y)} \right)^2}{f(x+y)}$$ holds for all positive rational numbers $x, y$.

1993 French Mathematical Olympiad, Problem 3
Tags: functional equation , fe , function , algebra
Let $f$ be a function from $\mathbb Z$ to $\mathbb R$ which is bounded from above and satisfies $f(n)\le\frac12(f(n-1)+f(n+1))$ for all $n$. Show that $f$ is constant.

2023 Princeton University Math Competition, 1
Tags: fe , algebra
1. Given $n \geq 1$, let $A_{n}$ denote the set of the first $n$ positive integers. We say that a bijection $f: A_{n} \rightarrow A_{n}$ has a hump at $m \in A_{n} \backslash\{1, n\}$ if $f(m)>f(m+1)$ and $f(m)>f(m-1)$. We say that $f$ has a hump at 1 if $f(1)>f(2)$, and $f$ has a hump at $n$ if $f(n)>f(n-1)$. Let $P_{n}$ be the probability that a bijection $f: A_{n} \rightarrow A_{n}$, when selected uniformly at random, has exactly one hump. For how many positive integers $n \leq 2020$ is $P_{n}$ expressible as a unit fraction?

2017-IMOC, N3
Tags: algebra , fe , functional equation , number theory
Find all functions $f:\mathbb N\to\mathbb N_0$ such that for all $m,n\in\mathbb N$, \begin{align*} f(mn)&=f(m)f(n)\\ f(m+n)&=\min(f(m),f(n))\qquad\text{if }f(m)\ne f(n)\end{align*}

1988 Bulgaria National Olympiad, Problem 6
Tags: polynomial , fe , functional equation , algebra
Find all polynomials $p(x)$ satisfying $p(x^3+1)=p(x+1)^3$ for all $x$.

2014 Contests, 1
Tags: function , algebra , functional equation , fe , real
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 $x,y \in \mathbb{R}$.

1987 IMO, 1
Tags: algebra , function , functional equation , fe
Prove that there is no function $f$ from the set of non-negative integers into itself such that $f(f(n))=n+1987$ for all $n$.

2019 APMO, 5
Tags: wrapped , fe , algebra
Determine all the functions $f : \mathbb{R} \to \mathbb{R}$ such that \[ f(x^2 + f(y)) = f(f(x)) + f(y^2) + 2f(xy) \] for all real numbers $x$ and $y$.

Russian TST 2018, P1
Tags: function , algebra , functional equation , fe
Functions $f,g:\mathbb{Z}\to\mathbb{Z}$ satisfy $$f(g(x)+y)=g(f(y)+x)$$ for any integers $x,y$. If $f$ is bounded, prove that $g$ is periodic.

1976 Bulgaria National Olympiad, Problem 2
Tags: polynomial , fe , functional equation , algebra
Find all polynomials $p(x)$ satisfying the condition: $$p(x^2-2x)=p(x-2)^2.$$

2000 Brazil Team Selection Test, Problem 2
Tags: functional equation , fe
Find all functions $f:\mathbb R\to\mathbb R$ such that (i) $f(0)=1$; (ii) $f(x+f(y))=f(x+y)+1$ for all real $x,y$; (iii) there is a rational non-integer $x_0$ such that $f(x_0)$ is an integer.

2020 Jozsef Wildt International Math Competition, W44
Tags: functional equation , fe
We consider a function $f:\mathbb R\to\mathbb R$ such that $$f(x+y)+f(xy-1)=f(x)f(y)+f(x)+f(y)+1$$ for each $x,y\in\mathbb R$. i) Calculate $f(0)$ and $f(-1)$. ii) Prove that $f$ is an even function. iii) Give an example of such a function. iv) Find all monotone functions with the above property. [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

2017-IMOC, N5
Tags: number theory , fe , functional equation
Find all functions $f:\mathbb N\to\mathbb N$ such that $$f(x)+f(y)\mid x^2-y^2$$holds for all $x,y\in\mathbb N$.

2024 Rioplatense Mathematical Olympiad, 5
Tags: algebra , functional equation , fe , rioplatense
Let $S = \{2, 3, 4, \dots\}$ be the set of positive integers greater than 1. Find all functions $f : S \to S$ that satisfy \[ \text{gcd}(a, f(b)) \cdot \text{lcm}(f(a), b) = f(ab) \] for all pairs of integers $a, b \in S$. Clarification: $\text{gcd}(a,b)$ is the greatest common divisor of $a$ and $b$, and $\text{lcm}(a,b)$ is the least common multiple of $a$ and $b$.

2020 Hong Kong TST, 4
Tags: functional equation , algebra , hong kong tst , fe , real
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 $y$.

2000 Mongolian Mathematical Olympiad, Problem 4
Tags: functional equation , fe , functional inequality , algebra
Suppose that a function $f:\mathbb R\to\mathbb R$ satisfies the following conditions: (i) $\left|f(a)-f(b)\right|\le|a-b|$ for all $a,b\in\mathbb R$; (ii) $f(f(f(0)))=0$. Prove that $f(0)=0$.

2005 Federal Math Competition of S&M, Problem 3
Tags: functional equation , fe , algebra , polynomial
Determine all polynomials $p$ with real coefficients for which $p(0)=0$ and $$f(f(n))+n=4f(n)\qquad\text{for all }n\in\mathbb N,$$where $f(n)=\lfloor p(n)\rfloor$.

1999 Mongolian Mathematical Olympiad, Problem 1
Tags: fe , functional equation , algebra
Suppose that a function $f:\mathbb R\to\mathbb R$ is such that for any real $h$ there exist at most $19990509$ different values of $x$ for which $f(x)\ne f(x+h)$. Prove that there is a set of at most $9995256$ real numbers such that $f$ is constant outside of this set.

2014 BMT Spring, 8
Tags: functional equation , fe , algebra
Suppose an integer-valued function $f$ satisfies $$\sum_{k=1}^{2n+1}f(k)=\ln|2n+1|-4\ln|2n-1|\enspace\text{and}\enspace\sum_{k=0}^{2n}f(k)=4e^n-e^{n-1}$$ for all non-negative integers $n$. Determine $\sum_{n=0}^\infty\frac{f(n)}{2^n}$.

2000 Moldova National Olympiad, Problem 4
Tags: functional equation , fe , polynomial , algebra
Find all polynomials $P(x)$ with real coefficients that satisfy the relation $$1+P(x)=\frac{P(x-1)+P(x+1)}2.$$

2019-IMOC, A4
Tags: fe , functional equation , algebra
Find all functions $f:\mathbb N\to\mathbb N$ so that $$f^{2f(b)}(2a)=f(f(a+b))+a+b$$ holds for all positive integers $a,b$.

2023 Brazil Team Selection Test, 4
Tags: functional equation , fe
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]