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: 1513

2017 Peru IMO TST, 14
Tags: sum of digits , number theory , polynomial , functional equation
For any positive integer $k$, denote the sum of digits of $k$ in its decimal representation by $S(k)$. Find all polynomials $P(x)$ with integer coefficients such that for any positive integer $n \geq 2016$, the integer $P(n)$ is positive and $$S(P(n)) = P(S(n)).$$ [i]Proposed by Warut Suksompong, Thailand[/i]

2008 National Olympiad First Round, 19
Tags: algebra , functional equation , quadratic , function
Let $f:(0,\infty) \rightarrow (0,\infty)$ be a function such that \[ 10\cdot \frac{x+y}{xy}=f(x)\cdot f(y)-f(xy)-90 \] for every $x,y \in (0,\infty)$. What is $f(\frac 1{11})$? $ \textbf{(A)}\ 1 \qquad\textbf{(B)}\ 11 \qquad\textbf{(C)}\ 21 \qquad\textbf{(D)}\ 31 \qquad\textbf{(E)}\ \text{There is more than one solution} $

2018 IFYM, Sozopol, 6
Tags: function , algebra , functional equation
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$, such that $f(x+y) = f(y) f(x f(y))$ for every two real numbers $x$ and $y$.

2024-IMOC, N7
Tags: functional equation , number theory
Find all functions $f:\mathbb{N}\to\mathbb{N}$ such that $$|xf(y)-yf(x)|$$ is a perfect square for every $x,y \in \mathbb{N}$

2022 Ecuador NMO (OMEC), 2
Tags: algebra , functional equation , function
Determine all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that for all real numbers $x, y$ \[f(x + y)=f(f(x)) + y + 2022\]

1978 IMO Longlists, 24
Tags: sequence , partition , algebra , functional equation
Let $0<f(1)<f(2)<f(3)<\ldots$ a sequence with all its terms positive$.$ The $n-th$ positive integer which doesn't belong to the sequence is $f(f(n))+1.$ Find $f(240).$

2018 Saudi Arabia BMO TST, 4
Tags: functional , functional equation , algebra
Find all functions $f : Z \to Z$ such that $x f (2f (y) - x) + y^2 f (2x - f (y)) = \frac{(f (x))^2}{x} + f (y f (y))$ , for all $x, y \in Z$, $x \ne 0$.

2017 Iran MO (3rd round), 1
Tags: functional equation , algebra
Let $\mathbb{R}^{\ge 0}$ be the set of all nonnegative real numbers. Find all functions $f:\mathbb{R}^{\ge 0} \to \mathbb{R}^{\ge 0}$ such that $$ x+2 \max\{y,f(x),f(z)\} \ge f(f(x))+2 \max\{z,f(y)\}$$ for all nonnegative real numbers $x,y$ and $z$.

2023 Korea Summer Program Practice Test, P2
Tags: function , functional equation , algebra
Find all functions $f : \mathbb{R} \to \mathbb{R}$ such that $$f(f(x)^2 + |y|) = x^2 + f(y)$$

2016 IFYM, Sozopol, 3
Tags: algebra , function , functional equation
Let $f: \mathbb{R}^2\rightarrow \mathbb{R}$ be a function for which for arbitrary $x,y,z\in \mathbb{R}$ we have that $f(x,y)+f(y,z)+f(z,x)=0$. Prove that there exist function $g:\mathbb{R}\rightarrow \mathbb{R}$ for which: $f(x,y)=g(x)-g(y),\, \forall x,y\in \mathbb{R}$.

2013 Korea Junior Math Olympiad, 7
Tags: functional equation
Let $f:\mathbb{N} \longrightarrow \mathbb{N}$ be such that for every positive integer $n$, followings are satisfied. i. $f(n+1) > f(n)$ ii. $f(f(n)) = 2n+2$ Find the value of $f(2013)$. (Here, $\mathbb{N}$ is the set of all positive integers.)

The Golden Digits 2024, P2
Tags: functional equation , algebra
Let $n\in\mathbb{Z}$, $n\geq 2$. Find all functions $f:\mathbb{R}_{>0}\rightarrow\mathbb{R}_{>0}$ such that $$f(x_1+\dots +x_n)^2=\sum_{i=1}^nf(x_i) ^2+ 2\sum_{i<j}f(x_ix_j),$$ for all $x_1,\dots ,x_n\in\mathbb{R}_{>0}$. [i]Proposed by Andrei Vila[/i]

VMEO III 2006 Shortlist, A6
Tags: functional , algebra , functional equation
The symbol $N_m$ denotes the set of all integers not less than the given integer $m$. Find all functions $f: N_m \to N_m$ such that $f(x^2+f(y))=y^2+f(x)$ for all $x,y \in N_m$.

2018 Abels Math Contest (Norwegian MO) Final, 3a
Tags: functional equation , algebra , polynomial
Find all polynomials $P$ such that $P(x)+3P(x+2)=3P(x+1)+P(x+3)$ for all real numbers $x$.

2022 Iran Team Selection Test, 12
Tags: interval , functional equation , algebra
suppose that $A$ is the set of all Closed intervals $[a,b] \subset \mathbb{R}$. Find all functions $f:\mathbb{R} \rightarrow A$ such that $\bullet$ $x \in f(y) \Leftrightarrow y \in f(x)$ $\bullet$ $|x-y|>2 \Leftrightarrow f(x) \cap f(y)=\varnothing$ $\bullet$ For all real numbers $0\leq r\leq 1$, $f(r)=[r^2-1,r^2+1]$ Proposed by Matin Yousefi

2019 Azerbaijan IMO TST, 1
Tags: functional equation , algebra
Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[ f(xy) = yf(x) + x + f(f(y) - f(x)) \] for all $x,y \in \mathbb{R}$.

2022 Latvia Baltic Way TST, P3
Tags: functional equation
Let $\mathbb R$ be the set of real numbers. Determine all functions $f:\mathbb R\to\mathbb R$ that satisfy the equation\[ f(f(x))+yf(xy+1) = f(x-f(y)) + xf(y)^2. \]for all real numbers $x$ and $y$.

Revenge EL(S)MO 2024, 4
Tags: functional equation , algebra
Determine all triples of positive integers $(A,B,C)$ for which some function $f \colon \mathbb Z_{\geq 0} \to \mathbb Z_{\geq 0}$ satisfies \[ f^{f(y)} (y + f(2x)) + f^{f(y)} (2y) = (Ax+By)^{C} \] for all nonnegative integers $x$ and $y$, where $f^k$ as usual denotes $f$ composed $k$ times. Proposed by [i]Benny Wang[/i]

2017 Thailand Mathematical Olympiad, 3
Tags: functional equation , functional , algebra
Determine all functions $f : R \to R$ satisfying $f(f(x) - y) \le xf(x) + f(y)$ for all real numbers $x, y$.

2014 Contests, 2
Tags: algebra , function , functional equation
Determine all the functions $f : \mathbb{R}\rightarrow\mathbb{R}$ that satisfies the following. $f(xf(x)+f(x)f(y)+y-1)=f(xf(x)+xy)+y-1$

Kvant 2019, M2562
Tags: functional equation , algebra
Each point $A$ in the plane is assigned a real number $f(A).$ It is known that $f(M)=f(A)+f(B)+f(C),$ whenever $M$ is the centroid of $\triangle ABC.$ Prove that $f(A)=0$ for all points $A.$

2003 IMO Shortlist, 2
Tags: functional equation , function , algebra
Find all nondecreasing functions $f: \mathbb{R}\rightarrow\mathbb{R}$ such that (i) $f(0) = 0, f(1) = 1;$ (ii) $f(a) + f(b) = f(a)f(b) + f(a + b - ab)$ for all real numbers $a, b$ such that $a < 1 < b$. [i]Proposed by A. Di Pisquale & D. Matthews, Australia[/i]

2020-IMOC, N5
Tags: functional equation , number theory
$\textbf{N5.}$ Find all $f: \mathbb{N} \rightarrow \mathbb{N}$ such that for all $a,b,c \in \mathbb{N}$ $f(a)+f(b)+f(c)-ab-bc-ca \mid af(a)+bf(b)+cf(c)-3abc$

2023 ELMO Shortlist, A1
Tags: polynomial , functional equation , algebra
Find all polynomials \(P(x)\) with real coefficients such that for all nonzero real numbers \(x\), \[P(x)+P\left(\frac1x\right) =\frac{P\left(x+\frac1x\right) +P\left(x-\frac1x\right)}2.\] [i]Proposed by Holden Mui[/i]

2019 Estonia Team Selection Test, 3
Tags: algebra , functional , functional equation
Find all functions $f : R \to R$ which for all $x, y \in R$ satisfy $f(x^2)f(y^2) + |x|f(-xy^2) = 3|y|f(x^2y)$.