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: 313

MathLinks Contest 2nd, 3.1
Tags: functional , inequalities , algebra
Determine all values of $a \in R$ such that there exists a function $f : [0, 1] \to R$ fulfilling the following inequality for all $x \ne y$: $$|f(x) - f(y)| \ge a.$$

2004 Estonia Team Selection Test, 1
Tags: algebra , functional equation , functional , polynomial
Let $k > 1$ be a fixed natural number. Find all polynomials $P(x)$ satisfying the condition $P(x^k) = (P(x))^k$ for all real numbers $x$.

2009 Ukraine Team Selection Test, 4
Tags: algebra , functional equation , functional
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 $x, y$

2013 NZMOC Camp Selection Problems, 5
Tags: function , algebra , functional , functional equation
Consider functions $f$ from the whole numbers (non-negative integers) to the whole numbers that have the following properties: $\bullet$ For all $x$ and $y$, $f(xy) = f(x)f(y)$, $\bullet$ $f(30) = 1$, and $\bullet$ for any $n$ whose last digit is $7$, $f(n) = 1$. Obviously, the function whose value at $n$ is $ 1$ for all $n$ is one such function. Are there any others? If not, why not, and if so, what are they?

2019 Greece Team Selection Test, 4
Tags: algebra , functional equation , functional
Find all functions $f:(0,\infty)\mapsto\mathbb{R}$ such that $\displaystyle{(y^2+1)f(x)-yf(xy)=yf\left(\frac{x}{y}\right),}$ for every $x,y>0$.

VMEO II 2005, 11
Tags: functional equation , functional , algebra , polynomial
Given $P$ a real polynomial with degree greater than $ 1$. Find all pairs $(f,Q)$ with function $f : R \to R$ and the real polynomial $Q$ satisfying the following two conditions: i) for all $x, y \in R$, we have $f(P(x) + f(y)) = y + Q(f(x))$. ii) there exists $x_0 \in R$ such that $f(P(x_0)) = Q(f(x_0))$.

2018 Costa Rica - Final Round, F2
Tags: sequence , functional , algebra , functional equation , combinatorics
Consider $f (n, m)$ the number of finite sequences of $ 1$'s and $0$'s such that each sequence that starts at $0$, has exactly n $0$'s and $m$ $ 1$'s, and there are not three consecutive $0$'s or three $ 1$'s. Show that if $m, n> 1$, then $$f (n, m) = f (n-1, m-1) + f (n-1, m-2) + f (n-2, m-1) + f (n-2, m-2)$$

2015 Belarus Team Selection Test, 1
Tags: algebra , functional equation , functional
Do there exist numbers $a,b \in R$ and surjective function $f: R \to R$ such that $f(f(x)) = bx f(x) +a$ for all real $x$? I.Voronovich

2017 QEDMO 15th, 11
Tags: functional equation , algebra , functional
Let $G$ be a finite group and $f: G \to G$ a map, such that $f (xy) = f (x) f (y)$ for all $x, y \in G$ and $f (x) = x^{-1}$ for more than $\frac34$ of all $x \in G$ is fulfilled. Show that $f (x) =x^{-1}$ even holds for all $x \in G$ holds.

2020 Grand Duchy of Lithuania, 1
Tags: functional equation , functional , algebra
Find all functions $f: R \to R$, such that equality $f (xf (y) - yf (x)) = f (xy) - xy$ holds for all $x, y \in R$.

2005 Thailand Mathematical Olympiad, 3
Tags: algebra , functional equation , functional
Does there exist a function $f : Z^+ \to Z^+$ such that $f(f(n)) = 2n$ for all positive integers $n$? Justify your answer, and if the answer is yes, give an explicit construction.

2007 Thailand Mathematical Olympiad, 9
Tags: functional , algebra , functional equation
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 $f(85)$?

2009 Postal Coaching, 6
Tags: algebra , functional , functional equation
Find all functions $f : N \to N$ such that $$\frac{f(x+y)+f(x)}{2x+f(y)}= \frac{2y+f(x)}{f(x+y)+f(y)}$$ , for all $x, y$ in $N$.

2006 Thailand Mathematical Olympiad, 6
Tags: algebra , trigonometry , functional , functional equation
A function $f : R \to R$ has $f(1) < 0$, and satisfy the functional equation $$f(\cos (x + y)) = (\cos x)f(\cos y) + 2f(\sin x)f(\sin y)$$ for all reals $x, y$. Compute $f \left(\frac{2006}{2549 }\right)$

2007 Switzerland - Final Round, 5
Tags: functional , functional equation , algebra
Determine all functions $f : R_{\ge 0} \to R_{\ge 0}$ with the following properties: (a) $f(1) = 0$, (b) $f(x) > 0$ for all $x > 1$, (c) For all $x, y\ge 0$ with $x + y > 0$ holds $$f(xf(y))f(y) = f\left( \frac{xy}{x + y}\right)$$

2021 Francophone Mathematical Olympiad, 4
Tags: functional , algebra , number theory , functional equation
Let $\mathbb{N}_{\ge 1}$ be the set of positive integers. Find all functions $f \colon \mathbb{N}_{\ge 1} \to \mathbb{N}_{\ge 1}$ such that, for all positive integers $m$ and $n$: (a) $n = \left(f(2n)-f(n)\right)\left(2 f(n) - f(2n)\right)$, (b)$f(m)f(n) - f(mn) = \left(f(2m)-f(m)\right)\left(2 f(n) - f(2n)\right) + \left(f(2n)-f(n)\right)\left(2 f(m) - f(2m)\right)$, (c) $m-n$ divides $f(2m)-f(2n)$ if $m$ and $n$ are distinct odd prime numbers.

2011 Indonesia TST, 1
Tags: functional equation in q , functional equation , functional , algebra
Let $Q^+$ denote the set of positive rationals. Determine all functions $f : Q^+ \to Q^+$ that satisfy both of these conditions: (i) $f(x)$ is an integer if and only if $x$ is an integer; (ii) $f(f(xf(y)) + x) = yf(x) + x$ for all $x, y \in Q^+$.

VMEO I 2004, 5
Tags: algebra , functional , functional equation
Find all the functions $f:R \to R$ satisfying $$(x + y)(f (x)-f (y)) = f (x^2) - f (y^2),\, \forall x, y \in R$$

1998 Italy TST, 1
Tags: functional , function , functional equation , algebra
A real number $\alpha$ is given. Find all functions $f : R^+ \to R^+$ satisfying $\alpha x^2f\left(\frac{1}{x}\right) +f(x) =\frac{x}{x+1}$ for all $x > 0$.

2013 Thailand Mathematical Olympiad, 6
Tags: algebra , functional , functional equation
Determine all functions $f$ : $\mathbb R\to\mathbb R$ satisfying $(x^2+y^2)f(xy)=f(x)f(y)f(x^2+y^2)$ $\forall x,y\in\mathbb R$

2009 Thailand Mathematical Olympiad, 5
Tags: functional , algebra , functional equation
Determine all functions $f : R\to R$ satisfying: $$f(xy + 2x + 2y - 1) = f(x)f(y) + f(y) + x -2$$ for all real numbers $x, y$.

1997 All-Russian Olympiad Regional Round, 11.8
Tags: functional , algebra , functional equation
For which $a$, there is a function $f: R \to R$, different from a constant, such that $$f(a(x + y)) = f(x) + f(y) ?$$

2001 Estonia Team Selection Test, 3
Tags: algebra , functional equation , functional
Let $k$ be a fixed real number. Find all functions $f: R \to R$ such that $f(x)+ (f(y))^2 = kf(x + y^2)$ for all real numbers $x$ and $y$.

2012 CHMMC Fall, 2
Tags: algebra , continuous function , functional
Find all continuous functions $f : R \to R$ such that $$f(x + f(y)) = f(x + y) + y,$$ for all $x, y \in R$. No proof is required for this problem.

2015 Thailand Mathematical Olympiad, 9
Tags: functional , algebra , functional equation
Determine all functions $f : R \to R$ satisfying $f(f(x) + 2y)= 6x + f(f(y) -x)$ for all real numbers $x,y$