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

2018 Saudi Arabia BMO TST, 2
Tags: functional , algebra , functional equation
Find all functions $f : R \to R$ such that $f( 2x^3 + f (y)) = y + 2x^2 f (x)$ for all real numbers $x, y$.

2007 Cuba MO, 4
Tags: algebra , functional , functional equation
Find all functions $f : R_+ \to R_+$ such that $$x^2(f(x)+f(y)) = (x+y)f(f(x)y)$$ for all positive real $x, y$.

VMEO III 2006 Shortlist, A6
Tags: algebra , functional equation , functional
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 $x,y \in N_m$.

2022 Dutch BxMO TST, 1
Tags: number theory , functional
Find all functions $f : Z_{>0} \to Z_{>0}$ for which $f(n) | f(m) - n$ if and only if $n | m$ for all natural numbers $m$ and $n$.

2016 Thailand Mathematical Olympiad, 9
Tags: functional equation , functional , algebra
A real number $a \ne 0$ is given. Determine all functions $f : R \to R$ satisfying $f(x)f(y) + f(x + y) = axy$ for all real numbers $x, y$.

I Soros Olympiad 1994-95 (Rus + Ukr), 11.1
Tags: algebra , functional
Let the function $f:R \to R$ satisfies the following conditions: 1) for all $x, y\in R$, $ f(x +y) = f(x) +f(y)$ 2)$ f(1)=1$ 3) for all $x \ne 0$ , $ f(1/x) =\frac{f(x)}{x^2}$ Prove that for all $x \in R$, $f(x) = x$.

2008 Switzerland - Final Round, 2
Tags: algebra , inequalities , functional inequality , functional
Determine all functions $f : R^+ \to R^+$, so that for all $x, y > 0$: $$f(xy) \le \frac{xf(y) + yf(x)}{2}$$

2009 Switzerland - Final Round, 6
Tags: algebra , functional , functional equation
Find all functions $f : R_{>0} \to R_{>0}$, which for all $x > y > z > 0$ is the following equation holds $$f(x - y + z) = f(x) + f(y) + f(z) - xy - yz + xz.$$

2004 Swedish Mathematical Competition, 3
Tags: algebra , functional , functional equation
A function $f$ satisfies $f(x)+x f(1-x) = x^2$ for all real $x$. Determine $f$ .

2017 Puerto Rico Team Selection Test, 6
Tags: algebra , functional equation , functional , inequalities
Find all functions $f: R \to R$ such that $f (xy) \le yf (x) + f (y)$, for all $x, y\in R$.

2006 Thailand Mathematical Olympiad, 5
Tags: algebra , functional equation , function , functional
Let $f : Z_{\ge 0} \to Z_{\ge 0}$ satisfy the functional equation $$f(m^2 + n^2) =(f(m) - f(n))^2 + f(2mn)$$ for all nonnegative integers $m, n$. If $8f(0) + 9f(1) = 2006$, compute $f(0)$.

1994 Bulgaria National Olympiad, 2
Tags: functional equation , functional , algebra
Find all functions $f : R \to R$ such that $x f(x)-y f(y) = (x-y)f(x+y)$ for all $x,y \in R$.

1995 Israel Mathematical Olympiad, 8
Tags: functional equation , functional , function , 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 $x > 0$.

2010 Saudi Arabia BMO TST, 3
Tags: algebra , inequalities , functional inequality , functional
Let $a > 0$ be a real number and let $f : R \to R$ be a function satisfying $$f(x_1) + f(x_2) \ge a f(x_1 + x_2), \forall x_1 ,x_2 \in R.$$ Prove that $$f(x_1) + f(x_2) +(x_3) \ge \frac{3a^2}{a+2} f(x_1+ x_2 + x_3), \forall x_1 ,x_2,x_3 \in R$$.

2016 Switzerland - Final Round, 10
Tags: algebra , functional , functional equation
Find all functions $f : R \to R$ such that for all $x, y \in R$: $$f(x + yf(x + y)) = y^2 + f(xf(y + 1)).$$

2022 Switzerland - Final Round, 3
Tags: algebra , functional equation , functional
Let $N$ be the set of positive integers. Find all functions $f : N \to N$ such that both $\bullet$ $f(f(m)f(n)) = mn$ $\bullet$ $f(2022a + 1) = 2022a + 1$ hold for all positive integers $m, n$ and $a$.

1998 Belarus Team Selection Test, 3
Tags: continuous , functional equation , functional , algebra
Find all continuous functions $f: R \to R$ such that $g(g(x)) = g(x)+2x$ for all real $x$.

2009 Thailand Mathematical Olympiad, 2
Tags: injective , function , functional , functional equation , algebra
Is there an injective function $f : Z^+ \to Q$ satisfying the equation $f(xy) = f(x) + f(y)$ for all positive integers $x$ and $y$?

2012 Switzerland - Final Round, 2
Tags: algebra , functional , functional equation
Determine all functions $f : R \to R$ such that for all $x, y\in R$ holds $$f (f(x) + 2f(y)) = f(2x) + 8y + 6.$$

1986 Dutch Mathematical Olympiad, 1
Tags: algebra , functional
$f(x) = \frac{12x+9}{19x+86}, \,\, x \ne -\frac{86}{19}$ Prove that $\exists ! \,\,\, {x_o \in R} \,\,\, \forall h_1,h_2 \in R [f(x_0+h_1)f(x_0-h_1)=f(x_0+h_2)f(x_0-h_2)]$, and calculate $x_0$.

2020 Dutch IMO TST, 3
Tags: functional equation , functional equation in z , functional , algebra
Find all functions $f: Z \to Z$ that satisfy $$f(-f (x) - f (y))= 1 -x - y$$ for all $x, y \in Z$

1968 German National Olympiad, 3
Tags: algebra , functional , functional equation
Specify all functions $y = f(x)$, each in the largest possible domain (within the range of real numbers) of the equation $$a \cdot f(x^n) + f(-x^n) = bx$$ suffice, where $b$ is any real number, $n$ is any odd natural number and $a$ is a real number with $|a| \ne 1$.

2017 QEDMO 15th, 8
Tags: algebra , functional , functional equation
For a function $f: R\to R $ , $ f (2017)> 0$ as well as $f (x^2 + yf (z)) = xf (x) + zf (y)$ for all $x,y,z \in R$ is known. What is the value of $f (-42)$?

2004 Estonia Team Selection Test, 1
Tags: polynomial , functional equation , functional , algebra
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 $x$.

2017 Czech And Slovak Olympiad III A, 3
Tags: functional equation , functional , algebra
Find all functions $f: R \to R$ such that for all real numbers $x, y$ holds $f(y - xy) = f(x)y + (x - 1)^2 f(y)$