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

2017 Costa Rica - Final Round, F1
Tags: functional equation , algebra , functional
Let $f: Z ^+ \to R$, such that $f (1) = 2018$ and $f (1) + f (2) + ...+ f (n) = n^2f (n)$, for all $n> 1$. Find the value $f (2017)$.

VI Soros Olympiad 1999 - 2000 (Russia), 9.4
Tags: inequalities , algebra , functional , functional inequation
Is there a function $f(x)$, which satisfies both of the following conditions: a) if $x \ne y$, then $f(x)\ne f(y)$ b) for all real $x$, holds the inequality $f(x^2-1998x)-f^2(2x-1999)\ge \frac14$?

2004 Olympic Revenge, 4
Tags: functional equation , functional , algebra
Find all functions $f:R \rightarrow R$ such that for any reals $x,y$, $f(x^2+y)=f(x)f(x+1)+f(y)+2x^2y$.

2004 Estonia Team Selection Test, 1
Tags: algebra , functional equation , polynomial , functional
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 Saudi Arabia BMO TST, 2
Tags: algebra , function , functional inequality , functional
Let $R^+$ be the set of positive real numbers. Find all function $f : R^+ \to R$ such that, for all positive real number $x$ and $y$, the following conditions are satisfied: i) $2f (x) + 2f (y) \le f (x + y)$ ii) $(x + y)[y f (x) + x f (y)] \ge x y f (x + y)$

2009 QEDMO 6th, 12
Tags: functional equation , algebra , functional
Find all functions $f: R\to R$, which satisfy the equation $f (xy + f (x)) = xf (y) + f (x)$.

1997 Slovenia Team Selection Test, 2
Tags: polynomial , algebra , inequalities , functional , functional inequality
Find all polynomials $p$ with real coefficients such that for all real $x$ , $xp(x)p(1-x)+x^3 +100 \ge 0$.

OIFMAT II 2012, 2
Tags: number theory , functional , functional equation in n
Find all functions $ f: N \rightarrow N $ such that: $\bullet$ $ f (m) = 1 \iff m = 1 $; $\bullet$ If $ d = \gcd (m, n) $, then $ f (mn) = \frac {f (m) f (n)} {f (d)} $; and $\bullet$ $ \forall m \in N $, we have $ f ^ {2012} (m) = m $. Clarification: $f^n (a) = f (f^{n-1} (a))$

2005 Portugal MO, 6
Tags: algebra , functional equation , number theory , functional
Prove that there is a unique function $f: N\to N$, that verifies $$f(a + b)f(a - b) = f(a^2)$$, for any $a, b\in N$ such that $a > b$.

1995 Singapore Team Selection Test, 1
Tags: functional equation , algebra , functional
Let $N =\{1, 2, 3, ...\}$ be the set of all natural numbers and $f : N\to N$ be a function. Suppose $f(1) = 1$, $f(2n) = f(n)$ and $f(2n + 1) = f(2n) + 1$ for all natural numbers $n$. (i) Calculate the maximum value $M$ of $f(n)$ for $n \in N$ with $1 \le n \le 1994$. (ii) Find all $n \in N$, with 1 \le n \le 1994, such that $f(n) = M$.

2004 Switzerland Team Selection Test, 11
Tags: functional equation , functional
Find all injective functions $f : R \to R$ such that for all real $x \ne y$ , $f\left(\frac{x+y}{x-y}\right) = \frac{f(x)+ f(y)}{f(x)- f(y)}$

2004 Cuba MO, 8
Tags: algebra , functional , functional equation
Determine all functions $f : R_+ \to R_+$ such that: a) $f(xf(y))f(y) = f(x + y)$ for $x, y \ge 0$ b) $f(2) = 0$ c) $f(x) \ne 0$ for $0 \le x < 2$.

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

2021 Irish Math Olympiad, 5
Tags: functional equation , algebra , functional
The function $g : [0, \infty) \to [0, \infty)$ satisfies the functional equation: $g(g(x)) = \frac{3x}{x + 3}$, for all $x \ge 0$. You are also told that for $2 \le x \le 3$: $g(x) = \frac{x + 1}{2}$. (a) Find $g(2021)$. (b) Find $g(1/2021)$.

2008 Thailand Mathematical Olympiad, 6
Tags: algebra , functional , functional equation
Let $f : R^+ \to R^+$ satisfy $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$ with $x^2y^3 > 2008.$ Prove that $f(xy)^2 = f(x^2)f(y^2)$ for all positive reals $x, y$.

1990 Chile National Olympiad, 4
Tags: algebra , functional
The function $g$, with domain and real numbers, fulfills the following: $\bullet$ $g (x) \le x$, for all real $x$ $\bullet$ $g (x + y) \le g (x) + g (y)$ for all real $x,y$ Find $g (1990)$.

2003 Singapore Team Selection Test, 3
Tags: functional equation , algebra , functional
Determine all functions $f : Z\to Z$, where $Z$ is the set of integers, such that $$f(m + f(f(n))) = -f(f(m + 1)) - n$$ for all integers $m$ and $n$.

2014 Ukraine Team Selection Test, 11
Tags: functional , algebra , functional equation
Find all functions $f: R \to R$ that satisfy the condition $(f (x) - f (y)) (u - v) = (f (u) - f (v)) (x -y)$ for arbitrary real $x, y, u, v$ such that $x + y = u + v$.

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

1995 Bulgaria National Olympiad, 5
Tags: functional , function , algebra
Let $A = \{1,2,...,m + n\}$, where $m,n$ are positive integers, and let the function f : $A \to A$ be defined by: $f(m) = 1$, $f(m+n) = m+1$ and $f(i) = i+1$ for all the other $i$. (a) Prove that if $m$ and $n$ are odd, then there exists a function $g : A \to A$ such that $g(g(a)) = f(a)$ for all $a \in A$. (b) Prove that if $m$ is even, then there is a function $g : A\to A$ such that $g(g(a))=f(a)$ for all $a \in A$ is and only if $n = m$.

1996 Israel National Olympiad, 2
Tags: polynomial , algebra , functional , functional equation
Find all polynomials $P(x)$ satisfying $P(x+1)-2P(x)+P(x-1)= x$ for all $x$

2002 Singapore Team Selection Test, 3
Tags: functional equation , algebra , functional
Find all functions $f : [0,\infty) \to [0,\infty)$ such that $f(f(x)) +f(x) = 12x$, for all $x \ge 0$.

2004 Estonia National Olympiad, 4
Tags: functional equation , functional , algebra
Find all functions $f$ which are defined on all non-negative real numbers, take nonnegative real values only, and satisfy the condition $x \cdot f(y) + y\cdot f(x) = f(x) \cdot f(y) \cdot (f(x) + f(y))$ for all non-negative real numbers $x, y$.

2014 Grand Duchy of Lithuania, 1
Tags: functional equation , algebra , functional
Determine all functions $f : R \to R$ such that $f(xy + f(x)) = xf(y) + f(x)$ holds for any $x, y \in R$.

2014 Contests, 1b
Tags: algebra , function , functional equation , functional
Find all functions $f : R-\{0\} \to R$ which satisfy $(1 + y)f(x) - (1 + x)f(y) = yf(x/y) - xf(y/x)$ for all real $x, y \ne 0$, and which take the values $f(1) = 32$ and $f(-1) = -4$.