Olympiad problems

2009 Cuba MO, 4

Tags: algebra , functional

Determine all the functions $f : R \to R$ such that: $$x + f(xf(y)) = f(y) + yf(x)$$ for all $x, y \in R$.

2010 Saudi Arabia BMO TST, 3

Tags: functional inequality , functional , inequalities , algebra

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 $$f(x_1) + f(x_2) +(x_3) \ge \frac{3a^2}{a+2} f(x_1+ x_2 + x_3), \forall x_1 ,x_2,x_3 \in R$$.

2010 Saudi Arabia IMO TST, 2

Tags: functional , functional equation , algebra

Find all functions $f,g : N \to N$ such that for all $m ,n \in N$ the following relation holds: $$f(m ) - f(n) = (m - n)(g(m) + g(n))$$. Note: $N = \{0,1,2,...\}$

2017 Czech And Slovak Olympiad III A, 3

Tags: functional , functional equation , algebra

Find all functions $f: R \to R$ such that for all real numbers $x, y$ holds $f(y - xy) = f(x)y + (x - 1)^2 f(y)$

2013 HMIC, 2

Tags: functional

Find all functions $f : R \to R$ such that, for all real numbers $x, y,$ $$(x - y)(f(x) - f(y)) = f(x - f(y))f(f(x) - y).$$

2020 Estonia Team Selection Test, 3

Tags: functional equation , functional , algebra

Find all functions $f :R \to R$ such that for all real numbers $x$ and $y$ $$f(x^3+y^3)=f(x^3)+3x^3f(x)f(y)+3f(x)(f(y))^2+y^6f(y)$$

1990 Greece National Olympiad, 4

Tags: function , functional , algebra

Find all functions $f: \mathbb{R}^+\to\mathbb{R}$ such that $f(x+y)=f(x^2)+f(y^2)$ for any $x,y \in\mathbb{R}^+$

2022 Peru MO (ONEM), 3

Tags: functional , functional equation , algebra

Let $R$ be the set of real numbers and $f : R \to R$ be a function that satisfies: $$f(xy) + y + f(x + f(y)) = (y + 1)f(x),$$ for all real numbers $x, y$. a) Determine the value of $f(0)$. b) Prove that $f(x) = 2-x$ for every real number $x$.

2008 Cuba MO, 4

Tags: functional , algebra

Determine all functions $f : R \to R$ such that $f(xy + f(x)) =xf(y) + f(x)$ for all real numbers $x, y$.

2008 Mathcenter Contest, 2

Tags: functional equation , functional , algebra

Find all the functions $f:\mathbb{R}\to\mathbb{R}$ which satisfy the functional equation $$f(xy^2)+f(x^2y)=y^2f(x)+x^2f(y)$$ for every $x,y\in\mathbb{R}$ and $f(2008) =f(-2008)$ [i](nooonuii)[/i]

1982 Austrian-Polish Competition, 6

Tags: functional , functional equation , algebra

An integer $a$ is given. Find all real-valued functions $f (x)$ defined on integers $x \ge a$, satisfying the equation $f (x+y) = f (x) f (y)$ for all $x,y \ge a$ with $x + y \ge a$.

2022 Azerbaijan BMO TST, A2

Tags: functional , functional equation , algebra

Find all functions $f : R \to R$ with $f (x + yf(x + y))= y^2 + f(x)f(y)$ for all $x, y \in R$.

1972 Dutch Mathematical Olympiad, 2

Tags: functional inequality , functional equation , functional , algebra , inequalities

Prove that there exists exactly one function $ƒ$ which is defined for all $x \in R$, and for which holds: $\bullet$ $x \le y \Rightarrow f(x) \le f(y)$, for all $x, y \in R$, and $\bullet$ $f(f(x)) = x$, for all $x \in R$.

1963 Swedish Mathematical Competition., 4

Tags: analysis , functional equation , algebra , functional

Given the real number $k$, find all differentiable real-valued functions $f(x)$ defined on the reals such that $f(x+y) = f(x) + f(y) + f(kxy)$ for all $x, y$.

1993 Austrian-Polish Competition, 8

Tags: polynomial , functional , algebra , functional equation

Determine all real polynomials $P(z)$ for which there exists a unique real polynomial $Q(x)$ satisfying the conditions $Q(0)= 0$, $x + Q(y + P(x))= y + Q(x + P(y))$ for all $x,y \in R$.

2000 Belarus Team Selection Test, 1.3

Tags: algebra , functional equation , functional

Does there exist a function $f : N\to N$ such that $f ( f (n-1)) = f (n+1)- f (n)$ for all $n \ge 2$?

2020 Dutch BxMO TST, 3

Tags: functional , algebra , functional equation

Find all functions $f: R \to R$ that satisfy $$f (x^2y) + 2f (y^2) =(x^2 + f (y)) \cdot f (y)$$ for all $x, y \in R$

2011 Denmark MO - Mohr Contest, 4

Tags: functional , algebra

A function $f$ is given by $f(x) = x^2 - 2x$ . Prove that there exists a number a which satisfies $f(f(a)) = a$ without satisfying $f(a) = a$ .

2018 Costa Rica - Final Round, 4

Tags: algebra , functional , functional inequality

Determine if there exists a function f: $N^*\to N^*$ that satisfies that for all $n \in N^*$, $$10^{f (n)} <10n + 1 <10^{f (n) +1}.$$ Justify your answer. Note: $N^*$ denotes the set of positive integers.

2018 Saudi Arabia IMO TST, 1

Tags: number theory , divide , divisible , functional

Find all functions $f : Z^+ \to Z^+$ satisfying $f (1) = 2, f (2) \ne 4$, and max $\{f (m) + f (n), m + n\} |$ min $\{2m + 2n, f (m + n) + 1\}$ for all $m, n \in Z^+$.

I Soros Olympiad 1994-95 (Rus + Ukr), 10.1

Tags: functional , number theory , algebra

The function $f: Z \to Z$ satisfies the following conditions: 1) $f(f(n))=n$ for all integers $n$ 2) $f(f(n+2)+2) = n$ for all integers $n$ 3) $f(0)=1$. Find the value of $f(1995)$ and $f(-1994)$.

VMEO III 2006, 10.3

Tags: algebra , functional

Find all functions $f : R \to R$ that satisfy $f(x^2 + f(y) - y) = (f(x))^2$ for all $x,y \in R$.

2010 Thailand Mathematical Olympiad, 5

Tags: functional equation , functional , algebra

Determine all functions $f : R \times R \to R$ satisfying the equation $f(x - t, y) + f(x + t, y) + f(x, y - t) + f(x, y + t) = 2010$ for all real numbers $x, y$ and for all nonzero $t$

2002 Singapore MO Open, 4

Tags: functional equation , algebra , functional

Find all real-valued functions $f : Q \to R$ defined on the set of all rational numbers $Q$ satisfying the conditions $f(x + y) = f(x) + f(y) + 2xy$ for all $x, y$ in $Q$ and $f(1) = 2002.$ Justify your answers.

1998 Estonia National Olympiad, 3

Tags: functional equation , periodic , algebra , functional , function

A function $f$ satisfies the conditions $f (x) \ne 0$ and $f (x+2) = f (x-1) f (x+5)$ for all real x. Show that $f (x+18) = f (x)$ for any real $x$.