Olympiad problems

2017 QEDMO 15th, 8

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 $f (-42)$?

2000 Slovenia National Olympiad, Problem 2

Tags: functional equation , fe

Find all functions $f:\mathbb R\to\mathbb R$ such that for all $x,y\in\mathbb R$, $$f(x-f(y))=1-x-y.$$

2021-IMOC, A4

Tags: function , functional equation , algebra

Find all functions f : R-->R such that f (f (x) + y^2) = x −1 + (y + 1)f (y) holds for all real numbers x, y

1992 IMO Longlists, 24

Tags: function , algebra , functional equation

[i](a)[/i] Show that there exists exactly one function $ f : \mathbb Q^+ \to \mathbb Q^+$ satisfying the following conditions: [b](i)[/b] if $0 < q < \frac 12$, then $f(q)=1+f \left( \frac{q}{1-2q} \right);$ [b](ii)[/b] if $1 < q \leq 2$, then $f(q) = 1+f(q + 1);$ [b](iii)[/b] $f(q)f(1/q) = 1$ for all $q \in \mathbb Q^+.$ [i](b)[/i] Find the smallest rational number $q \in \mathbb Q^+$ such that $f(q) = \frac{19}{92}.$

2010 Germany Team Selection Test, 3

Tags: algebra , function , trigonometry , functional equation

Find all functions $f: \mathbb{R} \to \mathbb{R}$ such that \[f(x)f(y) = (x+y+1)^2 \cdot f \left( \frac{xy-1}{x+y+1} \right)\] $\forall x,y \in \mathbb{R}$ with $x+y+1 \neq 0$ and $f(x) > 1$ $\forall x > 0.$

2017 Brazil Team Selection Test, 4

Tags: algebra , functional equation

Find all functions $f:(0,\infty)\rightarrow (0,\infty)$ such that for any $x,y\in (0,\infty)$, $$xf(x^2)f(f(y)) + f(yf(x)) = f(xy) \left(f(f(x^2)) + f(f(y^2))\right).$$

2012 Grand Duchy of Lithuania, 1

Tags: functional , functional equation , algebra

Find all functions $g : R \to R$, for which there exists a strictly increasing function $f : R \to R$ such that $f(x + y) = f(x)g(y) + f(y)$.

2012 Singapore MO Open, 2

Tags: function , algebra , functional equation

Find all functions $f:\mathbb{R}\to\mathbb{R}$ so that $(x+y)(f(x)-f(y))=(x-y)f(x+y)$ for all $x,y$ that belongs to $\mathbb{R}$.

2021 Brazil Team Selection Test, 4

Tags: functional equation , nonnegative integers , number theory

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

1988 IMO Longlists, 49

Tags: function , functional equation , algebra

Let $ f(n)$ be a function defined on the set of all positive integers and having its values in the same set. Suppose that $ f(f(n) \plus{} f(m)) \equal{} m \plus{} n$ for all positive integers $ n,m.$ Find the possible value for $ f(1988).$

2001 Estonia Team Selection Test, 3

Tags: functional , functional equation , algebra

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 $y$.

1996 IMO Shortlist, 8

Tags: function , number theory , functional equation , algebra

Let $ \mathbb{N}_0$ denote the set of nonnegative integers. Find all functions $ f$ from $ \mathbb{N}_0$ to itself such that \[ f(m \plus{} f(n)) \equal{} f(f(m)) \plus{} f(n)\qquad \text{for all} \; m, n \in \mathbb{N}_0. \]

2016 Azerbaijan IMO TST First Round, 4

Tags: function , functional equation , algebra

Find the solution of the functional equation $P(x)+P(1-x)=1$ with power $2015$ P.S: $P(y)=y^{2015}$ is also a function with power $2015$

2021 Taiwan TST Round 3, 1

Tags: nonnegative integers , functional equation , number theory

Determine all functions $f$ defined on the set of all positive integers and taking non-negative integer values, satisfying the three conditions: [list] [*] $(i)$ $f(n) \neq 0$ for at least one $n$; [*] $(ii)$ $f(x y)=f(x)+f(y)$ for every positive integers $x$ and $y$; [*] $(iii)$ there are infinitely many positive integers $n$ such that $f(k)=f(n-k)$ for all $k<n$. [/list]

2024 Brazil National Olympiad, 5

Tags: functional equation , real number , algebra

Let $ \mathbb{R} $ be the set of real numbers. Determine all functions $ f: \mathbb{R} \to \mathbb{R} $ such that, for any real numbers $ x $ and $ y $, \[ f(x^2 y - y) = f(x)^2 f(y) + f(x)^2 - 1. \]

1994 Abels Math Contest (Norwegian MO), 3b

Tags: algebra , functional , functional equation

Prove that there is no function $f : Z \to Z$ such that $f(f(x)) = x+1$ for all $x$.

2011 Belarus Team Selection Test, 3

Tags: functional , algebra , functional equation

Find all functions $f:R\to R$ such that for all real $x,y$ with $y\ne 0$ $$f(x-f(x/y))=xf(1-f(1/y))$$ and a) $f(1-f(1))\ne 0$ b) $ f(1-f(1))= 0$ S. Kuzmich, I.Voronovich

2022 Auckland Mathematical Olympiad, 9

Tags: functional , algebra , functional equation , number theory

Does there exist a function $f(n)$, which maps the set of natural numbers into itself and such that for each natural number $n > 1$ the following equation is satisfied $$f(n) = f(f(n - 1)) + f(f(n + 1))?$$

2016 Brazil Team Selection Test, 1

Tags: functional equation , functional equation in n , algebra

Determine all functions $f$ from the set of non-negative integers to itself such that $f(a + b) = f(a) + f(b) + f(c) + f(d)$, whenever $a, b, c, d$, are non-negative integers satisfying $2ab = c^2 + d^2$.

2022 Nordic, 1

Tags: algebra , functional equation

Find all functions $f:\mathbb{R}\to\mathbb{R}$ such that $f(f(x)f(1-x))=f(x)$ and $f(f(x))=1-f(x)$, for all real $x$.

1988 IMO Longlists, 39

Tags: algebra , function , functional equation , polynomial

[b]i.)[/b] Let $g(x) = x^5 + x^4 + x^3 + x^2 + x + 1.$ What is the remainder when the polynomial $g(x^{12}$ is divided by the polynomial $g(x)$? [b]ii.)[/b] If $k$ is a positive number and $f$ is a function such that, for every positive number $x, f(x^2 + 1 )^{\sqrt{x}} = k.$ Find the value of \[ f( \frac{9 +y^2}{y^2})^{\sqrt{ \frac{12}{y} }} \] for every positive number $y.$ [b]iii.)[/b] The function $f$ satisfies the functional equation $f(x) + f(y) = f(x+y) - x \cdot y - 1$ for every pair $x,y$ of real numbers. If $f(1) = 1,$ then find the numbers of integers $n,$ for which $f(n) = n.$

PEN K Problems, 32

Tags: function , functional equation

Find all functions $f: \mathbb{Z}^{2}\to \mathbb{R}^{+}$ such that for all $i, j \in \mathbb{Z}$: \[f(i,j)=\frac{f(i+1, j)+f(i,j+1)+f(i-1,j)+f(i,j-1)}{4}.\]

2016 APMO, 1

Tags: reflection , functional equation , right triangle , circumcircle , geometry

We say that a triangle $ABC$ is great if the following holds: for any point $D$ on the side $BC$, if $P$ and $Q$ are the feet of the perpendiculars from $D$ to the lines $AB$ and $AC$, respectively, then the reflection of $D$ in the line $PQ$ lies on the circumcircle of the triangle $ABC$. Prove that triangle $ABC$ is great if and only if $\angle A = 90^{\circ}$ and $AB = AC$. [i]Senior Problems Committee of the Australian Mathematical Olympiad Committee[/i]

1987 Traian Lălescu, 1.2

Tags: function , functional equation , algebra

Let $ A $ be a subset of $ \mathbb{R} $ and let be a function $ f:A\longrightarrow\mathbb{R} $ satisfying $$ f(x)-f(y)=(y-x)f(x)f(y),\quad\forall x,y\in A. $$ [b]a)[/b] Show that if $ A=\mathbb{R}, $ then $ f=0. $ [b]b)[/b] Find $ f, $ provided that $ A=\mathbb{R}\setminus\{1\} . $

2010 Brazil National Olympiad, 1

Tags: algebra , function , vieta , functional equation

Find all functions $f$ from the reals into the reals such that \[ f(ab) = f(a+b) \] for all irrational $a, b$.