Olympiad problems

PEN K Problems, 6

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f^{(19)}(n)+97f(n)=98n+232.\]

PEN K Problems, 15

Tags: function , Functional Equations

Find all functions $f: \mathbb{Z}\to \mathbb{Z}$ such that for all $m,n\in \mathbb{Z}$: \[f(m+f(n))=f(m)-n.\]

PEN K Problems, 22

Tags: function , induction , Functional Equations

Find all functions $f:\mathbb{Q}^{+} \to \mathbb{Q}^{+}$ such that for all $x\in \mathbb{Q}^+$: [list] [*] $f(x+1)=f(x)+1$, [*] $f(x^2)=f(x)^2$. [/list]

1996 IMO Shortlist, 5

Tags: function , number theory , Functional Equations , IMO Shortlist

Show that there exists a bijective function $ f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $ m,n\in \mathbb{N}_{0}$: \[ f(3mn \plus{} m \plus{} n) \equal{} 4f(m)f(n) \plus{} f(m) \plus{} f(n). \]

2015 IFYM, Sozopol, 1

Tags: algebra , Functional Equations

Determine all functions $f:\mathbb{Z} \rightarrow \mathbb{Z}$ which satisfy the following equations: a) $f(f(n))=4n+3$ $\forall$ $n \in \mathbb{Z}$; b) $f(f(n)-n)=2n+3$ $\forall$ $n \in \mathbb{Z}$.

PEN K Problems, 20

Tags: function , induction , Functional Equations

Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y \in \mathbb{Q}$: \[f(x+y)+f(x-y)=2(f(x)+f(y)).\]

2005 VJIMC, Problem 3

Tags: Polynomials , fe , functional equation , Functional Equations , algebra

Find all reals $\lambda$ for which there is a nonzero polynomial $P$ with real coefficients such that $$\frac{P(1)+P(3)+P(5)+\ldots+P(2n-1)}n=\lambda P(n)$$for all positive integers $n$, and find all such polynomials for $\lambda=2$.

PEN K Problems, 3

Tags: function , induction , Functional Equations

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(n+1) > f(f(n)).\]

2023 Dutch BxMO TST, 2

Tags: Functional Equations , Functional inequality , algebra

Find all functions $f : \mathbb R \to \mathbb R$ for which \[f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),\] for all real numbers $a, b, c$ and $d$. Note that there is only one occurrence of $f$ on the right hand side!

PEN K Problems, 23

Tags: function , Functional Equations

Let ${\mathbb Q}^{+}$ be the set of positive rational numbers. Construct a function $f:{\mathbb Q}^{+}\rightarrow{\mathbb Q}^{+}$ such that \[f(xf(y)) = \frac{f(x)}{y}\] for all $x, y \in{\mathbb Q}^{+}$.

2017 Pakistan TST, Problem 3

Tags: functional equation , Functional Equations , algebra proposed , algebra , algebraic identities

Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for all distinct $x,y,z$ $f(x)^2-f(y)f(z)=f(x^y)f(y)f(z)[f(y^z)-f(z^x)]$

2024 Kyiv City MO Round 1, Problem 5

Tags: functional equation , number theory , Functional Equations

Find all functions $f : \mathbb{N} \to \mathbb{N}$ such that for any positive integers $m, n$ the number $$(f(m))^2+ 2mf(n) + f(n^2)$$ is the square of an integer. [i]Proposed by Fedir Yudin[/i]

2004 Germany Team Selection Test, 1

Tags: function , Functional Equations , Closed Formula , Germany , Team Selection Test , TST , 2004

A function $f$ satisfies the equation \[f\left(x\right)+f\left(1-\frac{1}{x}\right)=1+x\] for every real number $x$ except for $x = 0$ and $x = 1$. Find a closed formula for $f$.

PEN K Problems, 11

Tags: function , algebra , functional equation , Functional Equations

Find all functions $f: \mathbb{N}_{0}\to \mathbb{N}_{0}$ such that for all $m,n\in \mathbb{N}_{0}$: \[mf(n)+nf(m)=(m+n)f(m^{2}+n^{2}).\]

2004 Spain Mathematical Olympiad, Problem 3

Tags: algebra , Functional Equations

Represent for $\mathbb {Z}$ the set of all integers. Find all of the functions ${f:}$ $ \mathbb{Z} \rightarrow \mathbb{Z}$ such that for any ${x,y}$ integers, they satisfy: ${f(x + f(y)) = f(x) - y.}$

2016 IFYM, Sozopol, 3

Tags: algebra , Functional Equations , functions

Let $f: \mathbb{R}^2\rightarrow \mathbb{R}$ be a function for which for arbitrary $x,y,z\in \mathbb{R}$ we have that $f(x,y)+f(y,z)+f(z,x)=0$. Prove that there exist function $g:\mathbb{R}\rightarrow \mathbb{R}$ for which: $f(x,y)=g(x)-g(y),\, \forall x,y\in \mathbb{R}$.

PEN K Problems, 5

Tags: function , arithmetic sequence , Functional Equations , pen

Find all functions $f: \mathbb{N}\to \mathbb{N}$ such that for all $n\in \mathbb{N}$: \[f(f(m)+f(n))=m+n.\]

2020 Final Mathematical Cup, 1

Tags: algebra , functional equation , Functional Equations , Functional equation in R

Find all such functions $f:\mathbb{R} \to \mathbb{R}$ that for any real $x,y$ the following equation is true. $$f(f(x)+y)+1=f(x^2+y)+2f(x)+2y$$

2020 Jozsef Wildt International Math Competition, W44

Tags: Functional Equations , fe

We consider a function $f:\mathbb R\to\mathbb R$ such that $$f(x+y)+f(xy-1)=f(x)f(y)+f(x)+f(y)+1$$ for each $x,y\in\mathbb R$. i) Calculate $f(0)$ and $f(-1)$. ii) Prove that $f$ is an even function. iii) Give an example of such a function. iv) Find all monotone functions with the above property. [i]Proposed by Mihály Bencze and Marius Drăgan[/i]

1994 Iran MO (2nd round), 3

Tags: function , induction , Functional Equations

Find all functions $ f: \mathbb{Z}\setminus\{0\}\to \mathbb{Q}$ such that for all $ x,y \in \mathbb{Z}\setminus\{0\}$: \[ f \left( \frac{x+y}{3}\right) =\frac{f(x)+f(y)}{2}, \; \; x, y \in \mathbb{Z}\setminus\{0\}\]

2011 USA TSTST, 1

Tags: function , algebra , Functional Equations , median , Tstst , USA , Hi

Find all real-valued functions $f$ defined on pairs of real numbers, having the following property: for all real numbers $a, b, c$, the median of $f(a,b), f(b,c), f(c,a)$ equals the median of $a, b, c$. (The [i]median[/i] of three real numbers, not necessarily distinct, is the number that is in the middle when the three numbers are arranged in nondecreasing order.)

2023 Dutch BxMO TST, 2

Tags: Functional Equations , Functional inequality , algebra

Find all functions $f : \mathbb R \to \mathbb R$ for which \[f(a - b) f(c - d) + f(a - d) f(b - c) \leq (a - c) f(b - d),\] for all real numbers $a, b, c$ and $d$. Note that there is only one occurrence of $f$ on the right hand side!

2024 Thailand Mathematical Olympiad, 3

Tags: Functional Equations , function , algebra

Let $c$ be a positive real number. Find all functions $f:\mathbb{R}^+\to\mathbb{R}^+$ that satisfy $$x^2f(xf(y))f(x)f(y)=c$$ for all positive reals $x$ and $y$.

PEN K Problems, 33

Tags: function , induction , strong induction , algebra , functional equation , Functional Equations

Find all functions $f: \mathbb{Q}\to \mathbb{Q}$ such that for all $x,y,z \in \mathbb{Q}$: \[f(x+y+z)+f(x-y)+f(y-z)+f(z-x)=3f(x)+3f(y)+3f(z).\]

VMEO II 2005, 11

Tags: algebra , polynomial , Functional Equations , functional

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 $f(P(x_0)) = Q(f(x_0))$.