Olympiad problems

1994 Austrian-Polish Competition, 1

A function $f: R \to R$ satisfies the conditions: $f (x + 19) \le f (x) + 19$ and $f (x + 94) \ge f (x) + 94$ for all $x \in R$. Prove that $f (x + 1) = f (x) + 1$ for all $x \in R$.

1992 Austrian-Polish Competition, 6

Tags: divisor , functional , functional equation , algebra , number theory

A function $f: Z \to Z$ has the following properties: $f (92 + x) = f (92 - x)$ $f (19 \cdot 92 + x) = f (19 \cdot 92 - x)$ ($19 \cdot 92 = 1748$) $f (1992 + x) = f (1992 - x)$ for all integers $x$. Can all positive divisors of $92$ occur as values of f?

2004 Thailand Mathematical Olympiad, 8

Tags: functional equation , algebra , functional

Let $f : R \to R$ satisfy $f(x + f(y)) = 2x + 4y + 2547$ for all reals $x, y$. Compute $f(0)$.

2007 Estonia Team Selection Test, 5

Tags: continuous , functional , algebra , functional equation

Find all continuous functions $f: R \to R$ such that for all reals $x$ and $y$, $f(x+f(y)) = y+f(x+1)$.

1994 Bulgaria National Olympiad, 2

Tags: functional equation , algebra , functional

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

1992 Romania Team Selection Test, 1

Tags: functional , function , algebra

Suppose that$ f : N \to N$ is an increasing function such that $f(f(n)) = 3n$ for all $n$. Find $f(1992)$.

1993 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: functional , functional equation , number theory

Find all functions $f$ defined on the integers greater than or equal to $1$ that satisfy: (a) for all $n,f(n)$ is a positive integer. (b) $f(n + m) =f(n)f(m)$ for all $m$ and $n$. (c) There exists $n_0$ such that $f(f(n_0)) = [f(n_0)]^2$ .

2008 Indonesia TST, 2

Tags: functional , algebra , functional equation

Find all functions $f : R \to R$ that satisfies the condition $$f(f(x - y)) = f(x)f(y) - f(x) + f(y) - xy$$ for all real numbers $x, y$.

2013 Saudi Arabia GMO TST, 1

Tags: algebra , functional , functional equation

Find all functions $f : R \to R$ which satisfy $f \left(\frac{\sqrt3}{3} x\right) = \sqrt3 f(x) - \frac{2\sqrt3}{3} x$ and $f(x)f(y) = f(xy) + f \left(\frac{x}{y} \right) $ for all $x, y \in R$, with $y \ne 0$

2003 Cuba MO, 4

Tags: functional , number theory

Let $f : N \to N$ such that $f(p) = 1$ for all p prime and $f(ab) =bf(a) + af(b)$ for all $a, b \in N$. Prove that if $n = p^{a_1}_1 p^{a_1}_2... p^{a_1}_k$ is the canonical distribution of $n$ and $p_i$ does not divide $a_i$ ($i = 1, 2, ..., k$) then $\frac{n}{gcd(n,f(n))}$ is square free (not divisible by a square greater than $1$).

2020 Greece Team Selection Test, 1

Tags: functional equation , functional , algebra

Let $R_+=(0,+\infty)$. Find all functions $f: R_+ \to R_+$ such that $f(xf(y))+f(yf(z))+f(zf(x))=xy+yz+zx$, for all $x,y,z \in R_+$. by Athanasios Kontogeorgis (aka socrates)

2008 Postal Coaching, 4

Tags: algebra , functional , functional equation

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

2001 Czech And Slovak Olympiad IIIA, 1

Tags: polynomial , functional , functional equation , algebra

Determine all polynomials $P$ such that for every real number $x$, $P(x)^2 +P(-x) = P(x^2)+P(x)$

1994 Austrian-Polish Competition, 8

Tags: functional equation , functional , algebra

Given real numbers $a, b$, find all functions $f: R \to R$ satisfying $f(x,y) = af (x,z) + bf(y,z)$ for all $x,y,z \in R$.

VMEO III 2006 Shortlist, A1

Tags: functional , functional equation , algebra

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

2005 Cuba MO, 4

Tags: functional , algebra

Determine all functions $f : R_+ \to R$ such that:$$f(x)f(y) = f(xy) + \frac{1}{x} + \frac{1}{y}$$ for all $x, y$ positive reals.

2021 Swedish Mathematical Competition, 4

Tags: functional , functional inequality , algebra

Give examples of a function $f : R \to R$ that satisfies $0 < f(x) < f(x + f(x)) <\sqrt2 x$, for all positive $x$, and show that there is no function $f : R \to R$ that satisfies $x < f(x + f(x)) <\sqrt2 f(x)$, for all positive $x$.

1991 Poland - Second Round, 4

Tags: functional , functional equation , algebra

Find all monotone functions $ f: \mathbb{R} \to \mathbb{R} $ satisfying the equation $$ f(4x)-f(3x) = 2x \ \ \text{ for } \ \ x \in \mathbb{R}.$$

2015 Costa Rica - Final Round, F2

Tags: functional , algebra , functional equation

Find all functions $f: R \to R$ such that $f (f (x) f (y)) = xy$ and there is no $k \in R -\{0,1,-1\}$ such that $f (k) = k$.

1997 Rioplatense Mathematical Olympiad, Level 3, 6

Tags: number theory , functional , functional equation , algebra

Let $N$ be the set of positive integers. Determine if there is a function $f: N\to N$ such that $f(f(n))=2n$, for all $n$ belongs to $N$.

2015 Ukraine Team Selection Test, 8

Tags: algebra , functional , functional equation

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

MathLinks Contest 2nd, 4.1

Tags: functional , algebra

The real polynomial $f \in R[X]$ has an odd degree and it is given that $f$ is co-prime with $g(x) = x^2 - x - 1$ and $$f(x^2 - 1) = f(x)f(-x), \forall x \in R.$$ Prove that $f$ has at least two complex non-real roots.

2014 Costa Rica - Final Round, 5

Tags: functional equation , recurrence relation , functional , algebra

Let $f : N\to N$ such that $$f(1) = 0\,\, , \,\,f(3n) = 2f(n) + 2\,\, , \,\,f(3n-1) = 2f(n) + 1\,\, , \,\,f(3n-2) = 2f(n).$$ Determine the smallest value of $n$ so that $f (n) = 2014.$

2017 Thailand Mathematical Olympiad, 9

Tags: functional , functional equation , algebra

Determine all functions $f$ on the set of positive rational numbers such that $f(xf(x) + f(y)) = f(x)^2 + y$ for all positive rational numbers $x, y$.

2010 Saudi Arabia BMO TST, 3

Tags: functional , algebra , functional equation

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