Olympiad problems

2008 Canada National Olympiad, 4

Determine all functions $ f$ defined on the natural numbers that take values among the natural numbers for which \[ (f(n))^p \equiv n\quad {\rm mod}\; f(p) \] for all $ n \in {\bf N}$ and all prime numbers $ p$.

1995 APMO, 5

Tags: function , modular arithmetic , algebra

Find the minimum positive integer $k$ such that there exists a function $f$ from the set $\Bbb{Z}$ of all integers to $\{1, 2, \ldots k\}$ with the property that $f(x) \neq f(y)$ whenever $|x-y| \in \{5, 7, 12\}$.

2019 India IMO Training Camp, P2

Tags: function , algebra

Determine all functions $f:(0,\infty)\to\mathbb{R}$ satisfying $$\left(x+\frac{1}{x}\right)f(y)=f(xy)+f\left(\frac{y}{x}\right)$$ for all $x,y>0$.

2023 Israel Olympic Revenge, P3

Tags: functional equation , algebra , function

Find all (weakly) increasing $f\colon \mathbb{R}\to \mathbb{R}$ for which \[f(f(x)+y)=f(f(y)+x)\] holds for all $x, y\in \mathbb{R}$.

2010 Harvard-MIT Mathematics Tournament, 5

Tags: calculus , algebra , polynomial , function

Let the functions $f(\alpha,x)$ and $g(\alpha)$ be defined as \[f(\alpha,x)=\dfrac{(\frac{x}{2})^\alpha}{x-1}\qquad\qquad\qquad g(\alpha)=\,\dfrac{d^4f}{dx^4}|_{x=2}\] Then $g(\alpha)$ is a polynomial is $\alpha$. Find the leading coefficient of $g(\alpha)$.

2019 Danube Mathematical Competition, 2

Tags: function , find all functions , algebra , functional equation

Find all nondecreasing functions $ f:\mathbb{R}\longrightarrow\mathbb{R} $ that verify the relation $$ f\left( f\left( x^2 \right) +y+f(y) \right) =x^2+2f(y) , $$ for any real numbers $ x,y. $

1977 Miklós Schweitzer, 8

Tags: function , real analysis

Let $ p \geq 1$ be a real number and $ \mathbb{R}_\plus{}\equal{}(0, \infty)$. For which continuous functions $ g : \mathbb{R}_\plus{} \rightarrow \mathbb{R}_\plus{}$ are following functions all convex? \[ M_n(x)\equal{}\left[ \frac{\sum_{i\equal{}1}^n g(\frac{x_i}{x_{i\plus{}1}}) x_{i\plus{}1}^p}{\sum_{i\equal{}1}^n g(\frac{x_i}{x_{i\plus{}1}})} \right ]^\frac 1p ,\] \[ x\equal{}(x_1,\ldots, x_{n\plus{}1}) \in \mathbb{R}_\plus{} ^ {n\plus{}1} , \; n\equal{}1,2,\ldots\] [i]L. Losonczi[/i]

2000 Poland - Second Round, 5

Tags: algebra , number theory , function

Decide whether exists function $f: \mathbb{N} \rightarrow \mathbb{N}$, such that for each $n \in \mathbb{N}$ is $f(f(n) )= 2n$.

2011 N.N. Mihăileanu Individual, 3

Tags: function , real analysis , continuity

Let $ f:\mathbb{R}\longrightarrow\mathbb{R} $ be a function having the property that $$ f(f(x))=f(x)-\frac{1}{4}x +1, $$ for all real numbers $ x. $ [b]a)[/b] Prove that $ f $ is increasing. [b]b)[/b] Show that the equation $ f(x)=ax $ has at least a real solution in $ x, $ for any real number $ a\ge 1. $ [b]c)[/b] Calculate $ \lim_{x\to\infty } \frac{f(x)}{x} $ supposing that it exists, it's finite, and that $ \lim_{x\to\infty } f(f(x))=\infty . $

2021 JBMO TST - Turkey, 5

Tags: function , number theory

$d(n)$ shows the number of positive integer divisors of positive integer $n$. For which positive integers $n$ one cannot find a positive integer $k$ such that $\underbrace{d(\dots d(d}_{k\ \text{times}} (n) \dots )$ is a perfect square.

2012 ELMO Shortlist, 2

Tags: function , relatively prime , number theory

For positive rational $x$, if $x$ is written in the form $p/q$ with $p, q$ positive relatively prime integers, define $f(x)=p+q$. For example, $f(1)=2$. a) Prove that if $f(x)=f(mx/n)$ for rational $x$ and positive integers $m, n$, then $f(x)$ divides $|m-n|$. b) Let $n$ be a positive integer. If all $x$ which satisfy $f(x)=f(2^nx)$ also satisfy $f(x)=2^n-1$, find all possible values of $n$. [i]Anderson Wang.[/i]

2006 Switzerland Team Selection Test, 3

Tags: function , domain , limit , algebra

Find all the functions $f : \mathbb{R} \to \mathbb{R}$ satisfying for all $x,y \in \mathbb{R}$ $f(f(x)-y^2) = f(x)^2 - 2f(x)y^2 + f(f(y))$.

2008 District Round (Round II), 4

Tags: calculus , function , geometry

A semicircle has diameter $AB$ and center $S$,with a point $M$ on the circumference.$U,V$ are the incircles of sectors $ASM$ and $BSM$.Prove that circles $U,V$ can be seperated by a line perpendicular to $AB$.

2011 Mediterranean Mathematics Olympiad, 2

Tags: ratio , inequalities , function , geometric sequence , combinatorics

Let $A$ be a finite set of positive reals, let $B = \{x/y\mid x,y\in A\}$ and let $C = \{xy\mid x,y\in A\}$. Show that $|A|\cdot|B|\le|C|^2$. [i](Proposed by Gerhard Woeginger, Austria)[/i]

1991 Irish Math Olympiad, 3

Tags: function , algebra

Three operations $f,g$ and $h$ are defined on subsets of the natural numbers $\mathbb{N}$ as follows: $f(n)=10n$, if $n$ is a positive integer; $g(n)=10n+4$, if $n$ is a positive integer; $h(n)=\frac{n}{2}$, if $n$ is an [i]even[/i] positive integer. Prove that, starting from $4$, every natural number can be constructed by performing a finite number of operations $f$, $g$ and $h$ in some order. $[$For example: $35=h(f(h(g(h(h(4)))))).]$

2008 Saint Petersburg Mathematical Olympiad, 1

Tags: quadratic , geometry , geometric transformation , function , algebra

Replacing any of the coefficients of quadratic trinomial $f(x)=ax^2+bx+c$ with an $1$ will result in a quadratic trinomial with at least one real root. Prove that the resulting trinomial attains a negative value at at least one point. EDIT: Oops I failed, added "with a 1." Also, I am sorry for not knowing these are posted already, however, these weren't posted in the contest lab yet, which made me think they weren't translated yet. Note: fresh translation

2011 Romania National Olympiad, 4

Tags: function , integration , real analysis

Let $ f,F:\mathbb{R}\longrightarrow\mathbb{R} $ be two functions such that $ f $ is nondecreasing, $ F $ admits finite lateral derivates in every point of its domain, $$ \lim_{x\to y^-} f(x)\le\lim_{x\to y^-}\frac{F(x)-F\left( y \right)}{x-y} ,\lim_{x\to y^+} f(x)\ge\lim_{x\to y^+}\frac{F(x)-F\left( y \right)}{x-y} , $$ for all real numbers $ y, $ and $ F(0)=0. $ Prove that $ F(x)=\int_0^x f(t)dt, $ for all real numbers $ x. $

2020 LIMIT Category 2, 1

Tags: function , limit , involution , counting

Find the number of $f:\{1,\ldots, 5\}\to \{1,\ldots, 5\}$ such that $f(f(x))=x$ (A)$26$ (B)$41$ (C)$120$ (D)$60$

2008 All-Russian Olympiad, 4

Tags: geometry , 3d geometry , tetrahedron , sphere , circumcircle , function , analytic geometry

Each face of a tetrahedron can be placed in a circle of radius $ 1$. Show that the tetrahedron can be placed in a sphere of radius $ \frac{3}{2\sqrt2}$.

1993 AMC 12/AHSME, 26

Tags: function , algebra , domain , inequalities , parabola , calculus , conic

Find the largest positive value attained by the function \[ f(x)=\sqrt{8x-x^2}-\sqrt{14x-x^2-48}, \qquad x\ \text{a real number} \] $ \textbf{(A)}\ \sqrt{7}-1 \qquad\textbf{(B)}\ 3 \qquad\textbf{(C)}\ 2\sqrt{3} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{55}-\sqrt{5} $

2010 CIIM, Problem 4

Tags: function , calculus , derivative

Let $f:[0,1] \to [0,1]$ a increasing continuous function, diferentiable in $(0,1)$ and with derivative smaller than 1 in every point. The sequence of sets $A_1,A_2,A_3,\dots$ is define as: $A_1 = f([0,1])$, and for $n \geq 2, A_n = f(A_{n-1}).$ Prove that $\displaystyle \lim_{n\to+\infty} d(A_n) = 0$, where $d(A)$ is the diameter of the set $A$. Note: The diameter of a set $X$ is define as $d(X) = \sup_{x,y\in X} |x-y|.$