Olympiad problems

2009 China Team Selection Test, 1

Tags: logarithm , ceiling function , function , combinatorics , ratio , floor function

Let $ \alpha,\beta$ be real numbers satisfying $ 1 < \alpha < \beta.$ Find the greatest positive integer $ r$ having the following property: each of positive integers is colored by one of $ r$ colors arbitrarily, there always exist two integers $ x,y$ having the same color such that $ \alpha\le \frac {x}{y}\le\beta.$

2006 District Olympiad, 4

Tags: real analysis , function

We say that a function $f: \mathbb R \to \mathbb R$ has the property $(P)$ if, for any real numbers $x$, \[ \sup_{t\leq x} f(x) = x. \] a) Give an example of a function with property $(P)$ which has a discontinuity in every real point. b) Prove that if $f$ is continuous and satisfies $(P)$ then $f(x) = x$, for all $x\in \mathbb R$.

2013 India Regional Mathematical Olympiad, 6

Tags: modular arithmetic , function , combinatorics

For a natural number $n$, let $T(n)$ denote the number of ways we can place $n$ objects of weights $1,2,\cdots, n$ on a balance such that the sum of the weights in each pan is the same. Prove that $T(100) > T(99)$.

2018 International Zhautykov Olympiad, 5

Tags: inequalities , algebra , function , floor function

Find all real numbers $a$ such that there exist $f:\mathbb{R} \to \mathbb{R}$ with $$f(x-f(y))=f(x)+a[y]$$ for all $x,y\in \mathbb{R}$

2019 South East Mathematical Olympiad, 3

Tags: number theory , function

Let $f:\mathbb{N}\rightarrow \mathbb{N}$ be a function such that $f(ab)$ divides $\max \{f(a),b\}$ for any positive integers $a,b$. Must there exist infinitely many positive integers $k$ such that $f(k)=1$?

2014 ELMO Shortlist, 8

Tags: function , greatest common divisor , number theory

Let $\mathbb N$ denote the set of positive integers. Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that: (i) The greatest common divisor of the sequence $f(1), f(2), \dots$ is $1$. (ii) For all sufficiently large integers $n$, we have $f(n) \neq 1$ and \[ f(a)^n \mid f(a+b)^{a^{n-1}} - f(b)^{a^{n-1}} \] for all positive integers $a$ and $b$. [i]Proposed by Yang Liu[/i]

2012 Harvard-MIT Mathematics Tournament, 3

Tags: hmmt , function

Given points $a$ and $b$ in the plane, let $a\oplus b$ be the unique point $c$ such that $abc$ is an equilateral triangle with $a,b,c$ in the clockwise orientation. Solve $(x\oplus (0,0))\oplus(1,1)=(1,-1)$ for $x$.

2005 Bulgaria Team Selection Test, 4

Tags: inequalities , quadratic , function

Let $a_{i}$ and $b_{i}$, where $i \in \{1,2, \dots, 2005 \}$, be real numbers such that the inequality $(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})$ holds for all $x \in \mathbb{R}$ and all $i \in \{1,2, \dots, 2005 \}$. Find the maximum possible number of positive numbers amongst $a_{i}$ and $b_{i}$, $i \in \{1,2, \dots, 2005 \}$.

1999 Brazil Team Selection Test, Problem 4

Tags: logarithm , triangle inequality , ring theory , function , number theory , induction , inequalities

Let Q+ and Z denote the set of positive rationals and the set of inte- gers, respectively. Find all functions f : Q+ → Z satisfying the following conditions: (i) f(1999) = 1; (ii) f(ab) = f(a) + f(b) for all a, b ∈ Q+; (iii) f(a + b) ≥ min{f(a), f(b)} for all a, b ∈ Q+.

2005 Belarusian National Olympiad, 6

Tags: function , number theory

$f(n+f(n))=f(n)$ for every $n \in \mathbb{N}$. a)Prove, that if $f(n)$ is finite, then $f$ is periodic. b) Give example nonperiodic function. PS. $0 \not \in \mathbb{N}$

2006 Tuymaada Olympiad, 4

Tags: function , algebra , induction

Find all functions $f: (0,\infty)\rightarrow(0,\infty)$ with the following properties: $f(x+1)=f(x)+1$ and $f\left(\frac{1}{f(x)}\right)=\frac{1}{x}$. [i]Proposed by P. Volkmann[/i]

2014 India IMO Training Camp, 3

Tags: combinatorics , additive combinatorics , sequence , function

Let $r$ be a positive integer, and let $a_0 , a_1 , \cdots $ be an infinite sequence of real numbers. Assume that for all nonnegative integers $m$ and $s$ there exists a positive integer $n \in [m+1, m+r]$ such that \[ a_m + a_{m+1} +\cdots +a_{m+s} = a_n + a_{n+1} +\cdots +a_{n+s} \] Prove that the sequence is periodic, i.e. there exists some $p \ge 1 $ such that $a_{n+p} =a_n $ for all $n \ge 0$.

2007 Putnam, 3

Tags: induction , logarithm , function , calculus , integration , floor function

Let $ x_0 \equal{} 1$ and for $ n\ge0,$ let $ x_{n \plus{} 1} \equal{} 3x_n \plus{} \left\lfloor x_n\sqrt {5}\right\rfloor.$ In particular, $ x_1 \equal{} 5,\ x_2 \equal{} 26,\ x_3 \equal{} 136,\ x_4 \equal{} 712.$ Find a closed-form expression for $ x_{2007}.$ ($ \lfloor a\rfloor$ means the largest integer $ \le a.$)

2014 ELMO Shortlist, 5

Tags: combinatorics , function

Let $n$ be a positive integer. For any $k$, denote by $a_k$ the number of permutations of $\{1,2,\dots,n\}$ with exactly $k$ disjoint cycles. (For example, if $n=3$ then $a_2=3$ since $(1)(23)$, $(2)(31)$, $(3)(12)$ are the only such permutations.) Evaluate \[ a_n n^n + a_{n-1} n^{n-1} + \dots + a_1 n. \][i]Proposed by Sammy Luo[/i]

1983 National High School Mathematics League, 2

Tags: function

Function $f(x)$ is defined on $[0,1]$, $f(0)=f(1)$. For any $x_1,x_2\in [0,1], |f(x_1)-f(x_2)|<|x_1-x_2|(x_1\neq x_2)$. Prove that $|f(x_1)-f(x_2)|<\frac{1}{2}$.

2015 USA TSTST, 5

Tags: relatively prime , number theory , analytic number theory , function

Let $\varphi(n)$ denote the number of positive integers less than $n$ that are relatively prime to $n$. Prove that there exists a positive integer $m$ for which the equation $\varphi(n)=m$ has at least $2015$ solutions in $n$. [i]Proposed by Iurie Boreico[/i]

Taiwan TST 2015 Round 1, 2

Tags: function , functional equation , algebra

Given a positive integer $n \geq 3$. Find all $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$ such that for any $n$ positive reals $a_1,...,a_n$, the following condition is always satisfied: $\sum_{i=1}^{n}(a_i-a_{i+1})f(a_i+a_{i+1}) = 0$ where $a_{n+1} = a_1$.

2003 Romania National Olympiad, 4

Tags: function , combinatorial geometry

Let $ P$ be a plane. Prove that there exists no function $ f: P\rightarrow P$ such that for every convex quadrilateral $ ABCD$, the points $ f(A),f(B),f(C),f(D)$ are the vertices of a concave quadrilateral. [i]Dinu Şerbănescu[/i]

2000 Baltic Way, 10

Tags: combinatorics , function

Two positive integers are written on the blackboard. Initially, one of them is $2000$ and the other is smaller than $2000$. If the arithmetic mean $ m$ of the two numbers on the blackboard is an integer, the following operation is allowed: one of the two numbers is erased and replaced by $ m$. Prove that this operation cannot be performed more than ten times. Give an example where the operation is performed ten times.

2019 India PRMO, 12

Tags: natural number , function

A natural number $k > 1$ is called [i]good[/i] if there exist natural numbers $$a_1 < a_2 < \cdots < a_k$$ such that $$\dfrac{1}{\sqrt{a_1}} + \dfrac{1}{\sqrt{a_2}} + \cdots + \dfrac{1}{\sqrt{a_k}} = 1$$. Let $f(n)$ be the sum of the first $n$ [i][good[/i] numbers, $n \geq$ 1. Find the sum of all values of $n$ for which $f(n+5)/f(n)$ is an integer.

2016 USAMTS Problems, 4:

Tags: function

Find all functions $f(x)$ from nonnegative reals to nonnegative reals such that $f(f(x))=x^4$ and $f(x)\leq Cx^2$ for some constant $C$.

2009 Postal Coaching, 3

Tags: function , sum , combination , recurrence relation

Let $N_0$ denote the set of nonnegative integers and $Z$ the set of all integers. Let a function $f : N_0 \times Z \to Z$ satisfy the conditions (i) $f(0, 0) = 1$, $f(0, 1) = 1$ (ii) for all $k, k \ne 0, k \ne 1$, $f(0, k) = 0$ and (iii) for all $n \ge 1$ and $k, f(n, k) = f(n -1, k) + f(n- 1, k - 2n)$. Find the value of $$\sum_{k=0}^{2009 \choose 2} f(2008, k)$$

1999 Ukraine Team Selection Test, 9

Tags: algebra , function , functional equation

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

1990 IMO Longlists, 53

Tags: function , algebra

Find the real solution(s) for the system of equations \[\begin{cases}x^3+y^3 &=1\\x^5+y^5 &=1\end{cases}\]

2011 ELMO Shortlist, 1

Tags: function , combinatorics , modular arithmetic

Let $S$ be a finite set, and let $F$ be a family of subsets of $S$ such that a) If $A\subseteq S$, then $A\in F$ if and only if $S\setminus A\notin F$; b) If $A\subseteq B\subseteq S$ and $B\in F$, then $A\in F$. Determine if there must exist a function $f:S\to\mathbb{R}$ such that for every $A\subseteq S$, $A\in F$ if and only if \[\sum_{s\in A}f(s)<\sum_{s\in S\setminus A}f(s).\] [i]Evan O'Dorney.[/i]