Olympiad problems

2015 IFYM, Sozopol, 4

Let $k$ be a natural number. For each natural number $n$ we define $f_k (n)$ to be the least number, greater than $kn$, for which $nf_k (n)$ is a perfect square. Prove that $f_k (n)$ is injective.

2019 Polish MO Finals, 4

Tags: number theory , injective function , function , algebra

Let $n, k, \ell$ be positive integers and $\sigma : \lbrace 1, 2, \ldots, n \rbrace \rightarrow \lbrace 1, 2, \ldots, n \rbrace$ an injection such that $\sigma(x)-x\in \lbrace k, -\ell \rbrace$ for all $x\in \lbrace 1, 2, \ldots, n \rbrace$. Prove that $k+\ell|n$.

1989 IMO Longlists, 71

Tags: injective function , combinatorial inequality , counting , combinatorics

A permutation $ \{x_1, x_2, \ldots, x_{2n}\}$ of the set $ \{1,2, \ldots, 2n\}$ where $ n$ is a positive integer, is said to have property $ T$ if $ |x_i \minus{} x_{i \plus{} 1}| \equal{} n$ for at least one $ i$ in $ \{1,2, \ldots, 2n \minus{} 1\}.$ Show that, for each $ n$, there are more permutations with property $ T$ than without.

2016 Mathematical Talent Reward Programme, SAQ: P 5

Tags: injective function , surjective function , function

Let $\mathbb{N}$ be the set of all positive integers. $f,g:\mathbb{N} \to \mathbb{N}$ be funtions such that $f$ is onto and $g$ is one-one and $f(n)\geq g(n)$ for all positive integers $n$. Prove that $f=g$.

2019 Middle European Mathematical Olympiad, 1

Tags: algebra , functional equation , injective function

Find all functions $f:\mathbb{R} \to \mathbb{R}$ such that for any two real numbers $x,y$ holds $$f(xf(y)+2y)=f(xy)+xf(y)+f(f(y)).$$ [i]Proposed by Patrik Bak, Slovakia[/i]

2025 Romania National Olympiad, 3

Tags: function , injective function , number theory

Define the functions $g_k \colon \mathbb{Z} \to \mathbb{Z}$, $g_k(x) = x^k$, where $k$ is a positive integer. Find the set $M_k$ of positive integers $n$ for which there exist injective functions $f_1,f_2, \dots ,f_n \colon \mathbb{Z} \to \mathbb{Z}$ such that $g_k=f_1\cdot f_2 \cdot \ldots \cdot f_n$. (Here, $\cdot$ denotes component-wise function multiplication)

1989 IMO Shortlist, 23

Tags: combinatorics , counting , injective function , combinatorial inequality

A permutation $ \{x_1, x_2, \ldots, x_{2n}\}$ of the set $ \{1,2, \ldots, 2n\}$ where $ n$ is a positive integer, is said to have property $ T$ if $ |x_i \minus{} x_{i \plus{} 1}| \equal{} n$ for at least one $ i$ in $ \{1,2, \ldots, 2n \minus{} 1\}.$ Show that, for each $ n$, there are more permutations with property $ T$ than without.

2019 Canadian Mathematical Olympiad Qualification, 1

Tags: functional equation , function , injective function , injective , algebra

A function $f$ is called injective if when $f(n) = f(m)$, then $n = m$. Suppose that $f$ is injective and $\frac{1}{f(n)}+\frac{1}{f(m)}=\frac{4}{f(n) + f(m)}$. Prove $m = n$

1989 IMO, 6

Tags: combinatorics , counting , injective function , combinatorial inequality

A permutation $ \{x_1, x_2, \ldots, x_{2n}\}$ of the set $ \{1,2, \ldots, 2n\}$ where $ n$ is a positive integer, is said to have property $ T$ if $ |x_i \minus{} x_{i \plus{} 1}| \equal{} n$ for at least one $ i$ in $ \{1,2, \ldots, 2n \minus{} 1\}.$ Show that, for each $ n$, there are more permutations with property $ T$ than without.

1997 Nordic, 4

Tags: algebra , functional equation , injective function

Let f be a function defined in the set $\{0, 1, 2,...\}$ of non-negative integers, satisfying $f(2x) = 2f(x), f(4x+1) = 4f(x) + 3$, and $f(4x-1) = 2f(2x - 1) -1$. Show that $f $ is an injection, i.e. if $f(x) = f(y)$, then $x = y$.