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Stage 9 · Number theory

10 problems · IMO P2/P5; hard shortlist · mathsolympiadprep.com

The answer key prints on its own page at the end.

Let $n$ be a positive integer. Let $s: \mathbb{N} \to \{1, \dots, n\}$ be a function such that $n$ divides $m - s(m)$ for all positive integers $m$ . Let $a_0, a_1, a_2, \dots$ be a sequence such that $a_0 = 0$ and
$a_k = a_{k-1} + s(k) \text{ for all } k \ge 1.$
Find all $n$ for which this sequence contains all the residues modulo $(n+1)^2$ .

India — Team Selection Test

Number theory Solution and answer checking →
Let $n$ be a positive integer. In this problem, we consider labellings of the squares of a chessboard of size $n \times n$ with the natural numbers from $1$ to $n^2$ such that every number is used exactly once. Given such a labelling, we say a positive integer is a rook product if it is the product of the labels of $n$ squares which have the property that if you place a rook on each of them, no two rooks will attack each other.
(Two rooks are attacking each other, if and only if they are in the same row or column.)
a. Let $n = 8$ . Determine whether there exists a labelling of an $8 \times 8$ chessboard such that the following condition is fulfilled: The difference of any two rook products is always divisible by $65$ .
b. Let $n = 10$ . Determine whether there exists a labelling of a $10 \times 10$ chessboard such that the following condition is fulfilled: The difference of any two rook products is always divisible by $101$ .

Baltic Way Shortlist 2023

Number theory Solution and answer checking →
Let $a, b$ be integers, and let $P(x) = a x^{3} + b x$ . For any positive integer $n$ we say that the pair $(a, b)$ is $n$ -good if $n \mid P(m) - P(k)$ implies $n \mid m - k$ for all integers $m, k$ . We say that $(a, b)$ is very good if $(a, b)$ is $n$ -good for infinitely many positive integers $n$ .
a. Find a pair $(a, b)$ which is $51$ -good, but not very good.
b. Show that all $2010$ -good pairs are very good.
(Turkey)

IMO Shortlist

Number theory Solution and answer checking →
Let $S \subset \{1, \dots, n\}$ be a nonempty set, where $n$ is a positive integer. We denote by $s$ the greatest common divisor of the elements of the set $S$ . We assume that $s \neq 1$ and let $d$ be its smallest divisor greater than $1$ . Let $T \subset \{1, \dots, n\}$ be a set such that $S \subset T$ and $|T| \geq 1 + \lfloor \frac{n}{d} \rfloor$ . Prove that the greatest common divisor of the elements in $T$ is $1$ .
Let $n$ ( $n \geq 1$ ) be a positive integer and $U = \{1, \dots, n\}$ . Let $S$ be a nonempty subset of $U$ and let $d$ ( $d \neq 1$ ) be the smallest common divisor of all elements of the set $S$ . Find the smallest positive integer $k$ such that for any subset $T$ of $U$ , consisting of $k$ elements, with $S \subset T$ , the greatest common divisor of all elements of $T$ is equal to $1$ .

BMO Shortlist 2019

Number theory Solution and answer checking →
Let $n$ be a positive integer not divisible by the cube of a prime. Given an integer-coefficient polynomial $f(x)$ , define its signature modulo $n$ to be the (ordered) sequence $f(1), \dots, f(n)$ modulo $n$ . Of the $n^n$ such $n$ -term sequences of integers modulo $n$ , how many are the signature of some polynomial $f(x)$ ?

United States — Team Selection Test

Number theory Solution and answer checking →
Does there exist an irrational number $x$ such that there are at most finitely many positive integers $n$ satisfying
$\{kx\} \geq \frac{1}{n+1}$
for every $k \in \{1, \dots, n\}$ ?
Note: Here, for a positive real number $y$ , $\{y\}$ denotes the fractional part of $y$ .

China-TST-B

Number theory Solution and answer checking →
Find all positive integers $m$ such that there exists an infinite set $A$ of positive integers satisfying: for any $m$ distinct elements $a_1, a_2, \dots, a_m$ in $A$ , both $a_1+a_2+\dots+a_m$ and $a_1a_2\cdots a_m$ are square-free.
*Note: A positive integer $n$ is called square-free if it is not divisible by the square of any prime number.*

International Mathematical Olympiad China National Team Selection Test 2025

Number theory Solution and answer checking →
a) Find all positive integers $k$ with the property $T(20)$ , where a positive integer $k$ has the property $T(m)$ if for any positive integer $a$ , there exists a positive integer $n$ such that
$1^k + 2^k + \dots + n^k \equiv a \pmod{m}.$
b) Find the smallest positive integer $k$ with the property $T(20^{15})$ .

IMO Team Selection Tests 2015

Number theory Solution and answer checking →
Given an integer $n > 1$ and an integer $a$ that is coprime with $n$ . There is a country consisting of $n$ islands $D_1, D_2, \dots, D_n$ . For any two different islands $D_i$ and $D_j$ , there is a one-way ferry from $D_i$ to $D_j$ if and only if $ij \equiv ia \pmod{n}$ . A tourist hopes to visit as many islands as possible. He can first fly to any island he chooses to start the tour, and afterwards can only use the one-way ferry to tour freely between islands in this country. Find the maximum possible number of different islands that the tourist can visit.

Chinese IMO National Team Selection Test 2023

Number theory Solution and answer checking →
Let $n$ be a positive integer. We say that a polynomial $P$ with integer coefficients is $n$ -good if there exists a polynomial $Q$ of degree 2 with integer coefficients such that $Q(k)(P(k)+Q(k))$ is never divisible by $n$ for any integer $k$ .
Determine all integers $n$ such that every polynomial with integer coefficients is an $n$ -good polynomial.

IMO Shortlist

Number theory Solution and answer checking →

Answer key — Stage 9 · Number theory

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