Olympiad problems

2013 Saudi Arabia IMO TST, 3

For a positive integer $n$, we consider all its divisors (including $1$ and itself). Suppose that $p\%$ of these divisors have their unit digit equal to $3$. (For example $n = 117$, has six divisors, namely $1,3,9,13,39,117$. Two of these divisors namely $3$ and $13$, have unit digits equal to $3$. Hence for $n = 117$, $p =33.33...$). Find, when $n$ is any positive integer, the maximum possible value of $p$.

1982 Bundeswettbewerb Mathematik, 1

Tags: algebra , divisor , decimal representation

Max divided a natural number $p$ by a natural number $q \leq 100$. In the decimal representation of the quotient he calculated, the sequence of digits $1982$ occurs somewhere after the decimal point. Show that Max made a computational mistake.

2016 Dutch Mathematical Olympiad, 3

Tags: divisor , greatest common divisor , gcd , divide , number theory

Find all possible triples $(a, b, c)$ of positive integers with the following properties: • $gcd(a, b) = gcd(a, c) = gcd(b, c) = 1$, • $a$ is a divisor of $a + b + c$, • $b$ is a divisor of $a + b + c$, • $c$ is a divisor of $a + b + c$. (Here $gcd(x,y)$ is the greatest common divisor of $x$ and $y$.)

1955 Moscow Mathematical Olympiad, 316

Tags: polynomial , algebra , divisor , integer polynomial , rational

Prove that if $\frac{p}{q}$ is an irreducible rational number that serves as a root of the polynomial $f(x) = a_0x^n + a_1x^{n-1} + ... + a_n$ with integer coefficients, then $p - kq$ is a divisor of $f(k)$ for any integer $k$.

2015 Latvia Baltic Way TST, 15

Tags: prime divisor , divisor , number theory

Let $w (n)$ denote the number of different prime numbers by which $n$ is divisible. Prove that there are infinitely many natural numbers $n$ such that $w(n) < w(n + 1) < w(n + 2)$.

2020 Dürer Math Competition (First Round), P1

Tags: divisor , number theory

a) Is it possible that the sum of all the positive divisors of two different natural numbers are equal? b) Is it possible that the product of all the positive divisors of two different natural numbers are equal?

2009 Flanders Math Olympiad, 2

Tags: divisor , number theory

A natural number has four natural divisors: $1$, the number itself, and two real divisors. That number plus $9$ is equal to seven times the sum of the true divisors. Determine that number and prove that it is unique.

1974 Bundeswettbewerb Mathematik, 4

Tags: number theory , divisor , odd , remainder

Peter and Paul gamble as follows. For each natural number, successively, they determine its largest odd divisor and compute its remainder when divided by $4$. If this remainder is $1$, then Peter gives Paul a coin; otherwise, Paul gives Peter a coin. After some time they stop playing and balance the accounts. Prove that Paul wins.

2022 China Team Selection Test, 6

Tags: divisibility , divisor , number theory

Given a positive integer $n$, let $D$ be the set of all positive divisors of $n$. The subsets $A,B$ of $D$ satisfies that for any $a \in A$ and $b \in B$, it holds that $a \nmid b$ and $b \nmid a$. Show that \[ \sqrt{|A|}+\sqrt{|B|} \le \sqrt{|D|}. \]

2020 Durer Math Competition Finals, 9

Tags: number theory , divisor

On a piece of paper, we write down all positive integers $n$ such that all proper divisors of $n$ are less than $18$. We know that the sum of all numbers on the paper having exactly one proper divisor is $666$. What is the sum of all numbers on the paper having exactly two proper divisors? We say that $k$ is a [i]proper divisor [/i]of the positive integer $n$ if $k | n$ and $1 < k < n$.

2018 Pan-African Shortlist, N6

Tags: number theory , arithmetic mean , geometric mean , divisor

Prove that there are infinitely many integers $n$ such that both the arithmetic mean of its divisors and the geometric mean of its divisors are integers. (Recall that for $k$ positive real numbers, $a_1, a_2, \dotsc, a_k$, the arithmetic mean is $\frac{a_1 +a_2 +\dotsb +a_k}{k}$, and the geometric mean is $\sqrt[k]{a_1 a_2\dotsb a_k}$.)

2024 Brazil Cono Sur TST, 3

Tags: divisor , sigma function , tau function , number theory , ceiling function

Given a positive integer $n$, define $\tau(n)$ as the number of positive divisors of $n$ and $\sigma(n)$ as the sum of those divisors. For example, $\tau(12) = 6$ and $\sigma(12) = 28$. Find all positive integers $n$ that satisfy: \[ \sigma(n) = \tau(n) \cdot \lceil \sqrt{n} \rceil \]

2024 Iberoamerican, 1

Tags: divisor , gcd and lcm , number theory

For each positive integer $n$, let $d(n)$ be the number of positive integer divisors of $n$. Prove that for all pairs of positive integers $(a,b)$ we have that: \[ d(a)+d(b) \le d(\gcd(a,b))+d(\text{lcm}(a,b)) \] and determine all pairs of positive integers $(a,b)$ where we have equality case.

2016 Saudi Arabia GMO TST, 3

Tags: polynomial , divide , divisor , algebra

Find all polynomials $P,Q \in Z[x]$ such that every positive integer is a divisor of a certain nonzero term of the sequence $(x_n)_{n=0}^{\infty}$ given by the conditions: $x_0 = 2016$, $x_{2n+1} = P(x_{2n})$, $x_{2n+2} = Q(x_{2n+1})$ for all $n \ge 0$

2004 Estonia Team Selection Test, 5

Tags: number theory , lcm , power of 2 , divisor

Find all natural numbers $n$ for which the number of all positive divisors of the number lcm $(1,2,..., n)$ is equal to $2^k$ for some non-negative integer $k$.

2019 Philippine TST, 3

Tags: divisor , bounding , infinite sequence , number theory

Let $a_1, a_2, a_3,\ldots$ be an infinite sequence of positive integers such that $a_2 \ne 2a_1$, and for all positive integers $m$ and $n$, the sum $m + n$ is a divisor of $a_m + a_n$. Prove that there exists an integer $M$ such that for all $n > M$, we have $a_n \ge n^3$.

2020 European Mathematical Cup, 2

Tags: number theory , fibonacci , divisor , sequence

A positive integer $k\geqslant 3$ is called[i] fibby[/i] if there exists a positive integer $n$ and positive integers $d_1 < d_2 < \ldots < d_k$ with the following properties: \\ $\bullet$ $d_{j+2}=d_{j+1}+d_j$ for every $j$ satisfying $1\leqslant j \leqslant k-2$, \\ $\bullet$ $d_1, d_2, \ldots, d_k$ are divisors of $n$, \\ $\bullet$ any other divisor of $n$ is either less than $d_1$ or greater than $d_k$. Find all fibby numbers. \\ \\ [i]Proposed by Ivan Novak.[/i]

2018 Switzerland - Final Round, 7

Tags: divisor , number theory , consecutive

Let $n$ be a natural integer and let $k$ be the number of ways to write $n$ as the sum of one or more consecutive natural integers. Prove that $k$ is equal to the number of odd positive divisors of $n$. Example: $9$ has three positive odd divisors and $9 = 9$, $9 = 4 + 5$, $9 = 2 + 3 + 4$.

2013 India Regional Mathematical Olympiad, 2

Tags: divisor , prime

Determine the smallest prime that does not divide any five-digit number whose digits are in a strictly increasing order.

2018 Danube Mathematical Competition, 2

Tags: divisor , infinitely many , number theory

Prove that there are infinitely many pairs of positive integers $(m, n)$ such that simultaneously $m$ divides $n^2 + 1$ and $n$ divides $m^2 + 1$.

2013 Austria Beginners' Competition, 1

Tags: number theory , divisor

Find all natural numbers $n> 1$ for which the following applies: The sum of the number $n$ and its second largest divisor is $2013$. (R. Henner, Vienna)

1993 Romania Team Selection Test, 2

Tags: algebra , polynomial , divisor

For coprime integers $m > n > 1$ consider the polynomials $f(x) = x^{m+n} -x^{m+1} -x+1$ and $g(x) = x^{m+n} +x^{n+1} -x+1$. If $f$ and $g$ have a common divisor of degree greater than $1$, find this divisor.

1998 IMO Shortlist, 6

Tags: number theory , number theoretic functions , prime factorization , divisor

For any positive integer $n$, let $\tau (n)$ denote the number of its positive divisors (including 1 and itself). Determine all positive integers $m$ for which there exists a positive integer $n$ such that $\frac{\tau (n^{2})}{\tau (n)}=m$.

2011 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: divisor , number theory

If $n$ is a positive integer and $n+1$ is divisible with $24$, prove that sum of all positive divisors of $n$ is divisible with $24$

2016 Dutch BxMO TST, 1

Tags: equation , divisor , odd , number theory

For a positive integer $n$ that is not a power of two, we define $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.