Olympiad problems

2012 Brazil Team Selection Test, 2

Let $a_1, a_2,..., a_n$ be positive integers and $a$ positive integer greater than $1$ which is a multiple of the product $a_1a_2...a_n$. Prove that $a^{n+1} + a - 1$ is not divisible by $(a + a_1 -1)(a + a_2 - 1) ... (a + a_n -1)$.

1984 All Soviet Union Mathematical Olympiad, 371

Tags: number theory , product , sum , divisible , combinatorics

a) The product of $n$ integers equals $n$, and their sum is zero. Prove that $n$ is divisible by $4$. b) Let $n$ is divisible by $4$. Prove that there exist $n$ integers such, that their product equals $n$, and their sum is zero.

2019 Ukraine Team Selection Test, 2

Tags: algebra , polynomial , divisible

Polynomial $p(x)$ with real coefficients, which is different from the constant, has the following property: [i] for any naturals $n$ and $k$ the $\frac{p(n+1)p(n+2)...p(n+k)}{p(1)p(2)...p(k)}$ is an integer.[/i] Prove that this polynomial is divisible by $x$.

2001 Chile National Olympiad, 4

Tags: number theory , multiple , divisible

Given a natural number $n$, prove that $2^{2n}-1$ is a multiple of $3$.

1995 Bulgaria National Olympiad, 1

Tags: number theory , divide , divisible

Find the number of integers $n > 1$ which divide $a^{25} - a$ for every integer $a$.

2012 NZMOC Camp Selection Problems, 2

Tags: number theory , consecutive , divide , divisible , sum

Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.

2020 Tournament Of Towns, 1

Tags: digit , number theory , divisible

Does there exist a positive integer that is divisible by $2020$ and has equal numbers of digits $0, 1, 2, . . . , 9$ ? Mikhail Evdokimov

2013 IMAC Arhimede, 1

Tags: number theory , multiple , divisible

Show that in any set of three distinct integers there are two of them, say $a$ and $b$ such that the number $a^5b^3-a^3b^5$ is a multiple of $10$.

2019 Bundeswettbewerb Mathematik, 2

Tags: digit , divisible , divisibility , number theory

The lettes $A,C,F,H,L$ and $S$ represent six not necessarily distinct decimal digits so that $S \ne 0$ and $F \ne 0$. We form the two six-digit numbers $SCHLAF$ and $FLACHS$. Show that the difference of these two numbers is divisible by $271$ if and only if $C=L$ and $H=A$. [i]Remark:[/i] The words "Schlaf" and "Flachs" are German for "sleep" and "flax".

2000 All-Russian Olympiad Regional Round, 9.2

Tags: divisible , divide , number theory

Are there different mutually prime natural numbers $a$, $b$ and $c$, greater than $1$, such that $2a + 1$ is divisible by $b$, $2b + 1$ is divisible by $c$ and $2c + 1$ is divisible by $a$?

2012 Dutch Mathematical Olympiad, 1

Tags: diophantine equation , divisible , number theory

Let $a, b, c$, and $d$ be four distinct integers. Prove that $(a-b)(a-c)(a-d)(b-c)(b-d)(c-d)$ is divisible by $12$.

2019 Ecuador NMO (OMEC), 3

Tags: divisible , divide , power of 2 , number theory

For every positive integer $n$, find the maximum power of $2$ that divides the number $$1 + 2019 + 2019^2 + 2019^3 +.. + 2019^{n-1}.$$

2022 Czech-Polish-Slovak Junior Match, 5

Tags: number theory , divisible

An integer $n\ge1$ is [i]good [/i] if the following property is satisfied: If a positive integer is divisible by each of the nine numbers $n + 1, n + 2, ..., n + 9$, this is also divisible by $n + 10$. How many good integers are $n\ge 1$?

2011 Danube Mathematical Competition, 3

Tags: number theory , prime number , divisible

Determine all positive integer numbers $n$ satisfying the following condition: the sum of the squares of any $n$ prime numbers greater than $3$ is divisible by $n$.

2006 Singapore Senior Math Olympiad, 1

Tags: mutliple , divide , divisible , prime , number theory

Let $a, d$ be integers such that $a,a + d, a+ 2d$ are all prime numbers larger than $3$. Prove that $d$ is a multiple of $6$.

2015 Puerto Rico Team Selection Test, 8

Tags: multiple , divisible , divide , number theory

Consider the $2015$ integers $n$, from $ 1$ to $2015$. Determine for how many values of $n$ it is verified that the number $n^3 + 3^n$ is a multiple of $5$.

1999 Tournament Of Towns, 3

Tags: number theory , divide , divisible , sum of squares

Find all pairs $(x, y)$ of integers satisfying the following condition: each of the numbers $x^3 + y$ and $x + y^3$ is divisible by $x^2 + y^2$ . (S Zlobin)

2019 Romanian Master of Mathematics Shortlist, C3

Tags: combinatorics , divisible , permutation

Fix an odd integer $n > 1$. For a permutation $p$ of the set $\{1,2,...,n\}$, let S be the number of pairs of indices $(i, j)$, $1 \le i \le j \le n$, for which $p_i +p_{i+1} +...+p_j$ is divisible by $n$. Determine the maximum possible value of $S$. Croatia

1949-56 Chisinau City MO, 7

Tags: factorial , number theory , divisible , prime , divide

Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.

2013 QEDMO 13th or 12th, 2

Tags: binomial , number theory , divisible , divide

Let $p$ be a prime number and $n, k$ and $q$ natural numbers, where $q\le \frac{n -1}{p-1}$ should be. Let $M$ be the set of all integers $m$ from $0$ to $n$, for which $m-k$ is divisible by $p$. Show that $$\sum_{m \in M} (-1) ^m {n \choose m}$$ is divisible by $p^q$.

2018 Saudi Arabia GMO TST, 2

Tags: number theory , digit , divisible

Two positive integers $m$ and $n$ are called [i]similar [/i] if one of them can be obtained from the other one by swapping two digits (note that a $0$-digit cannot be swapped with the leading digit). Find the greatest integer $N$ such that N is divisible by $13$ and any number similar to $N$ is not divisible by $13$.