Olympiad problems

1955 Kurschak Competition, 2

How many five digit numbers are divisible by $3$ and contain the digit $6$?

2019 Canadian Mathematical Olympiad Qualification, 8

Tags: number theory , divides , inequalities

For $t \ge 2$, define $S(t)$ as the number of times $t$ divides into $t!$. We say that a positive integer $t$ is a [i]peak[/i] if $S(t) > S(u)$ for all values of $u < t$. Prove or disprove the following statement: For every prime $p$, there is an integer $k$ for which $p$ divides $k$ and $k$ is a peak.

1927 Eotvos Mathematical Competition, 1

Tags: number theory , multiple , divides

Let the integers $a, b, c, d$ be relatively prime to $$m = ad - bc.$$ Prove that the pairs of integers $(x,y)$ for which $ax+by$ is a multiple of $m$ are identical with those for which $cx + dy$ is a multiple of $m$.

2001 Estonia National Olympiad, 2

Tags: max , number theory , divides

Find the maximum value of $k$ for which one can choose $k$ integers out of $1,2... ,2n$ so that none of them divides another one.

2000 Tournament Of Towns, 2

Tags: number theory , divides , divisible

Positive integers $a, b, c, d$ satisfy the inequality $ad - bc > 1$. Prove that at least one of the numbers $a, b, c, d$ is not divisible by $ad - bc$. (A Spivak)

1988 Poland - Second Round, 4

Tags: number theory , divides , divisible

Prove that for every natural number $ n $, the number $ n^{2n} - n^{n+2} + n^n - 1 $ is divisible by $ (n - 1 )^3 $.

2003 Chile National Olympiad, 5

Tags: number theory , divides , divisible

Prove that there is a natural number $N$ of the form $11...1100...00$ which is divisible by $2003$. (The natural numbers are: $1,2,3,...$)

2013 Saudi Arabia Pre-TST, 4.2

Tags: number theory , divides , divisible

Let $x, y$ be two integers. Prove that if $2013$ divides $x^{1433} + y^{1433}$ then $2013$ divides $x^7 + y^7$.

2008 China Northern MO, 3

Tags: number theory , Prime factor , divides

Prove that: (1) There are infinitely many positive integers $n$ such that the largest prime factor of $n^2+1$ is less than $n.$ (2) There are infinitely many positive integers $n$ such that $n^2+1$ divides $n!$.

1979 Czech And Slovak Olympiad IIIA, 6

Tags: number theory , divides , prime

Find all natural numbers $n$, $n < 10^7$, for which: If natural number $m$, $1 < m < n$, is not divisible by $n$, then $m$ is prime.

2000 Czech And Slovak Olympiad IIIA, 1

Tags: number theory , Even , divides , divisible

Let $n$ be a natural number. Prove that the number $4 \cdot 3^{2^n}+ 3 \cdot4^{2^n}$ is divisible by $13$ if and only if $n$ is even.

2019 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , coprime , divides

Let $n$ be a positive integer and $A$ a set containing $8n + 1$ positive integers co-prime with $6$ and less than $30n$. Prove that there exist $a, b \in A$ two different numbers such that $a$ divides $b$.

2021 Dutch IMO TST, 3

Tags: number theory , divides , divisible

Prove that for every positive integer $n$ there are positive integers $a$ and $b$ exist with $n | 4a^2 + 9b^2 -1$.

1983 Austrian-Polish Competition, 8

Tags: number theory , recurrence relation , Sequence , divides , factorial , divisible , Product

(a) Prove that $(2^{n+1}-1)!$ is divisible by $ \prod_{i=0}^n (2^{n+1-i}-1)^{2^i }$, for every natural number n (b) Define the sequence ($c_n$) by $c_1=1$ and $c_{n}=\frac{4n-6}{n}c_{n-1}$ for $n\ge 2$. Show that each $c_n$ is an integer.

2006 May Olympiad, 1

Tags: number theory , divides

Determine all pairs of natural numbers $a$ and $b$ such that $\frac{a+1}{b}$ and $\frac{b+1}{a}$ they are natural numbers.

2018 Czech-Polish-Slovak Junior Match, 1

Tags: number theory , divides , divisible

For natural numbers $a, b c$ it holds that $(a + b + c)^2 | ab (a + b) + bc (b + c) + ca(c + a) + 3abc$. Prove that $(a + b + c) |(a - b)^2 + (b - c)^2 + (c - a)^2$

1995 Chile National Olympiad, 1

Tags: number theory , divides , divisible

Let $a,b,c,d$ be integers. Prove that $ 12$ divides $ (a-b) (a-c) (a-d) (b- c) (b-d) (c-d)$.

2022 Saudi Arabia IMO TST, 1

Tags: number theory , divides , number theory with sequences , recurrence relation

Let $(a_n)$ be the integer sequence which is defined by $a_1= 1$ and $$ a_{n+1}=a_n^2 + n \cdot a_n \,\, , \,\, \forall n \ge 1.$$ Let $S$ be the set of all primes $p$ such that there exists an index $i$ such that $p|a_i$. Prove that the set $S$ is an infinite set and it is not equal to the set of all primes.

2013 Saudi Arabia GMO TST, 3

Tags: divides , divisible , number theory

Find the largest integer $k$ such that $k$ divides $n^{55} - n$ for all integer $n$.

2011 Indonesia TST, 4

Tags: number theory , prime , sum of divisors , divides

Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.

2003 Switzerland Team Selection Test, 4

Tags: divides , max , number theory

Find the largest natural number $n$ that divides $a^{25} -a$ for all integers $a$.

2020 Swedish Mathematical Competition, 1

Tags: number theory , divides , divisible

How many of the numbers $1\cdot 2\cdot 3$, $2\cdot 3\cdot 4$,..., $2020 \cdot 2021 \cdot 2022$ are divisible by $2020$?

2002 Singapore Team Selection Test, 3

Tags: number theory , divides

For every positive integer $n$, show that there is a positive integer $k$ such that $2k^2 + 2001k + 3 \equiv 0$ (mod $2^n$).

2015 Saudi Arabia Pre-TST, 3.3

Tags: number theory , divides

Let $(a_n)_{n\ge0}$ be a sequence of positive integers such that $a^2_n$ divides $a_{n-1}a_{n+1}$, for all $n \ge 1$. Prove that if there exists an integer $k \ge 2$ such that $a_k$ and $a_1$ are relatively prime, then $a_1$ divides $a_0$. (Malik Talbi)

1965 German National Olympiad, 2

Tags: number theory , divides

Determine which of the prime numbers $2,3,5,7,11,13,109,151,491$ divide $z =1963^{1965} -1963$.