Olympiad problems

2019 Auckland Mathematical Olympiad, 2

Prove that among any $43$ positive integers there exist two $a$ and $b$ such that $a^2 - b^2$ is divisible by $100$.

2019 Paraguay Mathematical Olympiad, 4

Tags: divisible , divide , number theory

Find the largest positive integer $n$ such that $n^2 + 10$ is divisible by $n-5$.

2013 Saudi Arabia GMO TST, 3

Tags: divide , divisible , number theory

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

2018 Stanford Mathematics Tournament, 1

Tags: number theory , divide , divisible

Prove that if $7$ divides $a^2 + b^2 + 1$, then $7$ does not divide $a + b$.

2019 New Zealand MO, 1

Tags: number theory , divisible

How many positive integers less than $2019$ are divisible by either $18$ or $21$, but not both?

1957 Moscow Mathematical Olympiad, 347

Tags: cubic , algebra , polynomial , divisible

a) Let $ax^3 + bx^2 + cx + d$ be divisible by $5$ for given positive integers $a, b, c, d$ and any integer $x$. Prove that $a, b, c$ and $d$ are all divisible by $5$. b) Let $ax^4 + bx^3 + cx^2 + dx + e$ be divisible by $7$ for given positive integers $a, b, c, d, e$ and all integers $x$. Prove that $a, b, c, d$ and $e$ are all divisible by $7$.

2015 Saudi Arabia IMO TST, 1

Tags: number theory , divide , divisible , gcd

Let $a, b,c,d$ be positive integers such that $ac+bd$ is divisible by $a^2 +b^2$. Prove that $gcd(c^2 + d^2, a^2 + b^2) > 1$. Trần Nam Dũng

1998 Singapore Senior Math Olympiad, 1

Tags: integer , divisible , divide , number theory

Prove that $1998! \left( 1+ \frac12 + \frac13 +...+\frac{1}{1998}\right)$ is an integer divisible by $1999$.

1953 Moscow Mathematical Olympiad, 236

Tags: number theory , divisible

Prove that $n^2 + 8n + 15$ is not divisible by $n + 4$ for any positive integer $n$.

2003 Bosnia and Herzegovina Junior BMO TST, 3

Tags: divide , divisible , number theory

Let $a, b, c$ be integers such that the number $a^2 +b^2 +c^2$ is divisible by $6$ and the number $ab + bc + ca$ is divisible by $3$. Prove that the number $a^3 + b^3 + c^3$ is divisible by $6$.

2008 Postal Coaching, 3

Tags: number theory , divide , divisible

Prove that for each natural number $m \ge 2$, there is a natural number $n$ such that $3^m$ divides $n^3 + 17$ but $3^{m+1}$ does not divide it.

2016 Grand Duchy of Lithuania, 4

Tags: divide , divisible , number theory

Determine all positive integers $n$ such that $7^n -1$ is divisible by $6^n -1$.

2018 Estonia Team Selection Test, 12

Tags: polynomial , divisible

We call the polynomial $P (x)$ simple if the coefficient of each of its members belongs to the set $\{-1, 0, 1\}$. Let $n$ be a positive integer, $n> 1$. Find the smallest possible number of terms with a non-zero coefficient in a simple $n$-th degree polynomial with all values at integer places are divisible by $n$.

1989 Tournament Of Towns, (210) 4

Tags: number theory , divide , divisible

Prove that if $k$ is an even positive integer then it is possible to write the integers from $1$ to $k-1$ in such an order that the sum of no set of successive numbers is divisible by $k$ .

1995 Czech And Slovak Olympiad IIIA, 4

Tags: digit , number theory , divisible

Do there exist $10000$ ten-digit numbers divisible by $7$, all of which can be obtained from one another by a reordering of their digits?

1953 Moscow Mathematical Olympiad, 250

Tags: number theory , combinatorics , divisible

Somebody wrote $1953$ digits on a circle. The $1953$-digit number obtained by reading these figures clockwise, beginning at a certain point, is divisible by $27$. Prove that if one begins reading the figures at any other place, one gets another $1953$-digit number also divisible by $27$.

2014 Switzerland - Final Round, 5

Tags: number theory , divide , divisible

Let $a_1, a_2, ...$ a sequence of integers such that for every $n \in N$ we have: $$\sum_{d | n} a_d = 2^n.$$ Show for every $n \in N$ that $n$ divides $a_n$. Remark: For $n = 6$ the equation is $a_1 + a_2 + a_3 + a_6 = 2^6.$

1995 Singapore MO Open, 4

Tags: number theory , divisible , divide

Let $a, b$ and $c$ be positive integers such that $1 < a < b < c$. Suppose that $(ab-l)(bc-1 )(ca-1)$ is divisible by $abc$. Find the values of $a, b$ and $c$. Justify your answer.

2013 Balkan MO Shortlist, N4

Tags: number theory , divisible , sum of powers , divisor

Let $p$ be a prime number greater than $3$. Prove that the sum $1^{p+2} + 2^{p+2} + ...+ (p-1)^{p+2}$ is divisible by $p^2$.

1972 Spain Mathematical Olympiad, 7

Tags: number theory , multiple , divide , divisible

Prove that for every positive integer $n$, the number $$A_n = 5^n + 2 \cdot 3^{n-1} + 1$$ is a multiple of $8$.

1975 Chisinau City MO, 102

Tags: combinatorics , number theory , divisible , divide , digit

Two people write a $2k$-digit number, using only the numbers $1, 2, 3, 4$ and $5$. The first number on the left is written by the first of them, the second - the second, the third - the first, etc. Can the second one achieve this so that the resulting number is divisible by $9$, if the first seeks to interfere with it? Consider the cases $k = 10$ and $k = 15$.

1940 Moscow Mathematical Olympiad, 070

Tags: divisible , number theory

How many positive integers $x$ less than $10 000$ are there such that $2^x - x^2$ is divisible by $7$ ?

2018 Costa Rica - Final Round, 5

Tags: number theory , divisible , divide

Let $a$ and $ b$ be even numbers, such that $M = (a + b)^2-ab$ is a multiple of $5$. Consider the following statements: I) The unit digits of $a^3$ and $b^3$ are different. II) $M$ is divisible by $100$. Please indicate which of the above statements are true with certainty.

2019 Durer Math Competition Finals, 5

Tags: divisible , divide , number theory

We want to write down as many distinct positive integers as possible, so that no two numbers on our list have a sum or a difference divisible by $2019$. At most how many integers can appear on such a list?

2017 Saudi Arabia IMO TST, 3

Tags: number theory , divide , divisible

Prove that there are infinitely many positive integers $n$ such that $n$ divides $2017^{2017^n-1} - 1$ but n does not divide $2017^n - 1$.