Olympiad problems

2021 Saudi Arabia Training Tests, 39

Determine if there exists pairwise distinct positive integers $a_1$, $a_2$,$ ...$, $a_{101}$, $b_1$, $b_2$,$ ...$, $b_{101}$ satisfying the following property: for each non-empty subset $S$ of $\{1, 2, ..., 101\}$ the sum $\sum_{i \in S} a_i$ divides $100! + \sum_{i \in S} b_i$.

1989 Tournament Of Towns, (205) 3

Tags: number theory , divisible , divide , digit

What digit must be put in place of the "$?$" in the number $888...88?999...99$ (where the $8$ and $9$ are each written $50$ times) in order that the resulting number is divisible by $7$? (M . I. Gusarov)

2015 Switzerland - Final Round, 9

Tags: divide , divisible , number theory

Let$ p$ be an odd prime number. Determine the number of tuples $(a_1, a_2, . . . , a_p)$ of natural numbers with the following properties: 1) $1 \le ai \le p$ for all $i = 1, . . . , p$. 2) $a_1 + a_2 + · · · + a_p$ is not divisible by $p$. 3) $a_1a_2 + a_2a_3 + . . . +a_{p-1}a_p + a_pa_1$ is divisible by $p$.

1999 Abels Math Contest (Norwegian MO), 2b

Tags: divide , divisible , number theory

If $a,b,c$ are positive integers such that $b | a^3, c | b^3$ and $a | c^3$ , prove that $abc | (a+b+c)^{13}$

1955 Kurschak Competition, 2

Tags: divide , digit , number theory

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

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?

1974 Dutch Mathematical Olympiad, 2

Tags: divide , divisible , number theory

$n>2$ numbers, $ x_1, x_2, ..., x_n$ are odd . Prove that $4$ divides $$ x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 -n.$$

2018 District Olympiad, 2

Tags: divide , number theory

Find the pairs of integers $(a, b)$ such that $a^2 + 2b^2 + 2a +1$ is a divisor of $2ab$.

2012 China Northern MO, 3

Tags: divide , number theory

Suppose $S= \{x|x=a^2+ab+b^2,a,b \in Z\}$. Prove that: (1) If $m \in S$, $3|m$ , then $\frac{m}{3} \in S$ (2) If $m,n \in S$ , then $mn\in S$.

2012 Brazil Team Selection Test, 4

Tags: divide , divisible , number theory

Let $p$ be a prime greater than $2$. Prove that there is a prime $q < p$ such that $q^{p-1} - 1$ is not divisible by $p^2$

1954 Moscow Mathematical Olympiad, 267

Tags: polynomial , divisible , algebra , divide

Prove that if $$x^4_0+ a_1x^3_0+ a_2x^2_0+ a_3x_0 + a_4 = 0 \ \ and \ \ 4x^3_0+ 3a_1x^2_0+ 2a_2x_0 + a_3 = 0,$$ then $x^4 + a_1x^3 + a_2x^2 + a_3x + a_4 $ is a mutliple of $(x - x_0)^2$.

1999 Singapore MO Open, 2

Tags: number theory , digit , divisible , divide

Call a natural number $n$ a [i]magic [/i] number if the number obtained by putting $n$ on the right of any natural number is divisible by $n$. Find the number of magic numbers less than $500$. Justify your answer

2012 Austria Beginners' Competition, 1

Tags: number theory , divide , divisible

Let $a, b, c$ and $d$ be four integers such that $7a + 8b = 14c + 28d$. Prove that the product $a\cdot b$ is always divisible by $14$.

2013 India PRMO, 13

Tags: maximum , divide , number theory

To each element of the set $S = \{1,2,... ,1000\}$ a colour is assigned. Suppose that for any two elements $a, b$ of $S$, if $15$ divides $a + b$ then they are both assigned the same colour. What is the maximum possible number of distinct colours used?

2007 Bosnia and Herzegovina Junior BMO TST, 2

Tags: divisible , divide , number theory

Find all pairs of relatively prime numbers ($x, y$) such that $x^2(x + y)$ is divisible by $y^2(y - x)^2$. .

2016 Costa Rica - Final Round, N2

Tags: divide , divisible , number theory

Let $x, y, z$ be positive integers and $p$ a prime such that $x <y <z <p$. Also $x^3, y^3, z^3$ leave the same remainder when divided by $p$. Prove that $x + y + z$ divides $x^2 + y^2 + z^2$.

2022 Switzerland - Final Round, 2

Tags: remainder , divide , number theory

Let $n$ be a positive integer. Prove that the numbers $$1^1, 3^3, 5^5, ..., (2n-1)^{2n-1}$$ all give different remainders when divided by $2^n$.

1998 Tournament Of Towns, 1

Tags: divisible , divide , number theory

Do there exist $10$ positive integers such that each of them is divisible by none of the other numbers but the square of each of these numbers is divisible by each of the other numbers? (Folklore)

1999 Singapore Team Selection Test, 3

Tags: prime number , divide , number theory

Let $f(x) = x^{1998} - x^{199}+x^{19}+ 1$. Prove that there is an infinite set of prime numbers, each dividing at least one of the integers $f(1), f(2), f(3), f(4), ...$

1939 Eotvos Mathematical Competition, 2

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

Determine the highest power of $2$ that divides $2^n!$.

1970 Poland - Second Round, 3

Tags: divide , divisible , number theory

Prove the theorem: There is no natural number $ n > 1 $ such that the number $ 2^n - 1 $ is divisible by $ n $.

2002 Kazakhstan National Olympiad, 7

Tags: prime divisor , divisor , divide , number theory

Prove that for any integers $ n> m> 0 $ the number $ 2 ^n-1 $ has a prime divisor not dividing $ 2 ^m-1 $.

2003 Switzerland Team Selection Test, 9

Tags: combinatorics , divide , number theory

Given integers $0 < a_1 < a_2 <... < a_{101} < 5050$, prove that one can always choose for different numbers $a_k,a_l,a_m,a_n$ such that $5050 | a_k +a_l -a_m -a_n$

2007 Switzerland - Final Round, 9

Tags: integer , divide , divisible , number theory

Find all pairs $(a, b)$ of natural numbers such that $$\frac{a^3 + 1}{2ab^2 + 1}$$ is an integer.

2015 Gulf Math Olympiad, 1

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

a) Suppose that $n$ is an odd integer. Prove that $k(n-k)$ is divisible by $2$ for all positive integers $k$. b) Find an integer $k$ such that $k(100-k)$ is not divisible by $11$. c) Suppose that $p$ is an odd prime, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by $p$. d) Suppose that $p,q$ are two different odd primes, and $n$ is an integer. Prove that there is an integer $k$ such that $k(n-k)$ is not divisible by any of $p,q$.