This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 408

2000 Estonia National Olympiad, 2

The first of an infinite triangular spreadsheet the line contains one number, the second line contains two numbers, the third line contains three numbers, and so on. In doing so is in any $k$-th row ($k = 1, 2, 3,...$) in the first and last place the number $k$, each other the number in the table is found, however, than in the previous row the least common of the two numbers above it multiple (the adjacent figure shows the first five rows of this table). We choose any two numbers from the table that are not in their row in the first or last place. Prove that one of the selected numbers is divisible by another. [img]https://cdn.artofproblemsolving.com/attachments/3/7/107d8999d9f04777719a0f1b1df418dbe00023.png[/img]

1982 Polish MO Finals, 5

Tags: divisible , sequence , sum
Integers $x_0,x_1,...,x_{n-1}, x_n = x_0, x_{n+1} = x_1$ satisfy the inequality $(-1)^{x_k} x_{k-1}x_{k+1} >0$ for $k = 1,2,...,n$. Prove that the difference $\sum_{k=0}^{n-1}x_k -\sum_{k=0}^{n-1}|x_k|$ is divisible by $4$.

2000 Bundeswettbewerb Mathematik, 2

A $5$-tuple $(1,1,1,1,2)$ has the property that the sum of any three of them is divisible by the sum of the remaining two. Is there a $5$-tuple with this property whose all terms are distinct?

2021 Austrian MO Regional Competition, 4

Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)

1979 Polish MO Finals, 1

Let be given a set $\{r_1,r_2,...,r_k\}$ of natural numbers that give distinct remainders when divided by a natural number $m$. Prove that if $k > m/2$, then for every integer $n$ there exist indices $i$ and $j$ (not necessarily distinct) such that $r_i +r_j -n$ is divisible by $m$.

2016 Saudi Arabia Pre-TST, 1.4

Let $p$ be a given prime. For each prime $r$, we defind the function as following $F(r) =\frac{(p^{rp} - 1) (p - 1)}{(p^r - 1) (p^p - 1)}$. 1. Show that $F(r)$ is a positive integer for any prime $r \ne p$. 2. Show that $F(r)$ and $F(s)$ are coprime for any primes $r$ and $s$ such that $r \ne p, s \ne p$ and $r \ne s$. 3. Fix a prime $r \ne p$. Show that there is a prime divisor $q$ of $F(r)$ such that $p| q - 1$ but $p^2 \nmid q - 1$.

2012 India Regional Mathematical Olympiad, 2

Let $a,b,c$ be positive integers such that $a|b^4, b|c^4$ and $c|a^4$. Prove that $abc|(a+b+c)^{21}$

1968 Czech and Slovak Olympiad III A, 2

Show that for any integer $n$ the number \[a_n=\frac{\bigl(2+\sqrt3\bigr)^n-\bigl(2-\sqrt3\bigr)^n}{2\sqrt3}\] is also integer. Determine all integers $n$ such that $a_n$ is divisible by 3.

2000 Tournament Of Towns, 3

The least common multiple of positive integers $a, b, c$ and $d$ is equal to $a + b + c + d$. Prove that $abcd$ is divisible by at least one of $3$ and $5$. ( V Senderov)

2012 Brazil Team Selection Test, 4

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 $p^2$

2015 Indonesia MO Shortlist, N6

Defined as $N_0$ as the set of all non-negative integers. Set $S \subset N_0$ with not so many elements is called beautiful if for every $a, b \in S$ with $a \ge b$ ($a$ and $b$ do not have to be different), exactly one of $a + b$ or $a - b$ is in $S$. Set $T \subset N_0$ with not so many elements is called charming if the largest number $k$ such that up to 3$^k | a$ is the same for each element $a \in T$. Prove that each beautiful set must be charming.

2006 Thailand Mathematical Olympiad, 5

Show that there are coprime positive integers $m$ and $n$ such that $2549 | (25 \cdot 49)^m + 25^n - 2 \cdot 49^n$

2013 Costa Rica - Final Round, 5

Determine the number of polynomials of degree $5$ with different coefficients in the set $\{1, 2, 3, 4, 5, 6, 7, 8\}$ such that they are divisible by $x^2-x + 1$. Justify your answer.

2001 Tuymaada Olympiad, 2

Is it possible to arrange integers in the cells of the infinite chechered sheet so that every integer appears at least in one cell, and the sum of any $10$ numbers in a row vertically or horizontal, would be divisible by $101$?

1980 Czech And Slovak Olympiad IIIA, 1

Prove that for every nonnegative integer $ k$ there is a product $$(k + 1)(k + 2)...(k + 1980)$$ divisible by $ 1980^{197}$.

2016 Bosnia And Herzegovina - Regional Olympiad, 4

Let $a$ and $b$ be distinct positive integers, bigger that $10^6$, such that $(a+b)^3$ is divisible with $ab$. Prove that $ \mid a-b \mid > 10^4$

2005 All-Russian Olympiad Regional Round, 10.5

Arithmetic progression $a_1, a_2, . . . , $ consisting of natural numbers is such that for any $n$ the product $a_n \cdot a_{n+31}$ is divisible by $2005$. Is it possible to say that all terms of the progression are divisible by $2005$?

2000 Chile National Olympiad, 3

A number $N_k$ is defined as [i]periodic[/i] if it is composed in number base $N$ of a repeated $k$ times . Prove that $7$ divides to infinite periodic numbers of the set $N_1, N_2, N_3,...$

2014 Chile National Olympiad, 4

Prove that for every integer $n$ the expression $n^3-9n + 27$ is not divisible by $81$.

2016 Saudi Arabia GMO TST, 3

Find all positive integer $n$ such that there exists a permutation $(a_1, a_2,...,a_n)$ of $(1, 2,3,..., n)$ satisfying the condition: $a_1 + a_2 +... + a_k$ is divisible by $k$ for each $k = 1, 2,3,..., n$.

2000 Kazakhstan National Olympiad, 5

Let the number $ p $ be a prime divisor of the number $ 2 ^ {2 ^ k} + 1 $. Prove that $ p-1 $ is divisible by $ 2 ^ {k + 1} $.

2014 Dutch Mathematical Olympiad, 4

A quadruple $(p, a, b, c)$ of positive integers is called a Leiden quadruple if - $p$ is an odd prime number, - $a, b$, and $c$ are distinct and - $ab + 1, bc + 1$ and $ca + 1$ are divisible by $p$. a) Prove that for every Leiden quadruple $(p, a, b, c)$ we have $p + 2 \le \frac{a+b+c}{3}$ . b) Determine all numbers $p$ for which a Leiden quadruple $(p, a, b, c)$ exists with $p + 2 = \frac{a+b+c}{3} $

2006 Estonia National Olympiad, 2

Let $a, b$ and $c$ be positive integers such that $ab + 1, bc + 1$ and $ca + 1$ are all integer squares. a) Give an example of such numbers $a, b$ and $c$. b) Prove that at least one of the numbers $a, b$ and $c$ is divisible by $4$

2020 Swedish Mathematical Competition, 5

Find all integers $a$ such that there is a prime number of $p\ge 5$ that divides ${p-1 \choose 2}$ $+ {p-1 \choose 3} a$ $+{p-1 \choose 4} a^2$+ ...+$ {p-1 \choose p-3} a^{p-5} .$

1986 Polish MO Finals, 3

$p$ is a prime and $m$ is a non-negative integer $< p-1$. Show that $ \sum_{j=1}^p j^m$ is divisible by $p$.