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.

AND:
OR:
NO:

Found problems: 408

2021 Dutch IMO TST, 3

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

2015 Switzerland - Final Round, 9

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 $p$.

2012 NZMOC Camp Selection Problems, 4

A pair of numbers are [i]twin primes[/i] if they differ by two, and both are prime. Prove that, except for the pair $\{3, 5\}$, the sum of any pair of twin primes is a multiple of $ 12$.

2013 IMAC Arhimede, 2

For all positive integer $n$, we consider the number $$a_n =4^{6^n}+1943$$ Prove that $a_n$ is dividible by $2013$ for all $n\ge 1$, and find all values of $n$ for which $a_n - 207$ is the cube of a positive integer.

1953 Moscow Mathematical Olympiad, 234

Find the smallest number of the form $1...1$ in its decimal expression which is divisible by $\underbrace{\hbox{3...3}}_{\hbox{100}}$,.

1993 All-Russian Olympiad Regional Round, 9.2

Find the largest natural number which cannot be turned into a multiple of $11$ by reordering its (decimal) digits.

2011 Saudi Arabia Pre-TST, 3.1

Let $n$ be a positive integer such that $2011^{2011}$ divides $n!$. Prove that $2011^{2012} $divides $n!$ .

2015 Gulf Math Olympiad, 1

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 $p,q$.

2012 Dutch Mathematical Olympiad, 1

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 $12$.

1947 Kurschak Competition, 1

Prove that $46^{2n+1} + 296 \cdot 13^{2n+1}$ is divisible by $1947$.

1969 Swedish Mathematical Competition, 5

Let $N = a_1a_2...a_n$ in binary. Show that if $a_1-a_2 + a_3 -... + (-1)^{n-1}a_n = 0$ mod $3$, then $N = 0$ mod $3$.

2022 APMO, 1

Find all pairs $(a,b)$ of positive integers such that $a^3$ is multiple of $b^2$ and $b-1$ is multiple of $a-1$.

2005 Estonia National Olympiad, 2

Let $a, b$ and $c$ be arbitrary integers. Prove that $a^2 + b^2 + c^2$ is divisible by $7$ when $a^4 + b^4 + c^4$ divisible by $7$.

2000 All-Russian Olympiad Regional Round, 9.2

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 Austria Beginners' Competition, 1

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 $14$.

1987 ITAMO, 1

Show that $3x^5 +5x^3 -8x$ is divisible by $120$ for any integer $x$

1975 Chisinau City MO, 87

Prove that among any $100$ natural numbers there are two numbers whose difference is divisible by $99$.

2017 India PRMO, 1

How many positive integers less than $1000$ have the property that the sum of the digits of each such number is divisible by $7$ and the number itself is divisible by $3$?

2000 Tournament Of Towns, 2

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)

1952 Moscow Mathematical Olympiad, 226

Seven chips are numbered $1, 2, 3, 4, 5, 6, 7$. Prove that none of the seven-digit numbers formed by these chips is divisible by any other of these seven-digit numbers.

2016 IMAR Test, 1

Fix an integer $n \ge 3$ and let $a_0 = n$. Does there exist a permutation $a_1, a_2,..., a_{n-1}$ of the fi rst $n-1$ positive integers such that $\Sigma_{j=0}^{k-1} a_j$ is divisible by $a_k$ for all indices $k < n$?

1999 Denmark MO - Mohr Contest, 5

Is there a number whose digits are only $1$'s and which is divided by $1999$?

1972 Putnam, A5

Prove that there is no positive integer $n>1$ such that $n\mid2^{n} -1.$

2015 Caucasus Mathematical Olympiad, 1

Is there an eight-digit number without zero digits, which when divided by the first digit gives the remainder $1$, when divided by the second digit will give the remainder $2$, ..., when divided by the eighth digit will give the remainder $8$?

2007 Switzerland - Final Round, 9

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