Found problems: 408
2006 MOP Homework, 6
Let $m$ and $n$ be positive integers with $m > n \ge 2$. Set $S =\{1,2,...,m\}$, and set $T = \{a_1,a_2,...,a_n\}$ is a subset of $S$ such that every element of $S$ is not divisible by any pair of distinct elements of $T$. Prove that
$$\frac{1}{a_1}+\frac{1}{a_2}+ ...+ \frac{1}{a_n} < \frac{m+n}{m}$$
2020 Malaysia IMONST 2, 3
Given integers $a$ and $b$ such that $a^2+b^2$ is divisible by $11$. Prove that $a$ and $b$ are both divisible by $11$.
2009 Thailand Mathematical Olympiad, 10
Let $p > 5$ be a prime. Suppose that $$\frac{1}{2^2} + \frac{1}{4^2}+ \frac{1}{6^2}+ ...+ \frac{1}{(p -1)^2} =\frac{a}{b}$$ where $a/b$ is a fraction in lowest terms. Show that $p | a$.
2015 JBMO Shortlist, NT3
a) Show that the product of all differences of possible couples of six given positive integers is divisible by $960$
b) Show that the product of all differences of possible couples of six given positive integers is divisible by $34560$
PS. a) original from Albania
b) modified by problem selecting committee
1995 Bulgaria National Olympiad, 1
Find the number of integers $n > 1$ which divide $a^{25} - a$ for every integer $a$.
2005 Thailand Mathematical Olympiad, 12
Find the number of even integers n such that $0 \le n \le 100$ and $5 | n^2 \cdot 2^{{2n}^2}+ 1$.
1997 Bundeswettbewerb Mathematik, 1
Given $100$ integers, is it always possible to choose $15$ of them such that the difference of any two of the chosen numbers is divisible by $7$? What is the answer if $15$ is replaced by $16$?
1949 Moscow Mathematical Olympiad, 156
Prove that $27 195^8 - 10 887^8 + 10 152^8$ is divisible by $26 460$.
2009 Chile National Olympiad, 4
Find a positive integer $x$, with $x> 1$ such that all numbers in the sequence $$x + 1,x^x + 1,x^{x^x}+1,...$$ are divisible by $2009.$
1999 Tournament Of Towns, 1
For what values o f $n$ is it possible to place the integers from $1$ to $n$ inclusive on a circle (not necessarily in order) so that the sum of any two successive integers in the circle is divisible by the next one in the clockwise order?
(A Shapovalov)
2019 Ukraine Team Selection Test, 2
Polynomial $p(x)$ with real coefficients, which is different from the constant, has the following property:
[i] for any naturals $n$ and $k$ the $\frac{p(n+1)p(n+2)...p(n+k)}{p(1)p(2)...p(k)}$ is an integer.[/i]
Prove that this polynomial is divisible by $x$.
2013 Saudi Arabia Pre-TST, 1.2
Let $x, y$ be two non-negative integers. Prove that $47$ divides $3^x - 2^y$ if and only if $23$ divides $4x + y$.
1970 Poland - Second Round, 3
Prove the theorem:
There is no natural number $ n > 1 $ such that the number $ 2^n - 1 $ is divisible by $ n $.
2008 Hanoi Open Mathematics Competitions, 1
How many integers from $1$ to $2008$ have the sum of their digits divisible by $5$ ?
2011 IMAR Test, 4
Given an integer number $n \ge 3$, show that the number of lists of jointly coprime positive integer numbers that sum to $n$ is divisible by $3$.
(For instance, if $n = 4$, there are six such lists: $(3, 1), (1, 3), (2, 1, 1), (1, 2, 1), (1, 1, 2)$ and $(1, 1, 1, 1)$.)
2004 Estonia National Olympiad, 4
Prove that the number $n^n-n$ is divisible by $24$ for any odd integer $n$.
1989 Tournament Of Towns, (210) 4
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 $k$ .
1954 Moscow Mathematical Olympiad, 267
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 $(x - x_0)^2$.
2005 Thailand Mathematical Olympiad, 2
Let $S $ be a set of three distinct integers. Show that there are $a, b \in S$ such that $a \ne b$ and $10 | a^3b - ab^3$.
2001 Chile National Olympiad, 4
Given a natural number $n$, prove that $2^{2n}-1$ is a multiple of $3$.
2020 Israel Olympic Revenge, N
Let $a_1,a_2,a_3,...$ be an infinite sequence of positive integers.
Suppose that a sequence $a_1,a_2,\ldots$ of positive integers satisfies $a_1=1$ and \[a_{n}=\sum_{n\neq d|n}a_d\] for every integer $n>1$. Prove that the exist infinitely many integers $k$ such that $a_k=k$.
2014 Saudi Arabia GMO TST, 2
Let $p$ be a prime number. Prove that there exist infinitely many positive integers $n$ such that $p$ divides $1^n + 2^n +... + (p + 1)^n.$
VII Soros Olympiad 2000 - 01, 8.3
Find the sum of all such natural numbers from $1$ to $500$ that are not divisible by $5$ or $7$.
2022 New Zealand MO, 5
The sequence $x_1, x_2, x_3, . . .$ is defined by $x_1 = 2022$ and $x_{n+1}= 7x_n + 5$ for all positive integers $n$. Determine the maximum positive integer $m$ such that $$\frac{x_n(x_n - 1)(x_n - 2) . . . (x_n - m + 1)}{m!}$$ is never a multiple of $7$ for any positive integer $n$.
2017 Czech And Slovak Olympiad III A, 6
Given is a nonzero integer $k$.
Prove that equation $k =\frac{x^2 - xy + 2y^2}{x + y}$ has an odd number of ordered integer pairs $(x, y)$ just when $k$ is divisible by seven.