Found problems: 1766
2012 China Team Selection Test, 3
Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$. Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset $, and \[\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.\]
2008 Mongolia Team Selection Test, 2
Let $ a,b,c,d$ be the positive integers such that $ a > b > c > d$ and $ (a \plus{} b \minus{} c \plus{} d) | (ac \plus{} bd)$ . Prove that if $ m$ is arbitrary positive integer , $ n$ is arbitrary odd positive integer, then $ a^n b^m \plus{} c^m d^n$ is composite number
2005 Iran Team Selection Test, 1
Find all $f : N \longmapsto N$ that there exist $k \in N$ and a prime $p$ that:
$\forall n \geq k \ f(n+p)=f(n)$ and also if $m \mid n$ then $f(m+1) \mid f(n)+1$
2005 Romania Team Selection Test, 2
Let $n\geq 1$ be an integer and let $X$ be a set of $n^2+1$ positive integers such that in any subset of $X$ with $n+1$ elements there exist two elements $x\neq y$ such that $x\mid y$. Prove that there exists a subset $\{x_1,x_2,\ldots, x_{n+1} \} \in X$ such that $x_i \mid x_{i+1}$ for all $i=1,2,\ldots, n$.
2006 Iran Team Selection Test, 4
Let $n$ be a fixed natural number.
Find all $n$ tuples of natural pairwise distinct and coprime numbers like $a_1,a_2,\ldots,a_n$ such that for $1\leq i\leq n$ we have
\[ a_1+a_2+\ldots+a_n|a_1^i+a_2^i+\ldots+a_n^i \]
2004 China National Olympiad, 3
Prove that every positive integer $n$, except a finite number of them, can be represented as a sum of $2004$ positive integers: $n=a_1+a_2+\cdots +a_{2004}$, where $1\le a_1<a_2<\cdots <a_{2004}$, and $a_i \mid a_{i+1}$ for all $1\le i\le 2003$.
[i]Chen Yonggao[/i]
2007 Romania Team Selection Test, 3
Consider the set $E = \{1,2,\ldots,2n\}$. Prove that an element $c \in E$ can belong to a subset $A \subset E$, having $n$ elements, and such that any two distinct elements in $A$ do not divide one each other, if and only if \[c > n \left( \frac{2}{3}\right )^{k+1},\] where $k$ is the exponent of $2$ in the factoring of $c$.
2008 Bulgaria Team Selection Test, 1
For each positive integer $n$, denote by $a_{n}$ the first digit of $2^{n}$ (base ten). Is the number $0.a_{1}a_{2}a_{3}\cdots$ rational?
2011 Switzerland - Final Round, 9
For any positive integer $n$ let $f(n)$ be the number of divisors of $n$ ending with $1$ or $9$ in base $10$ and let $g(n)$ be the number of divisors of $n$ ending with digit $3$ or $7$ in base $10$. Prove that $f(n)\geqslant g(n)$ for all nonnegative integers $n$.
[i](Swiss Mathematical Olympiad 2011, Final round, problem 9)[/i]
1995 Iran MO (2nd round), 2
Let $n \geq 0$ be an integer. Prove that
\[ \lceil \sqrt n +\sqrt{n+1}+\sqrt{n+2} \rceil = \lceil \sqrt{9n+8} \rceil\]
Where $\lceil x \rceil $ is the smallest integer which is greater or equal to $x.$
2010 Slovenia National Olympiad, 4
Find the smallest three-digit number such that the following holds:
If the order of digits of this number is reversed and the number obtained by this is added to the original number, the resulting number consists of only odd digits.
2004 Balkan MO, 2
Solve in prime numbers the equation $x^y - y^x = xy^2 - 19$.
2008 Saint Petersburg Mathematical Olympiad, 5
Given are distinct natural numbers $a$, $b$, and $c$. Prove that
\[ \gcd(ab+1, ac+1, bc+1)\le \frac{a+b+c}{3}\]
2011 Indonesia MO, 2
For each positive integer $n$, let $s_n$ be the number of permutations $(a_1, a_2, \cdots, a_n)$ of $(1, 2, \cdots, n)$ such that $\dfrac{a_1}{1} + \dfrac{a_2}{2} + \cdots + \dfrac{a_n}{n}$ is a positive integer. Prove that $s_{2n} \ge n$ for all positive integer $n$.
2015 Thailand TSTST, 2
Fix a sequence $ a_1,a_2,a_3,... $ of integers satisfying the following condition:for all prime numbers $ p $ and all positive integers $ k $, we have $ a_{pk+1}=pa_k-3a_p+13 $.Determine all possible values of $ a_{2013} $.
1971 IMO Longlists, 4
Let $x_n=2^{2^{n}}+1$ and let $m$ be the least common multiple of $x_2, x_3, \ldots, x_{1971}.$ Find the last digit of $m.$
2010 Slovenia National Olympiad, 2
Let $a, b$ and $c$ be nonzero digits. Let $p$ be a prime number which divides the three digit numbers $\overline{abc}$ and $\overline{cba}.$ Show that $p$ divides at least one of the numbers $a+b+c, a-b+c$ and $a-c.$
2002 Iran MO (3rd Round), 3
$a_{n}$ is a sequence that $a_{1}=1,a_{2}=2,a_{3}=3$, and \[a_{n+1}=a_{n}-a_{n-1}+\frac{a_{n}^{2}}{a_{n-2}}\] Prove that for each natural $n$, $a_{n}$ is integer.
2013 Argentina Cono Sur TST, 4
Show that the number $\begin{matrix} \\ N= \end{matrix} \underbrace{44 \ldots 4}_{n} \underbrace{88 \ldots 8}_{n} - 1\underbrace{33 \ldots3 }_{n-1}2$ is a perfect square for all positive integers $n$.
1996 Iran MO (3rd Round), 6
Find all pairs $(p,q)$ of prime numbers such that
\[m^{3pq} \equiv m \pmod{3pq} \qquad \forall m \in \mathbb Z.\]
1996 All-Russian Olympiad, 5
Do there exist three natural numbers greater than 1, such that the square of each, minus one, is divisible by each of the others?
[i]A. Golovanov[/i]
2002 France Team Selection Test, 2
Consider the set $S$ of integers $k$ which are products of four distinct primes. Such an integer $k=p_1p_2p_3p_4$ has $16$ positive divisors $1=d_1<d_2<\ldots <d_{15}<d_{16}=k$. Find all elements of $S$ less than $2002$ such that $d_9-d_8=22$.
2010 Romania National Olympiad, 2
How many four digit numbers $\overline{abcd}$ simultaneously satisfy the equalities $a+b=c+d$ and $a^2+b^2=c^2+d^2$?
2011 Indonesia MO, 1
For a number $n$ in base $10$, let $f(n)$ be the sum of all numbers possible by removing some digits of $n$ (including none and all). For example, if $n = 1234$, $f(n) = 1234 + 123 + 124 + 134 + 234 + 12 + 13 + 14 + 23 + 24 + 34 + 1 + 2 + 3 + 4 = 1979$; this is formed by taking the sums of all numbers obtained when removing no digit from $n$ (1234), removing one digit from $n$ (123, 124, 134, 234), removing two digits from $n$ (12, 13, 14, 23, 24, 34), removing three digits from $n$ (1, 2, 3, 4), and removing all digits from $n$ (0). If $p$ is a 2011-digit integer, prove that $f(p)-p$ is divisible by $9$.
Remark: If a number appears twice or more, it is counted as many times as it appears. For example, with the number $101$, $1$ appears three times (by removing the first digit, giving $01$ which is equal to $1$, removing the first two digits, or removing the last two digits), so it is counted three times.
2014 Baltic Way, 17
Do there exist pairwise distinct rational numbers $x, y$ and $z$ such that \[\frac{1}{(x - y)^2}+\frac{1}{(y - z)^2}+\frac{1}{(z - x)^2}= 2014?\]