Found problems: 1362
2007 Finnish National High School Mathematics Competition, 1
Show: when a prime number is divided by $30,$ the remainder is either $1$ or a prime number. Is a similar claim true, when the divisor is $60$ or $90$?
2002 Tuymaada Olympiad, 4
A real number $a$ is given. The sequence $n_{1}< n_{2}< n_{3}< ...$ consists of all the positive integral $n$ such that $\{na\}< \frac{1}{10}$. Prove that there are at most three different numbers among the numbers $n_{2}-n_{1}$, $n_{3}-n_{2}$, $n_{4}-n_{3}$, $\ldots$.
[i]A corollary of a theorem from ergodic theory[/i]
2011 China Team Selection Test, 3
For any positive integer $d$, prove there are infinitely many positive integers $n$ such that $d(n!)-1$ is a composite number.
2004 India IMO Training Camp, 2
Find all triples $(x,y,n)$ of positive integers such that \[ (x+y)(1+xy) = 2^{n} \]
1980 IMO, 13
Prove that the integer $145^{n} + 3114\cdot 138^{n}$ is divisible by $1981$ if $n=1981$, and that it is not divisible by $1981$ if $n=1980$.
2000 Vietnam Team Selection Test, 2
Let $k$ be a given positive integer. Define $x_{1}= 1$ and, for each $n > 1$, set $x_{n+1}$ to be the smallest positive integer not belonging to the set $\{x_{i}, x_{i}+ik | i = 1, . . . , n\}$. Prove that there is a real number $a$ such that $x_{n}= [an]$ for all $n \in\mathbb{ N}$.
1996 Polish MO Finals, 2
Let $p(k)$ be the smallest prime not dividing $k$. Put $q(k) = 1$ if $p(k) = 2$, or the product of all primes $< p(k)$ if $p(k) > 2$. Define the sequence $x_0, x_1, x_2, ...$ by $x_0 = 1$, $x_{n+1} = \frac{x_np(x_n)}{q(x_n)}$. Find all $n$ such that $x_n = 111111$
2010 Contests, 1
Let $a,b,c\in\{0,1,2,\cdots,9\}$.The quadratic equation $ax^2+bx+c=0$ has a rational root. Prove that the three-digit number $abc$ is not a prime number.
1992 China Team Selection Test, 3
For any prime $p$, prove that there exists integer $x_0$ such that $p | (x^2_0 - x_0 + 3)$ $\Leftrightarrow$ there exists integer $y_0$ such that $p | (y^2_0 - y_0 + 25).$
2000 USA Team Selection Test, 5
Let $n$ be a positive integer. A $corner$ is a finite set $S$ of ordered $n$-tuples of positive integers such that if $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ are positive integers with $a_k \geq b_k$ for $k = 1, 2, \ldots, n$ and $(a_1, a_2, \ldots, a_n) \in S$, then $(b_1, b_2, \ldots, b_n) \in S$. Prove that among any infinite collection of corners, there exist two corners, one of which is a subset of the other one.
2008 All-Russian Olympiad, 7
For which integers $ n>1$ do there exist natural numbers $ b_1,b_2,\ldots,b_n$ not all equal such that the number $ (b_1\plus{}k)(b_2\plus{}k)\cdots(b_n\plus{}k)$ is a power of an integer for each natural number $ k$? (The exponents may depend on $ k$, but must be greater than $ 1$)
1970 IMO Longlists, 29
Prove that the equation $4^x +6^x =9^x$ has no rational solutions.
2003 Finnish National High School Mathematics Competition, 4
Find pairs of positive integers $(n, k)$ satisfying \[(n + 1)^k - 1 = n!\]
1990 China National Olympiad, 2
Let $x$ be a natural number. We call $\{x_0,x_1,\dots ,x_l\}$ a [i]factor link [/i]of $x$ if the sequence $\{x_0,x_1,\dots ,x_l\}$ satisfies the following conditions:
(1) $x_0=1, x_l=x$;
(2) $x_{i-1}<x_i, x_{i-1}|x_i, i=1,2,\dots,l$ .
Meanwhile, we define $l$ as the length of the [i]factor link [/i] $\{x_0,x_1,\dots ,x_l\}$. Denote by $L(x)$ and $R(x)$ the length and the number of the longest [i]factor link[/i] of $x$ respectively. For $x=5^k\times 31^m\times 1990^n$, where $k,m,n$ are natural numbers, find the value of $L(x)$ and $R(x)$.
2010 Vietnam National Olympiad, 4
Prove that for each positive integer n,the equation
$x^{2}+15y^{2}=4^{n}$
has at least $n$ integer solution $(x,y)$
2008 Rioplatense Mathematical Olympiad, Level 3, 1
Can the positive integers be partitioned into $12$ subsets such that for each positive integer $k$, the numbers $k, 2k,\ldots,12k$ belong to different subsets?
2003 Baltic Way, 19
Let $a$ and $b$ be positive integers. Show that if $a^3+b^3$ is the square of an integer, then $a + b$ is not a product of two different prime numbers.
2005 MOP Homework, 3
For any positive integer $n$, the sum $1+\frac{1}{2}+\frac{1}{3}+\cdots +\frac{1}{n}$ is written in the lowest form $\frac{p_n}{q_n}$; that is, $p_n$ and $q_n$ are relatively prime positive integers. Find all $n$ such that $p_n$ is divisible by $3$.
2014 Costa Rica - Final Round, 2
Find all positive integers $n$ such that $n!+2$ divides $(2n)!$.
1996 IberoAmerican, 1
Let $ n$ be a natural number. A cube of edge $ n$ may be divided in 1996 cubes whose edges length are also natural numbers. Find the minimum possible value for $ n$.
2011 Estonia Team Selection Test, 5
Prove that if $n$ and $k$ are positive integers such that $1<k<n-1$,Then the binomial coefficient $\binom nk$ is divisible by at least two different primes.
1978 Canada National Olympiad, 2
Find all pairs of $a$, $b$ of positive integers satisfying the equation $2a^2 = 3b^3$.
1991 Canada National Olympiad, 1
Show that the equation $x^2+y^5=z^3$ has infinitely many solutions in integers $x, y,z$ for which $xyz \neq 0$.
2009 All-Russian Olympiad, 3
Given are positive integers $ n>1$ and $ a$ so that $ a>n^2$, and among the integers $ a\plus{}1, a\plus{}2, \ldots, a\plus{}n$ one can find a multiple of each of the numbers $ n^2\plus{}1, n^2\plus{}2, \ldots, n^2\plus{}n$. Prove that $ a>n^4\minus{}n^3$.
2006 Bundeswettbewerb Mathematik, 4
A positive integer is called [i]digit-reduced[/i] if at most nine different digits occur in its decimal representation (leading $0$s are omitted.) Let $M$ be a finite set of [i]digit-reduced[/i] numbers. Show that the sum of the reciprocals of the elements in $M$ is less than $180$.