Found problems: 1362
2005 Junior Balkan Team Selection Tests - Moldova, 7
Let $p$ be a prime number and $a$ and $n$ positive nonzero integers. Prove that
if $2^p + 3^p = a^n$ then $n=1$
2003 Federal Competition For Advanced Students, Part 2, 3
For every lattice point $(x, y)$ with $x, y$ non-negative integers, a square of side $\frac{0.9}{2^x5^y}$ with center at the point $(x, y)$ is constructed. Compute the area of the union of all these squares.
1977 IMO Longlists, 35
Find all numbers $N=\overline{a_1a_2\ldots a_n}$ for which $9\times \overline{a_1a_2\ldots a_n}=\overline{a_n\ldots a_2a_1}$ such that at most one of the digits $a_1,a_2,\ldots ,a_n$ is zero.
2004 Finnish National High School Mathematics Competition, 4
The numbers $2005! + 2, 2005! + 3, ... , 2005! + 2005$ form a sequence of $2004$ consequtive integers, none of which is a prime number.
Does there exist a sequence of $2004$ consequtive integers containing exactly $12$ prime numbers?
2010 Moldova National Olympiad, 11.4
Let $ a_n\equal{}1\plus{}\dfrac1{2^2}\plus{}\dfrac1{3^2}\plus{}\cdots\plus{}\dfrac1{n^2}$
Find $ \lim_{n\to\infty}a_n$
1999 Kurschak Competition, 1
For any positive integer $m$, denote by $d_i(m)$ the number of positive divisors of $m$ that are congruent to $i$ modulo $2$. Prove that if $n$ is a positive integer, then
\[\left|\sum_{k=1}^n \left(d_0(k)-d_1(k)\right)\right|\le n.\]
2009 All-Russian Olympiad, 8
Let $ x$, $ y$ be two integers with $ 2\le x, y\le 100$. Prove that $ x^{2^n} \plus{} y^{2^n}$ is not a prime for some positive integer $ n$.
2009 India Regional Mathematical Olympiad, 6
In a book with page numbers from $ 1$ to $ 100$ some pages are torn off. The sum of the numbers on the remaining pages is $ 4949$. How many pages are torn off?
2006 Nordic, 3
A sequence $(a_n)$ of positive integers is defined by $a_0=m$ and $a_{n+1}= a_n^5 +487$ for all $n\ge 0$.
Find all positive integers $m$ such that the sequence contains the maximum possible number of perfect squares.
1994 Taiwan National Olympiad, 4
Prove that there are infinitely many positive integers $n$ with the following property: For any $n$ integers $a_{1},a_{2},...,a_{n}$ which form in arithmetic progression, both the mean and the standard deviation of the set $\{a_{1},a_{2},...,a_{n}\}$ are integers.
[i]Remark[/i]. The mean and standard deviation of the set $\{x_{1},x_{2},...,x_{n}\}$ are defined by $\overline{x}=\frac{x_{1}+x_{2}+...+x_{n}}{n}$ and $\sqrt{\frac{\sum (x_{i}-\overline{x})^{2}}{n}}$, respectively.
2001 Korea - Final Round, 1
Given an odd prime $p$, find all functions $f:Z \rightarrow Z$ satisfying the following two conditions:
(i) $f(m)=f(n)$ for all $m,n \in Z$ such that $m\equiv n\pmod p$;
(ii) $f(mn)=f(m)f(n)$ for all $m,n \in Z$.
1985 Kurschak Competition, 2
For every $n\in\mathbb{N}$, define the [i]power sum[/i] of $n$ as follows. For every prime divisor $p$ of $n$, consider the largest positive integer $k$ for which $p^k\le n$, and sum up all the $p^k$'s. (For instance, the power sum of $100$ is $2^6+5^2=89$.) Prove that the [i]power sum[/i] of $n$ is larger than $n$ for infinitely many positive integers $n$.
1999 Finnish National High School Mathematics Competition, 3
Determine how many primes are there in the sequence \[101, 10101, 1010101 ....\]
2004 China Team Selection Test, 3
Find all positive integer $ m$ if there exists prime number $ p$ such that $ n^m\minus{}m$ can not be divided by $ p$ for any integer $ n$.
1992 USAMO, 3
For a nonempty set $\, S \,$ of integers, let $\, \sigma(S) \,$ be the sum of the elements of $\, S$. Suppose that $\, A = \{a_1, a_2, \ldots, a_{11} \} \,$ is a set of positive integers with $\, a_1 < a_2 < \cdots < a_{11} \,$ and that, for each positive integer $\, n\leq 1500, \,$ there is a subset $\, S \,$ of $\, A \,$ for which $\, \sigma(S) = n$. What is the smallest possible value of $\, a_{10}$?
1994 Vietnam Team Selection Test, 2
Consider the equation
\[x^2 + y^2 + z^2 + t^2 - N \cdot x \cdot y \cdot z \cdot t - N = 0\]
where $N$ is a given positive integer.
a) Prove that for an infinite number of values of $N$, this equation has positive integral solutions (each such solution consists of four positive integers $x, y, z, t$),
b) Let $N = 4 \cdot k \cdot (8 \cdot m + 7)$ where $k,m$ are no-negative integers. Prove that the considered equation has no positive integral solutions.
2003 China Team Selection Test, 3
Let $x_0+\sqrt{2003}y_0$ be the minimum positive integer root of Pell function $x^2-2003y^2=1$. Find all the positive integer solutions $(x,y)$ of the equation, such that $x_0$ is divisible by any prime factor of $x$.
2006 Korea - Final Round, 3
A positive integer $N$ is said to be $n-$ good if
(i) $N$ has at least $n$ distinct prime divisors, and
(ii) there exist distinct positive divisors $1, x_{2}, . . . , x_{n}$ whose sum is $N$ .
Show that there exists an $n-$ good number for each $n\geq 6$.
2008 Germany Team Selection Test, 3
Prove there is an integer $ k$ for which $ k^3 \minus{} 36 k^2 \plus{} 51 k \minus{} 97$ is a multiple of $ 3^{2008.}$
1997 Poland - Second Round, 4
There is a set with three elements: (2,3,5). It has got an interesting property: (2*3) mod 5=(2*5) mod 3=(3*5) mod 2. Prove that it is the only one set with such property.
2002 Italy TST, 2
Prove that for each prime number $p$ and positive integer $n$, $p^n$ divides
\[\binom{p^n}{p}-p^{n-1}. \]
2011 Indonesia TST, 4
A prime number $p$ is a [b]moderate[/b] number if for every $2$ positive integers $k > 1$ and $m$, there exists k positive integers $n_1, n_2, ..., n_k $ such that \[ n_1^2+n_2^2+ ... +n_k^2=p^{k+m} \]
If $q$ is the smallest [b]moderate[/b] number, then determine the smallest prime $r$ which is not moderate and $q < r$.
2014 Contests, 3
For any positive integer $n$, let $D_n$ denote the greatest common divisor of all numbers of the form $a^n + (a + 1)^n + (a + 2)^n$ where $a$ varies among all positive integers.
(a) Prove that for each $n$, $D_n$ is of the form $3^k$ for some integer $k \ge 0$.
(b) Prove that, for all $k\ge 0$, there exists an integer $n$ such that $D_n = 3^k$.
2006 Romania Team Selection Test, 1
Let $n$ be a positive integer of the form $4k+1$, $k\in \mathbb N$ and $A = \{ a^2 + nb^2 \mid a,b \in \mathbb Z\}$. Prove that there exist integers $x,y$ such that $x^n+y^n \in A$ and $x+y \notin A$.
1985 IMO Longlists, 58
Prove that there are infinitely many pairs $(k,N)$ of positive integers such that $1 + 2 + \cdots + k = (k + 1) + (k + 2)+\cdots + N.$