Found problems: 1362
2010 Princeton University Math Competition, 3
Show that, if $n \neq 2$ is a positive integer, that there are $n$ triangular numbers $a_1$, $a_2$, $\ldots$, $a_n$ such that $\displaystyle{\sum_{i=1}^n \frac1{a_i} = 1}$ (Recall that the $k^{th}$ triangular number is $\frac{k(k+1)}2$).
2012 Tuymaada Olympiad, 1
Solve in positive integers the following equation:
\[{1\over n^2}-{3\over 2n^3}={1\over m^2}\]
[i]Proposed by A. Golovanov[/i]
2013 South East Mathematical Olympiad, 6
$n>1$ is an integer. The first $n$ primes are $p_1=2,p_2=3,\dotsc, p_n$. Set $A=p_1^{p_1}p_2^{p_2}...p_n^{p_n}$.
Find all positive integers $x$, such that $\dfrac Ax$ is even, and $\dfrac Ax$ has exactly $x$ divisors
2005 South africa National Olympiad, 1
Five numbers are chosen from the diagram below, such that no two numbers are chosen from the same row or from the same column. Prove that their sum is always the same.
\[\begin{array}{|c|c|c|c|c|}\hline
1&4&7&10&13\\ \hline
16&19&22&25&28\\ \hline
31&34&37&40&43\\ \hline
46&49&52&55&58\\ \hline
61&64&67&70&73\\ \hline
\end{array}\]
1976 IMO Longlists, 18
Prove that the number $19^{1976} + 76^{1976}$:
$(a)$ is divisible by the (Fermat) prime number $F_4 = 2^{2^4} + 1$;
$(b)$ is divisible by at least four distinct primes other than $F_4$.
1984 IMO Longlists, 46
Let $(a_n)_{n\ge 1}$ and $(b_n)_{n\ge 1}$ be two sequences of natural numbers such that $a_{n+1} = na_n + 1, b_{n+1} = nb_n - 1$ for every $n\ge 1$. Show that these two sequences can have only a finite number of terms in common.
1989 IMO Longlists, 88
Prove that the sequence $ (a_n)_{n \geq 0,}, a_n \equal{} [n \cdot \sqrt{2}],$ contains an infinite number of perfect squares.
1995 South africa National Olympiad, 2
Find all pairs $(m,n)$ of natural numbers with $m<n$ such that $m^2+1$ is a multiple of $n$ and $n^2+1$ is a multiple of $m$.
2002 Irish Math Olympiad, 4
The sequence $ (a_n)$ is defined by $ a_1\equal{}a_2\equal{}a_3\equal{}1$ and $ a_{n\plus{}1}a_{n\minus{}2}\minus{}a_n a_{n\minus{}1}\equal{}2$ for all $ n \ge 3.$ Prove that $ a_n$ is a positive integer for all $ n \ge 1$.
1990 China Team Selection Test, 3
Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.
2007 India National Olympiad, 2
Let $ n$ be a natural number such that $ n \equal{} a^2 \plus{} b^2 \plus{}c^2$ for some natural numbers $ a,b,c$. Prove that
\[ 9n \equal{} (p_1a\plus{}q_1b\plus{}r_1c)^2 \plus{} (p_2a\plus{}q_2b\plus{}r_2c)^2 \plus{} (p_3a\plus{}q_3b\plus{}r_3c)^2\]
where $ p_j$'s , $ q_j$'s , $ r_j$'s are all [b]nonzero[/b] integers. Further, if $ 3$ does [b]not[/b] divide at least one of $ a,b,c,$ prove that $ 9n$ can be expressed in the form $ x^2\plus{}y^2\plus{}z^2$, where $ x,y,z$ are natural numbers [b]none[/b] of which is divisible by $ 3$.
1998 Polish MO Finals, 1
Define the sequence $a_1, a_2, a_3, ...$ by $a_1 = 1$, $a_n = a_{n-1} + a_{[n/2]}$. Does the sequence contain infinitely many multiples of $7$?
1993 China Team Selection Test, 1
Find all integer solutions to $2 x^4 + 1 = y^2.$
2009 Croatia Team Selection Test, 4
Determine all triplets off positive integers $ (a,b,c)$ for which $ \mid2^a\minus{}b^c\mid\equal{}1$
2014 Contests, 2
Do there exist positive integers $a$ and $b$ such that $a^n+n^b$ and $b^n+n^a$ are relatively prime for all natural $n$?
2011 China Girls Math Olympiad, 6
Do there exist positive integers $m,n$, such that $m^{20}+11^n$ is a square number?
2004 Tournament Of Towns, 4
A positive integer $a > 1$ is given (in decimal notation). We copy it twice and obtain a number $b = \overline{aa}$ which happens to be a multiple of $a^2$. Find all possible values of $b/a^2$.
2006 India IMO Training Camp, 2
Let $u_{jk}$ be a real number for each $j=1,2,3$ and each $k=1,2$ and let $N$ be an integer such that
\[\max_{1\le k \le 2} \sum_{j=1}^3 |u_{jk}| \leq N\]
Let $M$ and $l$ be positive integers such that $l^2 <(M+1)^3$. Prove that there exist integers $\xi_1,\xi_2,\xi_3$ not all zero, such that
\[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]
2006 Moldova National Olympiad, 10.8
Let $M=\{x^2+x \mid x\in \mathbb N^{\star} \}$. Prove that for every integer $k\geq 2$ there exist elements $a_{1}, a_{2}, \ldots, a_{k},b_{k}$ from $M$, such that $a_{1}+a_{2}+\cdots+a_{k}=b_{k}$.
2018 Latvia Baltic Way TST, P15
Determine whether there exists a positive integer $n$ such that it is possible to find at least $2018$ different quadruples $(x,y,z,t)$ of positive integers that simultaneously satisfy equations
$$\begin{cases}
x+y+z=n\\
xyz = 2t^3.
\end{cases}$$
2010 India IMO Training Camp, 5
Given an integer $k>1$, show that there exist an integer an $n>1$ and distinct positive integers $a_1,a_2,\cdots a_n$, all greater than $1$, such that the sums $\sum_{j=1}^n a_j$ and $\sum_{j=1}^n \phi (a_j)$ are both $k$-th powers of some integers.
(Here $\phi (m)$ denotes the number of positive integers less than $m$ and relatively prime to $m$.)
1994 Hong Kong TST, 3
Find all non-negative integers $x, y$ and $z$ satisfying the equation: \[7^{x}+1=3^{y}+5^z\]
2014 Dutch IMO TST, 4
Determine all pairs $(p, q)$ of primes for which $p^{q+1}+q^{p+1}$ is a perfect square.
2008 Vietnam National Olympiad, 5
What is the total number of natural numbes divisible by 9 the number of digits of which does not exceed 2008 and at least two of the digits are 9s?
1987 IMO Longlists, 1
Let $x_1, x_2,\cdots, x_n$ be $n$ integers. Let $n = p + q$, where $p$ and $q$ are positive integers. For $i = 1, 2, \cdots, n$, put
\[S_i = x_i + x_{i+1} +\cdots + x_{i+p-1} \text{ and } T_i = x_{i+p} + x_{i+p+1} +\cdots + x_{i+n-1}\]
(it is assumed that $x_{i+n }= x_i$ for all $i$). Next, let $m(a, b)$ be the number of indices $i$ for which $S_i$ leaves the remainder $a$ and $T_i$ leaves the remainder $b$ on division by $3$, where $a, b \in \{0, 1, 2\}$. Show that $m(1, 2)$ and $m(2, 1)$ leave the same remainder when divided by $3.$