Found problems: 1362
1977 IMO Longlists, 42
The sequence $a_{n,k} \ , k = 1, 2, 3,\ldots, 2^n \ , n = 0, 1, 2,\ldots,$ is defined by the following recurrence formula:
\[a_1 = 2,\qquad a_{n,k} = 2a_{n-1,k}^3, \qquad , a_{n,k+2^{n-1}} =\frac 12 a_{n-1,k}^3\]\[\text{for} \quad k = 1, 2, 3,\ldots, 2^{n-1} \ , n = 0, 1, 2,\ldots\]
Prove that the numbers $a_{n,k}$ are all different.
2010 China National Olympiad, 2
Let $k$ be an integer $\geq 3$. Sequence $\{a_n\}$ satisfies that $a_k = 2k$ and for all $n > k$, we have
\[a_n =
\begin{cases}
a_{n-1}+1 & \text{if } (a_{n-1},n) = 1 \\
2n & \text{if } (a_{n-1},n) > 1
\end{cases}
\]
Prove that there are infinitely many primes in the sequence $\{a_n - a_{n-1}\}$.
2008 Mexico National Olympiad, 1
Let $1=d_1<d_2<d_3<\dots<d_k=n$ be the divisors of $n$. Find all values of $n$ such that $n=d_2^2+d_3^3$.
2006 Junior Balkan MO, 1
If $n>4$ is a composite number, then $2n$ divides $(n-1)!$.
2010 Malaysia National Olympiad, 8
Find the last digit of \[7^1\times 7^2\times 7^3\times \cdots \times 7^{2009}\times 7^{2010}.\]
2007 Croatia Team Selection Test, 2
Prove that the sequence $a_{n}=\lfloor n\sqrt 2 \rfloor+\lfloor n\sqrt 3 \rfloor$ contains infintely many even and infinitely many odd numbers.
1996 Iran MO (2nd round), 2
Let $a,b,c,d$ be positive integers such that $ab\equal{}cd$. Prove that $a\plus{}b\plus{}c\plus{}d$ is a composite number.
2008 Federal Competition For Advanced Students, Part 2, 2
Which positive integers are missing in the sequence $ \left\{a_n\right\}$, with $ a_n \equal{} n \plus{} \left[\sqrt n\right] \plus{}\left[\sqrt [3]n\right]$ for all $ n \ge 1$? ($ \left[x\right]$ denotes the largest integer less than or equal to $ x$, i.e. $ g$ with $ g \le x < g \plus{} 1$.)
2009 Indonesia MO, 1
Find all positive integers $ n\in\{1,2,3,\ldots,2009\}$ such that
\[ 4n^6 \plus{} n^3 \plus{} 5\]
is divisible by $ 7$.
1985 IMO Longlists, 11
Let $a$ and $ b$ be integers and $n$ a positive integer. Prove that
\[\frac{b^{n-1}a(a + b)(a + 2b) \cdots (a + (n - 1)b)}{n!}\]
is an integer.
2014 India Regional Mathematical Olympiad, 3
let $m,n$ be natural number with $m>n$ . find all such pairs of $(m,n) $
such that
$gcd(n+1,m+1)=gcd(n+2,m+2) =..........=gcd(m, 2m-n) = 1 $
2005 Postal Coaching, 7
Fins all ordered triples $ \left(a,b,c\right)$ of positive integers such that $ abc \plus{} ab \plus{} c \equal{} a^3$.
2008 India Regional Mathematical Olympiad, 5
Three nonzero real numbers $ a,b,c$ are said to be in harmonic progression if $ \frac {1}{a} \plus{} \frac {1}{c} \equal{} \frac {2}{b}$. Find all three term harmonic progressions $ a,b,c$ of strictly increasing positive integers in which $ a \equal{} 20$ and $ b$ divides $ c$.
[17 points out of 100 for the 6 problems]
1993 Romania Team Selection Test, 2
$ x^2 \plus{} y^2 \plus{} z^2 \equal{} 1993$ then prove $ x \plus{} y \plus{} z$ can't be a perfect square:
2000 Turkey Team Selection Test, 1
$(a)$ Prove that for every positive integer $n$, the number of ordered pairs $(x, y)$ of integers satisfying $x^2-xy+y^2 = n$ is divisible by $3.$
$(b)$ Find all ordered pairs of integers satisfying $x^2-xy+y^2=727.$
2013 India Regional Mathematical Olympiad, 6
Let $n \ge 4$ be a natural number. Let $A_1A_2 \cdots A_n$ be a regular polygon and $X = \{ 1,2,3....,n \} $. A subset $\{ i_1, i_2,\cdots, i_k \} $ of $X$, with $k \ge 3$ and $i_1 < i_2 < \cdots < i_k$, is called a good subset if the angles of the polygon $A_{i_1}A_{i_2}\cdots A_{i_k}$ , when arranged in the increasing order, are in an arithmetic progression. If $n$ is a prime, show that a proper good subset of $X$ contains exactly four elements.
1999 Bulgaria National Olympiad, 3
Prove that $x^3+y^3+z^3+t^3=1999$ has infinitely many soln. over $\mathbb{Z}$.
2010 Malaysia National Olympiad, 3
Let $\gamma=\alpha \times \beta$ where \[\alpha=999 \cdots 9\] (2010 '9') and \[\beta=444 \cdots 4\] (2010 '4')
Find the sum of digits of $\gamma$.
2009 India Regional Mathematical Olympiad, 3
Show that $ 3^{2008} \plus{} 4^{2009}$ can be written as product of two positive integers each of which is larger than $ 2009^{182}$.
2008 Brazil National Olympiad, 1
A positive integer is [i]dapper[/i] if at least one of its multiples begins with $ 2008$. For example, $ 7$ is dapper because $ 200858$ is a multiple of $ 7$ and begins with $ 2008$. Observe that $ 200858 \equal{} 28694\times 7$.
Prove that every positive integer is dapper.
1998 Canada National Olympiad, 5
Let $m$ be a positive integer. Define the sequence $a_0, a_1, a_2, \cdots$ by $a_0 = 0,\; a_1 = m,$ and $a_{n+1} = m^2a_n - a_{n-1}$ for $n = 1,2,3,\cdots$.
Prove that an ordered pair $(a,b)$ of non-negative integers, with $a \leq b$, gives a solution to the equation
\[ {\displaystyle \frac{a^2 + b^2}{ab + 1} = m^2} \]
if and only if $(a,b)$ is of the form $(a_n,a_{n+1})$ for some $n \geq 0$.
1981 Bundeswettbewerb Mathematik, 4
Let $X$ be a non empty subset of $\mathbb{N} = \{1,2,\ldots \}$. Suppose that for all $x \in X$, $4x \in X$ and $\lfloor \sqrt{x} \rfloor \in X$. Prove that $X=\mathbb{N}$.
2001 All-Russian Olympiad, 1
The integers from $1$ to $999999$ are partitioned into two groups: the first group consists of those integers for which the closest perfect square is odd, whereas the second group consists of those for which the closest perfect square is even. In which group is the sum of the elements greater?
2007 Hong Kong TST, 6
[url=http://www.mathlinks.ro/Forum/viewtopic.php?t=107262]IMO 2007 HKTST 1[/url]
Problem 6
Determine all pairs $(x,y)$ of positive integers such that $\frac{x^{2}y+x+y}{xy^{2}+y+11}$ is an integer.
1995 China National Olympiad, 3
Let $n(n>1)$ be an odd. We define $x_k=(x^{(k)}_1,x^{(k)}_2,\cdots ,x^{(k)}_n)$ as follow:
$x_0=(x^{(0)}_1,x^{(0)}_2,\cdots ,x^{(0)}_n)=(1,0,\cdots ,0,1)$;
$ x^{(k)}_i =\begin{cases}0, \quad x^{(k-1)}_i=x^{(k-1)}_{i+1},\\ 1, \quad x^{(k-1)}_i\not= x^{(k-1)}_{i+1},\end{cases} $
$i=1,2,\cdots ,n$, where $x^{(k-1)}_{n+1}= x^{(k-1)}_1$.
Let $m$ be a positive integer satisfying $x_0=x_m$. Prove that $m$ is divisible by $n$.