Found problems: 1362
1972 IMO Longlists, 32
If $n_1, n_2, \cdots, n_k$ are natural numbers and $n_1+n_2+\cdots+n_k = n$,
show that
\[max(n_1n_2\cdots n_k)=(t + 1)^rt^{k-r},\]
where $t =\left[\frac{n}{k}\right]$ and $r$ is the remainder of $n$ upon division by $k$; i.e., $n = tk + r, 0 \le r \le k- 1$.
1992 IMTS, 5
An infinite checkerboard is divided by a horizontal line into upper and lower halves as shown on the right. A number of checkers are to be placed on the board below the line (within the squares). A "move" consists of one checker jumping horizontally or vertically over a second checker, and removing the second checker. What is the minimum value of $n$ which will allow the placement of the last checker in row 4 above the dividing horizontal line after $n-1$ moves? Describe the initial position of the checkers as well as each of the moves.
Picture: http://www.cms.math.ca/Competitions/IMTS/imts6.gif
1977 Bundeswettbewerb Mathematik, 1
Does there exist two infinite sets $A,B$ such that every number can be written uniquely as a sum of an element of $A$ and an element of $B$?
2010 Contests, 1
Let $a,b$ be two positive integers and $a>b$.We know that $\gcd(a-b,ab+1)=1$ and $\gcd(a+b,ab-1)=1$. Prove that $(a-b)^2+(ab+1)^2$ is not a perfect square.
2006 South East Mathematical Olympiad, 3
[b](1)[/b] Find the number of positive integer solutions $(m,n,r)$ of the indeterminate equation $mn+nr+mr=2(m+n+r)$.
[b](2)[/b] Given an integer $k (k>1)$, prove that indeterminate equation $mn+nr+mr=k(m+n+r)$ has at least $3k+1$ positive integer solutions $(m,n,r)$.
2001 Italy TST, 3
Find all pairs $ (p, q)$ of prime numbers such that $ p$ divides $ 5^q \plus{} 1$ and $ q$ divides $ 5^p \plus{} 1$.
2010 Tournament Of Towns, 4
Can it happen that the sum of digits of some positive integer $n$ equals $100$ while the sum of digits of number $n^3$ equals $100^3$?
2008 Nordic, 4
The difference between the cubes of two consecutive positive integers is equal to $n^2$ for a positive integer $n$. Show that $n$ is the sum of two squares.
2002 Vietnam Team Selection Test, 3
Prove that there exists an integer $n$, $n\geq 2002$, and $n$ distinct positive integers $a_1,a_2,\ldots,a_n$ such that the number $N= a_1^2a_2^2\cdots a_n^2 - 4(a_1^2+a_2^2+\cdots + a_n^2) $ is a perfect square.
1994 China Team Selection Test, 1
Find all sets comprising of 4 natural numbers such that the product of any 3 numbers in the set leaves a remainder of 1 when divided by the remaining number.
2003 All-Russian Olympiad, 2
Let $ a_0$ be a natural number. The sequence $ (a_n)$ is defined by $ a_{n\plus{}1}\equal{}\frac{a_n}{5}$ if $ a_n$ is divisible by $ 5$
and $ a_{n\plus{}1}\equal{}[a_n \sqrt{5}]$ otherwise . Show that the sequence $ a_n$ is increasing starting from some term.
2006 Moldova National Olympiad, 10.2
Let $n$ be a positive integer, $n\geq 2$. Let $M=\{0,1,2,\ldots n-1\}$. For an integer nonzero number $a$ we define the function $f_{a}: M\longrightarrow M$, such that $f_{a}(x)$ is the remainder when dividing $ax$ at $n$. Find a necessary and sufficient condition such that $f_{a}$ is bijective. And if $f_{a}$ is bijective and $n$ is a prime number, prove that $a^{n(n-1)}-1$ is divisible by $n^{2}$.
2009 Canada National Olympiad, 4
Find all ordered pairs of integers $(a,b)$ such that $3^a + 7^b$ is a perfect square.
2001 Tournament Of Towns, 2
There exists a block of 1000 consecutive positive integers containing no prime numbers, namely, $1001!+2,1001!+3,...,1001!+1001$. Does there exist a block of 1000 consecutive positive intgers containing exactly five prime numbers?
2006 Bulgaria Team Selection Test, 2
[b] Problem 5. [/b]Denote with $d(a,b)$ the numbers of the divisors of natural $a$, which are greater or equal to $b$. Find all natural $n$, for which
$d(3n+1,1)+d(3n+2,2)+\ldots+d(4n,n)=2006.$
[i]Ivan Landgev[/i]
2002 China Team Selection Test, 2
Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \cdots k_{2002}$ such that for any positive integer $ n \geq 2001$, one of $ k_12^n \plus{} 1, k_22^n \plus{} 1, \cdots, k_{2002}2^n \plus{} 1$ is prime?
2011 USA Team Selection Test, 3
Let $p$ be a prime. We say that a sequence of integers $\{z_n\}_{n=0}^\infty$ is a [i]$p$-pod[/i] if for each $e \geq 0$, there is an $N \geq 0$ such that whenever $m \geq N$, $p^e$ divides the sum
\[\sum_{k=0}^m (-1)^k {m \choose k} z_k.\]
Prove that if both sequences $\{x_n\}_{n=0}^\infty$ and $\{y_n\}_{n=0}^\infty$ are $p$-pods, then the sequence $\{x_ny_n\}_{n=0}^\infty$ is a $p$-pod.
2003 Turkey MO (2nd round), 1
Suppose that $2^{2n+1}+ 2^{n}+1=x^{k}$, where $k\geq2$ and $n$ are positive integers. Find all possible values of $n$.
2002 Vietnam Team Selection Test, 3
Let $m$ be a given positive integer which has a prime divisor greater than $\sqrt {2m} +1 $. Find the minimal positive integer $n$ such that there exists a finite set $S$ of distinct positive integers satisfying the following two conditions:
[b]I.[/b] $m\leq x\leq n$ for all $x\in S$;
[b]II.[/b] the product of all elements in $S$ is the square of an integer.
2010 Contests, 2
For any set $A=\{a_1,a_2,\cdots,a_m\}$, let $P(A)=a_1a_2\cdots a_m$. Let $n={2010\choose99}$, and let $A_1, A_2,\cdots,A_n$ be all $99$-element subsets of $\{1,2,\cdots,2010\}$. Prove that $2010|\sum^{n}_{i=1}P(A_i)$.
2010 Korea - Final Round, 6
An arbitrary prime $ p$ is given. If an integer sequence $ (n_1 , n_2 , \cdots , n_k )$ satisfying the conditions
- For all $ i\equal{} 1, 2, \cdots , k$, $ n_i \geq \frac{p\plus{}1}{2}$
- For all $ i\equal{} 1, 2, \cdots , k$, $ p^{n_i} \minus{} 1$ is divisible by $ n_{i\plus{}1}$, and $ \frac{p^{n_i} \minus{} 1}{n_{i\plus{}1}}$ is coprime to $ n_{i\plus{}1}$. Let $ n_{k\plus{}1} \equal{} n_1$.
exists not for $ k\equal{}1$, but exists for some $ k \geq 2$, then call the prime a good prime.
Prove that a prime is good iff it is not $ 2$.
2002 China Team Selection Test, 2
Does there exist $ 2002$ distinct positive integers $ k_1, k_2, \cdots k_{2002}$ such that for any positive integer $ n \geq 2001$, one of $ k_12^n \plus{} 1, k_22^n \plus{} 1, \cdots, k_{2002}2^n \plus{} 1$ is prime?
2006 Estonia National Olympiad, 1
Find all pairs of positive integers $ (a, b)$ such that
\[ ab \equal{} gcd(a, b) \plus{} lcm(a, b).
\]
2005 MOP Homework, 7
Find all positive integers $n$ for which there are distinct integers $a_1$, ..., $a_n$ such that
$\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1+a_2+...+a_n}{2}$.
2010 Finnish National High School Mathematics Competition, 2
Determine the least $n\in\mathbb{N}$ such that $n!=1\cdot 2\cdot 3\cdots (n-1)\cdot n$ has at least $2010$ positive factors.