Found problems: 1362
2011 Poland - Second Round, 3
Prove that $\forall x_{1},x_{2},\ldots,x_{2011},y_{1},y_{2},\ldots,y_{2011}\in\mathbb{Z_{+}}$ product:
\[(2x_{1}^{2}+3y_{1}^{2})(2x_{2}^{2}+3y_{2}^{2})\ldots(2x_{2011}^{2}+3y_{2011}^{2})\]
is not a perfect square.
2010 India IMO Training Camp, 8
Call a positive integer [b]good[/b] if either $N=1$ or $N$ can be written as product of [i]even[/i] number of prime numbers, not necessarily distinct.
Let $P(x)=(x-a)(x-b),$ where $a,b$ are positive integers.
(a) Show that there exist distinct positive integers $a,b$ such that $P(1),P(2),\cdots ,P(2010)$ are all good numbers.
(b) Suppose $a,b$ are such that $P(n)$ is a good number for all positive integers $n$. Prove that $a=b$.
2005 MOP Homework, 2
Let $a$, $b$, $c$, and $d$ be positive integers satisfy the following properties:
(a) there are exactly $2004$ pairs of real numbers $(x,y)$ with $0 \le x, y \le 1$ such that both $ax+by$ and $cx+dy$ are integers.
(b) $gcd(a,c)=6$.
Find $gcd(b,d)$.
1988 IMO Longlists, 29
Express the number 1988 as the sum of some positive integers in such a way that the product of these positive integers is maximal.
2009 Federal Competition For Advanced Students, P2, 5
Let $ n>1$ and for $ 1 \leq k \leq n$ let $ p_k \equal{} p_k(a_1, a_2, . . . , a_n)$ be the sum of the products of all possible combinations of k of the numbers $ a_1,a_2,...,a_n$. Furthermore let $ P \equal{} P(a_1, a_2, . . . , a_n)$ be the sum of all $ p_k$ with odd values of $ k$ less than or equal to $ n$.
How many different values are taken by $ a_j$ if all the numbers $ a_j (1 \leq j \leq n)$ and $ P$ are prime?
2006 Tuymaada Olympiad, 2
We call a sequence of integers a [i]Fibonacci-type sequence[/i] if it is infinite in both ways and $a_{n}=a_{n-1}+a_{n-2}$ for any $n\in\mathbb{Z}$. How many [i]Fibonacci-type sequences[/i] can we find, with the property that in these sequences there are two consecutive terms, strictly positive, and less or equal than $N$ ? (two sequences are considered to be the same if they differ only by shifting of indices)
[i]Proposed by I. Pevzner[/i]
1993 APMO, 4
Determine all positive integers $n$ for which the equation
\[ x^n + (2+x)^n + (2-x)^n = 0 \]
has an integer as a solution.
2008 Argentina National Olympiad, 5
Find all perfect powers whose last $ 4$ digits are $ 2,0,0,8$, in that order.
2002 China Team Selection Test, 3
For positive integers $a,b,c$ let $ \alpha, \beta, \gamma$ be pairwise distinct positive integers such that
\[ \begin{cases}{c} \displaystyle a &= \alpha + \beta + \gamma, \\
b &= \alpha \cdot \beta + \beta \cdot \gamma + \gamma \cdot \alpha, \\
c^2 &= \alpha\beta\gamma. \end{cases} \]
Also, let $ \lambda$ be a real number that satisfies the condition
\[\lambda^4 -2a\lambda^2 + 8c\lambda + a^2 - 4b = 0.\]
Prove that $\lambda$ is an integer if and only if $\alpha, \beta, \gamma$ are all perfect squares.
2012 Regional Olympiad of Mexico Center Zone, 2
Let $m, n$ integers such that:
$(n-1)^3+n^3+(n+1)^3=m^3$
Prove that 4 divides $n$
2010 Singapore Senior Math Olympiad, 4
An infinite sequence of integers, $a_0,a_1,a_2,\dots,$ with $a_0>0$, has the property that for $n\ge 0$, $a_{n+1}=a_n-b_n$, where $b_n$ is the number having the same sign as $a_n$, but having the digits written in the reverse order. For example if $a_0=1210,a_1=1089$ and $a_2=-8712$, etc. Find the smallest value of $a_0$ so that $a_n\neq 0$ for all $n\ge 1$.
1997 Hungary-Israel Binational, 1
Is there an integer $ N$ such that $ \left(\sqrt{1997}\minus{}\sqrt{1996}\right)^{1998}\equal{}\sqrt{N}\minus{}\sqrt{N\minus{}1}$?
2011 District Round (Round II), 1
Among all eight-digit multiples of four, are there more numbers with the digit $1$ or without the digit $1$ in their decimal representation?
1998 India National Olympiad, 6
It is desired to choose $n$ integers from the collection of $2n$ integers, namely, $0,0,1,1,2,2,\ldots,n-1,n-1$ such that the average of these $n$ chosen integers is itself an integer and as minimum as possible. Show that this can be done for each positive integer $n$ and find this minimum value for each $n$.
2006 Costa Rica - Final Round, 2
Let $n$ be a positive integer, and let $p$ be a prime, such that $n>p$.
Prove that :
\[ \displaystyle \binom np \equiv \left\lfloor\frac{n}{p}\right\rfloor \ \pmod p. \]
2012 Baltic Way, 16
Let $n$, $m$, and $k$ be positive integers satisfying $(n - 1)n(n + 1) = m^k$. Prove that $k = 1$.
1984 IMO Longlists, 56
Let $a, b, c$ be nonnegative integers such that $a \le b \le c, 2b \neq a + c$ and $\frac{a+b+c}{3}$ is an integer. Is it possible to find three nonnegative integers $d, e$, and $f$ such that $d \le e \le f, f \neq c$, and such that $a^2+b^2+c^2 = d^2 + e^2 + f^2$?
2006 Irish Math Olympiad, 4
Let $n$ be a positive integer.
Find the greatest common divisor of the numbers $\binom{2n}{1},\binom{2n}{3},\binom{2n}{5},...,\binom{2n}{2n-1}$.
1985 IberoAmerican, 2
To each positive integer $ n$ it is assigned a non-negative integer $f(n)$ such that the following conditions are satisfied:
(1) $ f(rs) \equal{} f(r)\plus{}f(s)$
(2) $ f(n) \equal{} 0$, if the first digit (from right to left) of $ n$ is 3.
(3) $ f(10) \equal{} 0$.
Find $f(1985)$. Justify your answer.
2007 Pre-Preparation Course Examination, 5
Prove that the equation
\[y^3=x^2+5\]
doesn't have any solutions in $Z$.
1976 IMO Longlists, 16
Prove that there is a positive integer $n$ such that the decimal representation of $7^n$ contains a block of at least $m$ consecutive zeros, where $m$ is any given positive integer.
2012 Kazakhstan National Olympiad, 3
Consider the equation $ax^{2}+by^{2}=1$, where $a,b$ are fixed rational numbers. Prove that either such an equation has no solutions in rational numbers, or it has infinitely many solutions.
1991 Cono Sur Olympiad, 2
Two people, $A$ and $B$, play the following game: $A$ start choosing a positive integrer number and then, each player in it's turn, say a number due to the following rule:
If the last number said was odd, the player add $7$ to this number;
If the last number said was even, the player divide it by $2$.
The winner is the player that repeats the first number said. Find all numbers that $A$ can choose in order to win. Justify your answer.
2002 Korea - Final Round, 1
For a prime $p$ of the form $12k+1$ and $\mathbb{Z}_p=\{0,1,2,\cdots,p-1\}$, let
\[\mathbb{E}_p=\{(a,b) \mid a,b \in \mathbb{Z}_p,\quad p\nmid 4a^3+27b^2\}\]
For $(a,b), (a',b') \in \mathbb{E}_p$ we say that $(a,b)$ and $(a',b')$ are equivalent if there is a non zero element $c\in \mathbb{Z}_p$ such that $p\mid (a' -ac^4)$ and $p\mid (b'-bc^6)$. Find the maximal number of inequivalent elements in $\mathbb{E}_p$.
2014 Contests, 4
(a) Let $a,x,y$ be positive integers. Prove: if $x\ne y$, the also
\[ax+\gcd(a,x)+\text{lcm}(a,x)\ne ay+\gcd(a,y)+\text{lcm}(a,y).\]
(b) Show that there are no two positive integers $a$ and $b$ such that
\[ab+\gcd(a,b)+\text{lcm}(a,b)=2014.\]