Found problems: 1362
2012 Federal Competition For Advanced Students, Part 2, 2
Solve over $\mathbb{Z}$:
\[ x^4y^3(y-x)=x^3y^4-216 \]
2005 India IMO Training Camp, 1
Let $0 <a <b$ be two rational numbers. Let $M$ be a set of positive real numbers with the properties:
(i) $a \in M$ and $b \in M$;
(ii) if $x$ $\in M$ and $y \in M$, then $\sqrt{xy} \in M$.
Let $M^*$denote the set of all irrational numbers in $M$. prove that every $c,d$ such that $a <c <d<b$, $M^*$ contains an element $m$ with property $c<m<d$
2013 Baltic Way, 16
We call a positive integer $n$ [i]delightful[/i] if there exists an integer $k$, $1 < k < n$, such that
\[1+2+\cdots+(k-1)=(k+1)+(k+2)+\cdots+n\]
Does there exist a delightful number $N$ satisfying the inequalities
\[2013^{2013}<\dfrac{N}{2013^{2013}}<2013^{2013}+4 ?\]
2010 Contests, 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 $n\ge 1$.
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.
2005 Postal Coaching, 1
Consider the sequence $<{a_n}>$ of natural numbers such that
{i} $a_n$ is a square numver for all $n$ ;
(ii) $a_{n+1} - a_n$ is either a prime or a square of a prime for each $n$.
Show that $<a_n>$ is a finite sequence. Determine the longest such sequence.
2001 Irish Math Olympiad, 1
Find all positive integer solutions $ (a,b,c,n)$ of the equation: $ 2^n\equal{}a!\plus{}b!\plus{}c!$.
2007 China Second Round Olympiad, 3
For positive integers $k,m$, where $1\le k\le 5$, define the function $f(m,k)$ as
\[f(m,k)=\sum_{i=1}^{5}\left[m\sqrt{\frac{k+1}{i+1}}\right]\]
where $[x]$ denotes the greatest integer not exceeding $x$. Prove that for any positive integer $n$, there exist positive integers $k,m$, where $1\le k\le 5$, such that $f(m,k)=n$.
1986 Federal Competition For Advanced Students, P2, 4
Find the largest $ n$ for which there is a natural number $ N$ with $ n$ decimal digits which are all different such that $ n!$ divides $ N$. Furthermore, for this largest $ n$ find all possible numbers $ N$.
2011 China Second Round Olympiad, 8
Given that $a_{n}= \binom{200}{n} \cdot 6^{\frac{200-n}{3}} \cdot (\dfrac{1}{\sqrt{2}})^n$ ($ 1 \leq n \leq 95$). How many integers are there in the sequence $\{a_n\}$?
2013 India Regional Mathematical Olympiad, 5
Let $a_1,b_1,c_1$ be natural numbers. We define \[a_2=\gcd(b_1,c_1),\,\,\,\,\,\,\,\,b_2=\gcd(c_1,a_1),\,\,\,\,\,\,\,\,c_2=\gcd(a_1,b_1),\] and \[a_3=\operatorname{lcm}(b_2,c_2),\,\,\,\,\,\,\,\,b_3=\operatorname{lcm}(c_2,a_2),\,\,\,\,\,\,\,\,c_3=\operatorname{lcm}(a_2,b_2).\] Show that $\gcd(b_3,c_3)=a_2$.
2011 Thailand Mathematical Olympiad, 5
Find all $n$ such that \[n = d (n) ^ 4\]
Where $d (n)$ is the number of divisors of $n$, for example $n = 2 \cdot 3\cdot 5\implies d (n) = 2 \cdot 2\cdot 2$.
1987 China Team Selection Test, 2
Find all positive integer $n$ such that the equation $x^3+y^3+z^3=n \cdot x^2 \cdot y^2 \cdot z^2$ has positive integer solutions.
2010 Kyrgyzstan National Olympiad, 7
Find all natural triples $(a,b,c)$, such that:
$a - )\,a \le b \le c$
$b - )\,(a,b,c) = 1$
$c - )\,\left. {{a^2}b} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{b^2}c} \right|{a^3} + {b^3} + {c^3}\,,\,\left. {{c^2}a} \right|{a^3} + {b^3} + {c^3}$.
1985 IMO Longlists, 67
Let $k \geq 2$ and $n_1, n_2, . . . , n_k \geq 1$ natural numbers having the property $n_2 | 2^{n_1} - 1, n_3 | 2^{n_2} -1 , \cdots, n_k | 2^{n_k-1}-1$, and $n_1 | 2^{n_k} - 1$. Show that $n_1 = n_2 = \cdots = n_k = 1.$
1976 IMO Longlists, 38
Let $x =\sqrt{a}+\sqrt{b}$, where $a$ and $b$ are natural numbers, $x$ is not an integer, and $x < 1976$. Prove that the fractional part of $x$ exceeds $10^{-19.76}$.
2014 Postal Coaching, 4
Denote by $F_n$ the $n^{\text{th}}$ Fibonacci number $(F_1=F_2=1)$.Prove that if $a,b,c$ are positive integers such that $a| F_b,b|F_c,c|F_a$,then either $5$ divides each of $a,b,c$ or $12$ divides each of $a,b,c$.
1998 Irish Math Olympiad, 5
If $ x$ is a real number such that $ x^2\minus{}x$ and $ x^n\minus{}x$ are integers for some $ n \ge 3$, prove that $ x$ is an integer.
2011 China Team Selection Test, 2
Let $a_1,a_2,\ldots,a_n,\ldots$ be any permutation of all positive integers. Prove that there exist infinitely many positive integers $i$ such that $\gcd(a_i,a_{i+1})\leq \frac{3}{4} i$.
1989 IMO Longlists, 19
Let $ a_1, \ldots, a_n$ be distinct positive integers that do not contain a $ 9$ in their decimal representations. Prove that the following inequality holds
\[ \sum^n_{i\equal{}1} \frac{1}{a_i} \leq 30.\]
1997 Vietnam Team Selection Test, 1
The function $ f : \mathbb{N} \to \mathbb{Z}$ is defined by $ f(0) \equal{} 2$, $ f(1) \equal{} 503$ and $ f(n \plus{} 2) \equal{} 503f(n \plus{} 1) \minus{} 1996f(n)$ for all $ n \in\mathbb{N}$. Let $ s_1$, $ s_2$, $ \ldots$, $ s_k$ be arbitrary integers not smaller than $ k$, and let $ p(s_i)$ be an arbitrary prime divisor of $ f\left(2^{s_i}\right)$, ($ i \equal{} 1, 2, \ldots, k$). Prove that, for any positive integer $ t$ ($ t\le k$), we have $ 2^t \Big | \sum_{i \equal{} 1}^kp(s_i)$ if and only if $ 2^t | k$.
2002 Baltic Way, 16
Find all nonnegative integers $m$ such that
\[a_m=(2^{2m+1})^2+1 \]
is divisible by at most two different primes.
2011 Albania Team Selection Test, 4
Find all prime numbers p such that $2^p+p^2 $ is also a prime number.
2012 Kazakhstan National Olympiad, 1
Solve the equation $p+\sqrt{q^{2}+r}=\sqrt{s^{2}+t}$ in prime numbers.
2004 Rioplatense Mathematical Olympiad, Level 3, 1
How many integers $n>1$ are there such that $n$ divides $x^{13}-x$ for every positive integer $x$?