Found problems: 1362
2011 Macedonia National Olympiad, 3
Find all natural numbers $n$ for which each natural number written with $~$ $n-1$ $~$ 'ones' and one 'seven' is prime.
2016 China Girls Math Olympiad, 8
Let $\mathbb{Q}$ be the set of rational numbers, $\mathbb{Z}$ be the set of integers. On the coordinate plane, given positive integer $m$, define $$A_m = \left\{ (x,y)\mid x,y\in\mathbb{Q}, xy\neq 0, \frac{xy}{m}\in \mathbb{Z}\right\}.$$
For segment $MN$, define $f_m(MN)$ as the number of points on segment $MN$ belonging to set $A_m$.
Find the smallest real number $\lambda$, such that for any line $l$ on the coordinate plane, there exists a constant $\beta (l)$ related to $l$, satisfying: for any two points $M,N$ on $l$, $$f_{2016}(MN)\le \lambda f_{2015}(MN)+\beta (l)$$
2003 Tournament Of Towns, 2
Prove that every positive integer can be represented in the form
\[3^{u_1} \ldots 2^{v_1} + 3^{u_2} \ldots 2^{v_2} + \ldots + 3^{u_k} \ldots 2^{v_k}\]
with integers $u_1, u_2, \ldots , u_k, v_1, \ldots, v_k$ such that $u_1 > u_2 >\ldots > u_k\ge 0$ and $0 \le v_1 < v_2 <\ldots < v_k$.
1982 IMO Longlists, 15
Show that the set $S$ of natural numbers $n$ for which $\frac{3}{n}$ cannot be written as the sum of two reciprocals of natural numbers ($S =\left\{n |\frac{3}{n} \neq \frac{1}{p} + \frac{1}{q} \text{ for any } p, q \in \mathbb N \right\}$) is not the union of finitely many arithmetic progressions.
2014 Contests, 1
Let be $n$ a positive integer. Denote all its (positive) divisors as $1=d_1<d_2<\cdots<d_{k-1}<d_k=n$.
Find all values of $n$ satisfying $d_5-d_3=50$ and $11d_5+8d_7=3n$.
(Day 1, 1st problem
author: Matúš Harminc)
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
\[\max_{1\le j \le 3}\xi_j \le M\ \ \ \ \text{and} \ \ \ \left|\sum_{j=1}^3 u_{jk}\xi_k\right| \le \frac{MN}{l} \ \ \ \ \text{for k=1,2}\]
1989 Vietnam National Olympiad, 2
The Fibonacci sequence is defined by $ F_1 \equal{} F_2 \equal{} 1$ and $ F_{n\plus{}1} \equal{} F_n \plus{}F_{n\minus{}1}$ for $ n > 1$. Let $ f(x) \equal{} 1985x^2 \plus{} 1956x \plus{} 1960$. Prove that there exist infinitely many natural numbers $ n$ for which $ f(F_n)$ is divisible by $ 1989$. Does there exist $ n$ for which $ f(F_n) \plus{} 2$ is divisible by $ 1989$?
2005 International Zhautykov Olympiad, 3
Find all prime numbers $ p,q < 2005$ such that $ q | p^{2} \plus{} 8$ and $ p|q^{2} \plus{} 8.$
1993 China Team Selection Test, 1
Find all integer solutions to $2 x^4 + 1 = y^2.$
2009 ISI B.Math Entrance Exam, 6
Let $a,b,c,d$ be integers such that $ad-bc$ is non zero. Suppose $b_1,b_2$ are integers both of which are multiples of $ad-bc$. Prove that there exist integers simultaneously satisfying both the equalities $ax+by=b_1, cx+dy=b_2$.
2011 Akdeniz University MO, 5
For all $n \in {\mathbb Z^+}$ we define
$$I_n=\{\frac{0}{n},\frac{1}{n},\frac{2}{n},\dotsm,\frac{n-1}{n},\frac{n}{n},\frac{n+1}{n},\dotsm\}$$
infinite cluster. For whichever $x$ and $y$ real number, we say $\mid{x-y}\mid$ is between distance of the $x$ and $y$.
[b]a[/b]) For all $n$'s we find a number in $I_n$ such that, the between distance of the this number and $\sqrt 2$ is less than $\frac{1}{2n}$
[b]b[/b]) We find a $n$ such that, between distance of the a number in $I_n$ and $\sqrt 2$ is less than $\frac{1}{2011n}$
2010 Paenza, 1
a) Replace each letter in the following sum by a digit from $0$ to $9$, in such a way that the sum is correct.
$\tab$ $\tab$ $ABC$
$\tab$ $\tab$ $DEF$
[u]$+GHI$[/u]
$\tab$ $\tab$ $\tab$ $J J J$
Different letters must be replaced by different digits, and equal letters must be replaced by equal digits. Numbers $ABC$, $DEF$, $GHI$ and $JJJ$ cannot begin by $0$.
b) Determine how many triples of numbers $(ABC,DEF,GHI)$ can be formed under the conditions given in a).
2009 Baltic Way, 7
Suppose that for a prime number $p$ and integers $a,b,c$ the following holds:
\[6\mid p+1,\quad p\mid a+b+c,\quad p\mid a^4+b^4+c^4.\]
Prove that $p\mid a,b,c$.
2004 Finnish National High School Mathematics Competition, 5
Finland is going to change the monetary system again and replace the Euro by the Finnish Mark.
The Mark is divided into $100$ pennies.
There shall be coins of three denominations only, and the number of coins a person has to carry in order to be able
to pay for any purchase less than one mark should be minimal.
Determine the coin denominations.
2009 Mexico National Olympiad, 2
In boxes labeled $0$, $1$, $2$, $\dots$, we place integers according to the following rules:
$\bullet$ If $p$ is a prime number, we place it in box $1$.
$\bullet$ If $a$ is placed in box $m_a$ and $b$ is placed in box $m_b$, then $ab$ is placed in the box labeled $am_b+bm_a$.
Find all positive integers $n$ that are placed in the box labeled $n$.
2010 Postal Coaching, 5
Prove that there exist a set of $2010$ natural numbers such that product of any $1006 $ numbers is divisible by product of remaining $1004$ numbers.
2008 China National Olympiad, 1
Let $A$ be an infinite subset of $\mathbb{N}$, and $n$ a fixed integer. For any prime $p$ not dividing $n$, There are infinitely many elements of $A$ not divisible by $p$. Show that for any integer $m >1, (m,n) =1$, There exist finitely many elements of $A$, such that their sum is congruent to 1 modulo $m$ and congruent to 0 modulo $n$.
2009 Germany Team Selection Test, 1
Let $p > 7$ be a prime which leaves residue 1 when divided by 6. Let $m=2^p-1,$ then prove $2^{m-1}-1$ can be divided by $127m$ without residue.
2001 Tournament Of Towns, 7
Alex thinks of a two-digit integer (any integer between $10$ and $99$). Greg is trying to guess it. If the number Greg names is correct, or if one of its digits is equal to the corresponding digit of Alex’s number and the other digit differs by one from the corresponding digit of Alex’s number, then Alex says “hot”; otherwise, he says “cold”. (For example, if Alex’s number was $65$, then by naming any of $64, 65, 66, 55$ or $75$ Greg will be answered “hot”, otherwise he will be answered “cold”.)
[list][b](a)[/b] Prove that there is no strategy which guarantees that Greg will guess Alex’s number in no more than 18 attempts.
[b](b)[/b] Find a strategy for Greg to find out Alex’s number (regardless of what the chosen number was) using no more than $24$ attempts.
[b](c)[/b] Is there a $22$ attempt winning strategy for Greg?[/list]
1985 IMO Longlists, 90
Factorise $ 5^{1985}\minus{}1$ as a product of three integers, each greater than $ 5^{100}$.
2014 Kazakhstan National Olympiad, 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 $n$?
2005 Moldova Team Selection Test, 2
Let $m\in N$ and $E(x,y,m)=(\frac{72}x)^m+(\frac{72}y)^m-x^m-y^m$, where $x$ and $y$ are positive divisors of 72.
a) Prove that there exist infinitely many natural numbers $m$ so, that 2005 divides $E(3,12,m)$ and $E(9,6,m)$.
b) Find the smallest positive integer number $m_0$ so, that 2005 divides $E(3,12,m_0)$ and $E(9,6,m_0)$.
2008 Bulgaria National Olympiad, 1
Find the smallest natural number $ k$ for which there exists natural numbers $ m$ and $ n$ such that $ 1324 \plus{} 279m \plus{} 5^n$ is $ k$-th power of some natural number.
2004 CHKMO, 4
Find all integer numbers $a,b,c$ such that $\frac{(a+b)(b+c)(c+a)}{2}+(a+b+c)^{3}=1-abc$.
2005 MOP Homework, 7
Let $a$, $b$, and $c$ be pairwise distinct positive integers, which are side lengths of a triangle. There is a line which cuts both the area and the perimeter of the triangle into two equal parts. This line cuts the longest side of the triangle into two parts with ratio $2:1$. Determine $a$, $b$, and $c$ for which the product abc is minimal.