Found problems: 2008
1990 China Team Selection Test, 3
Prove that for every integer power of 2, there exists a multiple of it with all digits (in decimal expression) not zero.
2011 ELMO Shortlist, 4
Let $p>13$ be a prime of the form $2q+1$, where $q$ is prime. Find the number of ordered pairs of integers $(m,n)$ such that $0\le m<n<p-1$ and
\[3^m+(-12)^m\equiv 3^n+(-12)^n\pmod{p}.\]
[i]Alex Zhu.[/i]
[hide="Note"]The original version asked for the number of solutions to $2^m+3^m\equiv 2^n+3^n\pmod{p}$ (still $0\le m<n<p-1$), where $p$ is a Fermat prime.[/hide]
1989 IMO Shortlist, 11
Define sequence $ (a_n)$ by $ \sum_{d|n} a_d \equal{} 2^n.$ Show that $ n|a_n.$
2002 Putnam, 6
Let $p$ be a prime number. Prove that the determinant of the matrix \[ \begin{bmatrix}x & y & z\\ x^p & y^p & z^p \\ x^{p^2} & y^{p^2} & z^{p^2} \end{bmatrix} \] is congruent modulo $p$ to a product of polynomials of the form $ax+by+cz$, where $a$, $b$, and $c$ are integers. (We say two integer polynomials are congruent modulo $p$ if corresponding coefficients are congruent modulo $p$.)
2005 Finnish National High School Mathematics Competition, 4
The numbers $1, 3, 7$ and $9$ occur in the decimal representation of an integer.
Show that permuting the order of digits one can obtain an integer divisible by $7.$
2007 Princeton University Math Competition, 5
For how many integers $x \in [0, 2007]$ is $\frac{6x^3+53x^2+61x+7}{2x^2+17x+15}$ reducible?
2010 Contests, 3
Let $n$ be a positive integer. Let $a$ be an integer such that $\gcd (a,n)=1$. Prove that
\[\frac{a^{\phi (n)}-1}{n}=\sum_{i\in R}\frac{1}{ai}\left[\frac{ai}{n}\right]\pmod{n}\]
where $R$ is the reduced residue system of $n$ with each element a positive integer at most $n$.
2010 Slovenia National Olympiad, 1
Let $a,b,c$ be positive integers. Prove that $a^2+b^2+c^2$ is divisible by $4$, if and only if $a,b,c$ are even.
2014 Czech-Polish-Slovak Match, 5
Let all positive integers $n$ satisfy the following condition:
for each non-negative integers $k, m$ with $k + m \le n$,
the numbers $\binom{n-k}{m}$ and $\binom{n-m}{k}$ leave the same remainder when divided by $2$.
(Poland)
PS. The translation was done using Google translate and in case it is not right, there is the original text in Slovak
PEN S Problems, 36
For every natural number $n$, denote $Q(n)$ the sum of the digits in the decimal representation of $n$. Prove that there are infinitely many natural numbers $k$ with $Q(3^{k})>Q(3^{k+1})$.
1967 IMO Shortlist, 4
Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.
2012 Morocco TST, 1
Find all positive integers $n, k$ such that $(n-1)!=n^{k}-1$.
1989 IMO, 5
Prove that for each positive integer $ n$ there exist $ n$ consecutive positive integers none of which is an integral power of a prime number.
2010 International Zhautykov Olympiad, 2
In every vertex of a regular $n$ -gon exactly one chip is placed. At each $step$ one can exchange any two neighbouring chips. Find the least number of steps necessary to reach the arrangement where every chip is moved by $[\frac{n}{2}]$ positions clockwise from its initial position.
2013 Stars Of Mathematics, 3
Consider the sequence $(3^{2^n} + 1)_{n\geq 1}$.
i) Prove there exist infinitely many primes, none dividing any term of the sequence.
ii) Prove there exist infinitely many primes, each dividing some term of the sequence.
[i](Dan Schwarz)[/i]
2013 Baltic Way, 19
Let $a_0$ be a positive integer and $a_n=5a_{n-1}+4$ for all $n\ge 1$. Can $a_0$ be chosen so that $a_{54}$ is a multiple of $2013$?
2010 India IMO Training Camp, 3
For any integer $n\ge 2$, let $N(n)$ be the maximum number of triples $(a_j,b_j,c_j),j=1,2,3,\cdots ,N(n),$ consisting of non-negative integers $a_j,b_j,c_j$ (not necessarily distinct) such that the following two conditions are satisfied:
(a) $a_j+b_j+c_j=n,$ for all $j=1,2,3,\cdots N(n)$;
(b) $j\neq k$, then $a_j\neq a_k$, $b_j\neq b_k$ and $c_j\neq c_k$.
Determine $N(n)$ for all $n\ge 2$.
2017 China Team Selection Test, 1
Let $n$ be a positive integer. Let $D_n$ be the set of all divisors of $n$ and let $f(n)$ denote the smallest natural $m$ such that the elements of $D_n$ are pairwise distinct in mod $m$. Show that there exists a natural $N$ such that for all $n \geq N$, one has $f(n) \leq n^{0.01}$.
2008 CHKMO, 2
is there any polynomial of $deg=2007$ with integer coefficients,such that for any integer $n$,$f(n),f(f(n)),f(f(f(n))),...$ is coprime to each other?
2012 Online Math Open Problems, 26
Find the smallest positive integer $k$ such that
\[\binom{x+kb}{12} \equiv \binom{x}{12} \pmod{b}\]
for all positive integers $b$ and $x$. ([i]Note:[/i] For integers $a,b,c$ we say $a \equiv b \pmod c$ if and only if $a-b$ is divisible by $c$.)
[i]Alex Zhu.[/i]
[hide="Clarifications"][list=1][*]${{y}\choose{12}} = \frac{y(y-1)\cdots(y-11)}{12!}$ for all integers $y$. In particular, ${{y}\choose{12}} = 0$ for $y=1,2,\ldots,11$.[/list][/hide]
PEN D Problems, 19
Let $a_{1}$, $\cdots$, $a_{k}$ and $m_{1}$, $\cdots$, $m_{k}$ be integers with $2 \le m_{1}$ and $2m_{i}\le m_{i+1}$ for $1 \le i \le k-1$. Show that there are infinitely many integers $x$ which do not satisfy any of congruences \[x \equiv a_{1}\; \pmod{m_{1}}, x \equiv a_{2}\; \pmod{m_{2}}, \cdots, x \equiv a_{k}\; \pmod{m_{k}}.\]
2003 France Team Selection Test, 1
A lattice point in the coordinate plane with origin $O$ is called invisible if the segment $OA$ contains a lattice point other than $O,A$. Let $L$ be a positive integer. Show that there exists a square with side length $L$ and sides parallel to the coordinate axes, such that all points in the square are invisible.
2008 Junior Balkan MO, 3
Find all prime numbers $ p,q,r$, such that $ \frac{p}{q}\minus{}\frac{4}{r\plus{}1}\equal{}1$
2001 IberoAmerican, 1
We say that a natural number $n$ is [i]charrua[/i] if it satisfy simultaneously the following conditions:
- Every digit of $n$ is greater than 1.
- Every time that four digits of $n$ are multiplied, it is obtained a divisor of $n$
Show that every natural number $k$ there exists a [i]charrua[/i] number with more than $k$ digits.
2014 ISI Entrance Examination, 8
$n(>1)$ lotus leaves are arranged in a circle. A frog jumps from a particular leaf from another under the following rule:
[list]
[*]It always moves clockwise.
[*]From starting it skips one leaf and then jumps to the next. After that it skips two leaves and jumps to the following. And the process continues. (Remember the frog might come back on a leaf twice or more.)[/list]
Given that it reaches all leaves at least once. Show $n$ cannot be odd.