Show that $ 3^{2008} \plus{} 4^{2009}$ can be written as product of two positive integers each of which is larger than $ 2009^{182}$.

Let $p$ be a prime number. Prove that there is no number divisible by $p$ in the $n-th$ row of Pascal's triangle if and only if $n$ can be represented in the form $n = p^sq - 1$, where $s$ and $q$ are integers with $s \geq 0, 0 < q < p$.

Solve $2a^2+3a-44=3p^n$ in positive integers where $p$ is a prime.

On the blackboard are written the $2003$ integers from $1$ to $2003$. Lucas must delete $90$ numbers. Next, Mauro must choose $37$ from the numbers that remain written. If the $37$ numbers Mauro chooses form an arithmetic progression, Mauro wins. If not, Lucas wins. Decide if Lucas can choose the $90$ numbers he erases so that victory is assured.

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

Consider the set of all $10$-digit numbers expressible with the help of figures $1$ and $2$ only. Divide it into two subsets so that the sum of any two numbers of the same subset is a number which is written with not less than two $3$’s.

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.

Find all triplets $(p,a,b)$ of positive integers such that $$p=b\sqrt{\frac{a-8b}{a+8b}}$$ is prime

Show that there exist infinitely many pairs of positive integers $(m,n)$ such that $\binom m{n-1}=\binom{m-1}n$.

Let $m$ and $n$ be positive integers. Find the smallest positive integer $s$ for which there exists an $m \times n$ rectangular array of positive integers such that [list] [*]each row contains $n$ distinct consecutive integers in some order, [*]each column contains $m$ distinct consecutive integers in some order, and [*]each entry is less than or equal to $s$. [/list] [i]Proposed by Ankan Bhattacharya.[/i]

Suppose $a_1, a_2, \ldots$ is a sequence of integers, and $d$ is some integer. For all natural numbers $n$, \begin{align*}\text{(i)} |a_n| \text{ is prime;} && \text{(ii)} a_{n+2} = a_{n+1} + a_n + d. \end{align*} Show that the sequence is constant.

Prove that $2010$ cannot be represented as the difference between two square numbers. (B. Schmidt, Graz University of Technology)

Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$

Let $p$ be a prime and $m \geq 2$ be an integer. Prove that the equation \[ \frac{ x^p + y^p } 2 = \left( \frac{ x+y } 2 \right)^m \] has a positive integer solution $(x, y) \neq (1, 1)$ if and only if $m = p$. [i]Romania[/i]

Find all $7$-digit numbers which are multiples of $21$ and which have each digit $3$ or $7$.

Find all positive integers $x, y, z$ that satisfy the conditions: $$[x,y,z] =(x,y)+(y,z) + (z,x), x\le y\le z, (x,y,z) = 1$$ The symbols $[m,n]$ and $(m,n)$ respectively represent positive integers, the least common multiple and the greatest common divisor of $m$ and $n$.

An arithmetical progression (whose difference is not equal to zero) consists of natural numbers without any nines in its decimal notation. (a) Prove that the number of its terms is less than $100$. (b) Give an example of such a progression with $72$ terms. (c) Prove that the number of terms in any such progression does not exceed $72$. (V. Bugaenko and Tarasov, Moscow)

Let $p, q, r$ be primes and let $n$ be a positive integer such that $p^n + q^n = r^2$. Prove that $n = 1$. Laurentiu Panaitopol

There are $m \geq 3$ positive integers, not necessarily distinct, that are arranged in a circle so that any positive integer divides the sum of its neighbours. Show that if there is exactly one $1$, then for any positive integer $n$, there are at most $\phi(n)$ copies of $n$. [i]Proposed By- (usjl, adapted from 2014 Taiwan TST)[/i]

By default, iPhone passcodes consist of four base-$10$ digits. However, Freya decided to be unconventional and use hexadecimal (base-$16$) digits instead of base-$10$ digits! (Recall that $10_{16} = 16_{10}$.) She sets her passcode such that exactly two of the hexadecimal digits are prime. How many possible passcodes could she have set?

What is the sum of all positive integers $n$ such that $\text{lcm}(2n, n^2) = 14n - 24$?

For positive integer $n$, find all pairs of coprime integers $p$ and $q$ such that $p+q^2=(n^2+1)p^2+q$

Does there exist a sequence of $2005$ consecutive positive integers that contains exactly $25$ prime numbers?

$n \geq k$ -two natural numbers. $S$ -such natural, that have not less than $n$ divisors. All divisors of $S$ are written in descending order. What minimal number of divisors can have number from $k$-th place ?

A diabolical combination lock has $n$ dials (each with $c$ possible states), where $n,c>1$. The dials are initially set to states $d_1, d_2, \ldots, d_n$, where $0\le d_i\le c-1$ for each $1\le i\le n$. Unfortunately, the actual states of the dials (the $d_i$'s) are concealed, and the initial settings of the dials are also unknown. On a given turn, one may advance each dial by an integer amount $c_i$ ($0\le c_i\le c-1$), so that every dial is now in a state $d_i '\equiv d_i+c_i \pmod{c}$ with $0\le d_i ' \le c-1$. After each turn, the lock opens if and only if all of the dials are set to the zero state; otherwise, the lock selects a random integer $k$ and cyclically shifts the $d_i$'s by $k$ (so that for every $i$, $d_i$ is replaced by $d_{i-k}$, where indices are taken modulo $n$). Show that the lock can always be opened, regardless of the choices of the initial configuration and the choices of $k$ (which may vary from turn to turn), if and only if $n$ and $c$ are powers of the same prime. [i]Bobby Shen.[/i]