A positive integer $N$ is [i]rioplatense[/i] if it satifies the following conditions: 1 -There exist $34$ consecutive integers such that its product is divisible by $N$, but none of them is divisible by $N$. 2 - There [b]not[/b] exist $30$ consecutive integers such that its product is divisible by $N$, but none of them is divisible by $N$. Determine all rioplatense numbers.

For $ n\in\mathbb{N}$, determine the number of natural solutions $ (a,b)$ such that \[ (4a\minus{}b)(4b\minus{}a)\equal{}2010^n\] holds.

Find all primes $p$ and $q$, with $p \le q$, so that $$p (2q + 1) + q (2p + 1) = 2 (p^2 + q^2).$$

Let $M, k>0$ integers. Let $X(M,k)$ the (infinite) set of all integers that can be factored as ${p_1}^{e_1} \cdot {p_2}^{e_2} \cdot \ldots \cdot {p_r}^{e_r}$ where each $p_i$ is not smaller than $M$ and also each $e_i$ is not smaller than $k$. Let $Z(M,k,n)$ the number of elements of $X(M,k)$ not bigger than $n$. Show that there are positive reals $c(M,k)$ and $\beta(M,k)$ such that $$\lim_{n \rightarrow \infty}{\frac{Z(M,k,n)}{n^{\beta(M,k)}}} = c(M,k)$$ and find $\beta(M,k)$

Let Pascal triangle be an equilateral triangular array of number, consists of $2019$ rows and except for the numbers in the bottom row, each number is equal to the sum of two numbers immediately below it. How many ways to assign each of numbers $a_0, a_1,...,a_{2018}$ (from left to right) in the bottom row by $0$ or $1$ such that the number $S$ on the top is divisible by $1019$.

Find the number of positive integer $x$ that has $ {a}_{1},{a}_{2},\cdot \cdot \cdot {a}_{20} $ which follows the following ($x \ge 1000$) 1) $ {a}_{1}=2, {a}_{2}=1, {a}_{3}=x $ 2) for positive integer $n$, ($ 4 \le n \le 20 $), $ {a}_{n}={a}_{n-3}+\frac{(-2)^n}{{a}_{n-1}{a}_{n-2}} $

Find all non-negative integers $k,n$ which satisfy $2^{2k+1} + 9\cdot 2^k+5=n^2$.

Prove that for each positive integer $n$, there exists a number $M$, such that $M$ can be written as sum of $1,2,3,\dots, n$ distinct perfect squares.

Two positive integers $m$ and $n$ are both less than $500$ and $\text{lcm}(m,n) = (m-n)^2$. What is the maximum possible value of $m+n$?

Let $n_1,n_2,n_3,...,n_{2000}$ be $2000$ positive integers satisfying $n_1<n_2<n_3<...<n_{2000}$. Prove that $$\frac{n_1}{[n_1,n_2]}+\frac{n_1}{[n_2,n_3]}+\frac{n_1}{[n_3,n_4]}+...+\frac{n_1}{[n_{1999},n_{2000}]} \le 1 - \frac{1}{2^{1999}}$$ where $[a, b]$ denotes the least common multiple of $a$ and $b$.

Determine all integers $a$ such that $a^k + 1$ is divisible by $12321$ for some $k$

Joey wrote a system of equations on a blackboard, where each of the equations was of the form $a + b = c$ or $a \cdot b = c$ for some variables or integers $a, b, c$. Then Sean came to the board and erased all of the plus signs and multiplication signs, so that the board reads: $x\,\,\,\, z = 15$ $x\,\,\,\, y = 12$ $x\,\,\,\, x = 36$ If $x, y, z$ are integer solutions to the original system, find the sum of all possible values of $100x+10y+z$.

Find all positive integers m, n such that $m^3 - n^3 = 999$.

You come across an ancient mathematical manuscript. It reads, "To find out whether a number is divisible by seventeen, take the number formed by the last two digits of the number, subtract the number formed by the third- and fourth-to-last digits of the number, add the number formed by the fifth- and sixth-to-last digits of the number and so on. The resulting number is divisible by seventeen if and only if the original number is divisible by seventeen." What is the sum of the five smallest bases the ancient culture might have been using? (Note that "seventeen" is the number represented by $17$ in base $10$, not $17$ in the base that the ancient culture was using. Express your answer in base $10$.)

A right parallelepiped (i.e. a parallelepiped one of whose edges is perpendicular to a face) is given. Its vertices have integral coordinates, and no other points with integral coordinates lie on its faces or edges. Prove that the volume of this parallelepiped is a sum of three perfect squares. [i]Proposed by A. Golovanov[/i]

Given an integer $n > 1$, prove that there exist distinct positive integers $a, b, c$ and $d$ such that $a + b = c + d$ and $\frac{a}{b}=\frac{nc}{d}$.

Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.

Let $p$ and $q$ be relatively prime positive odd integers such that $1 < p < q$. Let $A$ be a set of pairs of integers $(a, b)$, where $0 \le a \le p - 1, 0 \le b \le q - 1$, containing exactly one pair from each of the sets $$\{(a, b),(a + 1, b + 1)\}, \{(a, q - 1), (a + 1, 0)\}, \{(p - 1,b),(0, b + 1)\}$$ whenever $0 \le a \le p - 2$ and $0 \le b \le q - 2$. Show that $A$ contains at least $(p - 1)(q + 1)/8$ pairs whose entries are both even. Agnijo Banerjee and Joe Benton, United Kingdom

Find all non-negative integers $n$ for which the equation \[ {\left( x^2 + y^2 \right)}^n = {(xy)}^{2018} \] admits positive integral solutions.

How many prime numbers less than $100$ can be represented as sum of squares of consequtive positive integers? $ \textbf{(A)}\ 3 \qquad \textbf{(B)}\ 4 \qquad \textbf{(C)}\ 5 \qquad \textbf{(D)}\ 6 \qquad \textbf{(E)}\ 7$

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

Show that $n!=a^{n-1}+b^{n-1}+c^{n-1}$ has only finitely many solutions in positive integers. [i]Proposed by Dorlir Ahmeti, Albania[/i]

For each positive integer $m$ let $t_m$ be the smallest positive integer not dividing $m$. Prove that there are infinitely many positive integers which can not be represented in the form $m + t_m$. [i](A. Golovanov)[/i]

Let $n$ be a positive integer. Does $n^2$ has more positive divisors of the form $4k+1$ or of the form $4k-1$?

Prove that for all rationals $x,y$, $x-\frac{1}{x}+y-\frac{1}{y}=4$ is not true.