If $n=\overline{a_1a_2\cdots a_{k-1}a_k}$, be $s(n)$ such that . If $k$ is even, $s(n)=\overline{a_1a_2}+\overline{a_3a_4}\cdots+\overline{a_{k-1}a_k}$ . If $k$ is odd, $s(n)=a_1+\overline{a_2a_3}\cdots+\overline{a_{k-1}a_k}$ For example $s(123)=1+23=24$ and $s(2021)=20+21=41$ Be $n$ is $digital$ if $s(n)$ is a divisor of $n$. Prove that among any 198 consecutive positive integers, all of them less than 2000021 there is one of them that is $digital$.

Find all pairs $(x, y)$ of integers numbers such that $y^3+5=x(y^2+2)$

Call an $n$-digit integer with distinct digits [i]mountainous [/i]if, for some integer $1 \le k \le n$, the first $k$ digits are in strictly ascending order and the following $n - k$ digits are in strictly descending order. How many $5$-digit mountainous integers with distinct digits are there?

The non-zero natural number n is a perfect square. By dividing $2023$ by $n$, we obtain the remainder $223- \frac{3}{2} \cdot n$. Find the quotient of the division.

Let $(a_1,a_2,a_3,a_4,a_5)$ be a random permutation of the integers from $1$ to $5$ inclusive. Find the expected value of $$\sum^5_{i=1} |a_i -i | = |a_1 -1|+|a_2 -2|+|a_3 -3|+|a_4 -4|+|a_5 -5|.$$ [i]Proposed by Muztaba Syed[/i]

For any positive integer $n$ let $f(n)$ be the number of divisors of $n$ ending with $1$ or $9$ in base $10$ and let $g(n)$ be the number of divisors of $n$ ending with digit $3$ or $7$ in base $10$. Prove that $f(n)\geqslant g(n)$ for all nonnegative integers $n$. [i](Swiss Mathematical Olympiad 2011, Final round, problem 9)[/i]

Does there exist positive integer $n$ such that sum of all digits of number $n(4n+1)$ is equal to $2017$

Find all $n\in\mathbb{Z}$ such that the number $\sqrt{\frac{4n-2}{n+5}}$ is rational.

a) A certain locker room contains $n$ lockers numbered $1,2,\ldots,n$ and all are originally locked. An attendant performs a sequence of operations $T_1, T_2 ,\ldots, T_n$, whereby with the operation $T_k$ the state of those lockers whose number is divisible by $k$ is swapped. After all $n$ operations have been performed, it is observed that all lockers whose number is a perfect square (and only those lockers) are open. Prove this. b) Investigate in a meaningful mathematical way a procedure or set of operations similar to those above which will produce the set of cubes, or the set of numbers of the form $2 m^2 $, or the set of numbers of the form $m^2 +1$, or some nontrivial similar set of your own selection.

Find all positive integers $a, b, c$ such that $a < b$, $a < 4c$, and $b c^3 \le a c^3 + b$.

Let $n$ be a positive integer with $k$ digits. A number $m$ is called an $alero$ of $n$ if there exist distinct digits $a_1$, $a_2$, $\dotsb$, $a_k$, all different from each other and from zero, such that $m$ is obtained by adding the digit $a_i$ to the $i$-th digit of $n$, and no sum exceeds 9. For example, if $n$ $=$ $2024$ and we choose $a_1$ $=$ $2$, $a_2$ $=$ $1$, $a_3$ $=$ $5$, $a_4$ $=$ $3$, then $m$ $=$ $4177$ is an alero of $n$, but if we choose the digits $a_1$ $=$ $2$, $a_2$ $=$ $1$, $a_3$ $=$ $5$, $a_4$ $=$ $6$, then we don't obtain an alero of $n$, because $4$ $+$ $6$ exceeds $9$. Find the smallest $n$ which is a multiple of $2024$ that has an alero which is also a multiple of $2024$.

Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that \begin{align*} a^2 + b^2 + 3 &= 4ab\\ c^2 + d^2 + 3 &= 4cd\\ 4c^3 - 3c &= a \end{align*} [i]Travis Hance.[/i]

Find the maximum number of pairwise disjoint sets of the form $S_{a,b} = \{n^{2}+an+b | n \in \mathbb{Z}\}$, $a, b \in \mathbb{Z}$.

Let $ n$ be a positive integer and let $ a_1,a_2,a_3,\ldots,a_k$ $ ( k\ge 2)$ be distinct integers in the set $ { 1,2,\ldots,n}$ such that $ n$ divides $ a_i(a_{i + 1} - 1)$ for $ i = 1,2,\ldots,k - 1$. Prove that $ n$ does not divide $ a_k(a_1 - 1).$ [i]Proposed by Ross Atkins, Australia [/i]

Determine all triplets off positive integers $ (a,b,c)$ for which $ \mid2^a\minus{}b^c\mid\equal{}1$

A clue “$k$ digits, sum is $n$” gives a number k and the sum of $k$ distinct, nonzero digits. An answer for that clue consists of $k$ digits with sum $n$. For example, the clue “Three digits, sum is $23$” has only one answer: $6,8,9$. The clue “Three digits, sum is $8$” has two answers: $1,3,4$ and $1,2,5$. If the clue “Four digits, sum is $n$” has the largest number of answers for any four-digit clue, then what is the value of $n$? How many answers does this clue have? Explain why no other four-digit clue can have more answers.

Let $m$ be the smallest odd positive integer for which $1+ 2 +...+ m$ is a square of an integer and let $n$ be the smallest even positive integer for which $1 + 2 + ... + n$ is a square of an integer. What is the value of $m + n$?

Find all integer triples $(a,b,c)$ with $a>0>b>c$ whose sum equal $0$ such that the number $$N=2017-a^3b-b^3c-c^3a$$ is a perfect square of an integer.

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.

Determine all integers $n \geq 2$, satisfying $$n=a^2+b^2,$$ where $a$ is the smallest divisor of $n$ different from $1$ and $b$ is an arbitrary divisor of $n$. [i]Proposed by Walther Janous[/i]

Find the number of the subsets $B$ of the set $\{1,2,\cdots, 2005 \}$ such that the sum of the elements of $B$ is congruent to $2006$ modulo $2048$

Find all positive integers $k$ such that there exists positive integers $a, b$ such that \[a^2+4=(k^2-4)b^2.\]

Find all integer pair $(m,n)$ such that $7^m=5^n+24$.

The function $f : \mathbb N \to \mathbb R$ satisfies $f(1) = 1, f(2) = 2$ and \[f (n+2) = f(n+2 - f(n+1) ) + f(n+1 - f(n) ).\] Show that $0 \leq f(n+1) - f(n) \leq 1$. Find all $n$ for which $f(n) = 1025$.

The sequence $(x_n)_{n\geqslant 0}$ is defined as such: $x_0=1, x_1=2$ and $x_{n+1}=4x_n-x_{n-1}$, for all $n\geqslant 1$. Determine all the terms of the sequence which are perfect squares. [i]George Stoica, Canada[/i]