Define the sequence $ a_1, a_2, a_3, ...$ as follows. $ a_1$ and $ a_2$ are coprime positive integers and $ a_{n \plus{} 2} \equal{} a_{n \plus{} 1}a_n \plus{} 1$. Show that for every $ m > 1$ there is an $ n > m$ such that $ a_m^m$ divides $ a_n^n$. Is it true that $ a_1$ must divide $ a_n^n$ for some $ n > 1$?

A quadruple $(a; b; c; d)$ of positive integers with $a \leq b \leq c \leq d$ is called good if we can colour each integer red, blue, green or purple, in such a way that $i$ of each $a$ consecutive integers at least one is coloured red; $ii$ of each $b$ consecutive integers at least one is coloured blue; $iii$ of each $c$ consecutive integers at least one is coloured green; $iiii$ of each $d$ consecutive integers at least one is coloured purple. Determine all good quadruples with $a = 2.$

Let $n$ be a "Good Number" if sum of all divisors of $n$ is less than $2n$ for $n\in \mathbb{Z}.$ Does there exist an infinite set $M$ that satisfies the following? For all $a,b\in M,$ $a+b$ is good number. ($a=b$ is allowed.)

Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

Find all positive integers $n$ that have exactly $16$ positive integral divisors $d_{1},d_{2} \cdots, d_{16}$ such that $1=d_{1}<d_{2}<\cdots<d_{16}=n$, $d_6=18$, and $d_{9}-d_{8}=17$.

A set S consisting of $2019$ (different) positive integers has the following property: [i]the product of every 100 elements of $S$ is a divisor of the product of the remaining $1919$ elements[/i]. What is the maximum number of prime numbers that $S$ can contain?

Define a $\textit{tasty residue}$ of $n$ to be an integer $1<a<n$ such that there exists an integer $m>1$ satisfying \[a^m\equiv a\pmod n.\] Find the number of tasty residues of $2016$.

Compute the sum of all positive integers $n$ such that $n^n$ has 325 positive integer divisors. (For example, $4^4=256$ has 9 positive integer divisors: 1, 2, 4, 8, 16, 32, 64, 128, 256.)

Find all natural numbers $n$ such that $1+[\sqrt{2n}]~$ divides $2n$. ( For any real number $x$ , $[x]$ denotes the largest integer not exceeding $x$. )

A sequence $a_n$ is determined by the rules $a_0 = 9$ and for any nonnegative $k$, $$a_{k+1}=3a_k^4+4a_k^3.$$ Prove that $a_{10}$ contains more than $1000$ nines in decimal notation. (Yao)

Let $n$ be an odd positive integer. Show that the equation $$ \frac{4}{n} =\frac{1}{x} + \frac{1}{y}$$ has a solution in the positive integers if and only if $n$ has a divisor of the form $4k+3$.

Find in explicit form all ordered pairs of positive integers $(m, n)$ such that $mn-1$ divides $m^2 + n^2$.

Prove that for any positive integer $n$, the least common multiple of the numbers $1,2,\ldots,n$ and the least common multiple of the numbers: \[\binom{n}{1},\binom{n}{2},\ldots,\binom{n}{n}\] are equal if and only if $n+1$ is a prime number. [i]Laurentiu Panaitopol[/i]

Let $d(n)$ denote the number of positive divisors of $n$. Find all triples $(n,k,p)$, where $n$ and $k$ are positive integers and $p$ is a prime number, such that \[n^{d(n)} - 1 = p^k.\]

[b]p1.[/b] An animal farm has geese and pigs with a total of $30$ heads and $84$ legs. Find the number of pigs and geese on this farm. [b]p2.[/b] What is the maximum number of $1 \times 1$ squares of a $7 \times 7$ board that can be colored black in such a way that the black squares don’t touch each other even at their corners? Show your answer on the figure below and explain why it is not possible to get more black squares satisfying the given conditions. [img]https://cdn.artofproblemsolving.com/attachments/d/5/2a0528428f4a5811565b94061486699df0577c.png[/img] [b]p3.[/b] Decide whether it is possible to divide a regular hexagon into three equal not necessarily regular hexagons? A regular hexagon is a hexagon with equal sides and equal angles. [img]https://cdn.artofproblemsolving.com/attachments/3/7/5d941b599a90e13a2e8ada635e1f1f3f234703.png[/img] [b]p4.[/b] A rectangle is subdivided into a number of smaller rectangles. One observes that perimeters of all smaller rectangles are whole numbers. Is it possible that the perimeter of the original rectangle is not a whole number? [b]p5.[/b] Place parentheses on the left hand side of the following equality to make it correct. $$ 4 \times 12 + 18 : 6 + 3 = 50$$ [b]p6.[/b] Is it possible to cut a $16\times 9$ rectangle into two equal parts which can be assembled into a square? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Given $m\in\mathbb{Z}^+$. Find all natural numbers $n$ that does not exceed $10^m$ satisfying the following conditions: i) $3|n.$ ii) The digits of $n$ in decimal representation are in the set $\{2,0,1,5\}$.

Minivan chooses a prime number. Then every second, he adds either the digit $1$ or the digit $3$ to the right end of his number (after the unit digit), such that the new number is also a prime. Can he continue indefinitely? [i](Proposed by Wong Jer Ren)[/i]

Prove that among $18$ consecutive three digit numbers there must be at least one which is divisible by the sum of its digits.

If number $\overline{aaaa}$ is divided by $\overline{bb}$, the quotient is a number between $140$ and $160$ inclusively, and the remainder is equal to $\overline{(a-b)(a-b)}$. Find all pairs of positive integers $(a,b)$ that satisfy this.

Let $\{a_n \}_{n=1}^{\infty}$ be a sequence such that $a_1=\frac 12$ and \[a_n=\biggl( \frac{2n-3}{2n} \biggr) a_{n-1} \qquad \forall n \geq 2.\] Prove that for every positive integer $n,$ we have $\sum_{k=1}^n a_k <1.$

Prove that for an arbitrary prime $p \ge 3$ the number of positive integers $n$, for which $p | n! +1$ does not exceed $cp^{2/3}$, where c is a constant that does not depend on $p$.

Let $n>1$ be a positive integer. Each cell of an $n\times n$ table contains an integer. Suppose that the following conditions are satisfied: [list=1] [*] Each number in the table is congruent to $1$ modulo $n$. [*] The sum of numbers in any row, as well as the sum of numbers in any column, is congruent to $n$ modulo $n^2$. [/list] Let $R_i$ be the product of the numbers in the $i^{\text{th}}$ row, and $C_j$ be the product of the number in the $j^{\text{th}}$ column. Prove that the sums $R_1+\hdots R_n$ and $C_1+\hdots C_n$ are congruent modulo $n^4$.

$f(n)$ is the sum of all integers less than $n$ and relatively prime to $n$. Find all integers $n$ such that there exist integers $k$ and $\ell$ such that $f(n^k) = n^{\ell}$.

Sevara writes in red $8$ distinct positive integers and then writes in blue the $28$ sums of each two red numbers. At most how many of the blue numbers can be prime? [i]Marin Hristov, Bulgaria[/i]

There are integers $a$ and $b$, such that $a>b>1$ and $b$ is the largest divisor of $a$ different from $a$. Prove that the number $a+b$ is not a power of $2$ with integer exponent.