Does there exist a $2n$-digit number $\overline{a_{2n}a_{2n-1}\cdots a_1}$(for an arbitrary $n$) for which the following equality holds: \[\overline{a_{2n}\cdots a_1}= (\overline{a_n \cdots a_1})^2?\]

Find all positive integers $n$, such that $\sigma(n) =\tau(n) \lceil {\sqrt{n}} \rceil$.

Let $n$ be a positive integer. Show that there is a prime $p$ and a sequence $\left(a_k\right)_{k\ge1}$ of positive integers such that the sequence $\left(p+na_k\right)_{k\ge1}$ consists of distinct primes.

Does there exist an integer $n > 1$ such that $2^{2^n-1} -7$ is not a perfect square?

Call a positive integer $x$ $n$-[i]cube-invariant[/i] if the last $n$ digits of $x$ are equal to the last $n$ digits of $x^3$. For example, $1$ is $n$-cube invariant for any integer $n$. How many $2015$-cube-invariant numbers $x$ are there such that $x < 10^{2015}$?

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

The sequence of positive integers $a_0, a_1, a_2, . . .$ is defined by $a_0 = 3$ and $$a_{n+1} - a_n = n(a_n - 1)$$ for all $n \ge 0$. Determine all integers $m \ge 2$ for which $gcd (m, a_n) = 1$ for all $n \ge 0$.

Find all pairs of positive integers $(x,y)$ so that $$\frac{(x^2-x+1)(y^2-y+1)}{xy}\in\mathbb N.$$

[u]Round 1[/u] [b]p1.[/b] A positive integer is said to be transcendent if it leaves a remainder of $1$ when divided by $2$. Find the $1010$th smallest positive integer that is transcendent. [b]p2.[/b] The two diagonals of a square are drawn, forming four triangles. Determine, in degrees, the sum of the interior angle measures in all four triangles. [b]p3.[/b] Janabel multiplied $2$ two-digit numbers together and the result was a four digit number. If the thousands digit was nine and hundreds digit was seven, what was the tens digit? [u]Round 2[/u] [b]p4.[/b] Two friends, Arthur and Brandon, are comparing their ages. Arthur notes that $10$ years ago, his age was a third of Brandon’s current age. Brandon points out that in $12$ years, his age will be double of Arthur’s current age. How old is Arthur now? [b]p5.[/b] A farmer makes the observation that gathering his chickens into groups of $2$ leaves $1$ chicken left over, groups of $3$ leaves $2$ chickens left over, and groups of $5$ leaves $4$ chickens left over. Find the smallest possible number of chickens that the farmer could have. [b]p6.[/b] Charles has a bookshelf with $3$ layers and $10$ indistinguishable books to arrange. If each layer must hold less books than the layer below it and a layer cannot be empty, how many ways are there for Charles to arrange his $10$ books? [u]Round 3[/u] [b]p7.[/b] Determine the number of factors of $2^{2019}$. [b]p8.[/b] The points $A$, $B$, $C$, and $D$ lie along a line in that order. It is given that $\overline{AB} : \overline{CD} = 1 : 7$ and $\overline{AC} : \overline{BD} = 2 : 5$. If $BC = 3$, find $AD$. [b]p9.[/b] A positive integer $n$ is equal to one-third the sum of the first $n$ positive integers. Find $n$. [u]Round 4[/u] [b]p10.[/b] Let the numbers $a,b,c$, and $d$ be in arithmetic progression. If $a +2b +3c +4d = 5$ and $a =\frac12$ , find $a +b +c +d$. [b]p11.[/b] Ten people playing brawl stars are split into five duos of $2$. Determine the probability that Jeff and Ephramare paired up. [b]p12.[/b] Define a sequence recursively by $F_0 = 0$, $F_1 = 1$, and for all $n\ge 2$, $$F_n = \left \lceil \frac{F_{n-1}+F_{n-2}}{2} \right \rceil +1,$$ where $\lceil r \rceil$ denotes the least integer greater than or equal to $r$ . Find $F_{2019}$. PS. You should use hide for answers. Rounds 5-8 have been posted [url=https://artofproblemsolving.com/community/c3h3166019p28809679]here [/url] and 9-12 [url=https://artofproblemsolving.com/community/c3h3166115p28810631]here[/url].Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Call a rational number [i]short[/i] if it has finitely many digits in its decimal expansion. For a positive integer $m$, we say that a positive integer $t$ is $m-$[i]tastic[/i] if there exists a number $c\in \{1,2,3,\ldots ,2017\}$ such that $\dfrac{10^t-1}{c\cdot m}$ is short, and such that $\dfrac{10^k-1}{c\cdot m}$ is not short for any $1\le k<t$. Let $S(m)$ be the set of $m-$tastic numbers. Consider $S(m)$ for $m=1,2,\ldots{}.$ What is the maximum number of elements in $S(m)$?

Positive integers $(p,a,b,c)$ called [i]good quadruple[/i] if a) $p $ is odd prime, b) $a,b,c $ are distinct , c) $ab+1,bc+1$ and $ca+1$ are divisible by $p $. Prove that for all good quadruple $p+2\le \frac {a+b+c}{3} $, and show the equality case.

Vukasin, Dimitrije, Dusan, Stefan and Filip asked their teacher to guess three consecutive positive integers, after these true statements: Vukasin: " The sum of the digits of one number is prime number. The sum of the digits of another of the other two is, an even perfect number.($n$ is perfect if $\sigma\left(n\right)=2n$). The sum of the digits of the third number equals to the number of it's positive divisors". Dimitrije:"Everyone of those three numbers has at most two digits equal to $1$ in their decimal representation". Dusan:"If we add $11$ to exactly one of them, then we have a perfect square of an integer" Stefan:"Everyone of them has exactly one prime divisor less than $10$". Filip:"The three numbers are square free". Professor found the right answer. Which numbers did he mention?

Let $M$ and $m$ be, respectively, the greatest and the least ten-digit numbers that are rearrangements of the digits $0$ through $9$ such that no two adjacent digits are consecutive. Find $M - m$.

Determine the minimum value of prime $p> 3$ for which there is no natural number $n> 0$ such that $2^n+3^n\equiv 0\pmod{p} $.

Let $ a, b, c, d$ be positive integers such that $ ab \equal{} cd$ and $ a\plus{}b \equal{} c \minus{} d.$ Prove that there exists a right-angled triangle the measure of whose sides (in some unit) are integers and whose area measure is $ ab$ square units.

Determine all composite positive integers $n$ such that, for each positive divisor $d$ of $n$, there are integers $k\geq 0$ and $m\geq 2$ such that $d=k^m+1$.

Given real numbers $a,\alpha,\beta, \sigma \ and \ \varrho$ s.t. $\sigma, \varrho > 0$ and $\sigma \varrho = \frac{1}{16}$, prove that there exist integers $x$ and $y$ s.t. \[ - \sigma \leq (x+\alpha_(ax + y + \beta ) \leq \varrho \]

Let $p\geqslant 5$ be a prime number. Prove that the set $\{1,2,\ldots,p - 1\}$ can be divided into two nonempty subsets so that the sum of all the numbers in one subset and the product of all the numbers in the other subset give the same remainder modulo $p{}$.

Determine the smallest natural number $n$ for which there exist distinct nonzero naturals $a, b, c$, such that $n=a+b+c$ and $(a + b)(b + c)(c + a)$ is a perfect cube.

Let $A$ be a set of real numbers satisfying the following: (a) $\sqrt(n^2+1) \in A$ for all positive integers $n$, (b) if $x \in A$ and $y \in A$, then $x-y \in A$. Prove that every integer can be written as a product of two different elements in $A$.

Solve the given equation in integers \begin{align*} y^3=8x^6+2x^3y-y^2 \end{align*}

Show that for every polynomial $f(x)$ with integer coefficients, there exists a integer $C$ such that the set $\{n \in Z :$ the sum of digits of $f(n)$ is $C\}$ is not finite.

Find the set of all ordered pairs $(s,t)$ of positive integers such that \[t^{2}+1=s(s+1).\]

For $m$ and $n$ positive integers that are prime to each other, determine the possible values ​​of $$\gcd (5^m + 7^m, 5^n + 7^n)$$

Petya and Vasya play the following game. Petya conceives a polynomial $P(x)$ having integer coefficients. On each move, Vasya pays him a ruble, and calls an integer $a$ of his choice, which has not yet been called by him. Petya has to reply with the number of distinct integer solutions of the equation $P(x)=a$. The game continues until Petya is forced to repeat an answer. What minimal amount of rubles must Vasya pay in order to win? [i](Anant Mudgal)[/i] (Translated from [url=http://sasja.shap.homedns.org/Turniry/TG/index.html]here.[/url])