For a prime $p$, a natural number $n$ and an integer $a$, we let $S_n(a,p)$ denote the exponent of $p$ in the prime factorisation of $a^{p^n} - 1$. For example, $S_1(4,3) = 2$ and $S_2(6,2) = 0$. Find all pairs $(n,p)$ such that $S_n(2013,p) = 100$.

For a positive integer $n$ that is not a power of two, we define $t(n)$ as the greatest odd divisor of $n$ and $r(n)$ as the smallest positive odd divisor of $n$ unequal to $1$. Determine all positive integers $n$ that are not a power of two and for which we have $n = 3t(n) + 5r(n)$.

Find the largest integer $n$ such that $n$ is divisible by all positive integers less than $\sqrt[3]{n}$.

Let $S(n)=1\varphi(1)+2\varphi(2) \ldots +n\varphi(n)$, where $\varphi(n)$ is the number of positive integers less than or equal to $n$ that are relatively prime to $n$. (For instance $\varphi(12)=4$ and $\varphi(20)=8$.) Prove that for all $n \geq 2018$, the following inequality holds: $$0.17n^3 \leq S(n) \leq 0.23n^3$$

Find all positive integers $a$ so that for any $\left \lfloor \frac{a+1}{2} \right \rfloor$-digit number that is composed of only digits $0$ and $2$ (where $0$ cannot be the first digit) is not a multiple of $a$.

It is possible to partition the set $\{100, 101,\cdots , 1000\}$ into two subsets so that for any two distinct elements $x$ and $y$ belonging to the same subset $ \sqrt[3]{x + y}$ is irrational?

Suppose that $2^{2n+1}+ 2^{n}+1=x^{k}$, where $k\geq2$ and $n$ are positive integers. Find all possible values of $n$.

Find all triples $(a,b,c)$ of natural numbers, such that $LCM(a,b,c)=a+b+c$

Prove that the number $\lfloor (2+\sqrt5)^{2019} \rfloor$ is not prime.

Let $ a$, $ b$, $ c$ and $ n$ be positive integers such that $ a^n$ is divisible by $ b$, such that $ b^n$ is divisible by $ c$, and such that $ c^n$ is divisible by $ a$. Prove that $ \left(a \plus{} b \plus{} c\right)^{n^2 \plus{} n \plus{} 1}$ is divisible by $ abc$. An even broader [i]generalization[/i], though not part of the QEDMO problem and not quite number theory either: If $ u$ and $ n$ are positive integers, and $ a_1$, $ a_2$, ..., $ a_u$ are integers such that $ a_i^n$ is divisible by $ a_{i \plus{} 1}$ for every $ i$ such that $ 1\leq i\leq u$ (we set $ a_{u \plus{} 1} \equal{} a_1$ here), then show that $ \left(a_1 \plus{} a_2 \plus{} ... \plus{} a_u\right)^{n^{u \minus{} 1} \plus{} n^{u \minus{} 2} \plus{} ... \plus{} n \plus{} 1}$ is divisible by $ a_1a_2...a_u$.

We consider the following operation applied to a positive integer: The integer is represented in an arbitrary base $b \ge 2$, in which it has exactly two digits and in which both digits are different from $0$. Then the two digits are swapped and the result in base $b$ is the new number. Is it possible to transform every number $> 10$ to a number $\le 10$ with a series of such operations? (Theresia Eisenkölbl)

Define the polynomial $f(x) = x^4 + x^3 + x^2 + x + 1$. Compute the number of positive integers $n$ less than equal to $2022$ such that $f(n)$ is $1$ more than multiple of $5$.

Find all positive integers $1 \leq k \leq 6$ such that for any prime $p$, satisfying $p^2=a^2+kb^2$ for some positive integers $a, b$, there exist positive integers $x, y$, satisfying $p=x^2+ky^2$. [hide=Remark on 10.4] It also appears as ARO 2010 10.4 with the grid changed to $10 \times 10$ and $17$ changed to $5$, so it will not be posted.

Determine the number of real solutions $a$ to the equation: \[ \left[\,\frac{1}{2}\;a\,\right]+\left[\,\frac{1}{3}\;a\,\right]+\left[\,\frac{1}{5}\;a\,\right] = a. \] Here, if $x$ is a real number, then $[\,x\,]$ denotes the greatest integer that is less than or equal to $x$.

The quadratic expression $ax^2+bx+c$ is the $4$-th power (of an integer) for any integer $x$. Prove that $a = b = 0$.

Show that there exists a unique sequence of decimal digits $p_0=5,p_1,p_2,\ldots$ such that, for any $k$, the square of any positive integer ending with $\overline{p_kp_{k-1}\cdots p_0}$ ends with the same digits.

Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)

A pair of integers $ (m,n)$ is called [i]good[/i] if \[ m\mid n^2 \plus{} n \ \text{and} \ n\mid m^2 \plus{} m\] Given 2 positive integers $ a,b > 1$ which are relatively prime, prove that there exists a [i]good[/i] pair $ (m,n)$ with $ a\mid m$ and $ b\mid n$, but $ a\nmid n$ and $ b\nmid m$.

What is the smallest positive integer (in base 10) that has a digit sum of 23 in base 20, and a digit sum of 20 in base 23? (The digit sums are in base 10.) [i]Team #8[/i]

Let's say that an ordered triple of positive integers $(a,b,c)$ is [i]$n$-powerful[/i] if $a\le b\le c,\gcd (a,b,c)=1$ and $a^n+b^n+c^n$ is divisible by $a+b+c$. For example, $(1,2,2)$ is $5$-powerful. $a)$ Determine all ordered triples (if any) which are $n$-powerful for all $n\ge 1$. $b)$ Determine all ordered triples (if any) which are $2004$-powerful and $2005$-powerful, but not $2007$-powerful.

Let $\alpha$ be the positive root of the quadratic equation $x^2 = 1990x + 1$. For any $m, n \in \mathbb N$, define the operation $m*n = mn + [\alpha m][ \alpha n]$, where $[x]$ is the largest integer no larger than $x$. Prove that $(p*q)*r = p*(q*r)$ holds for all $p, q, r \in \mathbb N.$

Prove that for any positive integer $t,$ \[1+2^t+3^t+\cdots+9^t - 3(1 + 6^t +8^t )\] is divisible by $18.$

[b]p1.[/b] Find all solutions $a, b, c, d, e, f$ if it is known that they represent distinct digits and satisfy the following: $\begin{tabular}{ccccc} & a & b & c & a \\ + & & d & d & e \\ & & & d & e \\ \hline d & f & f & d & d \\ \end{tabular}$ [b]p2.[/b] Snowhite wrote on a piece of paper a whole number greater than $1$ and multiplied it by itself. She obtained a number, all digits of which are $1$: $n^2 = 111...111$ Does she know how to multiply? [b]p3.[/b] Two players play the following game on an $8\times 8$ chessboard. The first player can put a bishop on an arbitrary square. Then the second player can put another bishop on a free square that is not controlled by the first bishop. Then the first player can put a new bishop on a free square that is not controlled by the bishops on the board. Then the second player can do the same, etc. A player who cannot put a new bishop on the board loses the game. Who has a winning strategy? [b]p4.[/b] Four girls Marry, Jill, Ann and Susan participated in the concert. They sang songs. Every song was performed by three girls. Mary sang $8$ songs, more then anybody. Susan sang $5$ songs less then all other girls. How many songs were performed at the concert? [b]p5.[/b] Pinocchio has a $10\times 10$ table of numbers. He took the sums of the numbers in each row and each such sum was positive. Then he took the sum of the numbers in each columns and each such sum was negative. Can you trust Pinocchio's calculations? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Together, Abe and Bob have less than or equal to \$ $100$. When Corey asks them how much money they have, Abe says that the reciprocal of his money added to Bob’s money is thirteen times as much as the sum of Abe’s money and the reciprocal of Bob’s money. If Abe and Bob both have integer amounts of money, how many possible values are there for Abe’s money?

For a positive integer \( n \), let \( \tau(n) \) denote the number of positive divisors of \( n \). Determine all positive integers \( K \) such that the equation \[ \tau(x) = \tau(y) = \tau(z) = \tau(2x + 3y + 3z) = K \] holds for some positive integers $x,y,z$.