For a positive integer $n$, we dene $D_n$ as the largest integer that is a divisor of $a^n + (a + 1)^n + (a + 2)^n$ for all positive integers $a$. 1. Show that for all positive integers $n$, the number $D_n$ is of the form $3^k$ with $k \ge 0$ an integer. 2. Show that for all integers $k \ge 0$ there exists a positive integer n such that $D_n = 3^k$.

For any positive integer $n$, let $r_n$ denote the greatest odd divisor of $n$. Compute $T =r_{100}+ r_{101} + r_{102}+...+r_{200}$

Find all primes $p$ such that $p^2 + 2543$ has at most $16$ divisors.

For every positive integer $n$ we take the greatest divisor $d$ of $n$ such that $d\leq \sqrt{n}$ and we define $a_n=\frac{n}{d}-d$. Prove that in the sequence $a_1,a_2,a_3,...$, any non negative integer $k$ its in the sequence infinitely many times.

Let $D_n$ be the set of divisors of $n$. Find all natural $n$ such that it is possible to split $D_n$ into two disjoint sets $A$ and $G$, both containing at least three elements each, such that the elements in $A$ form an arithmetic progression while the elements in $G$ form a geometric progression.

We denote the number of positive divisors of a positive integer $m$ by $d(m)$ and the number of distinct prime divisors of $m$ by $\omega(m)$. Let $k$ be a positive integer. Prove that there exist infinitely many positive integers $n$ such that $\omega(n) = k$ and $d(n)$ does not divide $d(a^2+b^2)$ for any positive integers $a, b$ satisfying $a + b = n$.

Determine all the natural numbers $n$ such that exactly one fifth of the natural numbers $1,2,...,n$ are divisors of $n$.

Find all odd positive integers $ n > 1$ such that if $ a$ and $ b$ are relatively prime divisors of $ n$, then $ a\plus{}b\minus{}1$ divides $ n$.

Let $\tau(n)$ denote the number of positive divisors of the positive integer $n$. Prove that there exist infinitely many positive integers $a$ such that the equation $ \tau(an)=n $ does not have a positive integer solution $n$.

Find all positive integers $n$ such that \[n=d_6^2+d_7^2-1,\] where $1 = d_1 < d_2 < \cdots < d_k = n$ are all positive divisors of the number $n.$

Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.

Find all odd natural numbers $n$ such that $d(n)$ is the largest divisor of the number $n$ different from $n$. ($d(n)$ is the number of divisors of the number n including $1$ and $n$ ).

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

Let $k\geqslant 2$ be a positive integer and $n>1$ be a composite integer. Let $d_1<\cdots<d_m$ be all the positive divisors of $n{}.$ Is it possible for $d_i+d_{i+1}$ to be a perfect $k$-th power, for every $1\leqslant i<m$? [i]Proposed by Pavel Ciurea[/i]

Every positive integer is either [i]nice [/i] or [i]naughty[/i], and the Oracle of Numbers knows which are which. However, the Oracle will not directly tell you whether a number is [i]nice [/i] or [i]naughty[/i]. The only questions the Oracle will answer are questions of the form “What is the sum of all nice divisors of $n$?,” where $n$ is a number of the questioner’s choice. For instance, suppose ([i]just [/i] for this example) that $2$ and $3$ are nice, while $1$ and $6$ are [i]naughty[/i]. In that case, if you asked the Oracle, “What is the sum of all nice divisors of $6$?,” the Oracle’s answer would be $5$. Show that for any given positive integer $n$ less than $1$ million, you can determine whether $n$ is [i]nice [/i] or [i]naughty [/i] by asking the Oracle at most four questions.

A set of $2003$ positive integers is given. Show that one can find two elements such that their sum is not a divisor of the sum of the other elements.

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

Let $a > 3$ be an odd integer. Show that for every positive integer $n$ the number $a^{2^n}- 1$ has at least $n + 1$ distinct prime divisors.

A positive integer $n$ has exactly $12$ positive divisors $1 = d_1 < d_2 < d_3 < ... < d_{12} = n$. Let $m = d_4 - 1$. We have $d_m = (d_1 + d_2 + d_4) d_8$. Find $n$.

Find all integers $k \ge 5$ for which there is a positive integer $n$ with exactly $k$ positive divisors $1 = d_1 <d_2 < ... <d_k = n$ and $d_2d_3 + d_3d_5 + d_5d_2 = n$.

Let $p, q$ be two distinct primes. Prove that there are positive integers $a, b$ such that the arithmetic mean of all positive divisors of the number $n = p^aq^b$ is an integer.

Determine all pairs $(n, k)$ of distinct positive integers such that there exists a positive integer $s$ for which the number of divisors of $sn$ and of $sk$ are equal.

Let $A$ be the set of positive integer divisors of $2025$. Let $B$ be a randomly selected subset of $A$. The probability that $B$ is a nonempty set with the property that the least common multiple of its element is $2025$ is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Find all positive integers $a,b$ with $a>1$ such that, $b$ is a divisor of $a-1$ and $2a+1$ is a divisor of $5b-3$.

Find all positive integers $n$ such that if $S=\{d_1,d_2,\cdots,d_k\}$ is the set of positive integer divisors of $n$, then $S$ is a complete residue system modulo $k$. (In other words, for every pair of distinct indices $i$ and $j$, we have $d_i\not\equiv d_j \pmod{k}$). Proposed by Heidar Shushtari