For each pair of positive integers $m$ and $n$, we define $f_m(n)$ as follows: $$ f_m(n) = \gcd(n, d_1) + \gcd(n, d_2) + \cdots + \gcd(n, d_k), $$ where $1 = d_1 < d_2 < \cdots < d_k = m$ are all the positive divisors of $m$. For example, $f_4(6) = \gcd(6,1) + \gcd(6,2) + \gcd(6,4) = 5$. $a)\:$ Find all positive integers $n$ such that $f_{2017}(n) = f_n(2017)$. $b)\:$ Find all positive integers $n$ such that $f_6(n) = f_n(6)$.

What is the smallest positive integer that is neither prime nor square and that has no prime factor less than 50? $\textbf{(A)}\hspace{.05in}3127 \qquad \textbf{(B)}\hspace{.05in}3133 \qquad \textbf{(C)}\hspace{.05in}3137 \qquad \textbf{(D)}\hspace{.05in}3139 \qquad \textbf{(E)}\hspace{.05in}3149 $

Let $(x_n)$ be a sequence of positive integers defined as follows: $x_1$ is a fixed six-digit number and for any $n \geq 1$, $x_{n+1}$ is a prime divisor of $x_n + 1$. Find $x_{19} + x_{20}$.

Show that there are an infinity of positive integers $n$ such that $2^{n}+3^{n}$ is divisible by $n^{2}$.

Let $p$ be an odd prime. Prove that the number $$\left\lfloor \left(\sqrt{5}+2\right)^{p}-2^{p+1}\right\rfloor$$ is divisible by $20p$.

Find the least natural number such that if the first digit (in the decimal system) is placed last, the new number is $7/2 $ times as large as the original number.

For a positive integer $n$ consider all its divisors $1 = d_1 < d_2 < \ldots < d_k = n$. For $2 \le i \le k-1$, let's call divisor $d_i$ good, if $d_{i-1}d_{i+1}$ isn't divisible by $d_i$. Find all $n$, such that the number of their good divisors is smaller than the number of their prime distinct divisors. [i]Proposed by Mykhailo Shtandenko[/i]

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.

The sets $A = \{z : z^{18} = 1\}$ and $B = \{w : w^{48} = 1\}$ are both sets of complex roots of unity. The set $C = \{zw : z \in A \ \text{and} \ w \in B\}$ is also a set of complex roots of unity. How many distinct elements are in $C$?

Prove that $$ (n+2)^n=\prod_{k=1}^{n+1} \sum_{l=1}^{n+1} le^{\frac{2i\pi k (n-l+1)}{n+2}} , $$ for any natural number $ n. $ [i]Mihai Piticari[/i]

$n$ and $d$ are positive integers such that $d$ divides $2n^2$. Prove that $n^2 + d$ cannot be a square.

Let $A=\{ x\in \mathbb{N},x=a^2+2b^2,a,b\in \mathbb{Z},ab\neq 0 \}$ and $p$ is a prime number. Prove that if $p^2\in A$, then $p\in A$.

Let $M \geq 1$ be a real number. Determine all natural numbers $n$ for which there exist distinct natural numbers $a$, $b$, $c > M$, such that $n = (a,b) \cdot (b,c) + (b,c) \cdot (c,a) + (c,a) \cdot (a,b)$ (where $(x,y)$ denotes the greatest common divisor of natural numbers $x$ and $y$).

Alison is eating $2401$ grains of rice for lunch. She eats the rice in a very peculiar manner: every step, if she has only one grain of rice remaining, she eats it. Otherwise, she nds the smallest positive integer $d > 1$ for which she can group the rice into equal groups of size $d$ with none left over. She then groups the rice into groups of size $d$, eats one grain from each group, and puts the rice back into a single pile. How many steps does it take her to finish all her rice?

Sophie wrote on a piece of paper every integer number from 1 to 1000 in decimal notation (including both endpoints). [b]a)[/b] Which digit did Sophie write the most? [b]b)[/b] Which digit did Sophie write the least?

Let $m, n,$ and $p$ be odd positive integers. Prove that the number $\sum\limits_{k=1}^{{{(n-1)}^{p}}}{{{k}^{m}}}$ is divisible by $n$

[b]p1.[/b] You are given three buckets with a capacity to hold $8$, $5$, and $3$ quarts of water, respectively. Initially, the first bucket is filled with $8$ quarts of water, while the remaining two buckets are empty. There are no markings on the buckets, so you are only allowed to empty a bucket into another one or to fill a bucket to its capacity using the water from one of the other buckets. (a) Describe a procedure by which we can obtain exactly $6$ quarts of water in the first bucket. (b) Describe a procedure by which we can obtain exactly $4$ quarts of water in the first bucket. [b]p2.[/b] A point in the plane is called a lattice point if its coordinates are both integers. A triangle whose vertices are all lattice points is called a lattice triangle. In each case below, give explicitly the coordinates of the vertices of a lattice triangle $T$ that satisfies the stated properties. (a) The area of $T$ is $1/2$ and two sides of $T$ have length greater than $2011$. (b) The area of $T$ is $1/2$ and the three sides of $T$ each have length greater than $2011$. [b]p3.[/b] Alice and Bob play several rounds of a game. In the $n$-th round, where $n = 1, 2, 3, ...$, the loser pays the winner $2^{n-1}$ dollars (there are no ties). After $40$ rounds, Alice has a profit of $\$2011$ (and Bob has lost $\$2011$). How many rounds of the game did Alice win, and which rounds were they? Justify your answer. [b]p4.[/b] Each student in a school is assigned a $15$-digit ID number consisting of a string of $3$’s and $7$’s. Whenever $x$ and $y$ are two distinct ID numbers, then $x$ and $y$ differ in at least three entries. Show that the number of students in the school is less than or equal to $2048$. [b]p5.[/b] A triangle $ABC$ has the following property: there is a point $P$ in the plane of $ABC$ such that the triangles $PAB$, $PBC$ and $PCA$ all have the same perimeter and the same area. Prove that: (a) If $P$ is not inside the triangle $ABC$, then $ABC$ is a right-angled triangle. (b) If $P$ is inside the triangle $ABC$, then $ABC$ is an equilateral triangle. PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

Let $k\ge0$ be a given integer. Suppose there exists positive integer $n,d$ and an odd integer $m>1$ with $d\mid m^{2^k}-1$ and $m\mid n^d+1$. Find all possible values of $\frac{m^{2^k}-1}d$.

Let $\Omega(n)$ be the number of prime factors of $n$. Define $f(1)=1$ and $f(n)=(-1)^{\Omega(n)}.$ Furthermore, let $$F(n)=\sum_{d|n} f(d).$$ Prove that $F(n)=0,1$ for all positive integers $n$. For which integers $n$ is $F(n)=1?$

Find the least natural number $n$, which has at least 6 different divisors $1=d_1<d_2<d_3<d_4<d_5<d_6<...$, for which $d_3+d_4=d_5+6$ and $d_4+d_5=d_6+7$.

Prove that for any positive integer $ m$ there exist an infinite number of pairs of integers $(x,y)$ such that $(\text{i})$ $x$ and $y$ are relatively prime; $(\text{ii})$ $x$ divides $y^2+m;$ $(\text{iii})$ $y$ divides $x^2+m.$

$a)$ Solve over the positive integers $3^x=x+2.$ $b)$ Find pairs $(x,y)\in\mathbb{N}\times\mathbb{N}$ such that $(x+3^y)$ and $(y+3^x)$ are consecutive.

Find all pairs of positive integers $\left(n;\;k\right)$ such that $n!=\left( n+1\right)^{k}-1$.

Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors. Suppose $M$ can be written as $\Pi_{k=1}^{n}p_k^{a_k}$ where the $p_k$’s are distinct primes and the $a_k$’s are positive integers. Find $\Sigma_{k=1}^{n}(p_k + a_k)$

Let $P$ and $Q$ be polynomials with integer coefficients. Suppose that the integers $a$ and $a+1997$ are roots of $P$, and that $Q(1998)=2000$. Prove that the equation $Q(P(x))=1$ has no integer solutions.