Prove that if $a$, $b$ and $c$ are integers and the sums $$\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \,\,\,\, and \,\,\,\, \frac{a}{c}+\frac{c}{b}+\frac{b}{a}$$ are also integers, then we have $|a| = |v| = |c|$. (A Gribalko)

Determine the last digit of number $18^1+18^2+...+18^{19}+18^{20}$

Let $x, y$ be distinct positive integers. Prove that the number $$\frac{(x+y)^2}{ x^3 + xy^2 - x^2y - y^3}$$ is not an integer

How many pairs $ (m,n)$ of positive integers with $ m < n$ fulfill the equation $ \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n$?

For a positive integer $k$, let $d(k)$ denote the number of divisors of $k$ and let $s(k)$ denote the digit sum of $k$. A positive integer $n$ is said to be [i]amusing[/i] if there exists a positive integer $k$ such that $d(k)=s(k)=n$. What is the smallest amusing odd integer greater than $1$?

For positive integers $a, b, c$ (not necessarily distinct), suppose that $a+bc, b+ac, c+ab$ are all perfect squares. Show that $$a^2(b+c)+b^2(a+c)+c^2(a+b)+2abc$$ can be written as sum of two squares.

We are given an $n \times n$ board. Rows are labeled with numbers $1$ to $n$ downwards and columns are labeled with numbers $1$ to $n$ from left to right. On each field we write the number $x^2 + y^2$ where $(x, y)$ are its coordinates. We are given a figure and can initially place it on any field. In every step we can move the figure from one field to another if the other field has not already been visited and if at least one of the following conditions is satisfied:[list] [*] the numbers in those $2$ fields give the same remainders when divided by $n$, [*] those fields are point reflected with respect to the center of the board.[/list]Can all the fields be visited in case: [list=a][*] $n = 4$, [*] $n = 5$?[/list] [i]Josip Pupić[/i]

Find all subsets of $\mathbb N$ like $S$ such that \[\forall m,n \in S \implies \dfrac{m+n}{\gcd(m,n)} \in S \]

For any positive integer n, let $\varphi(n)$ be the number of integers in the set $\{1, 2, \ldots , n\}$ whose greatest common divisor with $n$ is 1. Determine the maximum value of $\frac{n}{\varphi(n)}$ for $n$ in the set $\{2, \ldots, 1000\}$ and all values of $n$ for which this maximum is attained.

Let $a, b, c$ be positive integers such that the product $$\gcd(a,b) \cdot \gcd(b,c) \cdot \gcd(c,a) $$ is a perfect square. Prove that the product $$\operatorname{lcm}(a,b) \cdot \operatorname{lcm}(b,c) \cdot \operatorname{lcm}(c,a) $$ is also a perfect square.

Given positive integers $a,b,c,d$ such that $a\mid c^d$ and $b\mid d^c$. Prove that \[ ab\mid (cd)^{max(a,b)} \]

Let $a_1, a_2, a_3, a_4, a_5$ be positive integers such that $a_1, a_2, a_3$ and $a_3, a_4, a_5$ are both geometric sequences and $a_1, a_3, a_5$ is an arithmetic sequence. If $a_3 = 1575$, find all possible values of $\vert a_4 - a_2 \vert$.

Consider the number $\left(101^2-100^2\right)\cdot\left(102^2-101^2\right)\cdot\left(103^2-102^2\right)\cdot...\cdot\left(200^2-199^2\right)$. [list=a] [*] Determine its units digit. [*] Determine its tens digit. [/list]

Suppose that $m,n\ge2$ are integers such that $m+n-1$ divides $m^2+n^2-1$. Prove that the number $m+n-1$ is not prime.

Let $a$ and $b$ be positive integers and $K=\sqrt{\frac{a^2+b^2}2}$, $A=\frac{a+b}2$. If $\frac KA$ is a positive integer, prove that $a=b$.

Prove that there exist infinitely many positive integers $n$ such that $\frac{4^n+2^n+1}{n^2+n+1}$ is a positive integer.

[b](a)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n + 1 = x^2$. [b](b)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n = x^2 + 1$.

Find all natural values of $n$ and $m$, such that $(n -1)2^{n - 1} + 5 = m^2 + 4m$.

Show that there do not exist natural numbers $a_1, a_2, \dots, a_{2018}$ such that the numbers \[ (a_1)^{2018}+a_2, (a_2)^{2018}+a_3, \dots, (a_{2018})^{2018}+a_1 \] are all powers of $5$ [i]Proposed by Tejaswi Navilarekallu[/i]

Prove that there exist infinitely many positive integers $n$ such that the largest prime divisor of $n^4 + n^2 + 1$ is equal to the largest prime divisor of $(n+1)^4 + (n+1)^2 +1$.

For some positive integer $n$, consider the usual prime factorization \[n = \displaystyle \prod_{i=1}^{k} p_{i}^{e_{i}}=p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k},\] where $k$ is the number of primes factors of $n$ and $p_{i}$ are the prime factors of $n$. Define $Q(n), R(n)$ by \[ Q(n) = \prod_{i=1}^{k} p_{i}^{p_{i}} \text{ and } R(n) = \prod_{i=1}^{k} e_{i}^{e_{i}}. \] For how many $1 \leq n \leq 70$ does $R(n)$ divide $Q(n)$?

a) Given that integers $a$ and $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer. b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?

Let $k$ and $m$ be integers with $1 < k < m$. For a positive integer $i$, let $L_i$ be the least common multiple of $1,2,\ldots,i$. Prove that $k$ is a divisor of $L_i \cdot [\binom{m}{i} - \binom{m-k}{i}]$ for all $i \geq 1$. [Here, $\binom{n}{i} = \frac{n!}{i!(n-i)!}$ denotes a binomial coefficient. Note that $\binom{n}{i} = 0$ if $n < i$.]

For a natural number $n$, set $S_n$ is defined as: \[S_n = \left \{ {n\choose n}, {2n \choose n}, {3n\choose n},..., {n^2 \choose n} \right \}.\] a) Prove that there are infinitely many composite numbers $n$, such that the set $S_n$ is not complete residue system mod $n$; b) Prove that there are infinitely many composite numbers $n$, such that the set $S_n$ is complete residue system mod $n$.

A binary string is a word containing only $0$s and $1$s. In a binary string, a $1-$run is a non extendable substring containing only $1$s. Given a positive integer $n$, let $B(n)$ be the number of $1-$runs in the binary representation of $n$. For example, $B(107)=3$ since $107$ in binary is $1101011$ which has exactly three $1-$runs. What is the following expression equal to? $$B(1)+B(2)+B(3)+ \dots + B(255)$$