A positive integer is called [i]beautiful[/i] if it can be represented in the form $\dfrac{x^2+y^2}{x+y}$ for two distinct positive integers $x,y$. A positive integer that is not beautiful is [i]ugly[/i]. a) Prove that $2014$ is a product of a beautiful number and an ugly number. b) Prove that the product of two ugly numbers is also ugly.

Given a positive integer $m$, let $d(m)$ be the number of positive divisors of $m$. Determine all positive integers $n$ such that $d(n) +d(n+ 1) = 5$.

Let be $n\geq 3$ fixed positive integer.Let be real numbers $a_1,a_2,...,a_n,b_1,b_2,...,b_n$ such that satisfied this conditions: [b]$i)$[/b] $ $ $a_n\geq a_{n-1}$ and $b_n\geq b_{n-1}$ [b]$ii)$[/b] $ $ $0<a_1\leq b_1\leq a_2\leq b_2\leq ... \leq a_{n-1}\leq b_{n-1}$ [b]$iii)$[/b] $ $ $a_1+a_2+...+a_n=b_1+b_2+...+b_n$ [b]$iv)$[/b] $ $ $a_{1}\cdot a_2\cdot ...\cdot a_n=b_1\cdot b_2\cdot ...\cdot b_n$ Show that $a_i=b_i$ for all $i=1,2,...,n$

Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.

Find the smallest positive integer for which when we move the last right digit of the number to the left, the remaining number be $\frac 32$ times of the original number.

[b]p1.[/b] Suppose that $a \oplus b = ab - a - b$. Find the value of $$((1 \oplus 2) \oplus (3 \oplus 4)) \oplus 5.$$ [b]p2.[/b] Neethin scores a $59$ on his number theory test. He proceeds to score a $17$, $23$, and $34$ on the next three tests. What score must he achieve on his next test to earn an overall average of $60$ across all five tests? [b]p3.[/b] Consider a triangle with side lengths $28$ and $39$. Find the number of possible integer lengths of the third side. [b]p4.[/b] Nithin is thinking of a number. He says that it is an odd two digit number where both of its digits are prime, and that the number is divisible by the sum of its digits. What is the sum of all possible numbers Nithin might be thinking of? [b]p5.[/b] Dora sees a fire burning on the dance floor. She calls her friends to warn them to stay away. During the first pminute Dora calls Poonam and Serena. During the second minute, Poonam and Serena call two more friends each, and so does Dora. This process continues, with each person calling two new friends every minute. How many total people would know of the fire after $6$ minutes? [b]p6.[/b] Charlotte writes all the positive integers $n$ that leave a remainder of $2$ when $2018$ is divided by $n$. What is the sum of the numbers that she writes? [b]p7.[/b] Consider the following grid. Stefan the bug starts from the origin, and can move either to the right, diagonally in the positive direction, or upwards. In how many ways can he reach $(5, 5)$? [img]https://cdn.artofproblemsolving.com/attachments/9/9/b9fdfdf604762ec529a1b90d663e289b36b3f2.png[/img] [b]p8.[/b] Let $a, b, c$ be positive numbers where $a^2 + b^2 + c^2 = 63$ and $2a + 3b + 6c = 21\sqrt7$. Find $\left( \frac{a}{c}\right)^{\frac{a}{b}} $. [b]p9.[/b] What is the sum of the distinct prime factors of $12^5 + 12^4 + 1$? [b]p10.[/b] Allen starts writing all permutations of the numbers $1$, $2$, $3$, $4$, $5$, $6$ $7$, $8$, $9$, $10$ on a blackboard. At one point he writes the permutation $9$, $4$, $3$, $1$, $2$, $5$, $6$, $7$, $8$, $10$. David points at the permutation and observes that for any two consecutive integers $i$ and $i+1$, all integers that appear in between these two integers in the permutation are all less than $i$. For example, $4$ and $5$ have only the numbers $3$, $1$, $2$ in between them. How many of the $10!$ permutations on the board satisfy this property that David observes? [b]p11.[/b] (Estimation) How many positive integers less than $2018$ can be expressed as the sum of $3$ square numbers? PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].

There are $n$ light bulbs in a circle and one of them is marked. Let operation $A$: Take a positive divisor $d$ of the number $n,$ starting with the light bulb marked and clockwise, we count around the circumference from $1$ to $dn$, changing the state (on or off) to those light bulbs that correspond to the multiples of $d$. Let operation $B$ be: Apply operation$ A$ to all positive divisors of $n$ (to the first divider that is applied is with all the light bulbs off and the remaining divisors is with the state resulting from the previous divisor). Determine all the positive integers $n$, such that when applying the operation on $B$, all the light bulbs are on.

Two operations are allowed: (i) to write two copies of number $1$; (ii) to replace any two identical numbers $n$ by $(n + 1)$ and $(n - 1)$. Find the minimal number of operations that required to produce the number $2005$ (at the beginning there are no numbers). [i](8 points)[/i]

Prove that you can represent an arbitrary number not exceeding $n!$ as a sum of $k$ different numbers ($k\le n$) that are divisors of $n!$.

Find all positive integers $a$, $b$ and $c$ such that $ab$ is a square, and \[a+b+c-3\sqrt[3]{abc}=1.\] [i]Proposed by usjl[/i]

Let $k$ be a positive integer. A sequence $a_0,a_1,...,a_n,n>0$ of positive integers satisfies the following conditions: $(i)$ $a_0=a_n=1$; $(ii)$ $2\leq a_i\leq k$ for each $i=1,2,...,n-1$; $(iii)$For each $j=2,3,...,k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0,a_1,...,a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$; $(iv)$For any $i=1,2,...,n-1$, $\gcd(a_i,a_{i-1})=1=\gcd(a_i,a_{i+1})$, and $a_i$ divides $a_{i-1}+a_{i+1}$. Suppose there is another sequence $b_0,b_1,...,b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}}>\frac{b_i}{a_i}$ for all $i=0,1,...,n-1$. Find the minimum value of $b_n-b_0$.

Let $\nu$ be an irrational positive number, and let $m$ be a positive integer. A pair of $(a,b)$ of positive integers is called [i]good[/i] if \[a \left \lceil b\nu \right \rceil - b \left \lfloor a \nu \right \rfloor = m.\] A good pair $(a,b)$ is called [i]excellent[/i] if neither of the pair $(a-b,b)$ and $(a,b-a)$ is good. Prove that the number of excellent pairs is equal to the sum of the positive divisors of $m$.

Prove that the integral part of the decimal representation of the number $(3+\sqrt{5})^n$ is odd, for every positive integer $n.$

Is there a number whose digits are only $1$'s and which is divided by $1999$?

Find all pairs $(x,y)$ of positive integers that satisfy $$2^x+17=y^4$$.

For all $n\ge2$, let $a_n$ be the product of all coprime natural numbers less than $n$. Prove that (a) $n\mid a_n+1\Leftrightarrow n=2,4,p^\alpha,2p^\alpha$ (b) $n\mid a_n-1\Leftrightarrow n\ne2,4,p^\alpha,2p^\alpha$ Here $p$ is an odd prime number and $\alpha\in\mathbb N$.

Find the least positive integer whose digits add to a multiple of 27 yet the number itself is not a multiple of 27. For example, 87999921 is one such number.

Find all the prime number $p, q$ and r with $p < q < r$, such that $25pq + r = 2004$ and $pqr + 1 $ is a perfect square.

Let $a_{1}, a_{2}, \cdots, a_{k}$ be relatively prime positive integers. Determine the largest integer which cannot be expressed in the form \[x_{1}a_{2}a_{3}\cdots a_{k}+x_{2}a_{1}a_{3}\cdots a_{k}+\cdots+x_{k}a_{1}a_{2}\cdots a_{k-1}\] for some nonnegative integers $x_{1}, x_{2}, \cdots, x_{k}$.

Prove that there is no positive integer $n>1$ such that $n\mid2^{n} -1.$

Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.

Given a prime number $p\ge 3,$ and an integer $d \ge 1$. Prove that there exists an integer $n\ge 1,$ such that $\gcd(n,d) = 1,$ and the product \[P=\prod\limits_{1 \le i < j < p} {({i^{n + j}} - {j^{n + i}})} \text{ is not divisible by } p^n.\]

Let $N>2^{5000}$ be a positive integer. Prove that if $1\leq a_1<\cdots<a_k<100$ are distinct positive integers then the number \[\prod_{i=1}^{k}\left(N^{a_i}+a_i\right)\] has at least $k$ distinct prime factors. Note. Results with $2^{5000}$ replaced by some other constant $N_0$ will be awarded points depending on the value of $N_0$. [i]Proposed by Evan Chen[/i]

Do there exist positive integers $a>b>1$ such that for each positive integer $k$ there exists a positive integer $n$ for which $an+b$ is a $k$-th power of a positive integer?

Are there $10$ different integers such that all the sums made up of $9$ of them are perfect squares?