Students in the class of Peter practice the addition and multiplication of integer numbers.The teacher writes the numbers from $1$ to $9$ on nine cards, one for each number, and places them in an ballot box. Pedro draws three cards, and must calculate the sum and the product of the three corresponding numbers. Ana and Julián do the same, emptying the ballot box. Pedro informs the teacher that he has picked three consecutive numbers whose product is $5$ times the sum. Ana informs that she has no prime number, but two consecutive and that the product of these three numbers is $4$ times the sum of them. What numbers did Julian remove?

Two positive integers are coprime if their greatest common divisor is $1$. Let $C$ be the set of all divisors of the number $8775$ that are greater than $ 1$. A set of $k$ consecutive positive integers satisfies that each of them is coprime with some element of $C$. Determine the largest possible value of $K$.

In how many different ways can you write $2016$ as the sum of a sequence of consecutive natural numbers?

The number $1000$ can be written as the sum of $16$ consecutive natural numbers: $$1000 = 55 + 56 + ... + 70.$$ Determines all natural numbers that cannot be written as the sum of two or more consecutive natural numbers .

a) Prove that of $5$ consecutive positive integers one that is relatively prime with the other $4$ can always be selected. b) Prove that of $10$ consecutive positive integers one that is relatively prime with the other $9$ can always be selected.

Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.

A positive integer $n$ is called [i]naughty[/i] if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$. Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?

Consider any three consecutive natural numbers. Prove that the cube of the largest cannot be the sum of the cubes of the other two.

The lengths of the sides of an acute angled triangle are successive integers. Prove that the altitude to the second longest side divides this side into two segments whose difference in length equals $4$.

A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number $1$. For example, $1900$ is lucky, as $1900 \to 82 \to 68 \to 100 \to 1$. Find infinitely many pairs of consecutive numbers each of which is lucky.

Vukasin, Dimitrije, Dusan, Stefan and Filip asked their teacher to guess three consecutive positive integers, after these true statements: Vukasin: " The sum of the digits of one number is prime number. The sum of the digits of another of the other two is, an even perfect number.($n$ is perfect if $\sigma\left(n\right)=2n$). The sum of the digits of the third number equals to the number of it's positive divisors". Dimitrije:"Everyone of those three numbers has at most two digits equal to $1$ in their decimal representation". Dusan:"If we add $11$ to exactly one of them, then we have a perfect square of an integer" Stefan:"Everyone of them has exactly one prime divisor less than $10$". Filip:"The three numbers are square free". Professor found the right answer. Which numbers did he mention?

Find the least positive integer $n$ with the property: Among arbitrarily $n$ selected consecutive positive integers, all smaller than $2018$, there is at least one that is divisible by its sum of digits .

Let $n$ be a natural number, $n \ge 5$, and $a_1, a_2, . . . , a_n$ real numbers such that all possible sums $a_i + a_j$, where $1 \le i < j \le n$, form $\frac{n(n-1)}{2}$ consecutive members of an arithmetic progression when taken in some order. Prove that $a_1 = a_2 = . . . = a_n$.

Find $n$ such that each of the numbers $n,(n+1),...,(n+20)$ has the common divider greater than one with the number $30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13$.

Prove that the sum of the squares of five consecutive integers cannot be a perfect square.

Prove that the product of four consecutive natural numbers cannot be the square of an integer.

Prove that there is a natural number $n > 1$ such that the product of some $n$ consecutive natural numbers is equal to the product of some $n + 100$ consecutive natural numbers.

An integer $N$ is the product of two consecutive integers. (a) Prove that we can add two digits to the right of this number and obtain a perfect square. (b) Prove that this can be done in only one way if $N > 12$

The values of the polynomial $P(x) = 2x^3-30x^2+cx$ for any three consecutive integers are also three consecutive integers. Find these values.

A positive integer $n$ is called [i]naughty[/i] if it can be written in the form $n=a^b+b$ with integers $a,b \geq 2$. Is there a sequence of $102$ consecutive positive integers such that exactly $100$ of those numbers are naughty?

Are there $14$ consecutive positive integers, each of which has a divisor other than $1$ and not exceeding $11$?

A positive integer $n$ is said to possess Property ($A$) if there exists a positive integer $N$ such that $N^2$ can be written as the sum of the squares of $n$ consecutive positive integers. Is it true that there are infinitely many positive integers which possess Property ($A$)? Justify your answer. (As an example, the number $n = 2$ possesses Property ($A$) since $5^2 = 3^2 + 4^2$).

A math teacher chose $10$ consequtive numbers and submitted them to Pete and Basil. Each boy should split these numbers in pairs and calculate the sum of products of numbers in pairs. Prove that the boys can pair the numbers differently so that the resulting sums are equal.

There are $2n$ consecutive integers on a board. It is permitted to split them into pairs and simultaneously replace each pair by their difference (not necessarily positive) and their sum. Prove that it is impossible to obtain any $2n$ consecutive integers again. Alexandr Gribalko

Is it possible to place a positive integer in every cell of a $10 \times 10$ array in such a way that both the following conditions are satisfied? $\bullet$ Each number (not in the top row) is a proper divisor of the number immediately above. $\bullet$ Each row consists of 1$0$ consecutive positive integers (but not necessarily in order).