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 sequence $(a_n)_{n=1}^{\infty}$ of positive integers satisfies the condition $a_{n+1} = a_n +\tau (n)$ for all positive integers $n$ where $\tau (n)$ is the number of positive integer divisors of $n$. Determine whether two consecutive terms of this sequence can be perfect squares.

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 .

The vertices of a $33$-gon are labelled with the integers from $1$ to $33$. Each edge is then labelled with the sum of the labels of its two vertices. Is it possible for the edge labels to consist of $33$ consecutive numbers?

Prove that the product of $8$ consecutive natural numbers can never be a fourth power of natural number.

Prove that among $39$ consecutive natural numbers, there is always a number that has sum of its digits divisible by $ 12$. Is it true if we replace $39$ with $38$?

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?

Each of $2n + 1$ students chooses a finite, nonempty set of consecutive integers . Two students are friends if they have chosen a common number. Everyone student is friends with at least $n$ other students. Show that there is a student who is friends with everyone else.

Is it possible for the least common multiple of five consecutive positive integers to be a perfect square?

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.

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 $a_1 = a_2 = . . . = a_n$.

In base $10$ there exist two-digit natural numbers that can be factorized into two natural factors such that the two digits and the two factors form a sequence of four consecutive integers (for example, $12 = 3 \cdot 4$). Determine all such numbers in all bases.

Find the smallest natural number that is the sum of $9$ consecutive natural numbers, is the sum of $10$ consecutive natural numbers and is also the sum of $11$ consecutive natural numbers.

Prove that the product of five consecutive positive integers cannot be a perfect square.

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?

An integer is called [i]autodivi [/i] if it is divisible by the two-digit number formed by its last two digits (tens and units). For example, $78013$ is autodivi as it is divisible by $13$, $8517$ is autodivi since it is divisible by $17$. Find $6$ consecutive integers that are autodivi and that have the digits of the units, tens and hundreds other than $0$.

Prove that there is only one triangle whose sides are consecutive natural numbers and one of the angles is twice the other angle.

Prove that there exists a sequence of $100$ different integers such that the sum of the squares of any two consecutive terms is a perfect square. (S Tokarev)

Prove that for any positive integer $n$ there is an equiangular hexagon whose sidelengths are $n + 1, n + 2 ,..., n + 6$ in some order.

Show that the product of three consecutive positive integers is never a perfect square.

Consider an infinite sequence $$1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, \ldots, \underbrace{n,\ldots,n}_{n\text{ times}},\ldots.$$ Find the 1000th term of the sequence.

Prove that the product of four consecutive integers plus $1$ is a perfect square.

Prove that for every prime number $p$ and positive integer $a$, there exists a natural number $n$ such that $p^n$ contains $a$ consecutive equal digits.

The numbers $0, 1, 2, . . . , 9$ are written (in some order) on the circumference. Prove that a) there are three consecutive numbers with the sum being at least $15$, b) it is not necessarily the case that there exist three consecutive numbers with the sum more than $15$.

Determine all triangles such that the lengths of the three sides and its area are given by four consecutive natural numbers.