Found problems: 110
1998 Bundeswettbewerb Mathematik, 2
Prove that there exists an infinite sequence of perfect squares with the following properties:
(i) The arithmetic mean of any two consecutive terms is a perfect square,
(ii) Every two consecutive terms are coprime,
(iii) The sequence is strictly increasing.
2017 Thailand Mathematical Olympiad, 5
Does there exist $2017$ consecutive positive integers, none of which could be written as $a^2 + b^2$ for some integers $a, b$? Justify your answer.
2009 Junior Balkan Team Selection Tests - Romania, 2
A positive integer is called [i]saturated [/i]i f any prime factor occurs at a power greater than or equal to $2$ in its factorisation. For example, numbers $8 = 2^3$ and $9 = 3^2$ are saturated, moreover, they are consecutive. Prove that there exist infinitely many saturated consecutive numbers.
2014 Contests, 3
Give are the integers $a_{1}=11 , a_{2}=1111, a_{3}=111111, ... , a_{n}= 1111...111$( with $2n$ digits) with $n > 8$ .
Let $q_{i}= \frac{a_{i}}{11} , i= 1,2,3, ... , n$ the remainder of the division of $a_{i}$ by$ 11$ .
Prove that the sum of nine consecutive quotients: $s_{i}=q_{i}+q_{i+1}+q_{i+2}+ ... +q_{i+8}$ is a multiple of $9$ for any $i= 1,2,3, ... , (n-8)$
2016 Portugal MO, 2
In how many different ways can you write $2016$ as the sum of a sequence of consecutive natural numbers?
1999 Estonia National Olympiad, 5
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 $15$.
1996 Tournament Of Towns, (504) 1
Do there exist $10$ consecutive positive integers such that the sum of their squares is equal to the sum of squares of the next $9$ integers?
(Inspired by a diagram in an old text book)
1926 Eotvos Mathematical Competition, 2
Prove that the product of four consecutive natural numbers cannot be the square of an integer.
2021 New Zealand MO, 6
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).
Indonesia MO Shortlist - geometry, g8
Prove that there is only one triangle whose sides are consecutive natural numbers and one of the angles is twice the other angle.
2019 Tournament Of Towns, 1
Consider a sequence of positive integers with total sum $20$ such that no number and no sum of a set of consecutive numbers is equal to $3$. Is it possible for such a sequence to contain more than $10$ numbers?
(Alexandr Shapovalov)
2018 Switzerland - Final Round, 7
Let $n$ be a natural integer and let $k$ be the number of ways to write $n$ as the sum of one or more consecutive natural integers. Prove that $k$ is equal to the number of odd positive divisors of $n$.
Example: $9$ has three positive odd divisors and $9 = 9$, $9 = 4 + 5$, $9 = 2 + 3 + 4$.
2006 Spain Mathematical Olympiad, 2
Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.
1995 North Macedonia National Olympiad, 3
Prove that the product of $8$ consecutive natural numbers can never be a fourth power of natural number.
2011 Saudi Arabia BMO TST, 1
Prove that for any positive integer $n$ there is an equiangular hexagon whose sidelengths are $n + 1, n + 2 ,..., n + 6$ in some order.
2015 Estonia Team Selection Test, 1
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 $a_1 = a_2 = . . . = a_n$.
2004 Abels Math Contest (Norwegian MO), 1a
If $m$ is a positive integer, prove that $2^m$ cannot be written as a sum of two or more consecutive natural numbers.
2006 Thailand Mathematical Olympiad, 1
Show that the product of three consecutive positive integers is never a perfect square.
2020 Tournament Of Towns, 6
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
2001 Singapore MO Open, 4
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 $5^2 = 3^2 + 4^2$).
2003 Austria Beginners' Competition, 3
a) Show that the product of $5$ consecutive even integers is divisible by $15$.
b) Determine the largest integer $D$ such that the product of $5$ consecutive even integers is always divisible by $D$.
2004 Tournament Of Towns, 2
Find all possible values of $n \ge 1$ for which there exist $n$ consecutive positive integers whose sum is a prime number.
2011 Junior Balkan Team Selection Tests - Romania, 1
It is said that a positive integer $n > 1$ has the property ($p$) if in its prime factorization $n = p_1^{a_1} \cdot ... \cdot p_j^{a_j}$ at least one of the prime factors $p_1, ... , p_j$ has the exponent equal to $2$.
a) Find the largest number $k$ for which there exist $k$ consecutive positive integers that do not have the property ($p$).
b) Prove that there is an infinite number of positive integers $n$ such that $n, n + 1$ and $n + 2$ have the property ($p$).
2006 BAMO, 2
Since $24 = 3+5+7+9$, the number $24$ can be written as the sum of at least two consecutive odd positive integers.
(a) Can $2005$ be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not.
(b) Can $2006$ be written as the sum of at least two consecutive odd positive integers? If yes, give an example of how it can be done. If no, provide a proof why not.
2017 Argentina National Olympiad, 2
In a row there are $51$ written positive integers. Their sum is $100$ . An integer is [i]representable [/i] if it can be expressed as the sum of several consecutive numbers in a row of $51$ integers. Show that for every $k$ , with $1\le k \le 100$ , one of the numbers $k$ and $100-k$ is representable.