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.
2010 Saudi Arabia Pre-TST, 2.2
Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.
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 $a_1 = a_2 = . . . = a_n$.
2023 Francophone Mathematical Olympiad, 4
Do there exist integers $a$ and $b$ such that none of the numbers $a,a+1,\ldots,a+2023,b,b+1,\ldots,b+2023$ divides any of the $4047$ other numbers, but $a(a+1)(a+2)\cdots(a+2023)$ divides $b(b+1)\cdots(b+2023)$?
2015 Germany Team Selection Test, 2
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?
2011 Portugal MO, 6
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 .
2008 Estonia Team Selection Test, 4
Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.
1980 Spain Mathematical Olympiad, 6
Prove that if the product of four consecutive natural numbers is added one unit, the result is a perfect square.
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).
1997 Singapore Team Selection Test, 2
Let $a_n$ be the number of n-digit integers formed by $1, 2$ and $3$ which do not contain any consecutive $1$’s. Prove that $a_n$ is equal to $$\left( \frac12 + \frac{1}{\sqrt3}\right)(\sqrt{3} + 1)^n$$ rounded off to the nearest integer.
1947 Moscow Mathematical Olympiad, 124
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.
1989 Tournament Of Towns, (224) 2
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 $4$.
2014 Federal Competition For Advanced Students, P2, 4
For an integer $n$ let $M (n) = \{n, n + 1, n + 2, n + 3, n + 4\}$. Furthermore, be $S (n)$ sum of squares and $P (n)$ the product of the squares of the elements of $M (n)$. For which integers $n$ is $S (n)$ a divisor of $P (n)$ ?
2014 Greece JBMO TST, 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 $i= 1,2,3, ... , (n-8)$
2004 Junior Balkan Team Selection Tests - Moldova, 4
Different non-zero natural numbers a$_1, a_2,. . . , a_{12}$ satisfy the condition:
all positive differences other than two numbers $a_i$ and $a_j$ form many $20$ consecutive natural numbers.
a) Show that $\max \{a_1, a_2,. . . , a_{12}\} - \min \{a_1, a_2,. . . , a_{12}\} = 20$.
b)Determine $12$ natural numbers with the property from the statement.
1977 Spain Mathematical Olympiad, 4
Prove that the sum of the squares of five consecutive integers cannot be a perfect square.
1926 Eotvos Mathematical Competition, 2
Prove that the product of four consecutive natural numbers cannot be the square of an integer.
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.
2015 JBMO Shortlist, NT1
What is the greatest number of integers that can be selected from a set of $2015$ consecutive numbers so that no sum of any two selected numbers is divisible by their difference?
1961 Poland - Second Round, 1
Prove that no number of the form $ 2^n $, where $ n $ is a natural number, is the sum of two or more consecutive natural numbers.
2000 Tournament Of Towns, 5
What is the largest number $N$ for which there exist $N$ consecutive positive integers such that the sum of the digits in the $k$-th integer is divisible by $k$ for $1 \le k \le N$ ?
(S Tokarev)
1981 All Soviet Union Mathematical Olympiad, 322
Find $n$ such that each of the numbers $n,(n+1),...,(n+20)$ has the common divider greater than one with the number $30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13$.
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.
1998 Belarus Team Selection Test, 2
The numbers $1,2,...,n$ ($n \ge 5$) are written on the circle in the clockwise order. Per move it is allowed to exchange any couple of consecutive numbers $a, b$ to the couple $\frac{a+b}{2}, \frac{a+b}{2}$.
Is it possible to make all numbers equal using these operations?
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 $i= 1,2,3, ... , (n-8)$