Found problems: 110
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.
1975 Dutch Mathematical Olympiad, 2
Let $T = \{n \in N|$n consists of $2$ digits $\}$ and $$P = \{x|x = n(n + 1)... (n + 7); n,n + 1,..., n + 7 \in T\}.$$
Determine the gcd of the elements of $P$.
2009 Danube Mathematical Competition, 2
Prove that all the positive integer numbers , except for the powers of $2$, can be written as the sum of (at least two) consecutive natural numbers .
1984 Tournament Of Towns, (071) T5
Prove that among $18$ consecutive three digit numbers there must be at least one which is divisible by the sum of its digits.
2014 Bosnia And Herzegovina - Regional Olympiad, 3
Find all integers $n$ such that $n^4-8n+15$ is product of two consecutive integers
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 $m$.
VMEO III 2006, 10.2
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 $38$?
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.
1954 Poland - Second Round, 2
Prove that among ten consecutive natural numbers there is always at least one, and at most four, numbers that are not divisible by any of the numbers $ 2 $, $ 3 $, $ 5 $, $ 7 $.
1936 Moscow Mathematical Olympiad, 026
Find $4$ consecutive positive integers whose product is $1680$.
1941 Moscow Mathematical Olympiad, 075
Prove that $1$ plus the product of any four consecutive integers is a perfect square.
2003 Junior Balkan Team Selection Tests - Romania, 2
Consider the prime numbers $n_1< n_2 <...< n_{31}$. Prove that if $30$ divides $n_1^4 + n_2^4+...+n_{31}^4$, then among these numbers one can find three consecutive primes.
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 $i= 1,2,3, ... , (n-8)$
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.
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?
1995 Argentina National Olympiad, 4
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.
2011 Tournament of Towns, 4
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?
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)
2006 Thailand Mathematical Olympiad, 1
Show that the product of three consecutive positive integers is never a perfect square.
2007 Peru MO (ONEM), 3
We say that a natural number of at least two digits $E$ is [i]special [/i] if each time two adjacent digits of $E$ are added, a divisor of $E$ is obtained. For example, $2124$ is special, since the numbers $2 + 1$, $1 + 2$ and $2 + 4$ are all divisors of $2124$. Find the largest value of $n$ for which there exist $n$ consecutive natural numbers such that they are all special.
2010 May Olympiad, 3
Find the minimum $k>2$ for which there are $k$ consecutive integers such that the sum of their squares is a square.
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 $m$.
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.
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.
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.