Olympiad problems

2015 Caucasus Mathematical Olympiad, 4

We call a number greater than $25$, [i] semi-prime[/i] if it is the sum of some two different prime numbers. What is the greatest number of consecutive natural numbers that can be [i]semi-prime[/i]?

2004 Abels Math Contest (Norwegian MO), 1a

Tags: consecutive , sum , number theory

If $m$ is a positive integer, prove that $2^m$ cannot be written as a sum of two or more consecutive natural numbers.

2003 Dutch Mathematical Olympiad, 3

Tags: number theory , diophantine equation , diophantine , consecutive

Determine all positive integers$ n$ that can be written as the product of two consecutive integers and as well as the product of four consecutive integers numbers. In the formula: $n = a (a + 1) = b (b + 1) (b + 2) (b + 3)$.

2016 Bundeswettbewerb Mathematik, 1

Tags: consecutive , sum , summation , number theory

There are $\tfrac{n(n+1)}{2}$ distinct sums of two distinct numbers, if there are $n$ numbers. For which $n \ (n \geq 3)$ do there exist $n$ distinct integers, such that those sums are $\tfrac{n(n-1)}{2}$ consecutive numbers?

2003 Austria Beginners' Competition, 3

Tags: number theory , consecutive , divide , divisible

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 $D$.

1997 Tournament Of Towns, (533) 5

Tags: number theory , perfect cube , consecutive

Prove that the number (a) $97^{97}$ (b) $1997^{17}$ cannot be equal to a sum of cubes of several consecutive integers. (AA Egorov)

1991 Mexico National Olympiad, 5

Tags: consecutive , sum of squares , number theory

The sum of squares of two consecutive integers can be a square, as in $3^2+4^2 =5^2$. Prove that the sum of squares of $m$ consecutive integers cannot be a square for $m = 3$ or $6$ and find an example of $11$ consecutive integers the sum of whose squares is a square.

1997 Tuymaada Olympiad, 6

Tags: consecutive , divisor , number theory

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

2019 Tournament Of Towns, 4

Tags: number theory , consecutive , multiple , sum of squares , circles

There are given $1000$ integers $a_1,... , a_{1000}$. Their squares $a^2_1, . . . , a^2_{1000}$ are written in a circle. It so happened that the sum of any $41$ consecutive numbers on this circle is a multiple of $41^2$. Is it necessarily true that every integer $a_1,... , a_{1000}$ is a multiple of $41$? (Boris Frenkin)

1984 Tournament Of Towns, (055) O3

Tags: combinatorics , square table , consecutive

Consider the $4(N-1)$ squares on the boundary of an $N$ by $N$ array of squares. We wish to insert in these squares $4 (N-1)$ consecutive integers (not necessarily positive) so that the sum of the numbers at the four vertices of any rectangle with sides parallel to the diagonals of the array (in the case of a “degenerate” rectangle, i.e. a diagonal, we refer to the sum of the two numbers in its corner squares) are one and the same number. Is this possible? Consider the cases (a) $N = 3$ (b) $N = 4$ (c) $N = 5$ (VG Boltyanskiy, Moscow)

2006 BAMO, 2

Tags: number theory , consecutive , sum , algebra

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.

1996 Tournament Of Towns, (504) 1

Tags: consecutive , sum of squares , algebra , number theory

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)

2009 Danube Mathematical Competition, 2

Tags: sum , consecutive , number theory

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 .

2018 Switzerland - Final Round, 7

Tags: number theory , consecutive , divisor

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 $9 = 9$, $9 = 4 + 5$, $9 = 2 + 3 + 4$.

2018 Costa Rica - Final Round, N1

Tags: consecutive , number theory

Prove that there are only two sets of consecutive positive integers that satisfy that the sum of its elements is equal to $100$.

2004 Cuba MO, 2

Tags: number theory , consecutive

When we write the number $n > 2$ as the sum of some integers consecutive positives (at least two addends), we say that we have an [i]elegant decomposition[/i] of $n$. Two [i]elegant decompositions[/i] will be different if any of them contains some term that does not contains the other. How many different elegant decompositions does the number $3^{2004}$ have?

2014 Bosnia And Herzegovina - Regional Olympiad, 3

Tags: consecutive , number theory

Find all integers $n$ such that $n^4-8n+15$ is product of two consecutive integers

1984 Tournament Of Towns, (060) A5

Tags: number theory , prime , consecutive

The two pairs of consecutive natural numbers $(8, 9)$ and $(288, 289)$ have the following property: in each pair, each number contains each of its prime factors to a power not less than $2$. Prove that there are infinitely many such pairs. (A Andjans, Riga)

2015 Germany Team Selection Test, 2

Tags: integer , positive , consecutive , form , number theory

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?

2019 Tournament Of Towns, 1

Tags: combinatorics , sum , consecutive

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)

2007 Peru MO (ONEM), 3

Tags: number theory , consecutive , divisor

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.

2015 Estonia Team Selection Test, 7

Tags: number theory , digit , consecutive , power

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.

2017 Ecuador Juniors, 5

Tags: number theory , consecutive , coprime , divisor

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 $K$.

1961 Polish MO Finals, 1

Tags: number theory , consecutive

Prove that every natural number which is not an integer power of $2$ is the sum of two or more consecutive natural numbers.

1973 Dutch Mathematical Olympiad, 2

Tags: consecutive , number theory , combinatorics

Prove that for every $n \in N$ there exists exactly one sequence of $2n + 1$ consecutive numbers, such that the sum of the squares of the first $n+1$ numbers is equal to the sum of the squares of the last $n$ numbers. Also express the smallest number of that sequence in terms of $n$.