Olympiad problems

2015 Estonia Team Selection Test, 7

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.

2015 JBMO Shortlist, NT1

Tags: integer , number theory , consecutive , divisible , difference , maximum

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?

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)

2014 Greece JBMO TST, 3

Tags: number theory , consecutive , sum , multiple

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 $i= 1,2,3, ... , (n-8)$

1995 North Macedonia National Olympiad, 3

Tags: number theory , consecutive , power

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

2010 Contests, 2

Tags: number theory , power of 2 , consecutive

A number is called polite if it can be written as $ m + (m+1)+...+ n$, for certain positive integers $ m <n$ . For example: $18$ is polite, since $18 =5 + 6 + 7$. A number is called a power of two if it can be written as $2^{\ell}$ for some integer $\ell \ge 0$. (a) Show that no number is both polite and a power of two. (b) Show that every positive integer is polite or a power of two.

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.

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?

1986 Brazil National Olympiad, 2

Tags: positive integer , number theory , combinatorics , consecutive

Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.

2015 Belarus Team Selection Test, 1

Tags: number theory , divisible , consecutive , prime

Given $m,n \in N$ such that $M>n^{n-1}$ and the numbers $m+1, m+2, ..., m+n$ are composite. Prove that exist distinct primes $p_1,p_2,...,p_n$ such that $M+k$ is divisible by $p_k$ for any $k=1,2,...,n$. Tuymaada Olympiad 2004, C.A.Grimm. USA

2010 Dutch Mathematical Olympiad, 2

Tags: number theory , power of 2 , consecutive

A number is called polite if it can be written as $ m + (m+1)+...+ n$, for certain positive integers $ m <n$ . For example: $18$ is polite, since $18 =5 + 6 + 7$. A number is called a power of two if it can be written as $2^{\ell}$ for some integer $\ell \ge 0$. (a) Show that no number is both polite and a power of two. (b) Show that every positive integer is polite or a power of two.

1992 Tournament Of Towns, (352) 1

Tags: number theory , consecutive , perfect square

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)

2005 May Olympiad, 2

Tags: number theory , consecutive

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 $0$.

2014 Federal Competition For Advanced Students, P2, 4

Tags: number theory , sum of squares , product , divisor , consecutive

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 $P (n)$ ?

1961 Czech and Slovak Olympiad III A, 1

Tags: sequence , consecutive , positive integer

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.

1949 Kurschak Competition, 3

Tags: number theory , consecutive

Which positive integers cannot be represented as a sum of (two or more) consecutive integers?

2013 Costa Rica - Final Round, LRP2

Tags: probability , consecutive , number theory

From a set containing $6$ positive and consecutive integers they are extracted, randomly and with replacement, three numbers $a, b, c$. Determine the probability that even $a^b + c$ generates as a result .

2023 Francophone Mathematical Olympiad, 4

Tags: number theory , divide , consecutive , product

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 $b(b+1)\cdots(b+2023)$?

2010 Saudi Arabia Pre-TST, 2.2

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

Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.

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: $n = a (a + 1) = b (b + 1) (b + 2) (b + 3)$.

2012 NZMOC Camp Selection Problems, 2

Tags: number theory , consecutive , divide , divisible , sum

Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.

1980 Spain Mathematical Olympiad, 6

Tags: perfect square , consecutive , number theory

Prove that if the product of four consecutive natural numbers is added one unit, the result is a perfect square.

2009 Regional Olympiad of Mexico Center Zone, 4

Tags: perfect square , consecutive , digit , number theory

Let $N = 2 \: \: \underbrace {99… 9} _{n \,\,\text {times}} \: \: 82 \: \: \underbrace {00… 0} _{n \,\, \text {times} } \: \: 29$. Prove that $N$ can be written as the sum of the squares of $3$ consecutive natural numbers.

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.

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)