Olympiad problems

1977 Spain Mathematical Olympiad, 4

Prove that the sum of the squares of five consecutive integers cannot be a perfect square.

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)

1997 Singapore Team Selection Test, 2

Tags: sequence , consecutive , number theory

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.

2015 Belarus Team Selection Test, 1

Tags: divisible , number theory , 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

1936 Moscow Mathematical Olympiad, 026

Tags: product , consecutive , algebra

Find $4$ consecutive positive integers whose product is $1680$.

2023 Francophone Mathematical Olympiad, 4

Tags: divide , product , number theory , consecutive

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

1997 Tuymaada Olympiad, 6

Tags: divisor , consecutive , number theory

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

2007 Peru MO (ONEM), 3

Tags: divisor , consecutive , number theory

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.

2002 Cono Sur Olympiad, 1

Tags: number theory , sums and products , consecutive

Students in the class of Peter practice the addition and multiplication of integer numbers.The teacher writes the numbers from $1$ to $9$ on nine cards, one for each number, and places them in an ballot box. Pedro draws three cards, and must calculate the sum and the product of the three corresponding numbers. Ana and Julián do the same, emptying the ballot box. Pedro informs the teacher that he has picked three consecutive numbers whose product is $5$ times the sum. Ana informs that she has no prime number, but two consecutive and that the product of these three numbers is $4$ times the sum of them. What numbers did Julian remove?

2018 Swedish Mathematical Competition, 4

Tags: consecutive , divisible , number theory , sum of digits

Find the least positive integer $n$ with the property: Among arbitrarily $n$ selected consecutive positive integers, all smaller than $2018$, there is at least one that is divisible by its sum of digits .

2011 Peru MO (ONEM), 1

Tags: consecutive , multiple , number theory

We say that a positive integer is [i]irregular [/i] if said number is not a multiple of none of its digits. For example, $203$ is irregular because $ 203$ is not a multiple of $2$, it is not multiple of $0$ and is not a multiple of $3$. Consider a set consisting of $n$ consecutive positive integers. If all the numbers in that set are irregular, determine the largest possible value of $n$.

2011 Dutch Mathematical Olympiad, 3

Tags: number theory , consecutive , combinatorics

In a tournament among six teams, every team plays against each other team exactly once. When a team wins, it receives $3$ points and the losing team receives $0$ points. If the game is a draw, the two teams receive $1$ point each. Can the final scores of the six teams be six consecutive numbers $a,a +1,...,a + 5$? If so, determine all values of $a$ for which this is possible.

1979 Bundeswettbewerb Mathematik, 3

Tags: other bases , consecutive , bases , number theory

In base $10$ there exist two-digit natural numbers that can be factorized into two natural factors such that the two digits and the two factors form a sequence of four consecutive integers (for example, $12 = 3 \cdot 4$). Determine all such numbers in all bases.

1981 All Soviet Union Mathematical Olympiad, 306

Tags: product , consecutive , number theory

Let us say, that a natural number has the property $P(k)$ if it can be represented as a product of $k$ succeeding natural numbers greater than $1$. a) Find k such that there exists n which has properties $P(k)$ and $P(k+2)$ simultaneously. b) Prove that there is no number having properties $P(2)$ and $P(4)$ simultaneously

2012 Balkan MO Shortlist, N1

Tags: sequence , consecutive , perfect square , number theory

A sequence $(a_n)_{n=1}^{\infty}$ of positive integers satisfies the condition $a_{n+1} = a_n +\tau (n)$ for all positive integers $n$ where $\tau (n)$ is the number of positive integer divisors of $n$. Determine whether two consecutive terms of this sequence can be perfect squares.

1961 Polish MO Finals, 1

Tags: consecutive , number theory

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

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

2015 JBMO Shortlist, NT1

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

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?

2020 Durer Math Competition Finals, 3

Tags: least common multiple , perfect square , consecutive , number theory

Is it possible for the least common multiple of five consecutive positive integers to be a perfect square?

1973 Dutch Mathematical Olympiad, 2

Tags: number theory , consecutive , 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 $n$.

1998 Belarus Team Selection Test, 2

Tags: consecutive , number theory , combinatorics

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?

2015 Estonia Team Selection Test, 7

Tags: number theory , digit , power , consecutive

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.

1975 Dutch Mathematical Olympiad, 2

Tags: number theory , consecutive

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 $P$.

2000 ITAMO, 1

Tags: consecutive , sum of squares , number theory

A possitive integer is called [i]special[/i] if all its decimal digits are equal and it can be represented as the sum of squares of three consecutive odd integers. (a) Find all $4$-digit [i]special[/i] numbers (b) Are there $2000$-digit [i]special[/i] numbers?

1980 Spain Mathematical Olympiad, 6

Tags: number theory , perfect square , consecutive

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