Olympiad problems

2018 Switzerland - Final Round, 7

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

2011 Peru MO (ONEM), 1

Tags: number theory , multiple , consecutive

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 $n$.

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?

2011 Saudi Arabia Pre-TST, 1.1

Tags: number theory , consecutive

Set $A$ consists of $7$ consecutive positive integers less than $2011$, while set $B$ consists of $11$ consecutive positive integers. If the sum of the numbers in $A$ is equal to the sum of the numbers in $B$ , what is the maximum possible element that $A$ could contain?

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)

1949-56 Chisinau City MO, 4

Tags: algebra , consecutive , perfect square

Prove that the product of four consecutive integers plus $1$ is a perfect square.

1981 All Soviet Union Mathematical Olympiad, 306

Tags: product , number theory , consecutive

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

2009 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , power , consecutive

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.

2011 Dutch Mathematical Olympiad, 3

Tags: combinatorics , number theory , consecutive

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.

2017 Thailand Mathematical Olympiad, 5

Tags: number theory , sum of squares , consecutive , perfect square

Does there exist $2017$ consecutive positive integers, none of which could be written as $a^2 + b^2$ for some integers $a, b$? Justify your answer.

2011 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , prime factorization , consecutive , prime

It is said that a positive integer $n > 1$ has the property ($p$) if in its prime factorization $n = p_1^{a_1} \cdot ... \cdot p_j^{a_j}$ at least one of the prime factors $p_1, ... , p_j$ has the exponent equal to $2$. a) Find the largest number $k$ for which there exist $k$ consecutive positive integers that do not have the property ($p$). b) Prove that there is an infinite number of positive integers $n$ such that $n, n + 1$ and $n + 2$ have the property ($p$).

Indonesia MO Shortlist - geometry, g8

Tags: integer , geometry , consecutive , equal angles

Prove that there is only one triangle whose sides are consecutive natural numbers and one of the angles is twice the other angle.

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?

1979 Bundeswettbewerb Mathematik, 3

Tags: other bases , number theory , consecutive , bases

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.

2019 Irish Math Olympiad, 6

Tags: number theory , sum of powers , consecutive

The number $2019$ has the following nice properties: (a) It is the sum of the fourth powers of fuve distinct positive integers. (b) It is the sum of six consecutive positive integers. In fact, $2019 = 1^4 + 2^4 + 3^4 + 5^4 + 6^4$ (1) $2019 = 334 + 335 + 336 + 337 + 338 + 339$ (2) Prove that $2019$ is the smallest number that satises [b]both [/b] (a) and (b). (You may assume that (1) and (2) are correct!)

2015 Caucasus Mathematical Olympiad, 4

Tags: number theory , sum , prime number , consecutive

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]?

1999 Switzerland Team Selection Test, 10

Tags: product , consecutive , perfect square , number theory

Prove that the product of five consecutive positive integers cannot be a perfect square.

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 $n$.

1999 Estonia National Olympiad, 5

Tags: number , consecutive , sum , number theory , combinatorics , circles

The numbers $0, 1, 2, . . . , 9$ are written (in some order) on the circumference. Prove that a) there are three consecutive numbers with the sum being at least $15$, b) it is not necessarily the case that there exist three consecutive numbers with the sum more than $15$.

2000 Tournament Of Towns, 5

Tags: consecutive , number theory , divide , divisible

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)

2015 Estonia Team Selection Test, 1

Tags: algebra , consecutive , arithmetic progression , arithmetic sequence

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 $a_1 = a_2 = . . . = a_n$.

2013 Switzerland - Final Round, 5

Tags: combinatorics , consecutive , number theory

Each of $2n + 1$ students chooses a finite, nonempty set of consecutive integers . Two students are friends if they have chosen a common number. Everyone student is friends with at least $n$ other students. Show that there is a student who is friends with everyone else.

2012 Balkan MO Shortlist, N1

Tags: number theory , sequence , consecutive , perfect square

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.

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.

2022 Saudi Arabia JBMO TST, 3

Tags: consecutive , combinatorics

$2000$ consecutive integers (not necessarily positive) are written on the board. A student takes several turns. On each turn, he partitions the $2000$ integers into $1000$ pairs, and substitutes each pair by the difference arid the sum of that pair (note that the difference does not need to be positive as the student may choose to subtract the greater number from the smaller one; in addition, all the operations are carried simultaneously). Prove that the student will never again write $2000$ consecutive integers on the board.