Olympiad problems

2015 Germany Team Selection Test, 2

Tags: form , integer , positive , consecutive , 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?

2003 Junior Balkan Team Selection Tests - Romania, 2

Tags: prime , consecutive , divide , sum of powers , number theory

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.

1941 Moscow Mathematical Olympiad, 075

Tags: number theory , perfect square , consecutive

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

1981 All Soviet Union Mathematical Olympiad, 322

Tags: consecutive , greatest common divisor , number theory

Find $n$ such that each of the numbers $n,(n+1),...,(n+20)$ has the common divider greater than one with the number $30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13$.

1954 Poland - Second Round, 2

Tags: consecutive , number theory

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 $.

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.

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.

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.

1961 Poland - Second Round, 1

Tags: power of 2 , consecutive , number theory

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.

2011 Tournament of Towns, 4

Tags: consecutive , sum , combinatorics

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?

1984 Tournament Of Towns, (055) O3

Tags: square table , consecutive , combinatorics

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)

2015 Germany Team Selection Test, 2

Tags: integer , positive , form , consecutive , 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?

2009 Regional Olympiad of Mexico Center Zone, 4

Tags: number theory , consecutive , digit , perfect square

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.

1995 Argentina National Olympiad, 4

Tags: consecutive , number theory

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.

1947 Moscow Mathematical Olympiad, 124

Tags: combinatorics , consecutive , relatively prime , number theory

a) Prove that of $5$ consecutive positive integers one that is relatively prime with the other $4$ can always be selected. b) Prove that of $10$ consecutive positive integers one that is relatively prime with the other $9$ can always be selected.

2019 Irish Math Olympiad, 6

Tags: consecutive , sum of powers , number theory

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!)

2011 Portugal MO, 6

Tags: consecutive , number theory

The number $1000$ can be written as the sum of $16$ consecutive natural numbers: $$1000 = 55 + 56 + ... + 70.$$ Determines all natural numbers that cannot be written as the sum of two or more consecutive natural numbers .

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.

1986 Brazil National Olympiad, 2

Tags: consecutive , positive integer , combinatorics , number theory

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

2011 Saudi Arabia Pre-TST, 1.1

Tags: consecutive , number theory

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?

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.

2019 Tournament Of Towns, 4

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

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)

1997 Tournament Of Towns, (533) 5

Tags: perfect cube , consecutive , number theory

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

2015 Estonia Team Selection Test, 1

Tags: consecutive , arithmetic progression , arithmetic sequence , algebra

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

2014 Federal Competition For Advanced Students, P2, 4

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

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 $P (n)$ ?