Olympiad problems

2011 Junior Balkan Team Selection Tests - Romania, 1

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 ($p$).

2008 Estonia Team Selection Test, 4

Tags: sequence , number theory , divisible , consecutive

Sequence $(G_n)$ is defined by $G_0 = 0, G_1 = 1$ and $G_n = G_{n-1} + G_{n-2} + 1$ for every $n \ge2$. Prove that for every positive integer $m$ there exist two consecutive terms in the sequence that are both divisible by $m$.

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

2017 Argentina National Olympiad, 2

Tags: sum , algebra , number theory , consecutive

In a row there are $51$ written positive integers. Their sum is $100$ . An integer is [i]representable [/i] if it can be expressed as the sum of several consecutive numbers in a row of $51$ integers. Show that for every $k$ , with $1\le k \le 100$ , one of the numbers $k$ and $100-k$ is representable.

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 $P$.

1936 Moscow Mathematical Olympiad, 026

Tags: consecutive , algebra , product

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

1984 Tournament Of Towns, (071) T5

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

Prove that among $18$ consecutive three digit numbers there must be at least one which is divisible by the sum of its digits.

2009 Regional Olympiad of Mexico Center Zone, 4

Tags: consecutive , digit , number theory , 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.

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!)

1998 Mexico National Olympiad, 1

Tags: number theory , digit , consecutive

A number is called lucky if computing the sum of the squares of its digits and repeating this operation sufficiently many times leads to number $1$. For example, $1900$ is lucky, as $1900 \to 82 \to 68 \to 100 \to 1$. Find infinitely many pairs of consecutive numbers each of which is lucky.

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 .

2022 Saudi Arabia JBMO TST, 3

Tags: combinatorics , consecutive

$2000$ 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.

2020 Tournament Of Towns, 6

Tags: consecutive , number theory , combinatorics

There are $2n$ consecutive integers on a board. It is permitted to split them into pairs and simultaneously replace each pair by their difference (not necessarily positive) and their sum. Prove that it is impossible to obtain any $2n$ consecutive integers again. Alexandr Gribalko

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.

2000 ITAMO, 1

Tags: number theory , consecutive , sum of squares

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?

2007 Bosnia and Herzegovina Junior BMO TST, 1

Tags: consecutive , sum , combinatorics , algebra

Write the number $1000$ as the sum of at least two consecutive positive integers. How many (different) ways are there to write it?

1949 Kurschak Competition, 3

Tags: number theory , consecutive

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

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.

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?

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.

2010 May Olympiad, 3

Tags: number theory , consecutive , perfect square

Find the minimum $k>2$ for which there are $k$ consecutive integers such that the sum of their squares is a square.

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.

2017 Latvia Baltic Way TST, 4

Tags: algebra , polynomial , consecutive

The values of the polynomial $P(x) = 2x^3-30x^2+cx$ for any three consecutive integers are also three consecutive integers. Find these values.

2006 Spain Mathematical Olympiad, 2

Tags: number theory , product , consecutive , perfect square , perfect cube

Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.

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