This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 110

VMEO III 2006, 10.2
Tags: number theory , consecutive , dividisible , divide
Prove that among $39$ consecutive natural numbers, there is always a number that has sum of its digits divisible by $ 12$. Is it true if we replace $39$ with $38$?

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 $0$.

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?

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.

2018 Costa Rica - Final Round, N1
Tags: number theory , consecutive
Prove that there are only two sets of consecutive positive integers that satisfy that the sum of its elements is equal to $100$.

2010 Saudi Arabia Pre-TST, 2.2
Tags: prime , consecutive , sum of squares , number theory
Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.

1989 Tournament Of Towns, (224) 2
Tags: geometry , consecutive , ratio
The lengths of the sides of an acute angled triangle are successive integers. Prove that the altitude to the second longest side divides this side into two segments whose difference in length equals $4$.

2008 Tournament Of Towns, 1
Tags: consecutive , perfect square , number theory
An integer $N$ is the product of two consecutive integers. (a) Prove that we can add two digits to the right of this number and obtain a perfect square. (b) Prove that this can be done in only one way if $N > 12$

1992 Tournament Of Towns, (352) 1
Tags: consecutive , perfect square , number theory
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)

2003 Dutch Mathematical Olympiad, 3
Tags: diophantine , consecutive , diophantine equation , number theory
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: $n = a (a + 1) = b (b + 1) (b + 2) (b + 3)$.

2008 Estonia Team Selection Test, 4
Tags: number theory , sequence , consecutive , divisible
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 $m$.

2012 NZMOC Camp Selection Problems, 2
Tags: number theory , divisible , consecutive , divide , sum
Show the the sum of any three consecutive positive integers is a divisor of the sum of their cubes.

1980 Spain Mathematical Olympiad, 8
Tags: consecutive , geometry
Determine all triangles such that the lengths of the three sides and its area are given by four consecutive natural numbers.

2017 Ecuador Juniors, 5
Tags: number theory , consecutive , coprime , divisor
Two positive integers are coprime if their greatest common divisor is $1$. Let $C$ be the set of all divisors of the number $8775$ that are greater than $ 1$. A set of $k$ consecutive positive integers satisfies that each of them is coprime with some element of $C$. Determine the largest possible value of $K$.

2008 Estonia Team Selection Test, 4
Tags: number theory , sequence , 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 $m$.

2013 Tournament of Towns, 2
Tags: number theory , consecutive , sum , product
A math teacher chose $10$ consequtive numbers and submitted them to Pete and Basil. Each boy should split these numbers in pairs and calculate the sum of products of numbers in pairs. Prove that the boys can pair the numbers differently so that the resulting sums are equal.

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]?

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

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.

2007 Bosnia and Herzegovina Junior BMO TST, 1
Tags: combinatorics , consecutive , sum , 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?

1909 Eotvos Mathematical Competition, 1
Tags: number theory , consecutive , perfect cube
Consider any three consecutive natural numbers. Prove that the cube of the largest cannot be the sum of the cubes of the other two.

1949-56 Chisinau City MO, 4
Tags: consecutive , perfect square , algebra
Prove that the product of four consecutive integers plus $1$ is a perfect square.

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.

2004 Junior Balkan Team Selection Tests - Moldova, 4
Tags: number theory , consecutive
Different non-zero natural numbers a$_1, a_2,. . . , a_{12}$ satisfy the condition: all positive differences other than two numbers $a_i$ and $a_j$ form many $20$ consecutive natural numbers. a) Show that $\max \{a_1, a_2,. . . , a_{12}\} - \min \{a_1, a_2,. . . , a_{12}\} = 20$. b)Determine $12$ natural numbers with the property from the statement.