Found problems: 110
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 $4$.
Consider any three consecutive natural numbers. Prove that the cube of the largest cannot be the sum of the cubes of the other two.
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?
Is it possible for the least common multiple of five consecutive positive integers to be a perfect square?
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.
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
Write the number $1000$ as the sum of at least two consecutive positive integers. How many (different) ways are there to write it?
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)
Show that the product of three consecutive positive integers is never a perfect square.
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.
Prove that every natural number which is not an integer power of $2$ is the sum of two or more consecutive natural numbers.
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?
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.
Find all possible values of $n \ge 1$ for which there exist $n$ consecutive positive integers whose sum is a prime number.
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.
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.
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?
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 $.
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 $38$?
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.
Prove that the product of four consecutive integers plus $1$ is a perfect square.
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?
Prove that there are only two sets of consecutive positive integers that satisfy that the sum of its elements is equal to $100$.
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.
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 $i= 1,2,3, ... , (n-8)$