Found problems: 110
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.
Determine all triangles such that the lengths of the three sides and its area are given by four consecutive natural numbers.
A positive integer $n$ is said to possess Property ($A$) if there exists a positive integer $N$ such that $N^2$ can be written as the sum of the squares of $n$ consecutive positive integers. Is it true that there are infinitely many positive integers which possess Property ($A$)? Justify your answer.
(As an example, the number $n = 2$ possesses Property ($A$) since $5^2 = 3^2 + 4^2$).
Find the least positive integer $n$ with the property:
Among arbitrarily $n$ selected consecutive positive integers, all smaller than $2018$, there is at least one that is divisible by its sum of digits .
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 satis es [b]both [/b] (a) and (b).
(You may assume that (1) and (2) are correct!)
Find the number of ways that a positive integer $n$ can be represented as a sum of one or more consecutive positive integers.
Do there exist integers $a$ and $b$ such that none of the numbers $a,a+1,\ldots,a+2023,b,b+1,\ldots,b+2023$ divides any of the $4047$ other numbers, but $a(a+1)(a+2)\cdots(a+2023)$ divides $b(b+1)\cdots(b+2023)$?
In how many different ways can you write $2016$ as the sum of a sequence of consecutive natural numbers?
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.
Prove that there is only one triangle whose sides are consecutive natural numbers and one of the angles is twice the other angle.
Prove that among $18$ consecutive three digit numbers there must be at least one which is divisible by the sum of its digits.
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.
Find the minimum $k>2$ for which there are $k$ consecutive integers such that the sum of their squares is a 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.
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.
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 $K$.
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 $0$.
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
Prove that the number
(a) $97^{97}$
(b) $1997^{17}$
cannot be equal to a sum of cubes of several consecutive integers.
(AA Egorov)
Are there $14$ consecutive positive integers, each of which has a divisor other than $1$ and not exceeding $11$?
Is it possible to place a positive integer in every cell of a $10 \times 10$ array in such a way that both the following conditions are satisfied?
$\bullet$ Each number (not in the top row) is a proper divisor of the number immediately above.
$\bullet$ Each row consists of 1$0$ consecutive positive integers (but not necessarily in order).
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.
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 $m$.
a) Show that the product of $5$ consecutive even integers is divisible by $15$.
b) Determine the largest integer $D$ such that the product of $5$ consecutive even integers is always divisible by $D$.
$2000$ 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.