Found problems: 521
Determine all positive integers $n$ such that $$n\cdot 2^{n-1}+1$$ is a perfect square.
Prove that for any positive integer $k$ there exist $k$ pairwise distinct integers for which the sum of their squares equals the sum of their cubes.
Determine whether there exists an infinite sequence of nonzero digits $a_1 , a_2 , a_3 , \cdots $ and a positive integer $N$ such that for every integer $k > N$, the number $\overline{a_k a_{k-1}\cdots a_1 }$ is a perfect square.
Let $S(n)$ denote the sum of all digits of natural number $n$. Determine all natural numbers $n$ for which both numbers $n + S(n)$ and $n - S(n)$ are square powers of non-zero integers.
There are pairs of square numbers with the following two properties:
(1) Their decimal representations have the same number of digits, with the first digit starting is different from $0$ .
(2) If one appends the second to the decimal representation of the first, the decimal representation results another square number.
Example: $16$ and $81$; $1681 = 41^2$.
Prove that there are infinitely many pairs of squares with these properties.
Find the probability that any given row in Pascal’s Triangle contains a perfect square.
[i] (.1 point)[/i]
An infinite arithmetic progression whose terms are positive integers contains the square of an integer and the cube of an integer. Show that it contains the sixth power of an integer.
Determine all sets $(a, b, c)$ of positive integers where one obtains $b^2$ by removing the last digit in $c^2$ and one obtains $a^2$ by removing the last digit in $b^2$.
.
Find all positive integers $n$ such that number $n^4-4n^3+22n^2-36n+18$ is perfect square of positive integer
Do there exist infinitely many positive integers $m$ such that the sum of the positive divisors of $m$ (including $m$ itself) is a perfect square?
[i]Proposed by Dylan Toh[/i]
For any non-negative integer $n$, we say that a permutation $(a_0,a_1,...,a_n)$ of $\{0,1,..., n\} $ is quadratic if $k + a_k$ is a square for $k = 0, 1,...,n$. Show that for any non-negative integer $n$, there exists a quadratic permutation of $\{0,1,..., n\}$.
Given the endless arithmetic progression with the positive integer members. One of those is an exact square. Prove that the progression contain the infinite number of the exact squares.
Let the sequences $(x_n)$ and $(y_n)$ be defined by $x_0 = 0$, $x_1 = 1$, $x_{n + 2} = 3x_{n + 1}-2x_n$ for $n = 0, 1, ...$ and $y_n = x^2_n+2^{n + 2}$ for $n = 0, 1, ...,$ respectively. Show that for all n> 0, and n is the square of a odd integer.
Prove that, for $k \in {\mathbb Z^+}$
$$k(k+1)(k+2)(k+3)$$
is not a perfect square.
Prove that you can pick $15$ distinct positive integers between $1$ and $2023$, such that each one of them and the sum between some of them is never a perfect square, nor a perfect cube or any other greater perfect power.
Determine if there is an infinite sequence $a_1,a_2,a_3,...,a_n$ of positive integers such that for all $n\ge 1$ the sum $a_1^2+a_2^2+a_3^2+...^2+a_n^2$ is a perfect square
Let $ b$ be an integer greater than $ 5$. For each positive integer $ n$, consider the number \[ x_n = \underbrace{11\cdots1}_{n \minus{} 1}\underbrace{22\cdots2}_{n}5, \] written in base $ b$.
Prove that the following condition holds if and only if $ b \equal{} 10$: [i]there exists a positive integer $ M$ such that for any integer $ n$ greater than $ M$, the number $ x_n$ is a perfect square.[/i]
[i]Proposed by Laurentiu Panaitopol, Romania[/i]
Positive integers $a, b, c$ satisfying the equality $a^2 + b^2 = c^2$.
Show that the number $\frac12(c - a) (c - b)$ is square of an integer.
Let $a$ and $b$ be positive integers. Show that if $a \cdot b$ is even, then there are positive integers $c$ and $d$ with $a^2 + b^2 + c^2 = d^2$; if, on the other hand, $a\cdot b$ is odd, there are no such positive integers $c$ and $d$.
A [i]permutation[/i] of the set of positive integers $[n] = \{1, 2, . . . , n\}$ is a sequence $(a_1 , a_2 , \ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P (n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P (n)$ is a multiple of $2010$.
For which integer \( h \), are there infinitely many positive integers \( n \) such that \( \lfloor \sqrt{h^2 + 1} \cdot n \rfloor \) is a perfect square? (Here \( \lfloor x \rfloor \) denotes the integer part of the real number \( x \)?
Prove there aren't positive integers $a, b, c, d$ forming an arithmetic progression such that $ ab + 1, ac + 1, ad + 1, bc + 1, bd + 1, cd + 1 $ are all perfect squares.
Determine whether there are infinitely many triples $(u,v,w)$ of positive integers such that $u,v,w$ form an arithmetic progression and the numbers $uv+1, vw+1$ and $wu+1$ are all perfect squares.
Proove that if for a positive integer $n$ , both $3n + 1$ and $10n + 1$ are perfect squares , then $29n + 11$ is not a prime number.
Show that there exist infinitely many integers that cannot be written as the difference between two perfect squares.