Found problems: 521
Let $\overline{abcd}$ be one of the 9999 numbers $0001, 0002, 0003, \ldots, 9998, 9999$. Let $\overline{abcd}$ be an [i]special[/i] number if $ab-cd$ and $ab+cd$ are perfect squares, $ab-cd$ divides $ab+cd$ and also $ab+cd$ divides $abcd$. For example 2016 is special. Find all the $\overline{abcd}$ special numbers.
[b]Note:[/b] If $\overline{abcd}=0206$, then $ab=02$ and $cd=06$.
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 $2010$.
Show that if $2 + 2\sqrt{28n^2 + 1}$ is an integer, then it is a square (for $n$ an integer).
The positive integer $a$ is greater than $10$, and all its digits are equal. Prove that $a$ is not a perfect square.
(A perfect square is a number which can be expressed as $n^2$ , where $n$ is an integer.)
Solve $ x^3-y^3=2xy+7 $ in integers.
Let $a, b, c, d$ be four positive integers such that each of them is a difference of two squares of positive integers. Prove that $abcd$ is also a difference of two squares of positive integers.
Find all the prime number $p, q$ and r with $p < q < r$, such that $25pq + r = 2004$ and $pqr + 1 $ is a perfect square.
The integers $a, b$ satisfy the following identity $$2a^2 + a = 3b^2 + b.$$ Prove that $a- b$, $2a + 2b + 1$, and $3a + 3b + 1$ are perfect squares.
Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that
\[\frac{a^p - a}{p}=b^2.\]
Let $m$ be the smallest odd positive integer for which $1+ 2 +...+ m$ is a square of an integer and let $n$ be the smallest even positive integer for which $1 + 2 + ... + n$ is a square of an integer. What is the value of $m + n$?
Prove that the sum of the squares of five consecutive integers cannot be a perfect square.
Determine all pairs $(p,q)$ of prime numbers for which $p^2+5pq+4q^2$ is a perfect square.
Prove that every non-negative integer $n$ is expressible in the form $n=t^2+u^2+v^2+w^2$, where $t,u,v,w$ are integers such that $t+u+v+w$ is a perfect square.
[i]* * *[/i]
What is the least positive integer by which $2^5 \cdot 3^6 \cdot 4^3 \cdot 5^3 \cdot 6^7$ should be multiplied so that, the product is a perfect square?
Prove that the number $\overline{40...0}9$ (with at least one zero) is not a perfect square.
(VA Senderov)
A [i]triangular number[/i] is a positive integer that can be expressed in the form $t_n = 1 + 2 + 3 +\cdots + n$, for some positive integer $n$. The three smallest triangular numbers that are also perfect squares are $t_1 = 1 = 1^2$, $t_8 = 36 = 6^2$, and $t_{49} = 1225 = 35^2$. What is the sum of the digits of the fourth smallest triangular number that is also a perfect square?
$\textbf{(A)} ~6 \qquad\textbf{(B)} ~9 \qquad\textbf{(C)} ~12 \qquad\textbf{(D)} ~18 \qquad\textbf{(E)} ~27 $
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.
Prove that exprssion $A=\frac{25}{2}(n+2-\sqrt{2n+3})$, $(n\in\mathbb{N})$ is a perfect square of an integer if exprssion $A$ is an integer .
Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$ and $ 8$, write the square and the cube of a positive integer. Determine what that number can be.
Prove that if the product of four consecutive natural numbers is added one unit, the result is a perfect square.
Is it possible to represent the number $1986$ as the sum of squares of $6$ odd integers?
Prove that in every $16$-digit number there is a chain of one or more consecutive digits such that the product of those digits is a perfect square.
For example, if the original number is $7862328578632785$ we can take the digits $6$, $2$ and $3$ whose product is $6^2$ (note that these appear consecutively in the number).
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]
Prove that if the natural numbers $ a $ and $ b $ satisfy the equation $ a^2+a = 3b^2 $, then the number $ a+1 $ is the square of an integer.
Can both $(x^2+y)$ and $(y^2+x)$ be exact squares for natural $x$ and $y$?