Found problems: 521
Positive integers $a$, $b$ and $c$ are positive integers with greatest common divisor equal to $1$ (i.e. they have no common divisors greater than $1$), and
$$\frac{ab}{a-b}=c$$
Prove that $a -b$ is a perfect square.
(SL Berlov)
Positive integers $a, b$ satisfy equality $b^2 = a^2 + ab + b$.
Prove that $b$ is a square of a positive integer.
(Patrik Bak)
Let $n$ be a positive integer. A function $f : \{1, 2, \dots, 2n\} \to \{1, 2, 3, 4, 5\}$ is [i]good[/i] if $f(j+2)$ and $f(j)$ have the same parity for every $j = 1, 2, \dots, 2n-2$.
Prove that the number of good functions is a perfect square.
Are there natural $a, b >1000$ , such that for any $c$ that is a perfect square, the three numbers $a, b$ and $c$ are not the lengths of the sides of a triangle?
Prove that number $2(1991m^2+1993mn+1995n^2)$ where $m,n$ are poitive integers, cannot be a square of an integer.
Let $m$ and $n$ be positive integers such that $m - n$ is odd. Show that $(m + 3n)(5m + 7n)$ is not a perfect square.
Prove that for every integer $k$, there exists a integer $n$ which can be expressed in at least $k$ different ways as the sum of a number of squares of integers (regardless of the order of additions) where the additions are all in different pairs.
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.
Find a four-digit perfect square whose first digit is the same as the second, and the third is the same as the fourth.
A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squares ($p = m^2 +n^2$ , with $m$ and $n$ integers) if and only if $p \equiv 1$ (mod $4$). Assuming this result, find which
primes $p > 2$ can be written in each of the following forms, using integers $x$ and $y$:
a) $x^2 +16y^2, $
b) $4x^2 +4xy+ 5y^2.$
Prove that the product of four consecutive natural numbers can not be neither square nor perfect cube.
Let \(a_0, a_1, a_2, \dots\) be an infinite sequence of positive integers with the following properties:
- \(a_0\) is a given positive integer;
- For each integer \(n \geq 1\), \(a_n\) is the smallest integer greater than \(a_{n-1}\) such that \(a_n + a_{n-1}\) is a perfect square.
For example, if \(a_0 = 3\), then \(a_1 = 6\), \(a_2 = 10\), \(a_3 = 15\), and so on.
(a) Let \(T\) be the set of numbers of the form \(a_k - a_l\), with \(k \geq l \geq 0\) integers.
Prove that, regardless of the value of \(a_0\), the number of positive integers not in \(T\) is finite.
(b) Calculate, as a function of \(a_0\), the number of positive integers that are not in \(T\).
Find all odd integers $n$ for which $\frac{2^{\phi (n)}-1}{n}$ is a perfect square.
Find all positive integers $k$ for which there is an irrational $\alpha>1$ and a positive integer $N$ such that $\left\lfloor\alpha^{n}\right\rfloor$ is of the form $m^2-k$ com $m \in \mathbb{Z}$ for every integer $n>N$.
Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$,
which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square .
Remark: $(7, 4) \ne (4, 7)$
We say that a positive integer $n$ is [i]memorable[/i] if it has a binary representation with strictly more $1$'s than $0$'s (for example $25$ is memorable because $25=(11001)_{2}$ has more $1$'s than $0$'s). Are there infinitely many memorable perfect squares?
[i]Proposed by Nikola Velov[/i]
Solve $ (n + 1)(n +10) = q^2$, for certain $q$ and maximum $n$.
Prove that $1$ plus the product of any four consecutive integers is a perfect square.
Determine all positive integers $n$ such that $2022 + 3^n$ is a perfect square.
Are there three pairwise distinct non-zero integers whose sum is zero and whose sum of thirteenth powers is the square of some natural number?
An integer $n$ is said to be [i]good[/i] if $|n|$ is not the square of an integer. Determine all integers $m$ with the following property: $m$ can be represented, in infinitely many ways, as a sum of three distinct good integers whose product is the square of an odd integer.
[i]Proposed by Hojoo Lee, Korea[/i]
A sequence of integers $(a_n)$ satisfies $a_{n+1} = a_n^3 + 1999$ for $n = 1,2,....$
Prove that there exists at most one $n$ for which $a_n$ is a perfect square.
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.
Proof that for every primes $p$, $q$
\[p^{q^2-q+1}+q^{p^2-p+1}-p-q\]
is never a perfect square.
[i]Proposed by chengbilly[/i]
Let $ a, b, c$ be integers and $ p$ an odd prime number. Prove that if $ f(x) \equal{} ax^2 \plus{} bx \plus{} c$ is a perfect square for $ 2p \minus{} 1$ consecutive integer values of $ x,$ then $ p$ divides $ b^2 \minus{} 4ac.$