Found problems: 521
If the natural numbers $a, b, c, d$ verify the relationships:
$$(a^2 + b^2)(c^2 + d^2) = (ab + cd)^2$$
$$(a^2 + d^2)(b^2 + c^2) = (ad + bc)^2$$
and the $gcd(a, b, c, d) = 1$, prove that $a + b + c + d$ is a perfect square.
Let $a, b$ and $c$ be positive integers such that $ab + 1, bc + 1$ and $ca + 1$ are all integer squares.
a) Give an example of such numbers $a, b$ and $c$.
b) Prove that at least one of the numbers $a, b$ and $c$ is divisible by $4$
The digits of a $3$-digit number are interchanged so that none of the digits retain their original position. The difference of the two numbers is a $2$-digit number and is a perfect square. Find the difference.
A natural number $n$ is called "$k$-squared" if it can be written as a sum of $k$ perfect squares not equal to 0.
a) Prove that 2020 is "$2$-squared" , "$3$-squared" and "$4$-squared".
b) Determine all natural numbers not equal to 0 ($a, b, c, d ,e$) $a<b<c<d<e$ that verify the following conditions simultaneously :
1) $e-2$ , $e$ , $e+4$ are all prime numbers.
2) $a^2+ b^2 + c^2 + d^2 + e^2$ = 2020.
Determine all integers $x$ such that $2x^2-x-36$ is a perfect square of a prime.
Georg investigates which integers are expressible in the form $$\pm 1^2 \pm 2^2 \pm 3^2 \pm \dots \pm n^2.$$
For example, the number $3$ can be expressed as $ -1^2 + 2^2$, and the number $-13$ can be expressed as
$+1^2 + 2^2 + 3^2 - 4^2 + 5^2 - 6^2$. Are all integers expressible in this form?
Can number $2012^n-3^n$ be perfect square, while $n$ is positive integer
Show that for each natural number $n$, there exist $n$ distinct natural numbers whose sum is a square and whose product is a cube.
Prove that the rectangle $m\times n$ table can be filled with exact squares so, that the sums in the rows and the sums in the columns will be exact squares also.
Let $n > 1$ be a positive integer and $p$ a prime number such that $n | (p - 1) $and $p | (n^6 - 1)$. Prove that at least one of the numbers $p- n$ and $p + n$ is a perfect square.
Let $k$ a positive integer number such that $\frac{k(k+1)}{3}$ is a perfect square. Show that $\frac{k}{3}$ and $k+1$ are both perfect squares.
Investigate whether for two positive integers $m$ and $n$ the numbers $m^2 + n$ and $n^2 + m$ can be both squares of integers.
You have a $4$-digit whole number that is a perfect square. Another number is built adding $ 1$ to the unit's digit, subtracting $ 1$ from the ten's digit, adding $ 1$ to the hundred's digit and subtracting $ 1$ from the ones digit of one thousand. If the number you get is also a perfect square, find the original number. It's unique?
Show that there is a set of $2002$ distinct positive integers such that the sum of one or more elements of the set is never a square, cube, or higher power.
Find the maximum value of $n$, such that there exist $n$ pairwise distinct positive numbers $x_1,x_2,\cdots,x_n$, satisfy $$x_1^2+x_2^2+\cdots+x_n^2=2017$$
For which $n$ can we find positive integers $a_1,a_2,\dots,a_n$ such that
\[
a_1^2+a_2^2+\cdots+a_n^2
\]
is a 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.
Let $n$ be a positive integer. Prove that the numbers $n^2(n^2 + 2)^2$ and $n^4(n^2 + 2)^2$ are written in base $n^2 +1$ with the same digits but in opposite order.
In a french village the number of inhabitants is a perfect square. If $100$ more people moved in, then the number of people would be $ 1$ bigger than a perfect square. If again $100$ more people moved in, then the number of people would be a perfect square again. How many people lives in the village if their number is the least possible?
For positive integer $n$, prove that at least one of the numbers $$A=2n-1 , B=5n-1, C=13n-1$$ is not perfect square
Let $n$ be a positive integer. For a positive integer $m$, we partition the set $\{1, 2, 3,...,m\}$ into $n$ subsets, so that the product of two different elements in the same subset is never a perfect square. In terms of $n$, fi nd the largest positive integer $m$ for which such a partition exists.
Define the sequence $a_1 = 1, a_2, a_3, ...$ by $$a_{n+1} = a_1^2 + a_2 ^2 + a_3^2 + ... + a_n^2 + n$$ Show that $1$ is the only square in the sequence.
Find all positive real numbers $c$ such that there are in finitely many pairs of positive integers $(n,m)$ satisfying the following conditions: $n \ge m+c\sqrt{m - 1}+1$ and among numbers $n. n+1,.... 2n-m$ there is no square of an integer.
(Slovakia)
Find the smallest positive integer $n \geq 10$ such that $n + 6$ is a prime and $9n + 7$ is a perfect square.
Prove that for every positive integer $m$ there exists positive integer $n$ such that $m+n+1$ is perfect square and $mn+1$ is perfect cube of some positive integers