Found problems: 521
A positive integer $m$ is called triangular if $m = 1 + 2 + ... + n$, for an integer $n$. Show that a positive integer $m$ is triangular if and only if $8m + 1$ is the square of an integer.
How many three-digit perfect squares are there such that if each digit is increased by one, the resulting number is also a perfect square?
(A): $1$, (B): $2$, (C): $4$, (D): $8$, (E) None of the above.
Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.
Find the smallest positive integer $n$ with the property that in the set $\{70, 71, 72,... 70 + n\}$ you can choose two different numbers whose product is the square of an integer.
Let $a,b,c$ natural numbers for which $a^2 + b^2 + c^2 = (a-b) ^2 + (b-c)^ 2 + (c-a) ^2$. Show that $ab, bc, ca$ and $ab + bc + ca$ are perfect squares .
Does there exist an integer n such that all three numbers
(a) $n - 96$, $n$ and $n + 96$
(b) $n - 1996$, $n$ and $n + 1996$
are positive prime numbers?
(V Senderov)
Find $26$ elements of $\{1, 2, 3, ... , 40\}$ such that the product of two of them is never a square.
Show that one cannot find $27$ such elements.
Suppose that the sequence $\{a_n\}$ of positive integers satisfies the following conditions:
[list]
[*]For an integer $i \geq 2022$, define $a_i$ as the smallest positive integer $x$ such that $x+\sum_{k=i-2021}^{i-1}a_k$ is a perfect square.
[*]There exists infinitely many positive integers $n$ such that $a_n=4\times 2022-3$.
[/list]
Prove that there exists a positive integer $N$ such that $\sum_{k=n}^{n+2021}a_k$ is constant for every integer $n \geq N$.
And determine the value of $\sum_{k=N}^{N+2021}a_k$.
Find the number of primes $p$ such that $p! + 25p$ is a perfect square.
(a) Let $a$ and $b$ be positive integers such that $M(a, b) = a - \frac1b +b(b + \frac3a)$ is an integer.
Prove that $M(a,b)$ is a square.
(b) Find nonzero integers $a$ and $b$ such that $M(a,b)$ is a positive integer, but not a square.
$2n$ students take part in a math competition. First, each of the students sends its task to the members of the jury, after which each of the students receives from the jury one of proposed tasks (all received tasks are different). Let's call the competition [i]honest[/i], if there are $n$ students who were given the tasks suggested by the remaining $n$ participants. Prove that the number of task distributions in which the competition is honest is a square of natural numbers.
Find all positive integer values of $n$ such that the value of the
$$\frac{2^{n!}-1}{2^n-1}$$
is a square of an integer.
Prove that $9$-digit number, that contains all the decimal digits except zero and does not ends with $5$ can not be exact square.
a) Prove that if the square of a number begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$, then the number itself begins with $0.\underbrace{\hbox{9...9}}_{\hbox{100}}$,.
b) Calculate $\sqrt{0.9...9}$ ($60$ nines) to $60$ decimal places
Let $N$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$, and let $M$ be the number of integral solutions of the equation
\[x^2 - y^2 = z^3 - t^3 + 1\]
satisfying the condition $0 \leq x, y, z, t \leq 10^6$. Prove that $N >M.$
Determine all integers $x,y,z$ such that \[x^2+y^2=3z^2.\]
Find all four-digit perfect squares such that:
$\bullet$ All your figures are less than $9$.
$\bullet$ By increasing each of its digits by one unit, the resulting number is again a perfect square.
Prove that the number $a =\overline{{1...1}{5...5}6}$ is a perfect square (where $1$s are $2012$ in total and $5$s are $2011$ in total)
Let $t$ be an integer. Suppose the equation $$x^2 + (4t - 1) x + 4t^2 = 0$$ has at least one positive integer solution $n$. Show that $n$ is a perfect square.
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.
Natural numbers $ x, y, z $ whose greatest common divisor is equal to 1 satisfy the equation
$$\frac{1}{x} + \frac{1}{y} = \frac{1}{z}$$
Prove that $ x + y $ is the square of a natural number.
Let $a{}$ and $b{}$ be positive integers, whose difference is a prime number. Prove that $(a^n+a+1)(b^n+b+1)$ is not a perfect square for infinitely many positive integers $n{}$.
[i]Proposed by Vlad Matei[/i]
Find all prime numbers $p$ such that $p^3-4p+9$ is a perfect square.
Can we find positive integers $a$ and $b$ such that both $(a^2 + b)$ and $(b^2 + a)$ are perfect squares?
Find all integers $n$ such that $2^n + n^2$ is a square of an integer.
[i](Tomas Jurik )[/i]