Found problems: 521
a) Do there exist positive integers $a$ and $b$ such that for any positive,integer $n$ the number $a \cdot 2^n+ b\cdot 5^n$ is a perfect square ?
b) Do there exist positive integers $a, b$ and $c$, such that for any positive integer $n$ the number $a\cdot 2^n+ b\cdot 5^n + c$ is a perfect square?
(M . Blotski)
Determine the number of integers $a$ satisfying $1 \le a \le 100$ such that $a^a$ is a perfect square.
(And prove that your answer is correct.)
All perfect squares, and all perfect squares multiplied by two, are written in a row in increasing order. let $f(n)$ be the $n$-th number in this sequence. (For instance, $f(1)=1,f(2)=2,f(3)=4,f(4)=8$.) Is there an integer $n$ such that all the numbers
\[f(n),f(2n),f(3n),\dots,f(10n^2)\]
are perfect squares?
The product $1\times 2\times 3\times ...\times n$ is written on the board. For what integers $n \ge 2$, we can add exclamation marks to some factors to convert them into factorials, in such a way that the final product can be a perfect square?
We call a natural number [i]almost a square[/i] if it can be represented as a product of two numbers that differ by no more than one percent of the larger of them. Prove that there are infinitely many consecutive quadruples of almost squares.
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]
Let $a_1, a_2, ..., a_n$ be a sequence of perfect squares such that $a_{i + 1}$ can be obtained by concatenating a digit to the right of $a_i$. Determine all such sequences that are of maximum length.
[i]Proposed by Ilija Jovčeski[/i]
Prove that there are no perfect squares $a,b,c$ such that $ab-bc = a$.
Let $n \ge 2003$ be a positive integer such that the number $1 + 2003n$ is a perfect square.
Prove that the number $n + 1$ is equal to the sum of $2003$ positive perfect squares.
Prove that for any prime number $p$ there are infinitely many fours $(x, y, z, t)$ pairwise distinct natural numbers such that the number $(x^2+p t^2)(y^2+p t^2)(z^2+p t^2)$ is a perfect square.
Two sequences $(a_k)_{k\ge 0}$ and $(b_k)_{k\ge 0}$ of integers are given by $b_k = a_k + 9$ and $a_{k+1} = 8b_k + 8$ for $k\ge 0$. Suppose that the number $1988$ occurs in one of these sequences. Show that the sequence $(a_k)$ does not contain any nonzero perfect square.
Prove that there exists a convex 1990-gon with the following two properties :
[b]a.)[/b] All angles are equal.
[b]b.)[/b] The lengths of the 1990 sides are the numbers $ 1^2$, $ 2^2$, $ 3^2$, $ \cdots$, $ 1990^2$ in some order.
How many natural numbers $n$ are there so that $n^2 + 2014$ is a perfect square?
(A): $1$, (B): $2$, (C): $3$, (D): $4$, (E) None of the above.
Prove that for every natural number $k$, at least one of the integers
\[ 2k-1, \quad 5k-1 \quad \text{and} \quad 13k-1\]
is not a perfect square.
Let $n$ be positive integer such that $\sqrt{1 + 12n^2}$ is an integer.
Prove that $2 + 2\sqrt{1 + 12n^2}$ is the square of an integer.
Prove that there is not a positive integer $n$ such that numbers $(n+1)2^n, (n+3)2^{n+2}$ are both perfect squares.
Consider the factorials of the first $100$ positive integers, namely, $1!, 2!$, $...$, $100!$. Is it possible to delete one of them so that the product of the remaining ones is a perfect square?
(S Tokarev)
Let n be a positive integer. Prove the following statement:
”If $2 + 2\sqrt{1 + 28n^2}$ is an integer, then it is the square of an 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]
Find all pairs $(m, n)$ of natural numbers for which the numbers $m^2 - 4n$ and $n^2 - 4m$ are both perfect squares.
For which positive integers $n \ge 2$ it is possible to represent the number $n^2$ as a sum of several distinct positive integers not exceeding $2n$?
Show that there are infinitely many triangles with side lengths $a$, $b$, $c$, where $a$ is a prime, $b$ is a power of $2$ and $c$ is the square of an odd integer.
Determine the largest possible number of groups one can compose from the integers $1,2,3,..., 19,20$, so that the product of the numbers in each group is a perfect square. (The group may contain exactly one number, in that case the product equals this number, each number must be in exactly one group.)
(E. Barabanov, I. Voronovich)
(a) Let $m$ and $n$ be positive integers and $p$ a positive rational number, with $m > n$, such that $\sqrt{m} -\sqrt{n}= p$. Prove that $m$ and $n$ are perfect squares.
(b) Find all four-digit numbers $\overline{abcd}$, where each letter $a, b, c$ and $d$ represents a digit, such that $\sqrt{\overline{abcd}} -\sqrt{\overline{acd}}= \overline{bb}$.
Find all pairs $(a,b)$ of prime positive integers $a,b$ such that number $A=3a^2b+16ab^2$ equals to a square of an integer.