Found problems: 113
Given the equation
$$
a^b\cdot b^c=c^a
$$
in positive integers $a$, $b$, and $c$.
[i](i)[/i] Prove that any prime divisor of $a$ divides $b$ as well.
[i](ii)[/i] Solve the equation under the assumption $b\ge a$.
[i](iii)[/i] Prove that the equation has infinitely many solutions.
[i](I. Voronovich)[/i]
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 number $1\underbrace{\hbox{0...0}}_{\hbox{49}}5\underbrace{\hbox{0...0}}_{\hbox{99}}1$ is not the cube of any integer.
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.
$a;b;c;d\in\mathbb{Z^+}$. Solve the equation: $$2^{a!}+2^{b!}+2^{c!}=d^3$$
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.
Find all positive integers $n$ for which both $837 + n$ and $837 - n$ are cubes of positive integers.
Let $a$, $b$ and $c$ are integers such that $\frac{a}{b}+\frac{b}{c}+\frac{c}{a}=3$. Prove that $abc$ is a perfect cube of an integer.
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
a) If positive integer $N$ is a perfect cube and is not divisible by $10$, then $N=(10m+n)^2$ where $m,n \in N$ with $1\le n\le 9$
b) Find all the positive integers $N$ which are perfect cubes, are not divisible by $10$, such that the number obtained by erasing the last three digits to be also also a perfect cube.
The sequence of positive integers $a_1, a_2, a_3, ...$ satisfies $a_{n+1} = a^2_{n} + 2018$ for $n \ge 1$.
Prove that there exists at most one $n$ for which $a_n$ is the cube of an integer.
Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes.
Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.
Show that the sum of three consecutive perfect cubes can always be written as the difference between two perfect squares.
For a positive integer $n$ we denote $D_2(n)$ to the number of divisors of $n$ which are perfect squares and $D_3(n)$ to the number of divisors of $n$ which are perfect cubes. Prove that there exists such that $D_2(n)=999D_3(n).$
Note. The perfect squares are $1^2,2^2,3^2,4^2,…$ , the perfect cubes are $1^3,2^3,3^3,4^3,…$ .
Let $a,b,c,n$ be positive integers such that the following conditions hold
(i) numbers $a,b,c,a+b+c$ are pairwise coprime,
(ii) number $(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)$ is a perfect $n$-th power.
Prove, that the product $abc$ can be expressed as a difference of two perfect $n$-th powers.
How many non-empty subsets of $\{1,2,\dots,11\}$ are there with the property that the product of its elements is the cube of an integer?
Find the minimum number of perfect cubes such that their sum is equal to $ 346^{346} . $
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ .
Show that for every integer $n > 1$, the following holds:
$n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.
Given $n \ge 2$ distinct positive integers $a_1, a_2, ..., a_n$ none of which is a perfect cube. Find the maximal possible number of perfect cubes among their pairwise products.
A natural number $n$ is called [i]perfect [/i] if it is equal to the sum of all its natural divisors other than $n$. For example, the number $6$ is perfect because $6 = 1 + 2 + 3$. Find all even perfect numbers that can be given as the sum of two cubes positive integers.
How many positive integer divisors of $201^9$ are perfect squares or perfect cubes (or both)?
$\textbf{(A) } 32 \qquad\textbf{(B) } 36 \qquad\textbf{(C) } 37 \qquad\textbf{(D) } 39 \qquad\textbf{(E) } 41$
Find all primes $p$ such that $p^2- p + 1$ is a perfect cube.
Prove that all numbers of the sequence \[ \frac{107811}{3}, \quad \frac{110778111}{3}, \frac{111077781111}{3}, \quad \ldots \] are exact cubes.
In a $99\times 101$ table , cubes of natural numbers, as shown in figure . Prove that the sum of all numbers in the table are divisible by $200$.
[img]https://cdn.artofproblemsolving.com/attachments/3/e/dd3d38ca00a36037055acaaa0c2812ae635dcb.png[/img]