Found problems: 916
Solve the system of equations in integers
$$x + y + z = 3$$
$$x^3 + y^3 + z^3 = 3$$
Let $m\ge10$ be any positive integer such that all its decimal digits are distinct. Denote $f(m)$ sum of positive integers created by all non-identical permutations of digits of $m,$ e.g. \[f(302)=320+023+032+230+203=808.\] Determine all positive integers $x$ such that \[f(x)=138\,012.\]
Determine all pairs of rational numbers $(x, y)$ such that \[x^{3}+y^{3}= x^{2}+y^{2}.\]
Determine all pairs $(p,q)$ of prime numbers with the following property: There are positive integers $a,b,c$ satisfying
\[\frac{p}{a}+\frac{p}{b}+\frac{p}{c}=1 \quad \text{and} \quad \frac{a}{p}+\frac{b}{p}+\frac{c}{p}=q+1.\]
Let $n\ge1$ and $r\ge2$ be positive integers. Prove that there is no integer $m$ such that $n(n+1)(n+2)=m^r$.
For which positive integers $m$ does the equation: $$(ab)^{2015} = (a^2 + b^2)^m$$ have positive integer solutions?
(a) Find all triples of integers $(a,b,c)$ for which $\frac14=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}$.
(b) Determine all positive integers $n$ for which there exist positive integers $x_1,x_2,\ldots,x_n$ such that $1=\frac1{x_1^2}+\frac1{x_2^2}+\ldots+\frac1{x_n^2}$.
Solve the equation for the positive integers: $$(x+2y)^2+2x+5y+9=(y+z)^2$$
How many pairs $(a, b)$ of positive integers $a,b$ solutions of the equation $(4a-b)(4b-a )=1770^n$ exist , if $n$ is a positive integer?
If $a$ and $b$ are integers and if the solutions of the system of equations
$$y - 2x - a = 0$$
$$y^2 - xy + x^2 - b = 0$$
are rational, prove that the solutions are integers.
Determine whether exist a prime number $p$ and natural number $n$ such that $n^2 + n + p = 1996$.
Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ .
a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$.
b) Show that $A$ is an infinite set.
Find all integers $x,y,z$ such that $x^3 +5y^3 = 9z^3$.
Find all pairs of integers $(a, b)$ such that $a^2 + ab + b^2 = 1$
Solve the following equations in the set of natural numbers:
a) $(5+11\sqrt2)^p=(11+5\sqrt2)^q$
b) $1005^x+2011^y=1006^z$
Let $n$ be a positive integer. Prove that the equation \[x+y+\frac{1}{x}+\frac{1}{y}=3n\] does not have solutions in positive rational numbers.
Find all positive integers $(m, n)$ such that $3 \cdot 2^n + 1 = m^2$.
Find all pairs of positive prime numbers $(p, q)$ such that
$$p^5 + p^3 + 2 = q^2 - q.$$
Prove that there is no pair of relatively prime positive integers $(a, b)$ that satisfy the equation
$$a^3 + 2017a = b^3 -2017b.$$
find all solutions, not necessarily positive integers for $(m^2+ n)(m+ n^2)= (m+ n)^3$
Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.
Determine, with proof, all the integer solutions of the equation $x^3 + 2y^3 + 4z^3 - 6xyz = 1$.
$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.
Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.
Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$
where $d$ is the smallest divisor of $n$ which is greater than $1$.