Found problems: 916
Find all pairs of $ (a, n) $ natural numbers such that $ \varphi (a ^ n + n) = 2 ^ n. $
($ \varphi (n) $ is the Euler function, that is, the number of integers from $1$ up to $ n $, relative prime to $ n $)
Solve in natural numbers the system of equations $3x^2+6y^2+5z^2=1997$ and $3x+6y+5z=161$ .
A triangle with integer sides is called Heronian if its area is an integer. Does there exist a Heronian triangle whose sides are the arithmetic, geometric and harmonic means of two positive integers?
Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy
\[(ab + 1)(bc + 1)(ca + 1) = 84.\]
Find all positive integers $m$ and $n$ satisfying $2^n+n=m!$.
Find all positive integers $x, y, z$ such that $2^x + 21^y = z^2$
Find all pairs $(m, n)$ of positive integers such that $m^2 + 2n^2 = 3(m + 2n)$
Determine all triples of natural numbers $(a,b, c)$ with $b> 1$ such that $2^c + 2^{2016} = a^b$.
Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$
Find all pairs of solutions $(x,y)$:
\[ x^3 + x^2y + xy^2 + y^3 = 8(x^2 + xy + y^2 + 1). \]
For which real numbers $a$, there exist pairwise different real numbers $x, y, z$ satisfying
$$\frac{x^3+a}{y+z}=\frac{y^3+a}{x+z}=\frac{z^3+a}{x+y}= -3.$$
Find all solutions in positive integers to $a^A = b^B = c^C = 1990^{1990}abc$, where $A = b^c, B = c^a, C = a^b$.
Determine all three integers $(x, y, z)$ that are solutions of the system
$$x + y -z = 6$$
$$x^3 + y^3 -z^3 = 414$$
Find the smallest positive integer $N$ for which the equation $(x^2 -1)(y^2 -1)=N$ is satis ed by at least two pairs of integers $(x, y)$ with $1 < x \le y$.
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 $1$.
Find all positive integers $n$ for which the equation \[a+b+c+d=n \sqrt{abcd}\] has a solution in positive integers.
Prove that, if $a$ and $b$ are given integers, the system of equatìons
$$x + y + 2z + 2t = a$$
$$2x - 2y + z- t = b$$
has a solution in integers $x, y,z,t$.
Consider the equation
\[(3x^3+xy^2)(x^2y+3y^3)=(x-y)^7\]
(a) Prove that there are infinitely many pairs $(x,y)$ of positive integers satisfying the equation.
(b) Describe all pairs $(x,y)$ of positive integers satisfying the equation.
Nam spent $20$ dollars for $20$ stationery items consisting of books, pens and pencils. Each book, pen, and pencil cost $3$ dollars, $1.5$ dollars and $0.5$ dollar respectively. How many dollars did Nam spend for books?
Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{2}\plus{}n\plus{}1$ are not more then $ \sqrt{n}$.
[hide] Stronger one.
Prove that there are infinitely many positive integers $ n$ for which all the prime divisors of $ n^{3}\minus{}1$ are not more then $ \sqrt{n}$.[/hide]
Determine all integers solutions of the equation $x + x^3 = 5y^2$.
Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that
\begin{align*}
a^2 + b^2 + 3 &= 4ab\\
c^2 + d^2 + 3 &= 4cd\\
4c^3 - 3c &= a
\end{align*}
[i]Travis Hance.[/i]
Prove that for all $n \in N$, $x^2 + y^2 = z^n$ has solutions with $x,y,z \in N$.
Let $x, y, p, n, k$ be positive integers such that $$x^n + y^n = p^k.$$
Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$.
Let $m, n$ be odd prime numbers.
Find all pairs of integers numbers $a, b$ for which the system of equations:
$x^m+y^m+z^m=a$,
$x^n+y^n+z^n=b$
has many solutions in integers $x, y, z$.