Found problems: 988
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.
Are there natural numbers $(m,n,k)$ that satisfy the equation $m^m+ n^n=k^k$ ?
For all $n \in \mathbb{N}$, show that the number of integral solutions $(x, y)$ of \[x^{2}+xy+y^{2}=n\] is finite and a multiple of $6$.
Find all primes $p,q$ such that $p ^3-q^7=p-q$.
Determine all real numbers a for which there exists positive reals $x_{1}, \ldots, x_{5}$ which satisfy the relations $ \sum_{k=1}^{5} kx_{k}=a,$ $ \sum_{k=1}^{5} k^{3}x_{k}=a^{2},$ $ \sum_{k=1}^{5} k^{5}x_{k}=a^{3}.$
a) Prove that the equation $2^x + 21^x = y^3$ has no solution in the set of natural numbers.
b) Solve the equation $2^x + 21^y = z^2y$ in the set of non-negative integer numbers.
Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$
Solve in integers the following equation \[n^{2002}=m(m+n)(m+2n)\cdots(m+2001n).\]
Determine all the ways in which the fraction $\frac{1}{11}$ can be written as $\frac{1}{n}+\frac{1}{m}$ , where $n$ and $m$ are two different positive integers.
Find all pairs $(m,n)$ of positive integers such that $m^2 + n^2 = 3(m + n)$.
Find all primes $p, q$ such that $p + q = (p-q)^3$.
Let $x,y,z$ be integer numbers satisfying the equality $yx^2+(y^2-z^2)x+y(y-z)^2=0$
a) Prove that number $xy$ is a perfect square.
b) Prove that there are infinitely many triples $(x,y,z)$ satisfying the equality.
I.Voronovich
Prove that for all rationals $x,y$, $x-\frac{1}{x}+y-\frac{1}{y}=4$ is not true.
Compute the number of ordered quadruples of positive integers $(a,b,c,d)$ such that
$$a!b!c!d!=24!$$$\textbf{(A)}~4$
$\textbf{(B)}~4!$
$\textbf{(C)}~4^4$
$\textbf{(D)}~\text{None of the above}$
Let $N$ be the number of ordered pairs $(x,y)$ of integers such that $x^2+xy+y^2 \le 2007$.
Remember, integers may be positive, negative, or zero!
(a) Prove that $N$ is odd.
(b) Prove that $N$ is not divisible by $3$.
Find all integer $n$ such that the equation $2x^2 + 5xy + 2y^2 = n$ has integer solution for $x$ and $y$.
Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$
Integers $a, b, c$ satisfy $a+b-c=1$ and $a^2+b^2-c^2=-1$. What is the sum of all possible values of $a^2+b^2+c^2$ ?
For what positive numbers $a$ is \[\sqrt[3]{2+\sqrt{a}}+\sqrt[3]{2-\sqrt{a}}\] an integer?
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?