Found problems: 988
Show that there is an infinite number of positive integers $t$ such that none of the equations $$ \begin{cases} x^2 + y^6 = t \\ x^2 + y^6 = t + 1 \\ x^2 - y^6 = t \\ x^2 - y^6 = t + 1 \end{cases}$$ has solutions $(x, y) \in Z \times Z$.
Suppose that $p$ and $q$ are two different positive integers and $x$ is a real number. Form the product $(x+p)(x+q).$ Find the sum $S(x,n) = \sum (x+p)(x+q),$ where $p$ and $q$ take values from 1 to $n.$ Does there exist integer values of $x$ for which $S(x,n) = 0.$
Determine the prime numbers $p, q, r$ with the property that: $p(p-7) + q (q-7) = r (r-7)$.
A positive integer $k$ is said to be [i]visionary[/i] if there are integers $a > 0$ and $b \geq 0$ such that $a \cdot k + b \cdot (k + 1) = 2020.$ How many visionary integers are there?
Determine the real numbers $ a,b, $ such that
$$ [ax+by]+[bx+ay]=(a+b)\cdot [x+y],\quad\forall x,y\in\mathbb{R} , $$
where $ [t] $ is the greatest integer smaller than $ t. $
Find all positive integers $m, n$ such that $\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}$.
Show that for every $n\in\mathbb{N}$, $n\sqrt{2}-\lfloor n\sqrt{2}\rfloor>\frac{1}{2n \sqrt{2}}$ and that for every $\epsilon >0$, there exists an $n\in\mathbb{N}$ such that $ n\sqrt{2}-\lfloor n\sqrt{2}\rfloor < \frac{1}{2n \sqrt{2}}+\epsilon$.
Four positive integers $x,y,z$ and $t$ satisfy the relations
\[ xy - zt = x + y = z + t. \]
Is it possible that both $xy$ and $zt$ are perfect squares?
Find all prime numbers $p, q$ such that$$p(p+1)(p^2+1) = q^2(q^2+q+1) + 2025.$$
[i]Proposed by Md. Fuad Al Alam[/i]
Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$
Find all $(x,y,z) \in {\mathbb{N}}^3$ such that $x^{4}-y^{4}=z^{2}$.
Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$.
Let $a$ be a prime number and $n > 2$ an integer.
Find all integer solutions of the equation $x^n +ay^n = a^2z^n$
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Find all positive integers $(m,n)$ such that $m^2 + n^2 + 3 = 4(m + n)$
Does there exist an infinite set of triples of integers $x, y, z$ (not necessarily positive) such that
$$x^2 + y^2 + z^2 = x^3 + y^3+z^3?$$
(NB Vassiliev)
How many integers $n$ are there those satisfy the following inequality $n^4 - n^3 - 3n^2 - 3n - 17 < 0$?
A. $4$ B. $6$ C. $8$ D. $10$ E. $12$
Find all integer solutions of the equation $p (x + y) = xy$, where $p$ is a prime number.
Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$
The positive integer $m$ and non-negative integers $x_0, x_1,..., x_{1001}$ satisfy the following equation: $$m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.$$ How many possibilities are there for the value of $m$?
Prove that the number $99999+111111\sqrt{3}$ cannot be written in the form $(A+B\sqrt{3})^2$, where $A$ and $B$ are integers.
Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.
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}.$
Find all triples ($a$,$b$,$n$) of positive integers such that $$a^3=b^2+2^n$$
Find all pairs of natural numbers $(\alpha,\beta)$ for which, if $\delta$ is the greatest common divisor of $\alpha,\beta$, and $\varDelta$ is the least common multiple of $\alpha,\beta$, then
\[ \delta + \Delta = 4(\alpha + \beta) + 2021\]
Find all pairs of natural numbers $(a, b)$ that fulfill $a^b=(a+b)^a$.