Found problems: 916
Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.
Consider the equation $x^3 + y^3 + z^3 = 2$.
a) Prove that it has infinitely many integer solutions $x,y,z$.
b) Determine all integer solutions $x, y, z$ with $|x|, |y|, |z| \leq 28$.
Find all positive integers $a, b, c$, such that $(8a-5b)^2 + (3b-2c)^2 + (3c-7a)^2 = 2$.
Let $n$ be a positive integer and $a_1, a_2, \dots , a_{2n}$ mutually distinct integers. Find all integers $x$ satisfying
\[(x - a_1) \cdot (x - a_2) \cdots (x - a_{2n}) = (-1)^n(n!)^2.\]
Find all pairs of primes $(p, q)$ such that $p^3 - q^5 = (p + q)^2$
.
Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.
Find the smallest value of the expression $|253^m - 40^n|$ over all pairs of positive integers $(m, n)$.
[i]Proposed by Oleksii Masalitin[/i]
Find the number of ordered triples of positive integers $(a, b, c)$ such that
\[6a + 10b + 15c = 3000.\]
Determine all prime numbers $ p $ and natural numbers $ x, y $ for which $ p^x-y^3 = 1 $.
(a) Let $k$ be an natural number so that the equation $ab + (a + 1) (b + 1) = 2^k$ does not have a positive integer solution $(a, b)$. Show that $k + 1$ is a prime number.
(b) Show that there are natural numbers $k$ so that $k + 1$ is prime numbers and equation $ab + (a + 1) (b + 1) = 2^k$ has a positive integer solution $(a, b)$.
Find all natural numbers $k$ such that there exist natural numbers $a_1,a_2,...,a_{k+1}$ with $ a_1!+a_2!+... +a_{k+1}!=k!$
Note that we do not consider $0$ to be a natural number.
Prove that for any integers $x$ and $y$ we have $x^5 + 3x^4y - 5x^3y^2 - 15x^2y^3 + 4xy^4 + 12y^5 \ne 33$.
Find all triples of positive integers $(x, y, z)$ such that \[(x+y)(1+xy)= 2^{z}.\]
Determine the pairs of positive integer numbers $m$ and $n$ that satisfy the equation $m^2=n^2 +m+n+2018$.
Find all pairs $(n, p)$ of positive integers such that $p$ is prime and
\[ 1 + 2 + \cdots + n = 3 \cdot (1^2 + 2^2 + \cdot + p^2). \]
Do there exist two right-angled triangles with integer length sides that have the lengths of exactly two sides in common?
Find a pair of relatively prime four digit natural numbers $A$ and $B$ such that for all natural numbers $m$ and $n$, $\vert A^m -B^n \vert \ge 400$.
Solve the equation in natural numbers $$
xy+yz+zx = xyz + 2.
$$
FIx positive integer $n$. Prove: For any positive integers $a,b,c$ not exceeding $3n^2+4n$, there exist integers $x,y,z$ with absolute value not exceeding $2n$ and not all $0$, such that $ax+by+cz=0$
Several pairs of positive integers $(m ,n )$ satisfy the condition $19m + 90 + 8n = 1998$. Of these, $(100, 1 )$ is the pair with the smallest value for $n$. Find the pair with the smallest value for $m$.
Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$
For each nonnegative integer $k$ find all nonnegative integers $x,y,z$ such that $x^2 +y^2 +z^2 = 8^k$
Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that:
\[a^p+b^p=p^c.\]
Find all triples of non-negative integer numbers $(E, C, U)$ such that $EC \ge 1$ and:
$$2^{3^E}+3^{2^C}=593 \cdot 5^U$$
Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$.
Daniel Harrer