Found problems: 436
Let $p$ be a prime number and let $m$ and $n$ be positive integers with $p^2 + m^2 = n^2$.
Prove that $m> p$.
(Karl Czakler)
Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$.
[i](Karl Czakler)[/i]
Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$
Are there integers $k$ and $m$ for which
$$\frac{(k-3)(k-2)(k-1)k+1}{(k+1)(k+2)(k+3)(k+4)+1}=m(m+1)+(m+1)(m+2)+(m+2)m \,\,
?$$
Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $.
Clarification: $1$ is not a prime number.
Two positive integers have the sum $2002$. Can $2002$ divide their product?
a) Given that integers $a$ and $b$ satisfy the equality $$a^2 - (b^2 - 4b + 1) a - (b^4 - 2b^3) = 0 \,\,\, (*)$$, prove that $b^2 + a$ is a square of an integer.
b) Do there exist an infinitely many of pairs $(a,b)$ satisfying (*)?
Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.
Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$ has positive integer solutions.
Find all the positive integers $a, b, c$ such that $$a! \cdot b! = a! + b! + c!$$
Find all integer solutions $(p, q, r)$ of the equation $r + p ^ 4 = q ^ 4$ with the following conditions:
$\bullet$ $r$ is a positive integer with exactly $8$ positive divisors.
$\bullet$ $p$ and $q$ are prime numbers.
For what pairs of integers $(x,y)$ does the inequality $x^2+5y^2-6\leq \sqrt{(x^2-2)(y^2-0.04)}$ hold?
Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds
The cost of $1000$ grams of chocolate is $x$ dollars and the cost of $1000$ grams of potatoes is $y$ dollars, the numbers $x$ and $y$ are positive integers and have not more than $2$ digits. Mother said to Maria to buy $200$ grams of chocolate and $1000$ grams of potatoes that cost exactly $N$ dollars. Maria got confused and bought $1000$ grams of chocolate and $200$ grams of potatoes that cost exactly $M$ dollars ($M >N$). It turned out that the numbers $M$ and $N$ have no more than two digits and are formed of the same digits but in a different order. Find $x$ and $y$.
Find all positive integers $x,y,z$ such that $x^3 - (x + y + z)^2 = (y + z)^3 + 34$
We say that a triple of integers $(x, y, z)$ is of [i]jenifer [/i] type if $x, y$, and $z$ are positive integers, with $y \ge 2$, and $$x^2 - 3y^2 = z^2 - 3.$$
a) Find a triple $(x, y, z)$ of the jenifer type with $x = 5$ and $x = 7$.
b) Show that for every $x \ge 5$ and odd there are at least two distinct triples $(x, y_1, z_1)$ and $(x, y_2, z_2)$ of jenifer type.
c) Find some triple $(x, y, z)$ of jenifer type with $x$ even.
Find the number of integer solutions of the equation
$x^{2016} + (2016! + 1!) x^{2015} + (2015! + 2!) x^{2014} + ... + (1! + 2016!) = 0$
For some integer n the equation $x^2 + y^2 + z^2 -xy -yz - zx = n$ has an integer solution $x, y, z$. Prove that the equation$ x^2 + y^2 - xy = n$ also has an integer solution $x, y$.
Alexandr Yuran
Show that the equation $x^3 +y^3 = 4(x^2y+xy^2 +1)$ has no integer solutions.
Find all integers $x,y$ satisfy the $x^3 + y^3 = 3xy$.
Find all ordered pairs of integers $(x, y)$ with $y \ge 0$ such that $x^2 + 2xy + y! = 131$.
Find the number of positive integer solutions to $(x_1 + x_2 + x_3)(y_1 + y_2 + y_3 + y_4) = 91$
Find all primes $p,$ $q,$ $r$ satisfying $p^{2}+2q^{2}=r^{2}.$
Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.
Determine the prime numbers $p$ and $q$ that satisfy the equality: $p^3 + 107 = 2q (17q + 24)$ .