Found problems: 436
Find all positive integers $m, n$ such that $n + (n+1) + (n+2) + ...+ (n+m) = 1000$.
Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$.
Find all positive integers $a, b,c,d$ such that $a + b + c + d - 3 = ab + cd$.
Let $a, b, c$ be integers not all the same with $a, b, c\ge 4$ that satisfy $$4abc = (a + 3) (b + 3) (c + 3).$$
Find the numerical value of $a + b + c$.
Show that for all positive integer $n$ the following inequality holds $3^{n^2} > (n!)^4$
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Find all the integers $x,y$ satisfying equation $x^2+x=y^4+y^3+y^2+y$.
Solve the equation $x^{2}+y^{2}=10xy$ for integers $x$ and $y$
Determine the integers $0 \le a \le b \le c \le d$ such that: $$2^n= a^2 + b^2 + c^2 + d^2.$$
Solve into $N$: $$a^2 = 2^b +c^4$$
Find all non-negative integers $a,b,c,d$ such that $7^a= 4^b + 5^c + 6^d$.
Let $a, b$ and $c$ be positive integers such that $a^{b+c} = b^{c} c$. Prove that b is a divisor of $c$, and that $c$ is of the form $d^b$ for some positive integer $d$.
Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$
Solve the diophantine equation $x^{2018}-y^{2018}=(xy)^{2017}$ when $x$ and $y$ are non-negative integers.
Determine all triples $(a, b, c)$ of positive integers such that
$$a! +b! = 2^{c!} .$$
Determine all pairs of integers $(x, y)$ that satisfy equation $(y - 2) x^2 + (y^2 - 6y + 8) x = y^2 - 5y + 62$.
Show that no pairs of integers $(m, n)$ satisfy $2560m^2 + 5m + 6 = n^5$.
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Determine all triples $(a, b, c)$ of nonnegative integers that satisfy: $$(c-1) (ab- b -a) = a + b-2$$
Positive integers $a$, $b$ and $c$ are positive integers with greatest common divisor equal to $1$ (i.e. they have no common divisors greater than $1$), and
$$\frac{ab}{a-b}=c$$
Prove that $a -b$ is a perfect square.
(SL Berlov)
Let $p$ be a prime number. Find all triples $(a, b, c)$ of integers (not necessarily positive) such that $a^bb^cc^a = p$.
Are there three integers $x,y,z$, such that $x^2 + y^3 = z^4$?
(a) Let $x$ and $y$ be two positive integers such that $\sqrt{x} +\sqrt{y}$ is an integer.
Show that $\sqrt{x}$ and $\sqrt{y}$ are both integers.
(b) Find all positive integers $x$ and $y$ such that $\sqrt{x} +\sqrt{y}=\sqrt{2007}$.
For each positive integer $n$ let $p(n)$ be the number of ordered pairs $(x,y)$ of positive integers such that$$\dfrac{1}{x}+\dfrac{1}{y} =\dfrac{1}{n}.$$For example, for $n=2$ the pairs are $(3,6),(4,4),(6,3)$. Therefore $p(2)=3$.
a) Determine $p(n)$ for all $n$ and calculate $p(1995)$.
b) Determine all pairs $n$ such that $p(n)=3$.
Find all solutions in positive integers to: $$\begin{cases} x_1^4 + 14x_1x_2 + 1 = y_1^4 \\ x_2^4 + 14x_2x_3 + 1 = y_2^4 \\ ... \\ x_n^4 + 14x_nx_1 + 1 = y_n^4 \end{cases}$$
Find all triples $(x, p, n)$ of non-negative integers such that $p$ is prime and $2x(x + 5) = p^n + 3(x - 1)$.
a) Find all positive integer solutions of the equation $x^y = y^x$ ($x \ne y$).
b) Find all positive rational solutions of the equation $x^y = y^x$ ($x \ne y$).