Found problems: 916
Suppose that the natural number $a, b, c, d$ satisfy the equation $a^ab^{a + b} = c^cd^{c + d}$.
(a) If gcd $(a, b) = $ gcd $(c, d) = 1$, prove that $a = c$ and $b = d$.
(b) Does the conclusion $a = c$ and $b = d$ apply, without the condition gcd $(a, b) = $ gcd $(c, d) = 1$?
Find all natural numbers $n$ such that
$$(n+3)^n=\sum_{k=3}^{n+2}k^n.$$
Find all integers $a$ for which $x^3 -x+a$ has three integer roots.
Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.
Find all pairs of integers $(x,y)$ satisfying the following equality $8x^2y^2 + x^2 + y^2 = 10xy$
Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$
Let $ p>3$ be a prime. If $ p$ divides $ x$, prove that the equation $ x^2-1=y^p$ does not have positive integer solutions.
Find all pairs of integers ($a, b$) such that $a^2 + b = b^{1999}$ .
Find all positive integers $(m, n)$ that satisfy $$m^2 =\sqrt{n} +\sqrt{2n + 1}.$$
Consider the set of solutions of the equation $$x^2+y^3=z^2.$$ in positive integers. Is it finite or infinite?
(Folklore)
How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 1$?
(A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.
Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied.
(Walther Janous)
Solve:
$a, b, m, n\in \mathbb{N}$
$a^2+b^2=m^2-n^2, ab=2mn$
Given positive integers $a,b,c$ which are pairwise coprime. Let $f(n)$ denotes the number of the non-negative integer solution $(x,y,z)$ to the equation
$$ax+by+cz=n.$$
Prove that there exists constants $\alpha, \beta, \gamma \in \mathbb{R}$ such that for any non-negative integer $n$,
$$|f(n)- \left( \alpha n^2+ \beta n + \gamma \right) | < \frac{1}{12} \left( a+b+c \right).$$
Solve the equation $xy =1997(x + y)$ in integers.
Show that there is no positive integer $k$ for which the equation \[(n-1)!+1=n^{k}\] is true when $n$ is greater than $5$.
$a)$ Solve over the positive integers $3^x=x+2.$
$b)$ Find pairs $(x,y)\in\mathbb{N}\times\mathbb{N}$ such that $(x+3^y)$ and $(y+3^x)$ are consecutive.
Two incongruent triangles $ABC$ and $XYZ$ are called a pair of [i]pals[/i] if they satisfy the following conditions:
(a) the two triangles have the same area;
(b) let $M$ and $W$ be the respective midpoints of sides $BC$ and $YZ$. The two sets of lengths $\{AB, AM, AC\}$ and $\{XY, XW, XZ\}$ are identical $3$-element sets of pairwise relatively prime integers.
Determine if there are infinitely many pairs of triangles that are pals of each other.
Find all nonnegative integers $a$ and $b$ that satisfy the equation $$3 \cdot 2^a + 1 = b^2.$$
Find all triples $(a, b, c)$ of integers that satisfy the equations
$ a + b = c$ and $a^2 + b^3 = c^2$
Determine all pairs of positive integers $(x, y)$ such that:
$$x^4 - 6x^2 + 1 = 7\cdot 2^y$$
Shenelle has some square tiles. Some of the tiles have side length $5\text{ cm}$ while the others have side length $3\text{ cm}$. The total area that can be covered by the tiles is exactly $2014\text{ cm}^2$. Find the least number of tiles that Shenelle can have.
Find all integers $a$ and $b$ satisfying the equality $3^a - 5^b = 2$.
(I. Gorodnin)
Find all pairs of integers $x,y$ for which
\[x^3+x^2+x=y^2+y.\]
Determine all positive integers $k$ for which there exist positive integers $n$ and $m, m\ge 2$, such that $3^k + 5^k = n^m$