Found problems: 988
Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2015$
Find all $(x,y,z) \in {\mathbb{Z}}^3$ such that $x^{3}+y^{3}+z^{3}=x+y+z=3$.
Does there exist an integer such that its cube is equal to $3n^2 +3n+7$, where $n$ is integer?
Let $n$ be a positive integer. Suppose that the diophantine equation
\[z^n = 8 x^{2009} + 23 y^{2009} \]
uniquely has an integer solution $(x,y,z)=(0,0,0)$. Find the possible minimum value of $n$.
Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$.
[i]Proposed by Lewis Chen[/i]
Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$.
Show that the equation ${n \choose k}=m^{l}$ has no integral solution with $l \ge 2$ and $4 \le k \le n-4$.
How many distinct integral solutions of the form $(x, y)$ exist to the equation$ 21x + 22y = 43$ such that $1 < x < 11$ and $y < 22$?
Find all pairs $(a, b)$ of positive integers that satisfy the equation $a^2 -3 \cdot 2^b = 1$.
Suppose that $A=1,2,$ or $3$. Let $a$ and $b$ be relatively prime integers such that $a^{2}+Ab^2 =s^3$ for some integer $s$. Then, there are integers $u$ and $v$ such that $s=u^2 +Av^2$, $a =u^3 - 3Avu^2$, and $b=3u^{2}v -Av^3$.
If $p$ is a prime number and $x, y$ are positive integers, find in terms of $p$, all pairs $(x, y)$ that satisfy the equation: $$p(x -2) = x(y -1).$$
If $x+y = 21$, find all triples $(x, y, p)$ that satisfy this equation.
Find all pairs $(m, n)$ of positive integers $m$ and $n$ for which one has $$\sqrt{ m^2 - 4} < 2\sqrt{n} - m < \sqrt{ m^2 - 2}$$
Find all nonnegative integers $k, n$ which satisfy $2^{2k+1} + 9\cdot 2^k + 5 = n^2.$
Positive integers $a$, $b$ and $c$ satisfy the system of equations
\begin{align*}
(ab-1)^2&=c(a^2+b^2)+ab+1,\\
a^2+b^2&=c^2+ab.
\end{align*}
a) Prove that $c+1$ is a perfect square.
b) Find all such triples $(a,b,c)$.
Let $A=\{a^2+13b^2 \mid a,b \in\mathbb{Z}, b\neq0\}$. Prove that there
a) exist
b) exist infinitely many
$x,y$ integer pairs such that $x^{13}+y^{13} \in A$ and $x+y \notin A$.
(proposed by B. Bayarjargal)
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.
Find prime numbers $p , q , r$ such that $p+q^2+r^3=200$. Give all the possibilities.
Remember that the number $1$ is not prime.
Solve, for integers $x$ and $y$ : $$2x^2y = (x+2)^2(y + 1), $$ provided that $(x+2)^2(y + 1)> 1000$.
Find all pairs of prime numbers $(p,q)$ for which
\[2^p = 2^{q-2} + q!.\]
Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.
Solve the equation in nonnegative integers $a,b,c$:
$3^a+2^b+2015=3c!$
I.Gorodnin
Solve in natural numbers $a,b,c$ the system \[\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc \\
a^2 = 2(a+b+c)\\
\end{array} \right.
\]
Solve, in the positive integers, the equation $5^m + n^2 = 3^p$ .
Find the sum of all integers $n$ with $2 \le n \le 999$ and the following property: if $x$ and $y$ are randomly selected without replacement from the set $\left\{ 1,2,\dots,n \right\}$, then $x+y$ is even with probability $p$, where $p$ is the square of a rational number.
[i]Proposed by Ivan Koswara[/i]
Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.