Found problems: 988
Show that the equation \[x^{3}+y^{3}+z^{3}+t^{3}=1999\] has infinitely many integral solutions.
Find all positive integer $n$ satisfying the conditions
$a)n^2=(a+1)^3-a^3$
$b)2n+119$ is a perfect square.
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\
a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$
Find all natural numbers $x,y$ such that $$x^5=y^5+10y^2+20y+1.$$
Let $n$ be a positive integer. Prove that there is no positive integer solution to thxe equation $(x + 2)^n - x^n = 1 + 7^n$.
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]
Number of solutions of the equation $3^x+4^x=8^x$ in reals is
[list=1]
[*] 0
[*] 1
[*] 2
[*] $\infty$
[/list]
Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?
Find all ordered pairs of integers $x,y$ such that $$xy(x^2y^2 - 12xy- 12x- 12y+2) = (2x + 2y)^2.$$
[i]Proposed by Henry Jiang[/i]
Determine all pairs of integers $(x, y)$ such that $2xy^2 + x + y + 1 = x^2 + 2y^2 + xy$.
Find the sum of all integers $n$ that fulfill the equation \[2^n(6-n)=8n.\]
a) Two positive integers are chosen. The sum is revealed to logician $A$, and the sum of squares is revealed to logician $B$. Both $A$ and $B$ are given this information and the information contained in this sentence. The conversation between $A$ and $B$ goes as follows: $B$ starts
B: ` I can't tell what they are.'
A: ` I can't tell what they are.'
B: ` I can't tell what they are.'
A: ` I can't tell what they are.'
B: ` I can't tell what they are.'
A: ` I can't tell what they are.'
B: ` Now I can tell what they are.'
What are the two numbers?
b) When $B$ first says that he cannot tell what the two numbers are, $A$ receives a large amount of information. But when $A$ first says that he cannot tell what the two numbers are, $B$ already knows that $A$ cannot tell what the two numbers are. What good does it do $B$ to listen to $A$?
Find all positive integers $k$ and $\ell$ such that $k^2 -\ell^2 = 1005$.
Solve the equation $2^a-5^b=3$ in positive integers $a,b$.
Solve the equation $x^2+y^4+1=6^z$ in the set of integers.
Solve in nonnegative integers the following equation :
$$21^x+4^y=z^2$$
Show that the following equation has finitely many solutions $(t,A,x,y,z)$ in positive integers
$$\sqrt{t(1-A^{-2})(1-x^{-2})(1-y^{-2})(1-z^{-2})}=(1+x^{-1})(1+y^{-1})(1+z^{-1})$$
Prove that the equation $x^2 + y^2 - z^2 = 1997$ has infinitely many solutions in integers $x$, $y$ and $z$.
(N Vassiliev)
Show that there exist no integer solutions $(x, y, z)$ to the equation
$$x^3+2y^3+4z^3=9$$
Let $ a, b \in \mathbb{Z}$ which are not perfect squares. Prove that if \[ x^2 \minus{} ay^2 \minus{} bz^2 \plus{} abw^2 \equal{} 0\] has a nontrivial solution in integers, then so does \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]
Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$.
[i](Karl Czakler)[/i]
[b]a)[/b] Prove that there exists two infinite sequences $ \left( a_n \right)_{n\ge 1} ,\left( b_n \right)_{n\ge 1} $ of nonnegative integers such that $ a_n>b_n $ and $ (2+\sqrt 3)^n =a_n (2+\sqrt 3) -b_n , $ for any natural numbers $ n. $
[b]b)[/b] Prove that the equation $ x^2-4xy+y^2=1 $ has infinitely many solutions in $ \mathbb{N}^2. $
[i]Florian Dumitrel[/i]
Find all nonnegative integer solutions $(x,y,z)$ of the equation
$\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}$
Determine the number of integer pairs $(x, y)$ such that $(|x| - 2)^2 + (|y| - 2)^2 < 5$ .
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)