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 \[ x^2 \minus{} ay^2 \minus{} bz^2 \equal{} 0.\]

Find all non-negative integers $n$, $a$, and $b$ satisfying \[2^a + 5^b + 1 = n!.\]

Determine all triples $(a, b, c)$ of nonnegative integers that satisfy: $$(c-1) (ab- b -a) = a + b-2$$

Let $a$ and $b$ be positive integers. As is known, the division of of $a \cdot b$ with $a + b$ determines integers $q$ and $r$ uniquely such that $a \cdot b = q (a + b) + r$ and $0 \le r <a + b$. Find all pairs $(a, b)$ for which $q^2 + r = 2011$.

Find all positive integers $m$ and $n$ satisfying $2^n+n=m!$.

Find all primes $p$ and all positive integers $a$ and $m$ such that $a\leq 5p^2$ and $(p-1)!+a=p^m$

Find all pairs of prime numbers $(p, q)$ for which $7pq^2 + p = q^3 + 43p^3 + 1$

Prove that the equation $\prod_{cyc} (x_1-x_2)= \prod_{cyc} (x_1-x_3)$ has a solution in natural numbers where all $x_i$ are different.

Find all triplets $(x, y, p)$ of positive integers such that $p$ is a prime number and $\frac{xy^3}{x+y}=p.$

One of Euler's conjectures was disproved in the $1980$s by three American Mathematicians when they showed that there is a positive integer $n$ such that \[n^{5}= 133^{5}+110^{5}+84^{5}+27^{5}.\] Find the value of $n$.

Find all pairs of integers $(x, y)$ satisfying the condition $12x^2 + 6xy + 3y^2 = 28(x + y)$.

Find all positive integers $a, b,c,d$ such that $a + b + c + d - 3 = ab + cd$.

Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$. Daniel Harrer

Find all triples of positive integers $(x, y, z)$ with $$\frac{xy}{z}+ \frac{yz}{x}+\frac{zx}{y}= 3$$

Find all couples $(x,y)$ over the positive integers such that: $7^x+x^4+47=y^2$

Are there any (not necessarily positive) integers $m$ and $n$ such that a) $\frac{1}{m}-\frac{1}{n}=\frac{1}{m-n}$ ? b) $\frac{1}{m}-\frac{1}{n}=\frac{1}{n-m}$

Let $a,b$ be nonnegative integers. Prove that $5a>7b$ if and only if there exist nonnegative integers $x,y,z,t$ such that \begin{align*} x+2y+3z+7t&=a,\\ y+2z+5t&=b. \end{align*}

Let $b$ be a positive integer. Determine all $2002$-tuples of non-negative integers $(a_{1}, a_{2}, \cdots, a_{2002})$ satisfying \[\sum^{2002}_{j=1}{a_{j}}^{a_{j}}=2002{b}^{b}.\]

Show that $\vert 12^m -5^n\vert \ge 7$ for all $m, n \in \mathbb{N}$.

Find all triples of positive integers $(m,p,q)$ such that $2^mp^2 + 27 = q^3$ and $p$ is a prime.

Suppose that $m$ and $n$ are integers, such that both the quadratic equations $x^2+mx-n=0$ and $x^2-mx+n=0$ have integer roots. Prove that $n$ is divisible by $6$.

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.

Find all $x, y, z \in N_0$ with $(2^x + 1) (2^y-1) = 2^z-1$.

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 pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$