In a herd consisting of horses and camels (some with one hump and some with two) there are a total of 200 humps. How many animals are in the herd, if the number of horses is equal to the number of camels with two humps?
Decipher the quote from "Alice in Wonderland" from the following matrix:
\[\begin{array}{@{}*{26}{c}@{}}
Y&q&o&l&u&e&c&d&a&i&n \\
w&a&r&l&a&w&e&a&t&y&k \\
s&n&t&c&a&e&k&c&e&a&m \\
t&o&d&r&w&e&a&t&a&h&r \\
a&c&n&t&n&e&o&d&t&r&h \\
n&i&d&n&l&g&m&e&x&s&z
\end{array}\]
Prove for any natural number \(n\) that \((n + 1)(n + 2). . .(2n)\) is divisible by \(2^n\).
Let \(p\) and \(q\) be two prime numbers such that \(q = p + 2\). Prove that \(p^q + q^p\) is divisible by \(p + q\).
Katie and Charlotte had \(4\) sheets of paper. They decided to cut some of the sheets into \(4\) pieces, then, some of the newly obtained papersheets they also cut into \(4\). In the end they counted the number of all sheets. Could this number be \(2024\)?
The distance between two villages equals \(999\) kilometres. When you go from one village to the other, every kilometre you see signs along the road, saying \(0 \mid 999, \, 1\mid 998, \, 2\mid 997, ..., 999\mid 0\). Find the number of signs, that contain only two different digits.