Robinson found a chest with books and instruments after the ship wreck. Not all the books were in readable condition, but some of the books he managed to read. One sentence read “72 chickens cost *619* p”. (The starred digits were not readable). He has not tasted a chicken for quite some time, and it was pleasant to imagine a properly cooked chicken in front of him. He also was able to decipher the cost of one chicken. Can you?
Notice that the square number 1089 \((=33^2)\) has two even and two odd digits in its decimal representation.
(a) Can you find a 6-digit square number with the same property (the number of odd digits equals the number of even digits)?
(b) What about such 100-digit square number?
Jack believes that he can place \(99\) integers in a circle such that for each pair of neighbours the ratio between the larger and smaller number is a prime. Can he be right?
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?
Prove there are no natural numbers \(a\) and \(b\), such as \(a^2 - 3b^2 = 8\).
Prove there are no integer solutions for the equation \(x^2 + 1990 = y^2\).
Solve the equation with natural numbers \(1 + x + x^2 + x^3 = 2y\).
Prove the magic trick for the number \(1089 = 33^2\): if you take any \(3\)-digit number \(\overline{abc}\) with digits coming in strictly descending order and subtract from it the number obtained by reversing the digits of the original number \(\overline{abc} - \overline{cba}\) you get another \(3\)-digit number, call it \(\overline{xyz}\). Then, no matter which number you started with, the sum \(\overline{xyz} + \overline{zyx} = 1089\).
Recall that a number \(\overline{abc}\) is divisible by \(11\) if and only if \(a-b+c\) also is.
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 cut some of the sheets into \(4\) pieces. They then cut some of the newly obtained papersheets also into \(4\) pieces. They did this several more times, cutting a piece of paper into \(4\). In the end they counted the number of sheets. Could this number be \(2024\)?