Problems

Draw a picture how Robinson used to tie the goat and the wolf in order for the goat to graze the grass in the shape of half a ring.

Find all the prime numbers $p$ such that there exist natural numbers $x$ and $y$ for which $p^x=y^3+1$.

Find all natural numbers $n$ for which there exist integers $a,b,c$ such that $a+b+c=0$ and the number $a^n+b^n+c^n$ is prime.

The dragon locked six dwarves in the cave and said, "I have seven caps of the seven colors of the rainbow. Tomorrow morning I will blindfold you and put a cap on each of you, and hide one cap. Then I’ll take off the blindfolds, and you can see the caps on the heads of others, but not your own and I won’t let you talk any more. After that, everyone will secretly tell me the color of the hidden cap. If at least three of you guess right, I’ll let you all go. If less than three guess correctly, I’ll eat you all for lunch." How can dwarves agree in advance to act in order to be saved?

Nick has written in some order all the numbers $1,2,...33$ at the vertices of a regular $33$-gon. His little sister Hannah assigned to each side of the $33$-gon the number equal to the sum of the numbers at the ends of that side. It turns out that Hannah obtained $33$ consecutive numbers in certain order. Can you find an arrangement of numbers as written by Nick which lead to this situation?

Alice the fox and Basilio the cat have grown $20$ counterfeit bills on a money tree and now write seven-digit numbers on them. Each bill has $7$ empty cells for numbers. Basilio calls out one digit "1" or "2" (he doesn’t know the others), and Alice writes the number into any empty cell of any bill and shows the result to Basilio. When all the cells are filled, Basilio takes as many bills with different numbers as possible (out of several with the same number, he takes only one), and the rest is taken by Alice. What is the largest number of bills Basilio can get, regardless of Alice’s actions?

Cut a $7\times 7$ square into $9$ rectangles, out of which you can construct any rectangle whose sidelengths are less than $7$. Show how to construct the rectangles.

There are six letters in the alphabet of the Bim-Bam tribe. A word is any sequence of six letters that has at least two identical letters. How many words are there in the language of the Bim-Bam tribe?

In how many ways can eight rooks be arranged on the chessboard in such a way that none of them can take any other. The color of the rooks does not matter, it’s everyone against everyone.

Find all the prime numbers $p$ such that the number $2p^2+1$ is also prime.