Problems

Suppose that $p$ is a prime number.

How many numbers are there less than $p$ that are relatively prime to $p$?

I have written $5$ composite (not prime and not $1$) numbers on a piece of paper and hidden it in a safe locker. Every pair of these numbers is relatively prime. Show that at least one of these numbers has to be larger than $100$.

Becky and Rishika play the following game: There are 21 biscuits on the table. Each girl is allowed to take 1, 2 or 3 biscuits at once. The girl who cannot take any more biscuits loses. Rishika starts – show that she can always win.

Alice and Bob play a game, Alice will go first. They have a strip divided into $2026$ identical squares. In each move, they put a $2\times 1$ domino block on the strip, covering two full squares. The person that is not able to make their move loses. Who has a winning strategy?

Ana and Daniel are playing a game that involves a chocolate bar. The top left square of the bar is poisoned. In each move, a player has to pick a square and take all the pieces contained in the rectangle whose top left corner is the selected square and the bottom right corner is the bottom right corner of the whole bar. The person who takes the poisoned square loses. Who has a winning strategy if Daniel starts?

Two pirates are playing a game. They have $42$ gold coins on a table. Each of them is allowed to take either $1$ or $5$ coins from the table. The pirate who takes the last coin wins. Who will win – the first pirate or the second pirate?

Rekha and Misha also play with coins. They have an unlimited supply of 10p coins and a perfectly round table. In each move, one of them places a coin somewhere on that table, but not on top of any other coins already there. A person that cannot place any more coins loses. Who will win, if Rekha goes first?

Varoon and Mahmoud are given two plates of fruit. On one plate, there are $13$ apples, on the other, there are $16$ pears. Each of the boys can take any number of fruit from one plate when he moves. The person who takes the last fruit wins. If Mahmoud starts, who will win?

This time, Sally and Fatima have some number of books on a shelf. Every turn, each of them is allowed to take 1, 3 or 4 books from the shelf. The girl that takes the last book wins, Sally goes first. Who will win if there are: a) 14, b) 16, c) 19 books on the shelf?

Danny and Robbie draw diagonals of a regular $2018$-gon. They can only draw a diagonal that does not cross any other diagonal that has been already drawn, neither it begins nor ends at a same point as any other drawn diagonal. Robbie starts – can he always win?