Last week eight heads of state, including 3 presidents, 3 prime ministers and 2 emperors got together for a conference. According to a reporter, each president shook hands with 6 heads of state, each prime minister shook hands with 4, and each emperor - with 1. Is this possible?
In the Kingdom of Rabbitland some cities are connected with each other by a railroad. There are two ways of getting from Longears city to Pufftown, one of them passes through an even number of other cities, and the other passes through an odd number of other cities. Show that in the Kingdom of Rabbitland there is a cycle of an odd number of distinct cities, such that each is connected to the next by railroads, and the last one is connected to the first one by railroads.
During a chess tournament some people played 5 games and some people played 6. Prove that the number of people who played 5 games is even.
There are \(6\) points in the three dimensional space. To connect some of the points we draw \(10\) segments. Show that there is at least one fully drawn triangle (with all vertices and sides).
There are ten islands in the Fantasia Archipelago. There used to be cruises connecting each island to each other island, each cruise served by a different ship. One day eight of those ships were taken over by pirates and do not serve anymore. Show that despite the efforts of pirates, it is still possible to get from each island to any other island, possibly changing ships on the way.
In Rabbitland you can get from any city to any other city. Bugs Bunny decided to find out what is the largest number of cities he can visit, while never visiting any city twice. He can start from any city and finish at any other. He realised that there are two longest paths he can take and they both involve the same number of cities. Show that those two paths have at least one city in common.
There are \(n\) people at a royal ball. Every time three of them meet and talk, it turns out some two of them didn’t know each other before. Show that the maximum number of connections between the guests is \(\frac14 n^2\). If any guest \(A\) knows guest \(B\), then \(B\) knows \(A\), and this counts as one connection.
An animal with \(n\) cells is a connected figure consisting of \(n\) equal-sized adjacent square cells. A dinosaur is an animal with at least 2007 cells. It is said to be primitive if its cells cannot be partitioned into two or more dinosaurs. Find with proof the maximum number of cells in a primitive dinosaur.
Remainder is the number that is "left over" from division. Even if a number \(n\) is not divisible by another number \(k\), it is still possible to divide \(n\) by \(k\), but with a remainder \(r\). Then, we can write \(n = qk +r\), where we always have \(0\leq r<k\). That is because we say that \(k\) goes into \(n\) \(q\) times, and a little bit is left. If that little bit was larger than \(k\), it could "go into" \(n\) once more. For example, a remainder of \(44\) in division by \(7\) is \(2\), because \(44 = 6 \times 7 + 2\).
The general rule is that a remainder of a sum, difference or a product of two remainders is equal to the remainder of a sum, difference or a product of the original numbers. What that means is if we want to find a remainder of a product of two numbers, we need to look at the individual remainders, multiply them, and then take a remainder. Let’s have a look on some examples:
What time is it going to be in \(2025\) hours from now?