Problems

In how many ways can you read the word TRAIN from the picture below, starting from T and going either down or right at each step?

There are $100$ people in a room. Each person knows at least $67$ others. Show that there is a group of four people in this room that all know each other. We assume that if person $A$ knows person $B$ then person $B$ also knows person $A$.

The numbers $a$ and $b$ are integers and the number $p\ge 3$ is prime. Suppose that $a+b$ and $a^2+b^2$ are divisible by $p$. Show that $a^2+b^2$ is divisible by $p^2$.

There are $33$ cities in the Republic of Farfarawayland. The delegation of senators wants to pick a new capital city. They want this city to be connected by roads to every other city in the Republic. They know for a fact that given any set of $16$ cities, there will always be some city that is connected by roads to all those selected cities. Show that there exists a suitable candidate for the capital.

John drunk a $\frac{1}{6}$ of a full cup of black coffee and then filled the cup back up with milk. Then he drunk a third of what he had in the cup. Then, he refilled it back to full with milk again, and after that, drunk a half of the cup. Finally, he once again refilled the cup with milk and drunk everything he had. What did he drink more of - coffee or milk?

A round necklace contains $45$ beads of two different colours: red and blue. Show that it is possible to find two beads of the same colour next to each other.

Cut this figure into $4$ identical shapes. (Note: you have to use the entire shape. Rotations and reflections count as identical shapes)

The diagram shows a $3\times 3$ square with one corner removed. Cut it into three pieces, not necessarily identical, which can be reassembled to make a square:

Find all possible non-zero digits $A$ for which the following holds $(AA+AA+1)\times A=AAA$. (Recall $AA$ means the two-digit number whose first and second digits are $A$)

A square has been divided into $4$ rectangles and a square. If the rectangle in the bottom left corner has dimensions $1\times 4$ and the one in the top right is $2\times 5$, what is the area of the small square in the middle?