Problems

The distance between two villages equals $999$ kilometres. When you go from one village to the other, every kilometre you see a sign on the road, saying $0\mid 999,1\mid 998,2\mid 997,...,999\mid 0$. The signs show the distances to the two villages. Find the number of signs that contain only two different digits. For example, the sign $0\mid 999$ contains only two digits, namely $0$ and $9$, whereas the sign $1\mid 998$ contains three digits, namely $1$, $8$ and $9$.

Michael made a cube with edge $1$ out of eight bars as in the picture. All $8$ bars have the same volume. The dimensions of the grey bars are the same as each other. Similarly, the dimensions of the white bars are the same as each other. Find the lengths of the edges of the white bars.

One cell was cut out of a $3\times 6$ rectangle, as seen in the diagram. How should you glue this cell in a different place to get a figure that can be cut into two identical ones? If needed, the resulting parts can be rotated and reflected.

In the sum below, different letters denote different digits and the same letters denote the same digit. $$P.Z+T.C+D.R+O.B+E.Y$$ None of the five terms are integers, but the sum itself is an integer. Find the possible sums of the expression. For each possible answer, write one example with these five terms. Explain why other sums cannot be obtained.

Peter went to the Museum of Modern Art and saw a square painting in a frame of an unusual shape. The frame consisted of $21$ congruent triangles. Peter was interested in what the angles of these triangles were equal to. Help him find these angles.

Red, blue and green chameleons live on an island. One day $35$ chameleons stood in a circle. A minute later, they all changed colour at the same time, each changing into the colour of one of their neighbours. A minute later, everyone again changed their colours at the same time into the colour of one of their neighbours. Is it ever possible that each chameleon was each of the colours red, blue and green at some point? For example, it’s allowed for a chameleon to start off blue, turn green after one minute, then turn red after the second minute. It’s not allowed for a chameleon to start off blue, turn green after one minute, but then turn back to blue after the second minute.

A useful common problem-solving strategy is to divide a problem into cases. We can divide the problem into familiar and unfamiliar cases; easy and difficult cases; typical and extreme cases etc. The division is sometimes suggested by the problem, but oftentimes requires a bit of work first.

If you are stuck on a problem or you are not sure where to begin, gathering data by trying out easy or typical cases first might help you with the following (this list is not exhaustive):

1. Gaining intuition of the problem

2. Isolating the difficulties

3. Quantifying progress on the problem

4. Setting up or completing inductive arguments

Let us take a look at this strategy in action.

Split the numbers from $1$ to $9$ into three triplets such that the sum of the three numbers in each triplet is prime. For example, if you split them into $124$, $356$ and $789$, then the triplet $124$ is correct, since $1+2+4=7$ is prime. But the other two triples are incorrect, since $3+5+6=14$ and $7+8+9=24$ are not prime.

A family is going on a big holiday, visiting Austria, Bulgaria, Cyprus, Denmark and Estonia. They want to go to Estonia before Bulgaria. How many ways can they visit the five countries, subject to this constraint?

Let $p$, $q$ and $r$ be distinct primes at least $5$. Can $p^2+q^2+r^2$ be prime? If yes, then give an example. If no, then prove it.