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 \(0\mid999\) contains only two digits, namely \(0\) and \(9\), whereas the sign \(1\mid998\) contains three digits, namely \(1\), \(8\) and \(9\).
Two players are playing a game. The first player is thinking of a finite sequence of positive integers \(a_1\), \(a_2\), ..., \(a_n\). The second player can try to find the first player’s sequence by naming their own sequence \(b_1\), \(b_2\), ..., \(b_n\). After this, the first player will give the result \(a_1b_1 + a_2b_2 + ...+a_nb_n\). Then the second player can say another sequence \(c_1\), \(c_2\), ..., \(c_n\) to get another answer \(a_1c_1+ a_2c_2 + ... +a_nc_n\) from the first player. Find the smallest number of sequences the second player has to name to find out the sequence \(a_1\), \(a_2\), ..., \(a_n\).
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.
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.
We wish to paint the \(15\) segments in the picture below in three colours. We want it such that no two segments of the same colour have a common end. For example, you cannot have both \(AB\) and \(BC\) blue since they share the end \(B\). Is such a painting possible?
In an \(n\times n\) table, two opposite corner squares are black and the rest are white. We wish to turn the whole \(n\times n\) table black in two stages. In the first stage, we paint black some of the squares that are white at the moment. In the second stage, we can perform the following two operations as much as we like. The row operation is to swap the colours of all the squares in a particular row. The column operation is to swap the colours of all the squares in a particular column. What is the fewest number of white squares that we can paint in the first stage?
An example of the row operation: let W stand for white and B stand for black and suppose that \(n=5\). Also suppose that a particular row has the colours WWBWB. Then performing the row operation would change this row to BBWBW.
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.
How many subsets of \(\{1,2,...,n\}\) (that is, the integers from \(1\) to \(n\)) have an even product? For the purposes of this question, take the product of the numbers in the empty set to be \(1\).