Problems

Age
Difficulty
Found: 1557

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.

image

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.

The letters \(A\), \(R\), \(S\) and \(T\) represent different digits from \(1\) to \(9\). The same letters correspond to the same digits, while different letters correspond to different digits.
Find \(ART\), given that \(ARTS+STAR=10,T31\).

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 \(1\).

How many subsets are there of \(\{1,2,...,n\}\) (the integers from \(1\) to \(n\) inclusive) containing no consecutive digits? That is, we do count \(\{1,3,6,8\}\) but do not count \(\{1,3,6,7\}\).
For example, when \(n=3\), we have \(8\) subsets overall but only \(5\) contain no consecutive integers. The \(8\) subsets are \(\varnothing\) (the empty set), \(\{1\}\), \(\{2\}\), \(\{3\}\), \(\{1,3\}\), \(\{1,2\}\), \(\{2,3\}\) and \(\{1,2,3\}\), but we exclude the final three of these

Let \(A\), \(B\), \(C\) and \(D\) be four points labelled clockwise on the circumference of a circle. The diagonals \(AC\) and \(BD\) intersect at the centre \(O\) of the circle. What can be deduced about the quadrilateral \(ABCD\)?