Problems

Due to a mistake in the bakery, a cake that was supposed to be shaped as two concentric pieces (like on the right diagram below) came out like the left diagram below. Find the smallest number of pieces the cake should be cut into in order to rearrange the pieces into the cake on the right side of the picture.

Note that the cake is \(\textit{not}\) tiered like a wedding cake, but is shaped like a cylinder with a flat top. Curved cuts are allowed.

Katie and Charlotte had \(4\) sheets of paper. They cut some of the sheets into \(4\) pieces. They then cut some of the newly obtained papersheets also into \(4\) pieces. They did this several more times, cutting a piece of paper into \(4\). In the end they counted the number of sheets. Could this number be \(2024\)?

One cell was cut out of a \(3\times6\) 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.

A monkey becomes happy when they eat three different fruits. What is the largest number of monkeys that can become happy with \(20\) pears, \(30\) bananas, \(40\) peaches and \(50\) tangerines?

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