Problems

In a parallelogram \(ABCD\), point \(E\) belongs to the side \(CD\) and point \(F\) belongs to the side \(BC\). Show that the total red area is the same as the total blue area:

The figure below is a regular pentagram. What is larger, the black area or the blue area?

There is a very, very fast way of computing the greatest common divisor of two positive integers. It was in fact known even to the Greeks two thousand years ago. This procedure is called the Euclidean algorithm, named after Euclid, a famous ancient Greek mathematician.

The algorithm works as follows. Take two positive integers \(a,b\). Let’s say \(a\geq b\).

1. Calculate the remainder of \(a\) when divided by \(b\). Call it \(r_1\).

2. Calculate the remainder of \(b\) when divided by \(r_1\). Call it \(r_2\).

3. Calculate the remainder of \(r_1\) when divided by \(r_2\). Call it \(r_3\).

4. Continue to divide the remainder from two steps prior by the remainder from the last step, until...

5. The remainder \(r_n\) is divisible by \(r_{n+1}\). The Euclidean algorithm stops now and \(r_{n+1}\) is \(\gcd(a,b)\).

Show that there is indeed some natural number \(n\) such that \(r_n\) is divisible by \(r_{n+1}\), so that the Euclidean algorithm must stop eventually. Furthermore, show that \(r_{n+1}\) is actually \(\gcd(a,b)\) (otherwise it is all in vain!).

Cut a packet of 4 cards. Is any of the cards in the same place as it was before?

We have a packet of 13 cards from Ace to King. Let Ace be the first card, 2 the second card and so on with King being the thirteenth card. How can you interchange 4 and 7 (and leave all other cards where they are) by only switching adjacent pairs of cards?

How many permutations of 13 cards leaves the third card where it started?

Does there exist an irreducible tiling with \(1\times2\) rectangles of a \(6\times 6\) rectangle?

Irreducibly tile a floor with \(1\times2\) tiles in a room that is a \(6\times8\) rectangle.

Let \(n\ge r\) be positive integers. What is \(F_n^2-F_{n-r}F_{n+r}\) in terms of \(F_r\)?

On the questioners’ planet (where everyone can only ask questions. Cricks can only ask questions to which the answer is yes, and Goops can only ask questions to which the answer is no), you meet 4 alien mathematicians.

They’re called Alexander Grothendieck, Nicolas Bourbaki, Henri Cartan and Daniel Kan (you may like to shorten their names to \(A\), \(B\), \(C\) and \(D\)).
Alexander asks the following question “Am I the kind who could ask whether Bourbaki could ask whether Cartan could ask whether Daniel is a Goop?"

Amongst the final three (that is, Bourbaki, Cartan and Daniel), are there an even or an odd number of Goops?