Problems

What is the ratio between the red and blue area? All shapes are semicircles.

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?

What’s the sum of the Fibonacci numbers \(F_0+F_1+F_2+...+F_n\)?

What’s the sum \(\frac{F_2}{F_1}+\frac{F_4}{F_2}+\frac{F_6}{F_3}+...+\frac{F_{18}}{F_9}+\frac{F_{20}}{F_{10}}\)?

Is \(F_{100}\) odd or even?

Is \(F_{100}\) a multiple of \(3\)?

We have a sequence where the first term (\(x_1\)) is equal to \(2\), and each term is \(1\) minus the reciprocal of the previous term (which we can write as \(x_{n+1}=1-\frac{1}{x_n}\)).

What’s \(x_{57}\)?

Let \(n\) be a positive integer. Can \(n^7-77\) ever be a Fibonacci number?

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