Problems
What is the following as a single fraction? \[\frac{1}{1\times2}+\frac{1}{2\times3}+\frac{1}{3\times4}+...+\frac{1}{98\times99}+\frac{1}{99\times100}.\]
Adi and Maxim play a game. There are \(100\) sweets in a bowl, and they each take in turns to take either \(2\), \(3\) or \(4\) sweets. Whoever cannot take any more sweets (since the bowl is empty, or there’s only \(1\) left) loses.
Maxim goes first - who has the winning strategy?
Michelle and Mondo play the following game, with Michelle going first. They start with a regular polygon, and take it in turns to move. A move is to pick two non-adjacent points in one polygon, connect them, and split that polygon into two new polygons. A player wins if their opponent cannot move - which happens if there are only triangles left. See the diagram below for an example game with a pentagon. Prove that Michelle has the winning strategy if they start with a decagon (\(10\)-sided polygon).
Let \(n\) be a positive integer. Show that \(1+3+3^2+...+3^{n-1}+3^n=\frac{3^{n+1}-1}{2}\).
Show that all integers greater than or equal to \(8\) can be written as a sum of some \(3\)s and \(5\)s. e.g. \(11=3+3+5\). Note that there’s no way to write \(7\) in such a way.
We want to prove Monsky’s theorem as a corollary of Sperner’s lemma:
it is not possible to dissect a square into an odd number of triangles
of equal area. After scaling we can consider the square with
coordinates: \((0,0),(0,1),(1,0),(1,1)\), which we want to
cut into \(n\) triangles with area
\(\frac{1}{n}\) each for an odd \(n\). Consider the following coloring of all
the points with rational coordinates \((\frac{p}{q},\frac{r}{s})\) inside the
square:
We look at the powers of \(2\) in the
fractions \((\frac{p}{q},\frac{r}{s})\), first of all
the numbers \(p,q\) are coprime, and
thus only one of them is divisible by \(2\), same with \(r,s\). Then the following possibilities
might occur:
Neither \(q\) nor \(s\) is divisible by \(2\). In this case we color the point red.
\(\frac{r}{s}\) is divisible by a larger or equal power of \(2\) than \(\frac{p}{q}\) and \(p\) is not divisible by \(2\). In this case we color the point blue.
\(\frac{p}{q}\) is divisible by a strictly larger power of \(2\) than \(\frac{r}{s}\) and \(r\) is not divisible by \(2\). In this case we color the point green.
Under an assumption (which you do not have to prove) that the area of any rainbow triangle is at least \(\frac{1}{2}\) prove the Monsky’s theorem.