I invite you all to look at the problems (try to ignore solutions for now) for the 49th UNSW School Mathematics Competition in this issue. How many could you do, in three hours?
In my previous column, I looked at what we can learn of the mathematical achievements of Pythagoras and Theano. I relied in particular on an article by the Leningrad-based mathematician Leonid Zhmud, although I found his account maddeningly incomplete in places.
The purpose of this paper is both to observe, understand and appreciate the link between the Fibonacci sequence and the ubiquitous mathematical constant, $\pi$. It proves the following series for $\pi$, making use of the Fibonacci numbers.
Problem 1
Find the set of all pairs of positive integers $(n,m)$ that satisfy
Competition Winners – Senior Division
First Prize
Q1331 Given any positive integers $m$ and $n$ prove
Q1321 Find the sum of the coefficients of those terms in the expansion of $$(x^{31}+x^5-1)^{2011}$$ which have an odd exponent in $x$
ANS:
Note that $$(x^{31}+x^5-1)^{2011}=a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$$