Dylan is a masters mathematics student at the University of Stellenbosch. When not doing mathematics, he enjoys sleeping, eating, and drinking coffee or wine. You can become one of his best friends, or possibly even one of his favourite people in the world, by facilitating any one or more of these activities.

## The 2018 South African Mathematics Olympiad — Problem 6

The final round of the South African Mathematics Olympiad will be taking place on Thursday, 28 July 2019. I have been writing about some of the problems from the senior paper from 2018. A list of all of the problems can be found here.

Today we will look at the sixth and final problem from the 2018 South African Mathematics Olympiad:

Let $n$ be a positive integer, and let $x_1, x_2, \dots, x_n$ be distinct positive integers with $x_1 = 1$. Construct an $n \times 3$ table where the entries of the $k$-th row are $x_k, 2x_k, 3x_k$ for $k = 1, 2, \dots, n$. Now follow a procedure where, in each step, two identical entries are removed from the table. This continues until there are no more identical entries in the table.

1. Prove that at least three entries remain at the end of the procedure.
2. Prove that there are infinitely many possible choices for $n$ and $x_1, x_2, \dots, x_n$ such that only three entries remain,

There are some heuristics that are often helpful when solving a problem, such as

• Looking at small cases:

This helps us to understand the problem and how the various pieces in the problem relate to each other.

## The 2018 South African Mathematics Olympiad — Problem 5

The final round of the South African Mathematics Olympiad will be taking place on Thursday, 28 July 2019. I have been writing about some of the problems from the senior paper from 2018. A list of all of the problems can be found here.

Today we will look at the fifth problem from the 2018 South African Mathematics Olympiad:

Determine all sequences $a_1, a_2, a_3, \ldots$ of nonnegative integers such that $a_1 < a_2 < a_3 < \ldots$, and $a_n$ divides $a_{n - 1} + n$ for all $n \geq 2$.

Since the sequence $a_1, a_2, \ldots$ is strictly increasing, we know that $a_n \geq n - 1$ for all positive integers $n$. (We could prove this rigorously by induction.) This means that $a_{n - 1} + n \leq (a_n - 1) + (a_n + 1) = 2a_n$ for all $n$, and so we know that $a_{n - 1} + n$ is equal to either $a_n$, or to $2a_n$ for all positive integers $n$. Perhaps we should try to figure out exactly when it is equal to $a_n$, and when it is equal to $2a_n$. If we knew, for example, that we always have that $a_{n - 1} + n = a_n$, then we have reduced the problem to solving this recurrence relation.…

## The 2018 South African Mathematics Olympiad — Problem 4

The final round of the South African Mathematics Olympiad will be taking place on Thursday, 28 July 2019. In the week leading up to the contest, I plan to take a look at some of the problems from the senior paper from 2018. A list of all of the posts can be found here.

Today we will look at the fourth problem from the 2018 South African Mathematics Olympiad:

Let $ABC$ be a triangle with circumradius $R$, and let $\ell_A, \ell_B, \ell_C$ be the altitudes through $A, B, C$ respectively. The altitudes meet at $H$. Let $P$ be an arbitrary point in the same plane as $ABC$. The feet of the perpendicular lines through $P$ onto $\ell_A, \ell_B, \ell_C$ are $D, E, F$ respectively. Prove that the areas of $DEF$ and $ABC$ satisfy the following equation:

$\displaystyle \text{area}(DEF) = \frac{{PH}^2}{4R^2} \cdot \text{area}(ABC).$

Once again, we begin by creating a diagram. Again, since I already know how the solution plays out, I’ve drawn in the circle that passes through $P, E, D, H$, and $F$. We do know yet that these points are concylic, however, as it is not given directly in the problem statement.…

## The 2018 South African Mathematics Olympiad — Problem 3

The final round of the South African Mathematics Olympiad will be taking place on Thursday, 28 July 2019. In the week leading up to the contest, I plan to take a look at some of the problems from the senior paper from 2018. A list of all the posts can be found here.

Today we will look at the third problem from the 2018 South African Mathematics Olympiad:

Determine the smallest positive integer $n$ whose prime factors are all greater than $18$, and that can be expressed as $n = a^3 + b^3$ with positive integers $a$ and $b$.

In many number theory problems, it helps to consider the prime factors of the numbers involved, and in this problem we are in fact forced to do so because the question itself is about the prime factors of a number. When dealing with factors of a number or an expression representing some number, it of course helps to consider whether we can factorise the given expression.…

## The 2018 South African Mathematics Olympiad — Problem 2

The final round of the South African Mathematics Olympiad will be taking place on Thursday, 28 July 2019. In the week and half leading up the the contest, I plan to take a look at some of the problems from the senior paper in 2018, and have already written about the first problem

The second problem from the 2018 South African Mathematics Olympiad was

In triangle $ABC$, $AB = AC$, and $D$ is on $BC$. A point $E$ is chosen on $AC$, and a point $F$ is chosen on $AB$, such that $DE = DC$ and $DF = DB$. It is given that $\frac{DC}{BD} = 2$ and $\frac{AF}{AE} = 5$. Determine the value of $\frac{AB}{BC}$.

The first step of solving any geometry problem should always be to draw a sketch. This helps you to understand how different parts of the figure relate to each other, and an accurate sketch may help you to form conjectures. Sometimes having a deliberately inaccurate sketch on hand is also helpful as it may help to avoid circular reasoning.…

## The 2018 South African Mathematics Olympiad — Problem 1

The final round of the South African Mathematics Olympiad will be taking place on Thursday, 28 July 2019. In the two weeks leading up to the contest, I plan to take a look at some of the problems from the senior paper from 2018.

The first problem from the 2018 South African Mathematics Olympiad was

One hundred empty glasses are arranged in a $10 \times 10$ array. Now we pick $a$ of the rows and pour blue liquid into all glasses in these rows, so that they are half full. The remaining rows are filled halfway with yellow liquid. Afterwards, we pick $b$ of the columns and fill them up with blue liquid. The remaining columns are filled with yellow liquid. The mixture of blue and yellow liquid turns green. If both halves have the same colour, then that colour remains as is.

1. Determine all possible combinations of values for $a$ and $b$ so that exactly half of the glasses contain green liquid at the end.

## The South African Mathematics Olympiad

The South African Mathematics Olympiad is an annual mathematics competition for high-school students in South Africa. The competition is organised by the South African Mathematics Foundation, and comprises three rounds which increase in difficulty. The final round of the 2019 South African Mathematics Olympiad will take place on Thursday, 25 July, and the top ten junior (Grade 8 and 9) and senior (Grades 10—12) competitors will be invited to a prize-giving evening taking place on 14 September 2019. At the same time, the problem selection committee will meet to start setting the 2020 papers.

According to the SAMF, nearly 100000 students participated in the 2017 edition of the competition. The numbers for the 2019 competition are likely to be similar. These students all write the first round of the competition, which learners write at their individual schools in March every year. The papers are marked at the school, and any student with more than 50% is invited to participate in the second round of the competition.…

## On Convergent Sequences and Prime Numbers

Ever since Euclid first proved that there are infinitely many prime numbers, mathematicians have found ever more creative ways to prove the same result, and also various stronger theorems that imply it. Dirichlet’s Theorem, for example, states that if$m$ and $n$ are relatively prime integers, then there are infinitely many prime numbers of the form $mk + n$ for some integer $k$. It is also known that the sum of the reciprocals of the prime numbers diverges, that the sum

$\displaystyle \sum_{\substack{p \leq n \\ p \text{ prime}}} \frac{1}{p} \sim \log(\log(n))$

and that the number of prime numbers less than $n$ is asymptotically equal to $\displaystyle \frac{n}{\log(n)}$. In this blog post, we will continue this proud tradition by proving that there are infinitely many prime numbers which have your phone number somewhere in their digits, and which simultaneously have a prime number of digits.

To do so, we will look at the convergence of two different sums: that of the reciprocals of the primes with a prime number of digits, and that of the reciprocals of the natural numbers which do not contain your phone number amongst their digits.…

## A Mathematics Problem from the SBITC

The Standard Bank IT Challenge (SBITC) is an annual coding competition for undergraduate and honours students in South Africa. The contest consists of two rounds: a regional event named the “heats”, and the final. In the heats, teams of up to four students each compete against other teams from the same university, and the winning team from each of the nine top-performing universities is invited to the final round in Johannesburg. Each member of the winning team wins a prize, and the winning university receives a large cash prize, but students mostly participate for the enjoyment that is to be obtained in solving the problems, and to test their skills against a set of problems that is designed to challenge the participants.

This year, the final problem from the heats (which took place on Saturday, 16 May) was fairly mathematical in nature; or more-so than the other problems at least. Essentially, the problem asks the following:

We consider generalised Fibonacci sequences $T_n$ which satisfy the same recurrence relation $T_{n + 2} = T_{n + 1} + T_n$ as the Fibonacci numbers, but with the first two terms $T_1$ and $T_2$ being arbitrary positive integers.…