Two year 13 students, Jozef and Jake, recently enjoyed outstanding success in the Senior Maths Challenge.

The UKMT Individual Maths Challenge attracts over 600,000 entries from over 4000 schools and colleges. The papers consist of lively and intriguing multiple choice questions which are designed to stimulate an interest in maths for large numbers of pupils. The following are two examples of the problems presented to the competitors:

**Example 1**

Five square tiles are put together side by side. A quarter circle is drawn on each tile to make a continuous curve as shown. Each of the smallest squares has a side-length 1. What is the total length of the curve?

A: 6π B: 6.5π C: 7π D: 7.5π E: 8π

**Example 2**

Let *n* be the smallest integer for which 7*n* has 2016 digits. What is the units digit of *n*?

A: 0 B: 1 C: 4 D: 6

Bournside achieved two gold, 2 silver and 10 bronze awards this year. Jozef and Jake scored 99 marks and qualified for the next round of the competition called the “Senior Kangaroo Challenge”. Below are examples of questions faced in the Kangaroo Challenge:

**Example 1**

Students in a class take turns to practise their arithmetic skills. Initially a board contains the integers from 1 to 10 inclusive, each written ten times. On each turn, a student first deletes two of the integers and then writes on the board the number that is one more than the sum of those two deleted integers. Turns are taken until there is only one number remaining on the board. Assuming no student makes a mistake, what is the remaining number?

**Example 2**

The sum of the squares of four consecutive positive integers is equal to the sum of the squares of the next three consecutive integers, What is the square of the smallest of these integers?

Both Josef and Jake have had a longstanding interest in mathematics, having excelled in the subject from an early age. Both students intent to go on to higher education with Josef attending Exeter University, and Jake planning on studying at Sheffield.