Two of our students, Isaac and Ying, were unusual this year – they had to sit exams for some of their qualifications!
Both boys took the fiendishly difficult “STEP II” and “STEP III” maths exams which they needed for their Cambridge University offers. These exams are only taken by the very best mathematicians worldwide and this year they were invigilated remotely from London. Ying scored “1”s on both of his papers which is the highest available numeric grade, so he has secured his place to read Mathematics at Maths at Clare College Cambridge.
Isaac was awarded the highly unusual “S” grade for his outstanding performance on both of the exams and will be reading Mathematics at Trinity College, Cambridge, following in the footsteps of his namesake Isaac Newton at probably the most competitive mathematics course in the world.
Both boys were also awarded straight “A*”s in their A Levels.
Alongside their talent for Maths, and their phenomenally hard work, both boys have also made the most of the other opportunities at Trinity. Isaac intends to continue his cross-country running and they may both end up playing in Cambridge University’s orchestra together – Isaac has a diploma in trombone and Ying on the violin.
The following STEP II question from 2018 is quite short and may be attempted by the reader; Mr Barlow is happy to mark any solutions!
(i) Find all pairs of positive integers (n,p) ,where p is a prime number, that satisfy:
n! + 5 = p
(ii) Find all pairs of positive integers (n,m) that satisfy:
1! x 3! x … x (2n-1)! = m!
For part (ii) you might like to use the following two theorems:
1. For all n>6, 1! x 3! x … x (2n-1)! > (4n)!
2. For every positive integer n, there is a prime number between 2n and 4n
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