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## Sunday, 15 June 2014

### Sunday Afternoon Maths XVII

Here's this week's collection. Answers & extensions tomorrow. Why not discuss the problems on Twitter using #SundayAfternoonMaths or on Reddit. This Tuesday is Maths Jam, make sure you're there!

#### Maths Jam

Maths Jam is always held on the second-to-last Tuesday of the month. This month, it will be held on the 17th. What is the earliest date in the month on which Maths Jam can fall and when will this next happen?

#### N

Consider three-digit integers $$N$$ such that:
(a) $$N$$ is not exactly divisible by 2, 3 or 5.
(b) No digit of $$N$$ is exactly divisible by 2, 3 or 5.
How many such integers $$N$$ are there?

#### Square Numbers

Towards the end of his life, Lewis Carroll recorded in his diary that he had discovered that double the sum of two square numbers could always be written as the sum of two square numbers. For example
$$2(3^2 +4^2 )=1^2 +7^2$$ $$2(5^2 +8^2 )=3^2 +13^2$$
Prove that this can be done for any two square numbers.

#### Differentiate This

$$f(x)=e^{x^{ \frac{\ln{\left(\ln{x}\right)}}{ \ln{x}}} }$$
Find $$f'(x)$$.