**Maths:****1.)** Find

in terms of

.

Solution 1 - Over9000

Solution 2 - dcc**2.)** Show that

.

Solution 1 - TrueTearsSolution 2 - NeobeoSolution 3 - dcc**3.)** Show

**4.)** Show that

(or perhaps the even more general result for

).

Solution 1 - /0**5.)** For a real number

, evaluate

.

(Source: 1995 Hosei University entrance exam/Business administration)Solution 1 - coblin**6.)** Find the minimum area of the part bounded by the parabola

and the line

.

(Source: 1963 Tokyo Metropolitan University entrance exam)Solution 1 - Neobeo

Solution 2 - TrueTears**7.)** Evaluate

(Source: 2008 Miyazaki University entrance exam/Agriculture)Solution 1 - kamil9876**8.)** Evaluate

(Source: Wikipedia)Solution 1 - TrueTearsSolution 2 - dcc**9.)** If you break a stick into 3 pieces what is the probability that the 3 pieces can form a triangle?

(Source: Neobeo)Solution 1 - /0Solution 2 - kamil9876**10.)** Show (

**without calculus**) that the minimum value of

is

.

Solution 1 - Damo17**11.)** Find the maximum value of

given that

.

Solution 1 - humph**12.)** Prove that

is irrational.

(Source: TrueTears)Solution 1 - Over9000**13.)** Show that

.

(Source: Damo17)Solution 1 - dccSolution 2 - Over9000Solution 3 - golden**14.)** Let N be the positive integer with 2008 decimal digits, all of them 1. That is,

, with 2008 occurrences of the digit 1. Find the 1005th digit after the decimal point in the decimal expansion of

.

(Source: /0 - Melbourne University/BHP Billiton Maths Competition 2008)Solution 1 - Over9000**15.)** Neobeo is walking around in Luna Park, and notices an alleyway called 'Infinite Ice Cream'. Neobeo notes that the 'Infinite Ice-Cream' appears to possess an infinitely large number of people selling ice-cream. Upon walking outside any particular shop, Neobeo feels a huge compulsion to purchase an ice-cream. For every shop that Neobeo visits, he is

less likely to purchase an ice-cream then at the previous shop. After purchasing an ice-cream, Neobeo leaves Luna Park. What is the probability of Neobeo purchasing an ice-cream at the second shop in 'Infinite Ice Cream'?

(Source: IRC & the recesses of my brain)Solution 1 - golden**16.)** Find

.

Solution 1 - TrueTears

Solution 2 - kamil9876

Solution 3 - dcc**17.)** Find

**18.)** Find

.

Do not use **L'Hospital's rule** to evaluate this, because that is boring. Try and use limit laws and properties, rather then mindless derivatives.**MORE TO COME, I WILL EDIT THIS POST**.