Maths:1.) Find
}{\sin(x)})
in terms of
.
Solution 1 - Over9000
Solution 2 - dcc2.) Show that
 = \arctan \left(\frac{3}{4}\right) + \arctan \left(\frac{1}{2}\right))
.
Solution 1 - TrueTearsSolution 2 - NeobeoSolution 3 - dcc3.) Show
 \ dt = \frac{48}{65} \left( \dfrac{e^{2\pi} + 1}{e^{2\pi} - 1} \right) \int_0^{\pi} e^{-2t} \sin^2 (t) \ dt)
4.) Show that
}{\sin(x)} = \dfrac{\sin\left(x\left(a + 1\right)\right) + \sin\left(x\left(a - 1\right)\right)}{\sin(2x)})
(or perhaps the even more general result for
}{\sin(bx)})
).
Solution 1 - /05.) For a real number
, evaluate
|\ dx)
.
(Source: 1995 Hosei University entrance exam/Business administration)Solution 1 - coblin6.) 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 - TrueTears7.) Evaluate
(Source: 2008 Miyazaki University entrance exam/Agriculture)Solution 1 - kamil98768.) Evaluate
(Source: Wikipedia)Solution 1 - TrueTearsSolution 2 - dcc9.) 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 - kamil987610.) Show (
without calculus) that the minimum value of
 = 4x^2 + \dfrac{64}{x} + 17\: (x > 0))
is
.
Solution 1 - Damo1711.) Find the maximum value of
 = 3f(x) + 4g(x) + 10h(x))
given that
^2 + g(x)^2 + h(x)^2 \leq 9)
.
Solution 1 - humph12.) Prove that
is irrational.
(Source: TrueTears)Solution 1 - Over900013.) Show that
.
(Source: Damo17)Solution 1 - dccSolution 2 - Over9000Solution 3 - golden14.) 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 - Over900015.) 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 - golden16.) Find
.
Solution 1 - TrueTears
Solution 2 - kamil9876
Solution 3 - dcc17.) Find
 - \sin x})
18.) Find
^{\frac{1}{x}})
.
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.
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