Nice Question: show that: The follow nice trigonometry $$\tan{\dfrac{2\pi}{13}}+4\sin{\dfrac{6\pi}{13}}=\sqrt{13+2\sqrt{13}}$$ This problem I have ugly so...

I have three points $A(-10,-12)$, $B(6,18)$, $C(-2,-14)$. I have to prove they exist on the same semicircle. How? If I try to draw the three points to inv...

Given: $$AB=\left( \begin{array}{ccc}2\\1\\3\end{array} \right) \;\;\;\; \text{and}\;\;\;\; CD=\left( \begin{array}{ccc}4\\3\\6\end{array} \right).$$ Just...

I was asked to prepare a set of notes for year 12 students. These notes are going to be used by these students to prepare for the examinations. Naturally,...

My goal is to prove the following inequality, known as the Dvoretsky-Kiefer-Wolfowitz inequality (1956) : Let $(X_i)_{i \geqslant}$ be iid random variable...

Given $Y\sim N(\mu, \sigma^2)$. I'm trying to find the moment generating function of $Z=\frac{Y-\mu}{\sigma}$ using the MGF transform method. Here�...

I am very confused with null sets. I get that a set which has no elements will be called a null set but I am not getting the examples given below. Please ...

You're trapped in a room with 2 doors, one on your left and one on your right. One door leads to heaven, one door leads to hell. There are two birds ...

Squares of consecutive numbers differ by the sum of those numbers, so $6^2 = 5^2 + 5 + 6$. Geometrically, this is because the difference between the two s...

How many digits are in the integer representation of 2 to the 30th power? Since I didn't really know any 'expert' way to approach this, I j...