UnWin Suggested This

What is $2 + 2 + 2$?

Question 2(e)

If $x$ is real and $e^x = 2^x$, what is $x$?

Showpan Showprano

What is the opus number of this late Chopin piece? Enter the opus number only. For example, if it's Op. 62 no 2, just enter 62.

Good luck using Shazam on this one :)

https://youtu.be/H8mtSjLao0g

NEW PHONE... SHINY...

You have just gotten a shiny new phone and a case for it and you are about to put the case on. Unfortunately, it seems like quite a tight fit and you might scratch your phone when putting it on.

The probability that you scratch the phone when putting the case on is $p$. The probability your phone makes an unpleasant sound when putting the case on is $q$, and is independent of whether the phone was actually scratched.

If it makes an unpleasant sound, you take the case off to check if your phone was indeed scratched. The probability that the phone is scratched by taking the case off is $r$.

After taking the case off, if you see that your phone was indeed scratched, you cry. Otherwise, you put the case back on. If it makes another unpleasant sound, you take the case off and check again, and so on, until you either cry or you manage to put the case on without the phone making an unpleasant sound.

Assume $(p, q, r) \neq (0, 1, 0)$.

What is the probability that you cry?

What is the probability that the phone is scratched, but goes unnoticed?

SOBBING...

Continuing from the previous question, what is the average number of times you check for scratches before you cry, given that you cry at all? You may assume $k \neq 0$ and $(p, q) \neq (0, 0)$.

Formally, let $X$ be a positive integer random variable where $\displaystyle P(X = k) = \frac{p_k}{p}$. Here, $p_k$ is the probability that you check $k$ times and cry on the $k$th time you check, and $p$ is the probability that you cry at all (the answer to the first part of the last question). Find the expectation \[ E(X) = \sum^\infty_{k=1} k P(X = k) \] which you may assume exists.

elections 101

Find a closed form of the expression \[ \sum_{k=1}^n k \binom{n}{k}^2. \] i.e. write an expression mathematically equivalent to the above, but without using summations or products. You can use $\mathrm{nCr}$.

Hint: Consider the classic “form a committee and pick a president" combinatorial argument. How many people are there in total?

the most complex question

Find the radius of convergence of the Taylor series of the complex function \[ f(z) = \csc(re^{it} z) \] centered at $z = a$, where $r \in \mathbb R$, $t \in \mathbb R$, and $a \in \mathbb R$. Here, $i = \sqrt{-1}$ (take that, electrical engineers).

Hint: Use a well-known complex analysis result about the radius of convergence.

prime rib... mmm....

Consider the vector space of quadratics over $\mathbb Z / p \mathbb Z$, where $p \geq 2$ is prime. That is, the vector space \[ V = \{ax^2 + bx + c : a, b, c \in \mathbb Z / p \mathbb Z\} \] over $\mathbb Z / p \mathbb Z$. Find the number of bases of $V$, i.e. how many subsets of $V$ are simultaneously linearly independent and span $V$.

Extra info for people who don't know abstract algebra / number theory: \[ \mathbb Z / p \mathbb Z = \{0, 1, 2, \cdots, p - 1\} \] is a field (i.e. $+$, $-$, $\times$, $\div$ are all defined), with $+$ and $\times$ being the same as normal integers, except you take mod $p$ after evaluating the expression, and $-$ and $\div$ being the inverses of the respective operations under this field.

my favourite display tech

Suppose that $x \in \mathbb R$, where $0 \leq x < 69$, satisfies \[ \left\{ \begin{alignedat}{2} \lfloor 69^n x \rfloor &\equiv a^n \ \ &&(\mathrm{mod}\ {3}),\\ \lfloor 69^n x \rfloor &\equiv b \ \ &&(\mathrm{mod}\ {23}) \end{alignedat} \right. \] for any integer $n \geq 0$, where $0 < a < 3$ and $0 \leq b < 23$ are integers. Find the value of $x$, which in fact exists and is unique. Your answer should be in terms of $a$ and $b$.