Arithmetic

What is $180 - 42 - 42$?

Timezones

What is the UNIX time (in seconds) of 22/5/2015 10:55:39pm HKT?

Spin

Suppose a rectangle spins clockwise at a rate of $2918572878278522^\circ$ per second.

After $2817274$ seconds, how many degrees would this rectangle be rotated clockwise from its original position? Give your answer as a positive integer that is as small as possible.

For example, if you think the answer is $362^\circ$, your answer should evaluate to 2.

Empirically Empirical

Suppose the $\mu$-ness of $\mathcal{R}$, denoted $\sigma(\mathcal{R})$ is related to its $v$-ness $\mu_v$, $\upsilon$-ness $\rho_v$, $i$-ness $v_\upsilon$ and highness $\upsilon_v$, and $D$-ness $\mathbb{D}$. They are empirically linked by the equation: \[ \sigma(\mathcal{R}) = \frac{\arctan(0.4412852^{\upsilon_v v_\upsilon^{1.002412}})}{\mathbb{D}\mu_{\upsilon} \rho_{v}} \] with the formulae $\mu_v = \frac{1}{\mu_\upsilon}$, $\rho_v = \frac{1}{\sqrt[0.441285]{\rho_v}}$. It is assumed that \[ -\frac{\pi}{2} < \sigma(\mathcal{R})\mathbb{D}\mu_{\upsilon} \rho_{v} < \frac{\pi}{2} \] Given that the $\mu$-ness of $\mathcal{R}$ is $0.69$, $i$-ness is $0.4$, $D$-ness is $0.6$ and $v$-ness is $0.4$, what is the highness of $\mathcal{R}$? (Please note that $v$ and $\upsilon$ are different characters; one is slightly wider than the other. Therefore, it is advised to use a magnifying glass and ruler for this question.)

(Hint: Any typo you think you see is probably intentional.)

Leap Years

Suppose we randomly pick a day between the first day of 2000 and the last day of 9999 inclusive. Assume all days have an equal probability of being picked. What is the probability we pick leap day, i.e. the 29th of February?

En Passant

In chess, a pawn can move ahead two squares on its first move. If it does so, and lands adjacent to an enemy pawn, it may be captured by that enemy pawn. This is known as en passant.

For this question, define two occurences of en passant to be different if any of the following are true:

• The pawn being captured is different. Two pawns are different if they originated from different files or are different colours.
• The capturing pawn is different.
• The pawn was captured from a different file. For example, suppose the black pawn on the $c$ file was captured en passant. Two different ways for this to happen is if the the white pawn was on the $b$ file or the $d$ file.

Assume a standard $8 \times 8$ chess board and standard chess rules. Assume no pawn promotions occur. How many different ways are there for en passant to theoretically occur? (Don't forget to account for both players capturing.)

Quadratic Roots

Consider the quadratic function: \[ f(x) = x^2 \sin^2 a + 2x \cos a - 1 \] What is the smallest positive value of $a$ such that $f(x)$ has a root at $x = 1$?

Red Panda

Suppose there are $27$ red pandas in a zoo and there are $3$ zookeepers who have been assigned to take care of these red pandas.

The zoo policy states each zookeeper has to take care of at least $3$ red pandas. How many ways are there to assign each red panda to a zookeeper so that this requirement is met?

(Hint: $|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C|$.)

Rotation Matrix

Consider the matrix for $(a, b) \neq (0, 0)$: \[ A = \begin{bmatrix} a & -b\\ b & a \end{bmatrix}. \] First, note that $A$ is a rotation matrix multiplied by some positive constant, i.e. \[ \begin{bmatrix} a & -b\\ b & a \end{bmatrix} = c \begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta & \cos \theta \end{bmatrix} \] for some $\theta$ and $c > 0$. Please give the value of that $c$ below in terms of $a$ and $b$.

Now, give a closed form expession for $A^n$ for integers $n$. Note $A^n$ is also a $2 \times 2$ matrix. You may want to use the function atan2(y, x).

Top left term of $A^n$:

Top right term of $A^n$:

Bottom left term of $A^n$:

Bottom right term of $A^n$:

Recursive Function

For $a, b \in \mathbb R$, consider the function $H$ defined on the non-negative integers $n, k \geq 0$: \[ H(n,k) = \begin{cases} 1 & n = 0, k = 0\\ 1 + aH(k-1, 0) + bH(0, k-1) & n = 0, k \neq 0\\ 1 + aH(n-1, 0) + bH(0, n-1) & n \neq 0, k = 0\\ 1 + aH(n-1, k) + bH(n, k-1) & \text{otherwise}. \end{cases} \] Find a closed form expression for $H(n, k)$ in terms of $n, k, a, b$. You may assume $a + b \neq 1$.

(Hint: Try solving it for $a = 1, b = 1$ first.)

HA-String

A $HA$-string is a string of uppercase characters that can be permuted into a sequence of zero or more $HA$, optionally followed by an $H$. For example, $AHAHAHAH$, $AAAHHHH$ and $H$ are all $HA$-strings, as they can be permuted into $HAHAHAHA$, $HAHAHAH$ and $H$ respectively. However, $HAHAAHAA$ is not a $HA$-string. The empty string is a $HA$-string.

How many $HA$-strings of length $n$ are there for $n \geq 0$? You can type floor(x) to represent $\lfloor x \rfloor$ and ceil(x) to represent $\lceil x \rceil$.

BAH-String

A $BAH$-string is a string of uppercase characters such that its Levenshtein distance from some $HA$-string is $1$. In other words, you can delete, insert, or modify exactly one character from any $BAH$-string such that the resulting string is a $HA$-string.

For example: $GAH$, $HA$ and $AAAGHHH$ are all $BAH$-strings.

How many $BAH$-strings of length $n$ are there for $n \geq 0$?

You may assume that $\binom{a}{b} = 0$ for $a, b \in \mathbb Z$ if $a < b$ or $b < 0$. You may also want to define constants using let (see the tutorial page).

Functional Equation

Suppose that for some fixed $C \in \mathbb R$, an (analytic) function $f(x)$ on $\mathbb{R}$ is characterised by \[ f(f(x)) - f(x) - x = C \] Find the $f(x)$ satisfying this constraint that has the largest $f'(0)$.

Also, how many such functions are there?

n-th Level Derivative

Let $f : \mathbb{R} \to \mathbb{R}$ be differentiable at $x_0$. Let: \[ \begin{align*} f_0(x, h) &= h\\ f_n(x, h) &= f(x + f_{n-1}(x, h)) - f(x) \end{align*} \]

Define the $n$-th level derivative of $f$ to be: \[ \frac{d_n}{dx}f(x) = \lim_{h \to 0} \frac{f_n(x, h)}{h} \] Express $\frac{d_n}{dx}f(x_0)$ in terms of $f'(x_0)$ for $n \geq 0$. You may use $F$ to denote $f'(x_0)$.

(You may be tempted to use L'hopital's rule. However, it requires that $f$ is differentiable on an open interval containing $x_0$, except possibly at $x_0$ itself, which is not a given condition.)

Checkers...?

Suppose you want to arrange red and green checkers on an $8 \times 8$ red and green checkboard in a certain way.

You can place red checkers on green tiles and green checkers on red tiles, but no green checker may be adjacent to a red checker.

How many valid ways are there to arrange the red and green checkers? Note that an empty board also counts as a valid arrangement.

Note: There is no known general formula.

Fibonacci Sum

Recall the Fibonacci numbers $F_n := 1, 1, 2, 3, 5, 8, 13, \cdots$.

Suppose two Fibonacci numbers $x_1, x_2$ are independently drawn at random with uniform probability. What is the probability that $x_1 + x_2$ shares a factor with $69$ other than $1$?

If you do not like the notion of drawing a random integer that could be arbitrarily large, find or bound the probability with the restriction $x_1, x_2 < N$ for any $N$ and take $N \to \infty$.