Santa's Christmas tree looks like an acute-angled scalene (i.e. non-isosceles) triangle $ABC$. Let $\omega$ be the circumcircle of $\triangle{ABC}$. $H$ and $O$ are the orthocentre and circumcentre of $\triangle{ABC}$ respectively. $AH$ meets $\omega$ again at $D$. Suppose $OH$ meets line segment $BC$ at $E$. $F$ is the circumcenter of $\triangle{BOH}$. Prove that $BF$ and $DE$ meet at $\omega$.

Santa has to deliver presents to infinitely many cities around the world. He labels the cities with positive integers starting from $1$ and every positive integer has a corresponding city. If Santa is in city $n$, he can take a reindeer sleigh ride and travel to a city labeled by numbers which can be represented as sum of distinct positive divisors of $n$. For example, if Santa is in city $9$, he can travel to cities $1, 3$, $4 (= 1 + 3)$, 9, $10 (= 1 + 9)$, $12 (= 3 + 9)$ or $13 (= 1 + 3 + 9)$, where $1, 3, 9$ are positive divisors of $9$. Suppose $m, k$ are positive integers, prove or disprove the following statements. (a) If $k > m > 1$, then Santa can travel from $m$ to $k$ with at most $2\lfloor \log_2 k\rfloor + 1$ rides. (b) If $m > k$, then Santa can travel from $m$ to $k$ with at most $\lfloor \log_2 k\rfloor + 2$ rides.

Santa asks Rudolph the Red-Nosed Reindeer and Frosty the Snowman to play a game. The winner can get an extra Christmas present, which is a cute Christmas frog! The rules of the game are as follows: Santa writes the following expression on the blackboard: $$x^{2024} +\_ \,\, x^{2023} + \_ \,\, x^{2022} + \_ \,\, x^{2021} + ... + \_ \,\, x^2 + \_ \,\, x +\_ \,\, $$Santa will first choose a positive integer $n \le 2024$. Rudolph will then choose $n$ consecutive blanks $(\_)$. Frosty will fill in the chosen blanks with real numbers. Finally Rudolph will fill in all the remaining blanks. If the resulting polynomial has $2024$ non-zero real roots, then Rudolph wins. Otherwise, Frosty wins. Find the value(s) of $n$ such that Rudolph has a winning strategy and the value(s) of $n$ such that Frosty has a winning strategy. What are their respective winning strategies?

Let $x, m, a, s$ be positive real numbers such that $xma+mas+asx+sxm =\sqrt3$. Prove that $$\frac{1}{\sqrt{x^2 + m^2 + a^2}}+\frac{1}{\sqrt{m^2 + a^2 + s^2}}+\frac{1}{\sqrt{a^2 + s^2 + x^2}} +\frac{1}{\sqrt{s^2 + x^2 + m^2}} \le \frac{1}{xmas}$$

Santa has to deliver Christmas presents to children in a town. The town is rectangular in shape and it can be divided into square regions. After the division, it looks like a grid with $101$ columns and infinitely many rows. Santa labeles the square regions with consecutive integers starting from $1$, so that the square on the $m^{th}$ row and $n^{th}$ column is labeled as $101m + n - 101$. Grinch hates Christmas, he plans to stop Santa by placing mischievous little monsters who love stealing Christmas presents in regions labeled with odd prime numbers. These regions are called “dangerous regions”. Those regions without monsters are called “safe regions”. Santa can move from one region to another region only if they share a common side. Suppose Santa wants to move from a dangerous region $p$ to another dangerous region $q$ (where $p$ and $q$ are odd primes), and he wants all the regions in between his way to be safe regions, is it always possible for Santa to do so? Prove your claim.

Frosty the Snowman has a $\ell \times \ell$ square backyard. He plans to put all the Christmas presents delivered by Santa into the backyard. The presents prepared by Santa are packed in the shape of a square with variable sizes. Frosty does not know the number nor the sizes of the presents before he receives it, but he does know that the sum of areas of the presents is $n$. After he receives one present, he will put it into the backyard, and then Santa will give him another present. This process continues until all the presents are delivered to Frosty. Since Frosty is lazy, once the presents are put into the backyard, he will not move them anymore. Frosty hopes that all the presents can be put into the backyard without overlapping, therefore he will have to arrange them wisely. (a) Prove that if $\frac{\ell^2}{ 2} < n \le \ell^2$, there does not exist a strategy such that Frosty’s goal can always be achieved. (b) Prove that if $0 < n \le \frac{\ell^2}{ 4}$ , there exists a strategy such that Frosty’s goal can always be achieved.