Let $a, b, c, d$ be odd positive integers and pairwise coprime. For a positive integer $n$, let $$f(n) = \left[\frac{n}{a} \right]+\left[\frac{n}{b}\right]+\left[\frac{n}{c}\right]+\left[\frac{n}{d}\right]$$Prove that $$\sum_{n=1}^{abcd}(-1)^{f(n)}=1$$

For a positive integer $n(\geq 2)$, there are $2n$ candies. Alice distributes $2n$ candies into $4n$ boxes $B_1, B_2, \dots, B_{4n}.$ Bob checks the number of candies that Alice puts in each box. After this, Bob chooses exactly $2n$ boxes $B_{k_1}, B_{k_2}, \dots, B_{k_{2n}}$ out of $4n$ boxes that satisfy the following condition, and takes all the candies. (Condition) $k_i - k_{i - 1}$ is either $1$ or $3$ for each $i = 1, 2, \dots, 2n$, and $k_{2n} = 4n$. ($k_0 = 0$) Alice takes all the candies in the $2n$ boxes that Bob did not choose. If Alice and Bob both use their best strategy to take as many candies as possible, how many candies can Alice take?

Find the smallest real number $p(\leq 1)$ that satisfies the following condition. (Condition) For real numbers $x_1, x_2, \dots, x_{2024}, y_1, y_2, \dots, y_{2024}$, if $0 \leq x_1 \leq x_2 \leq \dots \leq x_{2024} \leq 1$, $0 \leq y_1 \leq y_2 \leq \dots \leq y_{2024} \leq 1$, $\displaystyle \sum_{i=1}^{2024}x_i = \displaystyle \sum_{i=1}^{2024}y_i = 2024p$, then the inequality $\displaystyle \sum_{i=1}^{2024}x_i(y_{2025-i}-y_{2024-i}) \geq 1 - p$ holds.

For a triangle $ABC$, $O$ is the circumcircle and $D$ is a point on ray $BA$. $E$ and $F$ are points on $O$ so that $DE$ and $DF$ are tangent to $O$ and $EF$ cuts $AC$ at $T(\neq C)$. $P(\neq B,C)$ is a point on the arc $BC$ not containing $A$, and $DP$ cuts $O$ at $Q (\neq P)$. Let $BQ$ and $DT$ meets on $X (\neq Q)$, and $PT$ cuts $O$ at $Y (\neq P)$. Prove that $C,X,Y$ are collinear.

A positive integer $n (\ge 4)$ is given. Let $a_1, a_2, \cdots ,a_n$ be $n$ pairwise distinct positive integers where $a_i \le n$ for all $1 \le i \le n$. Determine the maximum value of $$\sum_{i=1}^{n}{|a_i - a_{i+1} + a_{i+2} - a_{i+3}|}$$where all indices are modulo $n$

Prove that there exists a positive integer $K$ that satisfies the following condition. Condition: For any prime $p > K$, the number of positive integers $a \le p$ that $p^2 \mid a^{p-1} - 1$ is less than $\frac{p}{2^{2024}}$