29th Philippine Mathematical Olympiad

Hi Robols Math community! I have made this thread to hopefully spark some discussion about the ongoing 29th PMO. As of this thread’s creation, the 29th PMO’s qualifying stage has concluded, with the area stage qualifiers to be announced in the near future. Problem discussions would be welcome! To start discussions, here are the part III problems from the qualifying stage. It would be interesting to see your solutions! (Note: The diagrams will be omitted, though they can be seen in the official Facebook post of the questions + answer sheet.) PART III The answer to each item is an integer from 1 to 999. Each correct answer is worth five points. \begin{enumerate} \item TACOCAT, CTC, and AA are palindromes since they read the same forward and backward. How many palindromes of at least one letter can be formed using the letters of TACOCAT, each of A, C, and T used at most twice and O used at most once? \item A 60-minute timer uses a seven-segment display to count up from 00:00 to 59:59. The ten digits are displayed below on the left, and an example on the right shows that at 01:42, there are $6+2+4+5=17$ segments turned on. At each displayed time from 00:00 through 59:59, count the lit segments in the four digits, excluding the colon, to form a list of 3600 numbers. Let $N$ be the average of the numbers in the list. What is $5N$? \item In triangle $ABC$ with circumcenter $O$, $AB=25\sqrt{2}$, $BC=40$, and $CA=35\sqrt{2}$. Let $E$ and $F$ be the feet of the altitudes from $B$ and $C$ to $AC$ and $AB$, respectively. If $M$ is the midpoint of $BC$ and $P$ is the intersection of $AO$ and $EF$, what is the area of triangle $PMO$? \item What is the largest composite integer $n\geq 100$ for which $(n-100)!$ is not divisible by $n$? \item How many positive integers $n\leq 625$ have the property that $n!$ ends in exactly $\left\lfloor\frac{n}{4}\right\rfloor$ zeroes? \item Triangle $ABC$ has sides $AB=15$, $BC=13$, $CA=4$. Let $P$ be a point in the interior of the triangle such that $\angle PAB=\angle PBC=\angle PCA=\theta$. The value of $\tan\theta$ can be written in the form $a/b$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$? \item Let $N$ be the largest positive integer for which there exist four pairwise distinct positive integers $a$, $b$, $c$, and $d$ such that $N=ab=cd$ and $|a-b|\leq |c-d|\leq 20$. What are the first three digits (from the left) in the decimal representation of $N$? \item Four married couples sit together around a square table. The table has eight seats, with two seats per side. Let $N$ be the number of possible seating arrangements such that no one sits on the same side as their spouse. What are the first three digits (from the left) of $N$? (Note that two arrangements are considered identical if one is a $90^\circ$, $180^\circ$, or $270^\circ$ rotation of the other.) \end{enumerate}