1. A palindrome is a word that reads the same backward as forward. Examples: OMO, radar, and racecar. How many palindromes can you make with the five letters $a, b, c, d$ and $e$ under the conditions:

- each letter may appear at most twice in each palindrome,
- the length of each palindrome is at least 3 letters.

(Any possible combination of letters is considered a word.)
1. A palindrome is fixed once the first half of the word (including the middle letter for an odd length) is chosen, and since each letter can be used at most twice, we can only choose each letter once for the first half. It is clear that a palindrome can have a maximum length of 10 letters under the given conditions. For a palindrome of length five, for example, we have 5 choices for the first letter, 4 choices for the second letter, and 3 choices for the third letter; the fourth and fifth letters are then fixed. This gives us $5 \times 4 \times 3$ possibilities. We can make a table:

lengte aantal  
3 \quad 5 \times 4 \quad 4 \quad 5 \times 4

5 \quad 5 \times 4 \times 3 \quad 6 \quad 5 \times 4 \times 3

7 \quad 5 \times 4 \times 3 \times 2 \quad 8 \quad 5 \times 4 \times 3 \times 2

9 \quad 5 \times 4 \times 3 \times 2 \times 1 \quad 10 \quad 5 \times 4 \times 3 \times 2 \times 1

In total, there are 640 words.

![](https://cdn.mathpix.com/cropped/2024_04_17_4b666a0774ccaad77cdbg-1.jpg?height=471&width=688&top_left_y=1078&top_left_x=381)

Let $c$ be the length of $AB$, let $a$ be the length of $BC$, and let $b$ be the length of $CA$.

Denote the area of triangle $ABC$ by $[ABC]$.

\[
\begin{aligned}
& {[ABC]=\frac{1}{2} a \times h_{A}=\frac{1}{2} b \times h_{B}=\frac{1}{2} c \times h_{C} .} \\
& {[ABC]=[BCP]+[CAP]+[ABP]=\frac{1}{2} a \times PD+\frac{1}{2} b \times PE+\frac{1}{2} c \times PF=} \\
& \frac{1}{2} a \times \frac{1}{3} h_{A}+\frac{1}{2} b \times \frac{1}{4} h_{B}+\frac{1}{2} c \times PF=\frac{7}{12}[ABC]+\frac{1}{2} c \times PF . \text { So } \frac{1}{2} c \times PF=\frac{5}{12}[ABC] . \\
& \text { So } PF=\frac{5}{12} h_{C} .
\end{aligned}
\]