[b]p1.[/b] Find the largest integer which is a factor of all numbers of the form $n(n +1)(n + 2)$ where $n$ is any positive integer with unit digit $4$. Prove your claims.


[b]p2.[/b] Each pair of the towns $A, B, C, D$ is joined by a single one way road. See example. Show that for any such arrangement, a salesman can plan a route starting at an appropriate town that: enables him  to call on a customer in each of the towns.

Note that it is not required that he return to his starting point.
[img]https://cdn.artofproblemsolving.com/attachments/6/5/8c2cda79d2c1b1c859825f3df0163e65da761b.png[/img]


[b]p3.[/b]  $A$ and $B$ are two points on a circular race track . One runner starts at $A$ running counter clockwise, and, at the same time, a second runner starts from $B$ running clockwise. They meet first $100$ yds from A, measured along the track. They meet a second time at $B$ and the third time at $A$. Assuming constant speeds, now long is the track?


[b]p4.[/b] $A$ and $B$ are points on the positive $x$ and positive $y$ axis, respectively, and $C$ is the point $(3,4)$. Prove that the perimeter of $\vartriangle ABC$ is greater than $10$.

Suggestion: Reflect!!


[b]p5.[/b] Let $A_1,A_2,...,A_8$ be a permutation of the integers $1,2,...,8$ so chosen that the eight sums $9 + A_1$, $10 + A_2$, $...$, $16 + A_8$ and the eight differences $9 -A_1$ , $10 - A_2$, $...$, $16 - A_8$ together comprise $16$ different numbers.
Show that the same property holds for the eight numbers in reverse order. That is, show that the $16$ numbers $9 + A_8$, $10 + A_7$, $...$, $16 + A_1$ and $9 -A_8$ , $10 - A_7$, $...$, $16 - A_1$ are also pairwise different.


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. To find the largest integer which is a factor of all numbers of the form \( n(n + 1)(n + 2) \) where \( n \) is any positive integer with unit digit \( 4 \), we need to analyze the divisibility properties of the product \( n(n + 1)(n + 2) \).

   - **Divisibility by 2**: Since \( n \) has a unit digit of 4, \( n \) is even. Therefore, \( n \) is divisible by 2.
   - **Divisibility by 3**: Among any three consecutive integers, one of them is always divisible by 3. Hence, \( n(n + 1)(n + 2) \) is divisible by 3.
   - **Divisibility by 4**: Since \( n \) is even, \( n \) or \( n + 2 \) is divisible by 4. Therefore, \( n(n + 1)(n + 2) \) is divisible by 4.
   - **Divisibility by 5**: Since \( n \) has a unit digit of 4, \( n + 1 \) has a unit digit of 5, which means \( n + 1 \) is divisible by 5. Therefore, \( n(n + 1)(n + 2) \) is divisible by 5.

   Combining these factors, the largest integer which is a factor of all numbers of the form \( n(n + 1)(n + 2) \) is:
   \[
   2 \times 3 \times 4 \times 5 = 120
   \]

   Therefore, the largest integer which is a factor of all such numbers is \( \boxed{120} \).