[b]p1.[/b] An airplane travels between two cities. The first half of the distance between the cities is traveled at a constant speed of $600$ mi/hour, and the second half of the distance is traveled at a a constant speed of $900$ mi/hour. Find the average speed of the plane.


[b]p2.[/b] The figure below shows two egg cartons, $A$ and $B$. Carton $A$ has $6$ spaces (cell) and has $3$ eggs. Carton $B$ has $12$ cells and $3$ eggs. Tow cells from the total of $18$ cells are selected at random and the contents of the selected cells are interchanged. (Not that one or both of the selected cells may be empty.)
[img]https://cdn.artofproblemsolving.com/attachments/6/7/2f7f9089aed4d636dab31a0885bfd7952f4a06.png[/img]
(a) Find the number of selections/interchanges that produce a decrease in the number of eggs in cartoon $A$- leaving carton $A$ with $2$ eggs.
(b) Assume that the total number of eggs in cartons $A$ and $B$ is $6$. How many eggs must initially be in carton $A$ and in carton $B$ so that the number of selections/interchanges that lead to an increase in the number of eggs in $A$ equals the number of selections/interchanges that lead to an increase in the number of eggs in $B$.
$\bullet$ In other words, find the initial distribution of $6$ eggs between $A$ and $B$ so that the likelihood of an increase in A equals the likelihood of an increase in $B$ as the result of a selection/interchange. Prove your answer.


[b]p3.[/b] Divide the following figure into four equal parts (parts should be of the same shape and of the same size, they may be rotated by different angles however they may not be disjoint and reconnected).
[img]https://cdn.artofproblemsolving.com/attachments/f/b/faf0adbf6b09b5aaec04c4cfd7ab1d6397ad5d.png[/img]


[b]p4.[/b] Find the exact numerical value of $\sqrt[3]{5\sqrt2 + 7}- \sqrt[3]{5\sqrt2 - 7}$
 (do not use a calculator and do not use approximations).


[b]p5.[/b] The medians of a triangle have length $9$, $12$ and $15$ cm respectively. Find the area of the triangle.


[b]p6. [/b]The numbers $1, 2, 3, . . . , 82$ are written in an arbitrary order. Prove that it is possible to cross out $72$ numbers in such a sway the remaining number will be either in increasing order or in decreasing order.


PS. You should use hide for answers.
**Solution for P5**

Given the lengths of the medians of a triangle are \(9\), \(12\), and \(15\) cm, we need to find the area of the triangle.

1. **Calculate the semi-perimeter of the medians:**
   \[
   S_m = \frac{m_1 + m_2 + m_3}{2} = \frac{9 + 12 + 15}{2} = 18
   \]

2. **Use the formula for the area of a triangle in terms of its medians:**
   The area \(A\) of a triangle with medians \(m_1\), \(m_2\), and \(m_3\) is given by:
   \[
   A = \frac{4}{3} \sqrt{S_m (S_m - m_1) (S_m - m_2) (S_m - m_3)}
   \]
   Substituting the values:
   \[
   A = \frac{4}{3} \sqrt{18 \cdot (18 - 9) \cdot (18 - 12) \cdot (18 - 15)}
   \]

3. **Simplify the expression inside the square root:**
   \[
   A = \frac{4}{3} \sqrt{18 \cdot 9 \cdot 6 \cdot 3}
   \]
   \[
   A = \frac{4}{3} \sqrt{18 \cdot 9 \cdot 6 \cdot 3} = \frac{4}{3} \sqrt{2916}
   \]

4. **Calculate the square root:**
   \[
   \sqrt{2916} = 54
   \]

5. **Calculate the final area:**
   \[
   A = \frac{4}{3} \cdot 54 = 72 \text{ cm}^2
   \]

The final answer is \(\boxed{72 \text{ cm}^2}\)