[b]p1.[/b] A [i]multimagic [/i] square is a $3 \times 3$ array of distinct positive integers with the property that the product of the $3$ numbers in each row, each column, and each of the two diagonals of the array is always the same.
(a) Prove that the numbers $1, 2, 3, . . . , 9$ cannot be used to form a multimagic square.
(b) Give an example of a multimagic square.


[b]p2.[/b] A sequence $a_1, a_2, a_3, ... , a_n$ of real numbers is called an arithmetic progression if $$a_1 - a_2 = a_2 - a_3 = ... = a_{n-1} - a_n.$$
Prove that there exist distinct positive integers $n_1, n_2, n_3, ... , n_{2014}$ such that $$\frac{1}{n_1},\frac{1}{n_2}, ... ,\frac{1}{n_{2014}}$$ is an arithmetic progression.


[b]p3.[/b] Let $\lfloor x \rfloor$  be the largest integer that is less than or equal to $x$. For example,  $\lfloor 3.9 \rfloor  = 3$ and  $\lfloor 4\rfloor  = 4$. Determine (with proof) all real solutions of the equation $$x^2 - 25 \lfloor x\rfloor + 100 = 0.$$


[b]p4.[/b] An army has $10$ cannons and $8$ carts. Each cart can carry at most one cannon. It takes one day for a cart to cross the desert. What is the least number of days that it takes to get the cannons across the desert? (Cannons can be left part way and picked up later during the procedure.) Prove that the amount of time that your solution requires to move the cannons across the desert is the smallest possible.


[b]p5.[/b] Let $C$ be a convex polygon with $4031$ sides. Let $p$ be the length of its perimeter and let $d$ be the sum of the lengths of its diagonals. Show that $$\frac{d}{p}> 2014.$$


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the equation \( x^2 - 25 \lfloor x \rfloor + 100 = 0 \), we need to consider the properties of the floor function and the quadratic equation.

1. **Express the equation in terms of \( \lfloor x \rfloor \):**
   \[
   x^2 - 25 \lfloor x \rfloor + 100 = 0
   \]
   Let \( \lfloor x \rfloor = n \), where \( n \) is an integer. Then the equation becomes:
   \[
   x^2 - 25n + 100 = 0
   \]

2. **Solve for \( x \) in terms of \( n \):**
   \[
   x^2 = 25n - 100
   \]
   \[
   x = \pm \sqrt{25n - 100}
   \]

3. **Determine the range of \( x \):**
   Since \( n \leq x < n+1 \), we have:
   \[
   n \leq \sqrt{25n - 100} < n+1
   \]
   and
   \[
   n \leq -\sqrt{25n - 100} < n+1
   \]
   The second inequality \( n \leq -\sqrt{25n - 100} \) is not possible because \( -\sqrt{25n - 100} \) is negative and \( n \) is non-negative.

4. **Solve the inequality \( n \leq \sqrt{25n - 100} < n+1 \):**
   \[
   n \leq \sqrt{25n - 100}
   \]
   Squaring both sides:
   \[
   n^2 \leq 25n - 100
   \]
   \[
   n^2 - 25n + 100 \leq 0
   \]
   Solving the quadratic inequality:
   \[
   n = \frac{25 \pm \sqrt{625 - 400}}{2} = \frac{25 \pm 15}{2}
   \]
   \[
   n = 20 \quad \text{or} \quad n = 5
   \]
   Therefore, \( 5 \leq n \leq 20 \).

5. **Solve the inequality \( \sqrt{25n - 100} < n+1 \):**
   \[
   25n - 100 < (n+1)^2
   \]
   \[
   25n - 100 < n^2 + 2n + 1
   \]
   \[
   n^2 - 23n + 101 > 0
   \]
   Solving the quadratic inequality:
   \[
   n = \frac{23 \pm \sqrt{529 - 404}}{2} = \frac{23 \pm 5}{2}
   \]
   \[
   n = 14 \quad \text{or} \quad n = 9
   \]
   Therefore, \( n < 9 \) or \( n > 14 \).

6. **Combine the inequalities:**
   \[
   5 \leq n < 9 \quad \text{or} \quad 14 < n \leq 20
   \]

7. **Check the valid values of \( n \):**
   The valid values of \( n \) are \( 5, 6, 7, 8, 15, 16, 17, 18, 19, 20 \).

8. **Find the corresponding \( x \) values:**
   For each valid \( n \):
   \[
   x = \sqrt{25n - 100}
   \]
   \[
   x = \sqrt{25 \times 5 - 100} = \sqrt{25} = 5
   \]
   \[
   x = \sqrt{25 \times 20 - 100} = \sqrt{400} = 20
   \]
   \[
   x = \sqrt{25 \times 18 - 100} = \sqrt{350}
   \]
   \[
   x = \sqrt{25 \times 19 - 100} = \sqrt{375}
   \]

The final answer is \( \boxed{ x = 5, 20, \sqrt{350}, \sqrt{375} } \).