[b]p1.[/b] Insert the pluses between (some of) the digits of $987654321$ to get the  total of $99$.


[b]p2.[/b] $30$ students from five grades donated $40$ toys. It is known that students from the same grade donated equal number of toys, while students from different grades donated different number of toys. How many students donated exactly one toy?


[b]p3.[/b] A nonstop train leaves New-York for Boston at $60$ miles per hour. Another nonstop train leaves Boston for New-York at $40$ miles per hour. How far apart are the trains $1$ hour before they pass each other? You may assume that the railroad is a straight segment.


[b]p4.[/b] A square carpet of the size $4 \times 4$ meters contains $15$ holes (you may assume that the holes are dots). Prove that one can cut out from it a carpet of the size $1 \times 1$ meter which does not contain holes inside.

PS. You should use hide for answers.
### Problem 1:
Insert the pluses between (some of) the digits of \(987654321\) to get the total of \(99\).

1. We need to find a way to insert plus signs between the digits of \(987654321\) such that the sum equals \(99\).
2. One possible way to achieve this is:
   \[
   9 + 8 + 7 + 6 + 5 + 43 + 21 = 99
   \]
3. Let's verify the calculation:
   \[
   9 + 8 = 17
   \]
   \[
   17 + 7 = 24
   \]
   \[
   24 + 6 = 30
   \]
   \[
   30 + 5 = 35
   \]
   \[
   35 + 43 = 78
   \]
   \[
   78 + 21 = 99
   \]
   The sum is indeed \(99\).

The final answer is \( \boxed{ 9 + 8 + 7 + 6 + 5 + 43 + 21 } \).