[b]p1.[/b] $17$ rooks are placed on an $8\times 8$ chess board. Prove that there must be at least one rook that is attacking at least $2$ other rooks.


[b]p2.[/b] In New Scotland there are three kinds of coins: $1$ cent, $6$ cent, and $36$ cent coins. Josh has $99$ of the $36$-cent coins (and no other coins). He is allowed to exchange a $36$ cent coin for $6$ coins of $6$ cents, and to exchange a $6$ cent coin for $6$ coins of $1$ cent. Is it possible that after several exchanges Josh will have $500$ coins?


[b]p3.[/b] Find all solutions $a, b, c, d, e, f, g, h, i$ if these letters represent distinct digits and the following multiplication is correct:
$\begin{tabular}{ccccc}
 &  & a & b & c \\
x &  &  & d & e \\
\hline
 & f & a & c & c \\
+ & g & h & i &  \\
\hline
f & f & f & c & c \\
\end{tabular}$


[b]p4.[/b] Pinocchio rode a bicycle for $3.5$ hours. During every $1$-hour period he went exactly $5$ km. Is it true that his average speed for the trip was $5$ km/h? Explain your reasoning.


[b]p5.[/b] Let $a, b, c$ be odd integers. Prove that the equation $ax^2 + bx + c = 0$ cannot have a rational solution.


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. Note that the concept of trading coins in for others doesn't mean much; it just means that we have $1$-cent, $6$-cent, and $36$-cent coins; the total amount of money is invariant under trades.
2. Let's say that it is possible for Josh to have $500$ coins. We will say he has $x$ $36$-cent coins, $y$ $6$-cent coins, and $500 - x - y$ $1$-cent coins.
3. The total value of these coins is $36 \cdot 99 = 3564$ cents.
4. So we have the equation:
   \[
   36x + 6y + 1(500 - x - y) = 3564
   \]
5. Simplifying the equation:
   \[
   36x + 6y + 500 - x - y = 3564
   \]
   \[
   35x + 5y + 500 = 3564
   \]
   \[
   35x + 5y = 3064
   \]
   \[
   7x + y = 612.8
   \]
6. Since $x$ and $y$ must be integers, the equation $7x + y = 612.8$ is clearly a contradiction because $612.8$ is not an integer.

Therefore, it is impossible for Josh to have $500$ coins.

The final answer is $\boxed{\text{no}}$