[u]Round 1[/u]


[b]p1.[/b] Six pirates – Captain Jack and his five crewmen – sit in a circle to split a treasure of $99$ gold coins. Jack must decide how many coins to take for himself and how many to give each crewman (not necessarily the same number to each). The five crewmen will then vote on Jack's decision. Each is greedy and will vote “aye” only if he gets more coins than each of his two neighbors. If a majority vote “aye”, Jack's decision is accepted. Otherwise Jack is thrown overboard and gets nothing. What is the most coins Captain Jack can take for himself and survive?


[b]p2[/b]. Rose and Bella take turns painting cells red and blue on an infinite piece of graph paper. On Rose's turn, she picks any blank cell and paints it red. Bella, on her turn, picks any blank cell and paints it blue. Bella wins if the paper has four blue cells arranged as corners of a square of any size with sides parallel to the grid lines. Rose goes first. Show that she cannot prevent Bella from winning.
[img]https://cdn.artofproblemsolving.com/attachments/d/6/722eaebed21a01fe43bdd0dedd56ab3faef1b5.png[/img]


[b]p3.[/b] A $25\times 25$ checkerboard is cut along the gridlines into some number of smaller square boards. Show that the total length of the cuts is divisible by $4$. For example, the cuts shown on the picture have total length $16$, which is divisible by $4$.
[img]https://cdn.artofproblemsolving.com/attachments/c/1/e152130e48b804fe9db807ef4f5cd2cbad4947.png[/img]


[b]p4.[/b] Each robot in the Martian Army is equipped with a battery that lasts some number of hours. For any two robots, one's battery lasts at least three times as long as the other's. A robot works until its battery is depleted, then recharges its battery until it is full, then goes back to work, and so on. A battery that lasts $N$ hours takes exactly $N$ hours to recharge. Prove that there will be a moment in time when all the robots are recharging (so you can invade the planet).


[b]p5.[/b] A casino machine accepts tokens of $32$ different colors, one at a time. For each color, the player can choose between two fixed rewards. Each reward is up to $\$10$ cash, plus maybe another token. For example, a blue token always gives the player a choice of getting either $\$5$ plus a red token or $\$3$ plus a yellow token; a black token can always be exchanged either for $\$10$ (but no token) or for a brown token (but no cash). A player may keep playing as long as he has a token. Rob and Bob each have one white token. Rob watches Bob play and win $\$500$. Prove that Rob can win at least $\$1000$.
[img]https://cdn.artofproblemsolving.com/attachments/6/6/e55614bae92233c9b2e7d66f5f425a18e6475a.png
[/img]


[u]Round 2[/u]


[b]p6.[/b] The sum of $2015$ rational numbers is an integer. The product of every pair of them is also an integer. Prove that they are all integers.
(A rational number is one that can be written as $m/n$, where $m$ and $n$ are integers and $n\ne 0$.)


[b]p7.[/b] An $N \times N$ table is filled with integers such that numbers in cells that share a side differ by at most $1$. Prove that there is some number that appears in the table at least $N$ times. For example, in the $5 \times 5$ table below the numbers $1$ and $2$ appear at least $5$ times.
[img]https://cdn.artofproblemsolving.com/attachments/3/8/fda513bcfbe6834d88fb8ca0bfcdb504d8b859.png[/img]


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
To solve the problem, we need to show that the total length of the cuts in a \(25 \times 25\) checkerboard, when cut into smaller square boards, is divisible by 4.

1. **Understanding the Problem:**
   - A \(25 \times 25\) checkerboard has 25 rows and 25 columns.
   - We need to cut this checkerboard along the gridlines into smaller square boards.
   - We need to show that the total length of these cuts is divisible by 4.

2. **General Approach:**
   - Each cut along the gridlines can be either horizontal or vertical.
   - The length of each cut is measured in terms of the number of unit squares it crosses.

3. **Counting the Cuts:**
   - Consider a general \(N \times N\) checkerboard.
   - To cut an \(N \times N\) checkerboard into smaller squares, we need to make cuts along the gridlines.
   - For an \(N \times N\) checkerboard, there are \(N-1\) horizontal cuts and \(N-1\) vertical cuts.

4. **Length of Each Cut:**
   - Each horizontal cut spans \(N\) unit squares.
   - Each vertical cut also spans \(N\) unit squares.

5. **Total Length of Cuts:**
   - The total length of the horizontal cuts is \((N-1) \times N\).
   - The total length of the vertical cuts is \((N-1) \times N\).
   - Therefore, the total length of all cuts is:
     \[
     2 \times (N-1) \times N
     \]

6. **Divisibility by 4:**
   - We need to show that \(2 \times (N-1) \times N\) is divisible by 4.
   - For \(N = 25\):
     \[
     2 \times (25-1) \times 25 = 2 \times 24 \times 25 = 1200
     \]
   - Since 1200 is divisible by 4, the total length of the cuts in a \(25 \times 25\) checkerboard is divisible by 4.

Conclusion:
\[
\boxed{1200}
\]