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2017-03-20 Intermediate

         

\[ \LARGE 3^{2^{85}} = \, ?\]

Note: Exponent towers are evaluated as \( \large a ^ {b^c} = a^{ \left( b^ c \right) } \).

There is a faulty ATM machine which only allows you to withdraw exactly $306 or deposit exactly $221 each time. If you try different amounts, it would reject your transaction.

You have $1000 in your bank account but $0 on you. You are desperate to get as much money out as possible. Using multiple transactions, what is the maximum net amount of money (in $) that you can obtain?

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Clarification: The answer is the sum of withdrawals minus the sum of deposits.

You would like to maximize the area of an orange square that you fit between the four blue unit squares that lie on a regular grid as shown:

What is the largest area you can achieve?

There are three schools, each with exactly 2017 students. Every student is friends with at least 2018 students from the other two schools.

Does there exist a group of 3 students--one from each of the three schools--that are all pairwise friends?

Note: If \(A\) is friends with \(B,\) then \(B\) is friends with \(A.\)

A cylindrical chamber of cross sectional area \(\SI{4}{\centi\meter\squared}\) is filled with an ideal gas, and a piston of mass \(\SI{2}{\kilo\gram}\) sits at the midpoint of the cylinder. The rest of the chamber has total mass \(M.\)

Now, the piston is pulled very slowly upward. If the temperature of the gas inside the chamber remains constant, find the maximum value of \(M\) \((\)in \(\text{kg})\) such that the chamber can be lifted off the ground in this way. Submit your answer to 2 decimal places.

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Details and Assumptions:

  • There is negligible friction in the system.
  • Atmospheric pressure is \(10^5 \text{ N/m}^2,\) and \(g=10 \text{ m/s}^2.\)
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