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2017-10-16 Intermediate

         

In a \(3 \times 3 \times 3\) tic-tac-toe game, a winning line can go through each layer of the three boards (in addition to the standard winning lines in any 2D plane of a single board). Two players play on this 3-layer board with normal tic-tac-toe rules.

Is it possible to tie in this version of tic-tac-toe?

It is \(\color{red}\text{X}\)'s turn.  \(\color{red}\text{X}\) can win by filling in the Top NW space.  The winning line will go through Top NW, Middle Center, and Bottom SE.

It is \(\color{red}\text{X}\)'s turn. \(\color{red}\text{X}\) can win by filling in the Top NW space. The winning line will go through Top NW, Middle Center, and Bottom SE.

10 identical circles of radius 2 are externally tangential to one another, as shown.

Connecting the centers of adjacent circles gives a 10-sided polygon, which divides the circles into two areas: blue and yellow. The positive difference between the blue and yellow areas is \( k \pi.\)

What is \(k?\)

Does there exist a positive integer \(n\) such that \[\sqrt{2 \times \sqrt{3 \times \sqrt{\dots \sqrt{(n-1)\times {\sqrt{n}}}}}}\ \ {\large{>3}\hspace{0.3mm}?}\]

Is it possible to fill a \(2017 \times 2017\) square grid with \(2017^2\) distinct positive integers such that the product of the numbers in each row and each column is the same?

Helium gas is enclosed in a gas-tight piston which is held in position by the clamping force of an elastic spring. In mechanical equilibrium, the gas pressure \(p = -\frac FA\) corresponds to the tensioning force \(F\) of the spring divided by the area \(A\). The gas is then brought to a temperature of \(T = 600 \,\text{K}\) by rapid heating from room temperature \(T = 300 \,\text{K}\).

What is the final temperature of the gas (in units of Kelvin) after setting a new mechanical balance with the spring?

\(\)
Details and Assumptions:

  • The volume of the gas is \(V = Ax,\) where \(A\) is the cross-sectional area and \(x\) is the piston position.
  • The mechanical force of the spring equals \(F = - k x,\) where \(k\) is the spring constant.
  • The piston does not move during the heating from \(300 \,\text{K}\) to \(600 \,\text{K}\).
  • Helium can be treated as an ideal gas.
  • Energy is transferred only by mechanical work. The piston is perfectly thermally insulated.
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