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