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2017-10-16 Intermediate

         

In a 3×3×33 \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 X\color{#D61F06}\text{X}'s turn. X\color{#D61F06}\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 kπ. k \pi.

What is k?k?

Does there exist a positive integer nn such that 2×3×(n1)×n  >3?\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 2017×20172017 \times 2017 square grid with 201722017^2 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=FAp = -\frac FA corresponds to the tensioning force FF of the spring divided by the area AA. The gas is then brought to a temperature of T=600KT = 600 \,\text{K} by rapid heating from room temperature T=300KT = 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,V = Ax, where AA is the cross-sectional area and xx is the piston position.
  • The mechanical force of the spring equals F=kx,F = - k x, where kk is the spring constant.
  • The piston does not move during the heating from 300K300 \,\text{K} to 600K600 \,\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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