Wild tic-tac-toe is like traditional tic-tac-toe except that on each turn, a player may place an X or an O. The first person to get a row, column, or diagonal of three Xs or Os wins the game.
If Wild tic-tac-toe is played optimally, who should win?
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\[\begin{align} 1+1 \times 2 \times 3 \times 4 &= 5^2 \\ 1+2 \times 3 \times 4 \times 5 &= 11^2 \\ 1 + 3 \times 4 \times 5 \times 6 &= 19^2 \end{align}\]
Is 1 plus the product of four consecutive integers always a perfect square?
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This bowl is in the shape of the paraboloid \(z=x^2+y^2\). A ball falls to the bottom such that it touches the lowest point \((0,0,0)\) of the bowl.
What is the maximum possible radius of the ball?
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Two \(\SI{1}{\kilo\gram}\) beads are at the ends of a compressed spring resting on a smooth parabolic wire \(\big(y = x^2\big).\) The spring has a natural length of \(\SI{1}{\meter}\) and a force constant of \(\SI[per-mode=symbol]{5}{\newton\per\meter}.\)
If the system is in static equilibrium, how far away from the \(y\)-axis is each bead (in meters)?
Note: The spring is horizontal and gravity is \(\SI[per-mode=symbol]{10}{\meter\per\second\squared}\) in the \(-y\) direction.
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\[ 012345678910{\color{blue}111}21314\ldots\]
The above is the beginning of all non-negative integers concatenated into a single infinite string.
\(\color{blue}111\) is the first sub-string with 3 identical digits in a row, with different digits preceding and following the run of the identical digits. The first \(\color{blue}1\) of \(\color{blue}111\) comes at position 13 (from left) in the infinite string.
Where is the first such sub-string with 4 identical digits?
Give the position of the first digit of the four.
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