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2018-05-14 Intermediate

         

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?

This particular game ends on victory for player 2. The moves (and which player made them) are marked.

This particular game ends on victory for player 2. The moves (and which player made them) are marked.

\[\begin{align} 1+1 \times 2 \times 3 \times 4 &= 5^2 \\ 1+2 \times 3 \times 4 \times 5 &= 11^2 \\ 1 + 3 \times 4 \times 5 \times 6 &= 19^2 \end{align}\]

Is 1 plus the product of four consecutive integers always a perfect square?

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?

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.

\[ 012345678910{\color{blue}111}21314\ldots\]

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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