MathDB

22

Part of 2021 AMC 12/AHSME Fall

Problems(2)

Non optimal Tic Tac Toe

Source: AMC 12A #22

11/11/2021
Azar and Carl play a game of tic-tac-toe. Azar places an X in one of the boxes in the 33-by-33 array of boxes, then Carl places an O in one of the remaining boxes. After that, Azar places an X in one of the remaining boxes, and so on until all 99 boxes are filled or one of the players has 33 of their symbols in a row — horizontal, vertical, or diagonal — whichever comes first, in which case that player wins the game. Suppose the players make their moves at random, rather than trying to follow a rational strategy, and that Carl wins the game when he places his third O. How many ways can the board look after the game is over?
<spanclass=′latex−bold′>(A)</span> 36<spanclass=′latex−bold′>(B)</span> 112<spanclass=′latex−bold′>(C)</span> 120<spanclass=′latex−bold′>(D)</span> 148<spanclass=′latex−bold′>(E)</span> 160<span class='latex-bold'>(A)</span>\ 36 \qquad<span class='latex-bold'>(B)</span>\ 112 \qquad<span class='latex-bold'>(C)</span>\ 120 \qquad<span class='latex-bold'>(D)</span>\ 148 \qquad<span class='latex-bold'>(E)</span>\ 160
2021 AMC Fall
Tangent circles to right triangle

Source: 2021 AMC 12B #22

11/17/2021
Right triangle ABCABC has side lengths BC=6BC=6, AC=8AC=8, and AB=10AB=10. A circle centered at OO is tangent to line BCBC at BB and passes through AA. A circle centered at PP is tangent to line ACAC at AA and passes through BB. What is OPOP?
<spanclass=′latex−bold′>(A)</span> 238<spanclass=′latex−bold′>(B)</span> 2910<spanclass=′latex−bold′>(C)</span> 3512<spanclass=′latex−bold′>(D)</span> 7325<spanclass=′latex−bold′>(E)</span> 3<span class='latex-bold'>(A)</span> ~\frac{23}{8}\qquad<span class='latex-bold'>(B)</span> ~\frac{29}{10}\qquad<span class='latex-bold'>(C)</span> ~\frac{35}{12}\qquad<span class='latex-bold'>(D)</span> ~\frac{73}{25}\qquad<span class='latex-bold'>(E)</span> ~3
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