2018 AMC 8 Problem 19
Attempt Problem 19 of the 2018 AMC 8 below, then check your answer against the video solution and professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2018 AMC 8 solutions, or check the answer key.
All problems are used with official legal permission of the Mathematical Association of America (MAA).
19.
In a sign pyramid a cell gets a “+” if the two cells below it have the same sign, and it gets a “-” if the two cells below it have different signs. The diagram below illustrates a sign pyramid with four levels. How many possible ways are there to fill the four cells in the bottom row to produce a “+” at the top of the pyramid?
Answer: C
Small hint:
If you know one lower cell and the cell above a pair, the other lower cell is forced.
Big hint:
Work downward from the top, counting how many choices appear at each new row.
Video solution:
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Written solution:
Suppose we have two cells and the cell above them. If we are given the bottom left cell and the top cell, we can always find the bottom right cell as follows:
If the top cell is then the bottom right cell must be the same as the bottom left cell, and if the top cell is the bottom right cell must be the opposite of the bottom left cell.
Now, suppose we are given a row. Then, suppose we choose a value for the cell below and to the left of the leftmost cell in our given row. We then can inductively determine the entire row below our given by first finding the bottom-right cell of the leftmost cell in our row, and using that newly found cell as the bottom-left reference for the second to the left cell in the given row to find its bottom-right counterpart. The process continues on until the row below the given row is fully solved.
Therefore, since we know that the top row has a cell labelled we have choices for the row below, depending on our choice of the bottom-left cell. Similarly, we have choices for the third row, and thus choices for the fourth row. This makes total choices for the bottom row of the sign pyramid.
Thus, the correct answer is C.
Problem 19 in other years
- 1985 AMC 8
- 1986 AMC 8
- 1987 AMC 8
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- 1989 AMC 8
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