1989 AHSME Problems/Problem 23
Problem
A particle moves through the first quadrant as follows. During the first minute it moves from the origin to . Thereafter, it continues to follow the directions indicated in the figure, going back and forth between the positive x and y axes, moving one unit of distance parallel to an axis in each minute. At which point will the particle be after exactly 1989 minutes?
Solution
Squares of size are successively enclosed between the path and the axes.
It takes minutes to enclose the first square, minutes to enclose the second, minutes to enclose the third, and so on. After odd squares, the particle is on the Y axis; after even squares, the particle is on the X axis.
First we find the highest integer such that . The sum is equal to so the highest value is for which the sum is .
After minutes the 43rd square is enclosed and the particle is at the point . During the 1936th minute it moves up to . At the end of the 1980th minute it has moved right to . After this it moves downward, and at the end of the 1989th minute it is at . The answer is .
See also
1989 AHSME (Problems • Answer Key • Resources) | ||
Preceded by Problem 22 |
Followed by Problem 24 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 | ||
All AHSME Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.