2024 AIME I Problems/Problem 12
Problem
Define and
. Find the number of intersections of the graphs of
Solution 1 (BASH, DO NOT ATTEMPT IF INSUFFICIENT TIME)
If we graph , we see it forms a sawtooth graph that oscillates between
and
(for values of
between
and
, which is true because the arguments are between
and
). Thus by precariously drawing the graph of the two functions in the square bounded by
,
,
, and
, and hand-counting each of the intersections, we get
(and yes, I did use this on the real AIME and it worked)
Note
While this solution might seem unreliable (it probably is), the only parts where counting the intersection might be tricky is near . Make sure to count them as two points and not one, or you'll get
.
See also
2024 AIME I (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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