2024 AIME I Problems/Problem 12
Problem
Define and
. Find the number of intersections of the graphs of
Graph
https://www.desmos.com/calculator/wml09giaun
Solution 1
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
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
.
Note 1
The answer should be 385 since there are 16 intersections in each of 24 smaller boxes of dimensions 1/6 x 1/4 and then another one at the corner (1,1).
Solution 2
We will denote for simplicity. Denote
as the first equation and
as the graph of the second. We notice that both
and
oscillate between 0 and 1. The intersections are thus all in the square
,
,
, and
. Every
wave going up and down crosses every
wave. Now, we need to find the number of times each wave touches 0 and 1.
We notice that occurs at
, and
occurs at
. A sinusoid passes through each point twice during each period, but it only passes through the extrema once.
has 1 period between 0 and 1, giving 8 solutions for
and 9 solutions for
, or 16 up and down waves.
has 1.5 periods, giving 12 solutions for
and 13 solutions for
, or 24 up and down waves. This amounts to
intersections.
However, we have to be very careful when counting around . At this point,
has an infinite downwards slope and
is slanted, giving us an extra intersection; thus, we need to add 1 to our answer to 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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.