Difference between revisions of "2024 AIME I Problems/Problem 12"

Revision as of 18:08, 3 February 2024

Problem

Define $f(x)=|| x|-\tfrac{1}{2}|$ and $g(x)=|| x|-\tfrac{1}{4}|$ . Find the number of intersections of the graphs of $y=4 g(f(\sin (2 \pi x))) \quad\text{ and }\quad x=4 g(f(\cos (3 \pi y))).$

Graph

https://www.desmos.com/calculator/wml09giaun

Solution 1

If we graph $4g(f(x))$ , we see it forms a sawtooth graph that oscillates between $0$ and $1$ (for values of $x$ between $-1$ and $1$ , which is true because the arguments are between $-1$ and $1$ ). Thus by precariously drawing the graph of the two functions in the square bounded by $(0,0)$ , $(0,1)$ , $(1,1)$ , and $(1,0)$ , and hand-counting each of the intersections, we get $\boxed{385}$

Note

While this solution might seem unreliable (it probably is), the only parts where counting the intersection might be tricky is near $(1,1)$ . Make sure to count them as two points and not one, or you'll get $384$ .

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).

@@ Line 11: / Line 11: @@
 While this solution might seem unreliable (it probably is), the only parts where counting the intersection might be tricky is near <math>(1,1)</math>. Make sure to count them as two points and not one, or you'll get <math>384</math>.
-== Note ==
+== 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).
 ==See also==
 {{AIME box|year=2024|n=I|num-b=11|num-a=13}}
 {{MAA Notice}}

2024 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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