2024 AIME I Problems/Problem 12

Problem

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

Solution 1 (BASH, DO NOT ATTEMPT IF INSUFFICIENT TIME)

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 $\boxed{384}$

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 $383$ .

2024 AIME I (Problems • Answer Key • Resources)
Preceded by Problem 11	Followed by Problem 13
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2024 AIME I Problems/Problem 12

Contents

Problem

Solution 1 (BASH, DO NOT ATTEMPT IF INSUFFICIENT TIME)

Note

See also