MathDB

12

Part of 2024 AIME

Problems(2)

Silly Extra Intersection

Source: 2024 AIME I Problem 12

2/2/2024
Define f(x)=∣∣x∣−12∣f(x)=|| x|-\tfrac{1}{2}| and g(x)=∣∣x∣−14∣g(x)=|| x|-\tfrac{1}{4}|. Find the number of intersections of the graphs of y=4 g(f(\sin (2 \pi x)))  \text{ and }  x=4 g(f(\cos (3 \pi y))).
AIMEAIME I
Family of segments

Source: 2024 AIME II P12

2/8/2024
Let O(0,0)O(0,0), A(12,0)A(\tfrac{1}{2},0), and B(0,32)B(0, \tfrac{\sqrt{3}}{2}) be points in the coordinate plane. Let F\mathcal{F} be the family of segments PQ‾\overline{PQ} of unit length lying in the first quadrant with PP on the xx-axis and QQ on the yy-axis. There is a unique point CC on AB‾\overline{AB}, distinct from AA and BB, that does not belong to any segment from F\mathcal{F} other than AB‾\overline{AB}. Then OC2=pqOC^2 = \tfrac{p}{q} where pp and qq are relatively prime positive integers. Find p+qp + q.
AIMEAIME IAIME II