MathDB

15

Part of 2024 AIME

Problems(2)

Box Inequalities

Source: 2024 AIME I #15

2/2/2024
Let B\mathcal{B} be the set of rectangular boxes that have volume 2323 and surface area 5454. Suppose rr is the least possible radius of a sphere that can fit any element of B\mathcal{B} inside it. Then r2r^{2} can be expressed as pq\tfrac{p}{q}, where pp and qq are relatively prime positive integers. Find p+qp+q.
geometry3D geometryprismAMCAIMEAIME Iinequalities
2002 I #5: Level 2

Source: 2024 AIME II Problem 15

2/8/2024
Find the number of rectangles that can be formed inside a fixed regular dodecagon (1212-gon) where each side of the rectangle lies on either a side or a diagonal of the dodecagon. The diagram below shows three of those rectangles.
[asy] unitsize(40); real r = pi/6; pair A1 = (cos(r),sin(r)); pair A2 = (cos(2r),sin(2r)); pair A3 = (cos(3r),sin(3r)); pair A4 = (cos(4r),sin(4r)); pair A5 = (cos(5r),sin(5r)); pair A6 = (cos(6r),sin(6r)); pair A7 = (cos(7r),sin(7r)); pair A8 = (cos(8r),sin(8r)); pair A9 = (cos(9r),sin(9r)); pair A10 = (cos(10r),sin(10r)); pair A11 = (cos(11r),sin(11r)); pair A12 = (cos(12r),sin(12r)); draw(A1--A2--A3--A4--A5--A6--A7--A8--A9--A10--A11--A12--cycle); filldraw(A2--A1--A8--A7--cycle, mediumgray, linewidth(1.2)); draw(A4--A11); draw(0.365*A3--0.365*A12, linewidth(1.2)); dot(A1); dot(A2); dot(A3); dot(A4); dot(A5); dot(A6); dot(A7); dot(A8); dot(A9); dot(A10); dot(A11); dot(A12); [/asy]
AMCAIMEAIME IIgeometryrectangle