MathDB

14

Part of 2023 AIME

Problems(2)

Time is ticking

Source: 2023 AIME I Problem 14

2/8/2023
The following analog clock has two hands that can move independently of each other. [asy] unitsize(2cm); draw(unitcircle,black+linewidth(2));
for (int i = 0; i < 12; ++i) { draw(0.9*dir(30*i)--dir(30*i)); } for (int i = 0; i < 4; ++i) { draw(0.85*dir(90*i)--dir(90*i),black+linewidth(2)); } for (int i = 0; i < 12; ++i) { label("\small" + (string) i, dir(90 - i * 30) * 0.75); } draw((0,0)--0.6*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); draw((0,0)--0.4*dir(90),black+linewidth(2),Arrow(TeXHead,2bp)); [/asy] Initially, both hands point to the number 12. The clock performs a sequence of hand movements so that on each movement, one of the two hands moves clockwise to the next number on the clock while the other hand does not move.
Let NN be the number of sequences of 144 hand movements such that during the sequence, every possible positioning of the hands appears exactly once, and at the end of the 144 movements, the hands have returned to their initial position. Find the remainder when NN is divided by 1000.
AMCAIMEAIME I
Cube on a Plane

Source: 2023 AIME II/14

2/17/2023
A cube-shaped container has vertices AA, BB, CC, and DD where AB‾\overline{AB} and CD‾\overline{CD} are parallel edges of the cube, and AC‾\overline{AC} and BD‾\overline{BD} are diagonals of the faces of the cube. Vertex AA of the cube is set on a horizontal plane P\mathcal P so that the plane of the rectangle ABCDABCD is perpendicular to P\mathcal P, vertex BB is 22 meters above P\mathcal P, vertex CC is 88 meters above P\mathcal P, and vertex DD is 1010 meters above P\mathcal P. The cube contains water whose surface is 77 meters above P\mathcal P. The volume of the water is mn\tfrac mn cubic meters, where mm and nn are relatively prime positive integers. Find m+nm+n. [asy] size(250); defaultpen(linewidth(0.6)); pair A = origin, B = (6,3), X = rotate(40)*B, Y = rotate(70)*X, C = X+Y, Z = X+B, D = B+C, W = B+Y; pair P1 = 0.8*C+0.2*Y, P2 = 2/3*C+1/3*X, P3 = 0.2*D+0.8*Z, P4 = 0.63*D+0.37*W; pair E = (-20,6), F = (-6,-5), G = (18,-2), H = (9,8); filldraw(E--F--G--H--cycle,rgb(0.98,0.98,0.2)); fill(A--Y--P1--P4--P3--Z--B--cycle,rgb(0.35,0.7,0.9)); draw(A--B--Z--X--A--Y--C--X^^C--D--Z); draw(P1--P2--P3--P4--cycle^^D--P4); dot("AA",A,S); dot("BB",B,S); dot("CC",C,N); dot("DD",D,N); label("P\mathcal P",(-13,4.5)); [/asy]
geometry3D geometryAMCAIMErectangle