MathDB

9

Part of 2015 AIME Problems

Problems(2)

Recursions going to zero

Source: 2015 AIME I Problem 9

3/20/2015
Let SS be the set of all ordered triples of integers (a1,a2,a3)(a_1,a_2,a_3) with 1≤a1,a2,a3≤101 \le a_1,a_2,a_3 \le 10. Each ordered triple in SS generates a sequence according to the rule an=an−1⋅∣an−2−an−3∣a_n=a_{n-1}\cdot | a_{n-2}-a_{n-3} | for all n≥4n\ge 4. Find the number of such sequences for which an=0a_n=0 for some nn.
AMCAIMEAIME I
Solid Cube Placed Within a Cylinder with Water

Source: 2015 AIME 2 Problem 9

3/26/2015
A cylindrical barrel with radius 44 feet and height 1010 feet is full of water. A solid cube with side length 88 feet is set into the barrel so that the diagonal of the cube is vertical. The volume of water thus displaced is vv cubic feet. Find v2v^2.
[asy] import three; import solids; size(5cm); currentprojection=orthographic(1,-1/6,1/6);
draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,360)),white,nolight);
triple A =(8*sqrt(6)/3,0,8*sqrt(3)/3), B = (-4*sqrt(6)/3,4*sqrt(2),8*sqrt(3)/3), C = (-4*sqrt(6)/3,-4*sqrt(2),8*sqrt(3)/3), X = (0,0,-2*sqrt(2));
draw(X--X+A--X+A+B--X+A+B+C); draw(X--X+B--X+A+B); draw(X--X+C--X+A+C--X+A+B+C); draw(X+A--X+A+C); draw(X+C--X+C+B--X+A+B+C,linetype("2 4")); draw(X+B--X+C+B,linetype("2 4"));
draw(surface(revolution((0,0,0),(-2,-2*sqrt(3),0)--(-2,-2*sqrt(3),-10),Z,0,240)),white,nolight); draw((-2,-2*sqrt(3),0)..(4,0,0)..(-2,2*sqrt(3),0)); draw((-4*cos(atan(5)),-4*sin(atan(5)),0)--(-4*cos(atan(5)),-4*sin(atan(5)),-10)..(4,0,-10)..(4*cos(atan(5)),4*sin(atan(5)),-10)--(4*cos(atan(5)),4*sin(atan(5)),0)); draw((-2,-2*sqrt(3),0)..(-4,0,0)..(-2,2*sqrt(3),0),linetype("2 4")); [/asy]
geometry3D geometryAIME