2030 AIME I
Work in progress
Problem 1
The stupid diagram below shows the circular face of a clock with radius 
cm and a circular disk with radius 
cm externally tangent to the clock face at 
o'clock. The disk has an arrow painted on it, initially pointing in the upward vertical direction. Let the disk roll clockwise around the clock face. At what point on the clock face will the disk be tangent when the arrow is next pointing in the upward vertical direction?
![[asy] size(170); defaultpen(linewidth(0.9)+fontsize(13pt)); draw(unitcircle^^circle((0,1.5),0.5)); path arrow = origin--(-0.13,-0.35)--(-0.06,-0.35)--(-0.06,-0.7)--(0.06,-0.7)--(0.06,-0.35)--(0.13,-0.35)--cycle; for(int i=1;i<=12;i=i+1) { draw(0.9*dir(90-30*i)--dir(90-30*i)); label("$"+(string) i+"$",0.78*dir(90-30*i)); } dot(origin); draw(shift((0,1.87))*arrow); draw(arc(origin,1.5,68,30),EndArrow(size=12)); [/asy]](http://latex.artofproblemsolving.com/1/a/0/1a0357714d7d41aaa5b649c49b5ad0a3505baf5d.png)
Problem 2
Prove that for any positive integer 
![\[\left(k^2\right)!\cdot\prod_{j=0}^{k-1}\frac{j!}{\left(j+k\right)!}\]](http://latex.artofproblemsolving.com/4/e/d/4ed23f598fe1573fe79161056cfcad1ffcb0f1fd.png)
is an integer.
Problem 3
(
) Let 
be a scalene triangle with circumcircle 
and incenter 
. Ray 
meets 
at 
and meets again at 
; the circle with diameter 
cuts again at 
. Lines 
and 
meet at 
, and 
is the midpoint of 
. The circumcircles of 
and 
intersect at points 
and 
. Prove that passes through the midpoint of either 
or 
.
Problem 4
Find the minimum possible value of ![\[\frac{a}{b^3+4}+\frac{b}{c^3+4}+\frac{c}{d^3+4}+\frac{d}{a^3+4}\]](http://latex.artofproblemsolving.com/4/d/2/4d21982ba412c0bb4a7985798e81bc5b01ac0e43.png)
given that 
, 
, 
, 
are nonnegative real numbers such that 
.
