2030 AIME I
Work in progress
Problem 1
The 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?
Problem 2
Prove that for any positive integer
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 given that
,
,
,
are nonnegative real numbers such that
.