such that 
.
be positive real numbers such that 
and 
,
be a graph colored using 
colors. We say that a vertex is forced if it has neighbors in all the other 
colors.
-
colors such that at least 
of the colors have a forced vertex of that color.
vertices of the same color may be adjacent.
be an acute, scalene triangle, and let 
,
,
be the midpoints of 
,
,
,
intersect ray 
in points 
and 
respectively, and let lines 
and 
intersect in point 
,
,
be a parallelogram with 
.
,
,
intersects diagonal 
at points 
and 
with 
,
,
,
.
,
and 
.
and 
intersect at points 
and 
.
is tangent to 
and 
,
closer to point 
passes through 
and intersecting 
.
,
are collinear, 
,
,
.
.
and 
intersect at 
,
intersect at 
,
and 
intersect at 
.
intersects line 
at 
intersects line 
and 
are collinear in that order. Prove that if lines 
and 
intersect at 
,
.
and 
are written on a blackboard, where 
and 
are relatively prime positive integers. At any point, Evan may pick two of the numbers 
and 
written on the board and write either their arithmetic mean 
or their harmonic mean 
on the board as well. Find all pairs 
such that Evan can write 1 on the board in finitely many steps.
![\[ \dots, \; P_{-3}, \; P_{-2},\; P_{-1},\; P_0,\; P_1,\; P_2,\; P_3,\; \dots \]](http://latex.artofproblemsolving.com/a/c/c/acc097f524dc9fe9bd85e8eb659ea4a75aec475d.png)
in the plane with the following property: For any three distinct integers 
and 
,
,
,
are collinear if and only if 
.
Y by Adventure10, Mango247
Given 
points 
on the 
-plane.
Let
. Denote by 
the distance between 
and the line 
. Let 
.
Answer the following questions:
(1) Express
in terms of 
.
(2) When
moves in the range of 
, express the maximum and minimum value of in terms of 
.
points 
on the 
-plane.Let
. Denote by 
the distance between 
and the line 
. Let 
.Answer the following questions:
(1) Express
in terms of 
.(2) When
moves in the range of 
, express the maximum and minimum value of in terms of 
.
and define 
as the perpendicular distance from point 
to 
.
on 
and 
;
;![$d_{k}^{2} = \frac{[1+\cot^{2}(\theta)]x_{k}^{2} + 2[\tan(\theta)+\cot(\theta)]x_{k}y_{k} + [1+\tan^{2}(\theta)]y_{k}^{2}}{[\tan(\theta)+\cot(\theta)]^2}$](http://latex.artofproblemsolving.com/7/c/2/7c263d679079d4bdd6b703dce356cc8a0e0d9d7c.png)
;![$d_{k}^{2} = \frac{\csc^{2}(\theta)x_{k}^{2} + 2[\tan(\theta)+\cot(\theta)]x_{k}y_{k} + \sec^{2}(\theta)y_{k}^{2}}{[\tan(\theta)+\cot(\theta)]^2} = \cos^{2}(\theta)x_{k}^{2} + 2\sin(\theta)\cos(\theta)x_{k}y_{k} +\sin^{2}(\theta)y_{k}^{2}$](http://latex.artofproblemsolving.com/1/0/4/104f062d859dc750fa96b4633e88e862789be897.png)
;
.
yields the critical values:
or 
or 
;
;
for 
(hence, a maximum) and 
for 
(hence, a minimum). Thus, we ultimately obtain:
;
.
