[/quote]
,
.
such that for all any 
and 
.
Is there any generalizations?
such that 
.
,
,
be non-negative real numbers, no two of which are zero. Such that 
Prove that :
be a convex quadrilateral and points 
and 
on sides 
such that![\[\tfrac{AB}{AE}=\tfrac{CD}{DF}=n\]](http://latex.artofproblemsolving.com/b/4/2/b42cd6305fab848a60c79689f17bdb55e1ba0d31.png)
is the area of 
show that 
and can teleport to other Cartesian coordinates with only 
of 
moves: 
,
when 
,
,
when 
.
be any polynomial with positive integer coefficients that passes through 
.
be a sequence of positive integers. Let 
be the number of 3-element subsequences 
with 
,
and 
.
,
is the midpoint of 
.
is an arbitrary point on 
.
be the intersection of 
and 
.
intersects with the circumcircle of 
,
.
is the circumcenter of triangle 
.
.
in the expansion of 
table. A cannon can fire at any 
cells of the board after that the tank will move to one of the adjacent cells (by side). Then the process is repeated. Find the smallest value of 
.
meets line 
,
such that 
.
intersects 
,
at point 
is tangent to 
.
Y by Resolut1on07, Rounak_iitr, GeoKing, pomodor_ap, Kizaruno, ehuseyinyigit
Given 
. Let the perpendicular line from 
to 
meets 
at points 
, respectively, and the foot from 
to 
is 
. 
intersects line 
at 
, 
intersects line at 
, and lines 
intersect at 
.
Prove that
is tangent to 
.
. Let the perpendicular line from 
to 
meets 
at points 
, respectively, and the foot from 
to 
is 
. 
intersects line 
at 
, 
intersects line at 
, and lines 
intersect at 
.Prove that
is tangent to 
.Attachments:
This post has been edited 1 time. Last edited by Itoz, Apr 18, 2025, 11:53 AM
to 
,
and 
.
and 
meet at a point 
is the tangent point. The rest is easy
