[/quote]
,
.
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
beats 
when, when 
.
What is 
satisfy 
.
and the incircle 
of 
at 
,
meets 
at 
and intersects 
at 
,
meets 
at 
.
(different from 
and 
.
.
is situated inside the parallelogram 
such that 
.
.
inscribed in 
of 
be the projection of 
.
that is not on 
intersects 
at 
and 
intersect at 
is an isosceles triangle.
.
on 
respectively. Circumcircle 
of triangle 
.
intersects 
at 
.
cuts 
respectively. Construct parallelogram 
.
are collinear.
a grid of 9*9, we choose a little square in 
is a strictly increasing sequence of positive integers such that the sub-sequences ![\[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\]](http://latex.artofproblemsolving.com/2/6/8/2686c6e4c1110bce322dd876aba37d5273bffe05.png)
are both arithmetic progressions. Prove that the sequence 
is itself an arithmetic progression.
:
meet the sides 
in 
respectively. Show that ![\[ \frac{AP}{PD} = \frac{AF}{FB} + \frac{AE}{EC}. \]](http://latex.artofproblemsolving.com/7/5/9/759482bace109eb619aaea692625f7854faa9fec.png)
meet the segment 
at its midpoint if and only if 
.
Prove that
Let 
Prove that
Let 
Prove that
-
,
,
be positive real numbers such that 
.![$$\sqrt[3]{(a^2+1)(b^2+1)(c^2+1)} \le \left(\frac{a+b+c}{3}\right)^2 + 1.$$](http://latex.artofproblemsolving.com/2/0/7/207934bca53047858a3820f37705171ff0abfd45.png)
Y by
An acute triangle 
is given on the plane. Let the points 
, 
and 
be on the sides 
, 
and 
, respectively. (, and are different from the vertices 
, 
and 
) Also the points 
, 
and 
are taken such that 
is an equilateral hexagon which the opposite sides are parallel. Suppose that the circumcenters of and 
are coincident. Then determine the least possible value of:
Note: 
area of 
is given on the plane. Let the points 
, 
and 
be on the sides 
, 
and 
, respectively. (, and are different from the vertices 
, 
and 
) Also the points 
, 
and 
are taken such that 
is an equilateral hexagon which the opposite sides are parallel. Suppose that the circumcenters of and 
are coincident. Then determine the least possible value of:
Note: 
area of 
This post has been edited 1 time. Last edited by Ege_Saribass, Apr 27, 2025, 11:10 AM
Reason: missing information
Reason: missing information
.
.
,
.
.
and 
.
.
hence 
thus 
.
.
.
via inversion.
,
,
.
and 
and 
.
.
are linear.![$[E'F']$](http://latex.artofproblemsolving.com/b/8/1/b8142d200cf7f09b6a3c009b17fe198a406538ec.png)
intersects 
,
and 
.
,
and 
.
which is a contradiction.
.
and 
.
.
and 
.
and circumradius of the 
.
and 
,
hence 
.
