[/quote]
,
,
,
terms is divisible by 
be a positive integer divisible by 3, and let 
be non-negative real numbers satisfying 
.![\[
M_n = \min_{1 \le i \le n} \{x_i x_{3i}\},
\]](http://latex.artofproblemsolving.com/d/b/b/dbbf0f6bf20f506d6cb63357b06ec6c52bebed10.png)
where the indices are taken modulo 
.
be odd natural number and 
be pairwise distinct numbers. Prove that someone can divide the difference of these number into two sets with equal sum.
)
,
,
positive real numbers. Prove that:![\[ \frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a} \geq 4+\frac{8(a-d)^2}{(a+b+c+d)^2}\]](http://latex.artofproblemsolving.com/f/b/5/fb5a179f427615c8bc2da46164ecd9a3484d6625.png)
![$a,b,c\in{[2-\sqrt{3},2+\sqrt{3}]},$](http://latex.artofproblemsolving.com/0/c/9/0c9d55a20ca1a60e4d0cd51e44171923fc44abf1.png)
t![\[2(\frac{a}{b}+\frac{b}{c}+\frac{c}{a})\geq \frac{b}{a}+\frac{c}{b}+\frac{a}{c}+3\]](http://latex.artofproblemsolving.com/4/c/0/4c092e0e8e46d1014b2cc80caee0f5959003c36f.png)
![$a,b,c,d\in{[\frac{1}{2},2]},$](http://latex.artofproblemsolving.com/b/6/9/b69e17212ca1f383bf3f1abf908169d012a8e12b.png)
t![\[2(\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a})\geq \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}+4\]](http://latex.artofproblemsolving.com/9/6/c/96cc857aeceb419f95f7fea36fcd44428fe26d55.png)
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
be an acute scalene triangle with circumcenter 
.
be the feet of the altitudes from 
to 
respectively. Let 
be the circle with diameter 
.
be the intersection of 
and 
.
be the orthocenter of 
be the intersection of 
and 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
.
be the intersection of 
and 
.
be the line through 
and let 
be the reflection of 
is tangent to 
.
and 
and 
are on the same side of 
be midpoint of segment 
be midpoint of segment 
be intersection of 
with 
and 
be the set of vertices of 
be the set of triangles with vertices all in 
of 
be the set of triangles of this type.
be the set of triangles of this type.
,
the area of ![\[\sum_{T \in \mathcal A} S_T \geq \sum_{T \in \mathcal B} S_T.\]](http://latex.artofproblemsolving.com/5/e/d/5ed261e1e812466ea3e78f3b681f6bbdb7ebccd1.png)
and orthocenter 
intersect 
respectively. Lines through 
intersects 
respectively. Point 
is a parallelogram. Prove that lines 
and 
.
![$f(x)\in\mathbb Z[x]$](http://latex.artofproblemsolving.com/2/9/5/2953d049559ec5d4020e6777d965bdfff9ebde0f.png)
have positive integer leading coefficient. Show that there exists infinte positive integer 
such that the number of digit that doesn'r equal to 
is no more than 
is parallel with line 
.
has at least one pair of positive integer solution 
for each positive integer 
.
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 
.
