AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
, which is a permutation of the natural numbers (that is, each natural number appears exactly once), such that for every 
, the sum of the first 
terms is divisible by ?
be a positive integer divisible by 3, and let 
be non-negative real numbers satisfying 
. Define![\[
M_n = \min_{1 \le i \le n} \{x_i x_{3i}\},
\]](http://latex.artofproblemsolving.com/d/b/b/dbbf0f6bf20f506d6cb63357b06ec6c52bebed10.png)
where the indices are taken modulo . Determine the maximum possible value of 
.
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)
then ![\[2(\frac{a}{b}+\frac{b}{c}+\frac{c}{a})\geq \frac{b}{a}+\frac{c}{b}+\frac{a}{c}+3\]](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)
then ![\[2(\frac{a}{b}+\frac{b}{c}+\frac{c}{d}+\frac{d}{a})\geq \frac{b}{a}+\frac{c}{b}+\frac{d}{c}+\frac{a}{d}+4\]](http://latex.artofproblemsolving.com/9/6/c/96cc857aeceb419f95f7fea36fcd44428fe26d55.png)
be an acute scalene triangle with circumcenter 
. Let 
be the feet of the altitudes from 
to 
respectively. Let 
be the midpoint of 
. Let 
be the circle with diameter 
. Let 
be the intersection of 
and 
. Let 
be the orthocenter of . Let 
be the intersection of 
and . Let 
be the lines through 
tangent to 
respectively. Let 
be the intersection of 
and 
. Let 
be the intersection of 
and 
. Let 
be the line through parallel to 
and let 
be the reflection of across . Prove that 
is tangent to 
.
be triangle with point 
and 
on 
and 
are on the same side of 
be midpoint of segment and 
be midpoint of segment 
be intersection of 
with 
and 
be a convex polygon with centroid , and let 
be the set of vertices of . Let 
be the set of triangles with vertices all in . We sort the elements 
of into the following three types: lies in the strict interior of ; let 
be the set of triangles of this type. lies in the strict exterior of ; let 
be the set of triangles of this type. lies on the boundary of .
, denote by 
the area of . Prove that ![\[\sum_{T \in \mathcal A} S_T \geq \sum_{T \in \mathcal B} S_T.\]](http://latex.artofproblemsolving.com/5/e/d/5ed261e1e812466ea3e78f3b681f6bbdb7ebccd1.png)
be triangle with circumcenter 
and orthocenter . 
intersect again at 
respectively. Lines through parallel to 
intersects at 
respectively. Point such that 
is a parallelogram. Prove that lines 
and are concurrent at a point on 
.
![$f(x)\in\mathbb Z[x]$](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 
be a triangle and let be its circumcenter and its incenter. be the radical center of its three mixtilinears and let be the isogonal conjugate of . be the Gergonne point of the triangle .
is parallel with line 
.
has at least one pair of positive integer solution 
for each positive integer 
.
given points 
and 
which are symmetric to its circumcenter 
with respect to 
and 
and the straight line 
which contains its altitude to 
meets the line 
at the orthocenter H. The circle through H centered meets the circle through H centered 
at C. The circle 
meets the perp line from C to at B.
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
