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.
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 
.
, 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 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 
.
be a positive integer. Prove that there exists a positive integer and distinct primes 
such that if 
denotes the number of positive integers less than or equal to and not divisible by any of , then
be a positive integer. Prove that there exists a positive integer and distinct primes such that if denotes the number of positive integers less than or equal to and not divisible by any of , then
, such that for all 
:![\[ f(x+f(y))=\dfrac{f(x)}{1+f(xy)}\]](http://latex.artofproblemsolving.com/1/4/1/1418fad9fd00b31c38bfbc5657e2a322d66ef450.png)
