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.
, 
is the midpoint of side 
, and 
and 
the feet of the altitudes from 
and 
, respectively. Let 
be the orthocenter of 
, 
the midpoint of 
, and 
the intersection of 
and . If 
is the intersection of the median 
and the circumcircle of 
, prove that 
.
and 
be positive reals such that 
.Then prove that ![\[a^2+b^2+c^2+a^2b^2+a^2c^2+b^2c^2\ge\frac{15}{16}\]](http://latex.artofproblemsolving.com/1/1/c/11ce50ac265197c872dfb6487ed491ea0d70f42a.png)
such that:
For all real numbers 
.
denote the set of all real numbers. Find all functions 
such that
for all real numbers 
, 
.
be a triangle such that 
. In the circumcircle of this triangle, let 
and be points such that: 
, 
and 
.
be the intersection of lines 
and 
and let 
be the circle that passes through and is tangent to the line 
at 
.
be the intersection point of and 
.
and are collinear.
such that 
and 
for some prime 
and positive integers 
.
be six positive real numbers. Prove that![\[a^2+b^2+c^2+\frac{4(ax+by+cz)\sqrt{ab+bc+ca}}{x+y+z} \geqslant 2(ab+bc+ca).\]](http://latex.artofproblemsolving.com/f/8/0/f808f7857c2150960f2a430837286487666797d8.png)
is inscribed in circle . Let and 
denote its orthocenter and circumcenter, respectively. Let and be the midpoints of sides 
and , respectively. Rays 
and 
meet at 
and 
, respectively. Lines 
and 
meet at 
. Prove that 
.
, such that for any positive integers , ,![\[f(x)^2+y^2\mid\sum_{i=0}^{2023}(xf(x))^{2023-i}\left(f^{(i)}(y)\right)^{2i}.\]](http://latex.artofproblemsolving.com/8/f/2/8f2160b66ce65f929f2b4a21d29d93f2d1fc57ed.png)
Created by Yuxing Ye
be a parallelogram with 
A point is chosen on the extension of ray past 
The circumcircle of 
meets the segment 
again at 
The circumcircle of triangle 
meets the segment 
at 
Prove that lines 
are concurrent.
is such that number 
has exactly 
positive divisors. Prove that GCD 
. Prove that ![$\dfrac{a+b+c}{3}+\dfrac{3}{\frac{1}{a}+\frac{1}{b}+\frac{1}{c}}\geq5\sqrt[3]{\dfrac{abc}{16}}$](http://latex.artofproblemsolving.com/d/1/3/d1387ade631d1af1a4e7304b2b41829507b045b8.png)
where 
and its cyclic permutation. One of my solution is directly guessing 
and 
by homogeneity. But I want a more elegant proof (use AM-GM, Cauchy, Schur, etc).
) stones. On their turn, each player must divide a pile. The player who can make all piles have at most 2 stones wins. Depending on n, determine which player has a winning strategy.
) stones. On their turn, each player must divide a pile. The player who can make all piles have at most 2 stones wins. Depending on n, determine which player has a winning strategy.
