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.
so that the sum of its edges' lengths is minumum.
find, in terms of , the minimum value of 
for which there is a set of 
distinct positive integers that has sum greater than but every subset of size has sum at most 
with 
equilateral. Let 
be the area of the triangle of sides 
, 
and 
. Show that 
where 
is the circumradius and 
is the volume of the tetrahedron. be circumscribed about circle 
, and let 
be the orthocenter of 
. The circle touches line 
at 
. The tangent to the circle 
at meets at . Let 
be the midpoint of 
, and let the line 
meet again at 
. The tangent to parallel to meets the line 
at 
. Prove that 
is tangent to .
be an equilateral triangle, 
be a point inside it, and 
be the intersections of 
with the sides of . If 
are the midpoints of , 
, 
, show that there is a triangle with sides 
, 
and 
.
be two integers and 
be a positive integer such that 
. Show that ![\[(n-t)(d-t)>\dbinom{n-t+r}{r+1},\]](http://latex.artofproblemsolving.com/e/5/f/e5f9486f980870ff72858b57f7b50cd2aa9d142a.png)
for 
, 
, 
we have ![\[ 3(a+b+c) \ge 8\sqrt[3]{abc} + \sqrt[3]{\frac{a^3+b^3+c^3}{3}}. \]](http://latex.artofproblemsolving.com/a/9/8/a98b6c29802355c491016bcce58228a020423526.png)
Prove that 
When does equality hold?
satisfy that: 
and 
. Find the maximum: 
be positive real numbers satisfying 
. Prove that![\[
3(x^{\frac{2}{3}} + y^{\frac{2}{3}}) \geq 4 + 2x^{\frac{1}{3}}y^{\frac{1}{3}}.
\]](http://latex.artofproblemsolving.com/f/3/c/f3cd114f909f388eacee142d3f90d07f6b149131.png)
be positive real numbers such that 
.![\[
\left( \frac{a + b}{c} \right) \left( \frac{b + c}{a} \right) \left( \frac{c + a}{b} \right) \geq 9.
\]](http://latex.artofproblemsolving.com/7/7/f/77f8bffb0d9b617655c7a8f4a8eb7245d60d8d9f.png)
of the convex quadrilateral 
be such that 
Let 
be the center of the circumcircle of the triangle 
. Prove that 
.
, we obtain 
. Additionally, since PB=PC and PD=PA, triangles PBD and PCA are congruent, leading to 
.
.So from 
. And since 
, we find 
.
be an acute triangle and and 
are the points such that 
, 
and 
, 
. Let be the circumcenter of 
. Then prove that 
. and 
be the second intersection points of 
, 
with . Since 
we have will have that 
. Also from 
we get that 
which implies that 
.
and 
Prove that
, such that 
Prove that :
equality cases : ?
