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 triangle and let 
be the midpoint of 
. A circle 
passing through 
and meets the sides 
and 
at points 
and 
respectively. Let 
be the point such that 
is a parallelogram. Suppose that lies on the circumcircle of . Determine all possible values of 
.
be a triangle with circumcircle 
. Let 
and 
respectively denote the midpoints of the arcs and that do not contain the third vertex. Let 
denote the midpoint of arc 
(the arc including 
). Let 
be the incenter of . Let 
be the circle that is tangent to and internally tangent to at , and let 
be the circle that is tangent to and internally tangent to at . Show that the line 
, and the lines through the intersections of and , meet on .
be a circle with centre , and 
a convex quadrilateral such that each of the segments 
and 
is tangent to . Let be the circumcircle of the triangle 
. The extension of 
beyond meets at 
, and the extension of 
beyond 
meets at 
. The extensions of 
and 
beyond 
meet at 
and , respectively. Prove that ![\[A D+D T+T X+X A=C D+D Y+Y Z+Z C.\]](http://latex.artofproblemsolving.com/8/2/6/8261276053071cc6cc06dda824a883426bf1d4ac.png)
such that for every 
is satisfied? Explain the answer.
be a random point on the smaller arc of the circumcircle of square 
, and let 
be the intersection point of segments and 
. The feet of the tangents from point to the circumcircle of the triangle 
are and , where 
is the center of the square. Prove that points , , and lie on a single circle.
and 
such that
be an integer, and let 
be positive real numbers such that 
. Prove that![\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]](http://latex.artofproblemsolving.com/7/7/f/77fb719e975225b59c64f1f3aa102c98a6c25f50.png)
colors, there exists a monochromatic regular hexagon.
is the smallest number such that if 
, for each coloring of the set 
with two colors there exists a monochromatic arithmetic progression of length 
. Prove that
.
points in plane. Prove that the number of isosceles triangles having their vertices among these points is 
. Find a configuration of points in plane such that the number of equilateral triangles with vertices among these points is 
.
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that Prove that
Equality holds iff 
or 
and any cyclic permutations.
be non-negative real numbers such that Prove that
be non-negative real numbers such that 
Prove that
be non-negative real numbers such that Prove that![\[\sum a^2+8\sum ab\ge 3\sum a\sum\sqrt{ab}.\]](http://latex.artofproblemsolving.com/2/b/a/2bac168124825dbc152f3085a0d3723fa34caf47.png)
Changing 
it becomes a fourth degree inequality, so is linear in 
. Thus it remains to check only the cases 
and 
which is easy.
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
be positive real variables with 
Prove that
Equality holds iff 
be non-negative real variables with 
Prove thatEquality holds iff or where ![\[\frac{12a(a+2b)}{4ab+bc+ca} \geqslant \frac{32a^3+3(33b+56c)a^2+3(26b^2+102bc+13c^2)a-4(4b+c)(b-2c)^2}{11[ab(a+b)+bc(b+c)+ca(c+a)]+51abc}.\]](http://latex.artofproblemsolving.com/a/b/0/ab0f41dc6c9cf3e143885b3e7e7079048d3841c3.png)
be three non-negative real numbers with Prove that
When does equality hold?
be three non-negative real numbers with Prove that
When does equality hold?
be non-negative real variables with Prove that
Equality holds iff or 
where 
,
for all 
,
such that: 
and 
.
and 
.
,
and 
.
and set 
The inequality becomes
ưhere
which ends the proofs 