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.
and a positive real number 
, non-negative real numbers 
satisfy 
. Define 
as the number of elements in the following collection of subsets:
.
, from 
outside the circle, draw tangents 
(
are tangential points) and secant 
(
lie on circle 
, 
is between and 
). 
intersects 
at 
; 
is midpoint of 
. The line crossing , paralleling with 
, intersects at 
. 
intersects 
at 
. Let 
lie inside the triangle 
such that: 
=
. From draw chords 
, 
, 
on circle . Prove that: 
is a isosceles triangle
with 
colors. We call a vertex rainbow if it's connected to all of the colors. At most how many rainbows can exist?
be an integer. You have two 
boards. Each board contains the numbers 
to 
inclusive, one number per square, arbitrarily arranged on each board. A move consists of exchanging two rows or two columns on the first board (no moves can be made on the second board). Show that it is possible to make a sequence of moves such that for all 
and 
, the number that is in the 
row and 
column of the first board is different from the number that is in the row and column of the second board.
be real numbers such that 
and 
Prove that
Equality holds when 
Equality holds when 
Equality holds when 
are pairwise distinct 
numbers and 
, find the minimum value of![\[
\left( \frac{a}{b - c} + \frac{b}{c - a} + \frac{c}{a - b} \right)^2 + a^2 + b^2 + c^2.
\]](http://latex.artofproblemsolving.com/3/6/9/3694d92e77e5049af2ec32f1dc1879dcc74f5936.png)
be a fixed integer and 
.
and 
, the number![\[\sum\limits_{k=1}^n \dfrac{|a-r\omega^k|^2}{|a|^2+r^2}\]](http://latex.artofproblemsolving.com/e/e/3/ee3b0568b9a758cf22c23f96dbb7e7afe3601964.png)
is an integer. Interpet this result geometrically.
dominos, then at least one of its sides has even length.
be a right triangle with a right angle in and a circumcenter 
. On the sides 
and , the points and 
lie in such a way that 
. Let 
and 
be the projection of and on 
, respectively. Prove that 
is half as long as .
with 
. Prove that:![\[
\sqrt{a + (b - c)^2} + \sqrt{b + (c - a)^2} + \sqrt{c + (a - b)^2} \geq \sqrt{3},
\]](http://latex.artofproblemsolving.com/c/b/5/cb551e6d6ce94902b7bcb78f6da210d6c27b96d5.png)
with equality if and only if 
.
and 
Show that
Where 
and 
Show that
Where
is given inside a triangle . Let 
be the midpoints of 
, and let 
be the intersection points of 
and 
and 
and 
, respectively.
is equal to one third of the area of triangle .
, and 
are concurrent.
belongs to a circle of diameter . Point divides the shorter arc in half. Lines 
and intersects at point . Find the length of the segment if 
, 
.
belongs to a circle of diameter . Point divides the shorter arc in half. Lines and intersects at point . Find the length of the segment if , .
.
, which means that 
, which is false since max 
value can be 
(approaching to it) .
, where is the centre of that circle.
.
.
.
, we get that 
be real numbers such that 
and 
Prove that
