AoPS Academy
+1 w
[/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.
has an incenter 
Suppose 
. Let 
be the midpoint of 
. Suppose that 
. 
meets 
again at point 
. Let points 
and 
be such that is the midpoint of 
and is the midpoint of 
. Point 
lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.
and 
meet 
at 
and 
. Prove that 
.
be a triangle. Let 
and 
be the intersections of the 
and 
angle bisectors with the opposite sides. Let 
. Let 
where 
is the major 
arc midpiont.
and are coconic on a conic 
intersects 
again at , not equal to 
or , then prove 
and 
concur on 
be a positive integer. Alex plays on a row of 9 squares as follows. Initially, all squares are empty. In each turn, Alex must perform exactly one of the following moves:
Choose a number of the form 
, with 
a non-negative integer, and place it in an empty square.
Choose two (not necessarily consecutive) squares containing the same number, say . Replace the number in one of the squares with 
and erase the number in the other square.
, while the other squares are empty. Determine, as a function of , the maximum number of turns Alex can make.
be an acute-angled triangle with circumcircle 
and circumcentre 
. Points 
and 
lie on such that 
and 
. Let 
meet 
at , and 
meet at .
and 
have an intersection lie on line 
.
be a convex hexagon with 
and 
. which is equidistant from sides 
and 
.
and 
are the centers of mass of 
and 
, show that 
.
be a point in the plane, distinct from the vertices of 
. Consider 
the reflections of with respect to lines 
and 
, in this order.
are collinear if and only if lies on the circumcircle of . does not lie on the circumcircle of and the centroids of triangles and 
coincide, prove that is equilateral.
given that there exist positive integers 
satisfying
and , we define 
as follows:
where 
are all the positive divisors of . For example,
.
Find all positive integers such that 
.
Find all positive integers such that 
.
be an odd prime number. How many -element subsets 
of 
are there, the sum of whose elements is divisible by ?
be real numbers satisfying
be a fixed square and 
a variable point on segment 
. The square 
is constructed such that is on segment 
and is on segment 
. Let be the intersection point of lines 
and 
. Find the locus of as varies along segment .
be a positive integer. In an 
board, two opposite sides have been joined, forming a cylinder. Determine whether it is possible to place queens on the board such that no two threaten each other when: 
. 
.
be positive reals such that 
Prove that 
, by the AM-GM Inequality we deduce that
which yields 
。Applying the Hölder's Inequality we get
As stated. 
and 
we obtain the quoted result.
be positive reals such that 
Find the maximum value of 
. Take 
where 
. This expression has no maximum.
, 
, 
be positive reals such that 
Find the minimum and maximum value of 
, , be positive reals such that Prove that 
, , be positive reals such that Find the minimum and maximum value of 
and![\[ f(x+y)+f(x^2+f(y))=f(f(x))^2+f(x)+f(y)+y,\forall x,y\in\mathbb{R}\]](http://latex.artofproblemsolving.com/c/8/9/c895ae7fdf8d7f284ac9fc94cc077d6edad6cbf0.png)
be positive real numbers . Find the minimum value of 
and 
Find the maximum value of 
.
.
.
Hence shifting any 
to 
is non-increasing, so we may assume 
.
may be done with standard calculus tricks.
Thus we just minimise the function 
over 
which can be done with standard calculus tricks.
implies that
where the last inequality holds because 
for all 
,
.
is 
.
Let 
Proof:
,
But 
when 
,
is 
.
is fixed positive real. Let 
in terms of 
,
But 
when 
,
is 
.
Let 
![\[a^3+ b^3-kab + \frac{k^3}{27} = \frac{(3b+3a+k)}{27} \cdot \frac{(6a-3b-k)^2+3(3b-k)^2}4.\]](http://latex.artofproblemsolving.com/2/8/5/2859187e2d5806549f3b7008886a518160c84b52.png)
So![\[a^3+ b^3-kab \ge -\frac{k^3}{27}.\]](http://latex.artofproblemsolving.com/e/8/f/e8fa110c024aca972b043a4d2e27e359eff97039.png)
![$$a^3+ b^3-kab\geq 3\sqrt[3]{a^3\cdot b^3\cdot\frac{k^3}{27}}-kab-\frac{k^3}{27}=-\frac{k^3}{27}.$$](http://latex.artofproblemsolving.com/c/5/f/c5ff0bed94583e9cba00e7118483debb07a75332.png)
The equality occurs when 
.
Show that
and 
,
thereby 
.
We are done. 
,
be positive reals such that 
Show that
be positive reals such that 
Then 
Prove that 
.
.
and the inequality is equivalent to 
which is obvious.
.
.
.
It's clear that for positive 
is increasing, so maximum of 
.
and 
Prove that
Old.
Prove that