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.
. At a maths camp, there are 
children, each of whom selects
tables, with children tables, with children
and 
.Prove that ![$$ a^2-ab+b^2 \geq \sqrt[3]{2} $$](http://latex.artofproblemsolving.com/7/9/8/798484c6b985ebbace2a3dde816f11a46635feba.png)
Let and 
.Prove that ![$$ 3\geq a^2+ab+b^2 \geq \sqrt[3]{4} $$](http://latex.artofproblemsolving.com/d/7/7/d77fbdc50a88395259e58d31413b9ac453e5dfd0.png)
of odd degree with real coefficients such that 
.
be non-negative real numbers such that 
Prove that
be an acute triangle with 
and 
. Let 
be the midpoint of the side 
. The circumcircle of triangle 
intersects the side 
again at 
and the circumcircle of triangle 
intersects the side 
again at 
. The point 
lies on the perpendicular bisector of the segment so that the points and 
lie on the same side of 
and 
. Prove that the circumcircles of triangles 
and 
are tangent.
Define a cell with 
property if after remove that cell from the board The board can be tile with 
tetromino. cell 
Note: tetromino can be rotated but not flipped
distinct primes?
with side length 
as its base and apex 
Segments 
and 
are perpendicular. What is the square of the height of the pyramid?
with side length as its base and apex Segments 
and 
are perpendicular. What is the square of the height of the pyramid?
and 
and 
(here 
is obivously the radius of a sidelength- regular octagon). Then we have![\[ AV\perp DV\implies (A-V)\cdot (D-V) = 0\implies (r,0,-h)\cdot (-r/\sqrt{2}, r/\sqrt{2}, -h) = 0\implies h^2 = \frac{r^2}{\sqrt{2}}.\]](http://latex.artofproblemsolving.com/9/f/0/9f0f04906c1d5c03679314ff48ef3a756897c5e6.png)
We can easily derive 
with the law of cosines, so solving for 
and plugging it into the above, we get![\[h^2 = \frac{1}{\sqrt{2}}\left(\frac{1}{2(1 - \frac{1}{\sqrt{2}})}\right) = \frac{1}{2(\sqrt{2} - 1)} = \boxed{\frac{\sqrt{2} + 1}{2}}.\]](http://latex.artofproblemsolving.com/e/0/b/e0babc096dffa982afc24b660bb900503d021c1b.png)
Prove that
.![$\sqrt[3]{ab}+ \sqrt[3]{cd} < \sqrt[3]{(a+b+c)(b+c+d)}$](http://latex.artofproblemsolving.com/0/5/0/0502ba65f0545b3af84c41b2549d6256550b473b.png)
.
be real numbers less than 2.Then prove that 
with 
Prove that
,
,
to meet at 
and 
are both right isosceles, so 
.
