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.
table is colored black or white. Every cell on table border is black. It is known, that in every 
square there are cells of two colors. Prove, that exist square that is colored in chess order.
+1 w
(use in a problem for junior) to practice. Can anyone share to me?
be a non-isosceles triangle with orthocenter 
, circumcenter 
. Let 
be midpoints of 
and 
be reflections of across 
. Prove that triangle 
have the same centroid.
be a quadrilateral such that 
and 
are acute and 
. Suppose that 
, prove that 
.
. From the fourth term of the sequence onwards, each term of the sequence is the average of the previous three. Given that the first six terms of the sequence are natural numbers and that the sixth number written was 
, what is the fifth term of the sequence?
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
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 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)
and 
, then we can deduce 
\le 2 $ ?
be the maximal number of the vertices of a convex lattice polygon with exactly 
lattice points in the interior. Show that:
for 
for some constant 
.
, points 
and 
are the centers of the sides 
and respectively. 
is such a point that 
. Prove that lies on the line 
.
is an isosceles trapezoid (
) therefore is cyclic
therefore lies on both perpendicular bisectors of diagonals 
and 
of cyclic , is the circumcenter of isosceles trapezoid ,
, which is line
and 
.
prove that![$\frac{a}{b} + \sqrt {\frac{b}{c}} + \sqrt[3]{{\frac{c}{a}}} > \frac{5}{2}$](http://latex.artofproblemsolving.com/6/a/6/6a657cb647e279b0745e3189c3f9d54aa25bf976.png)
as 
,
as 
,
,
and 
.
and 
(note that we can just equate the equations and multiply them by 
)
,
is denoted as 
.
,
.
and 
into the above equation, we get that it does indeed hold. (to do this neatly, you must once again multiply both sides by 
)