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 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)
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 
.
and a set consisting of 
disting positive integers smaller than 
are given. Prove that there exists a positive integer 
that can be written in the form 
, for 
in at least 
different ways.
be real numbers such that 
. Prove that
Let be real numbers such that 
. Prove that
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 a convex quadrilateral. Suppose that the circles with diameters 
and 
intersect at points and 
. Let 
and 
. Prove that the points 
, 
, and are concyclic. and are not the diagnols)
attaining all positive values and positive numbers 
are given. Numbers 
satisfy
for every 
. Prove that there exists a positive integer satisfying 
.
?
.
, I is the incircle center of , ⊙I is tangent to side AB, AC at D, E respectively. Line DE intersects line BI at F. We can prove 
.
.
are cyclic quadrilaterals.
(
)
and that's what we need to prove.
is equilateral, also 
, hence 
are concyclic and 
, similarly 
.
; due to symmetry, 
, so 
are collinear.
and 
. 
is median in right-angled 
, hence 
. Similarly 
; with midline in 
, we easily get 
isosceles, and 
, done.
.
be a triangle with incircle touching 
at 
. let 
be incentre and let 
intersect 
at 
. than 
.
we get 
which gives 
is cyclic quad.
is cyclic quad. with as diameter.
be midpoint of 
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)
be reals such that 
and 
.
.
is the incircle of 
at 
,
and 
,
since 
.![\[\measuredangle FIC = 60^\circ \qquad \measuredangle GIB = -60^\circ,\]](http://latex.artofproblemsolving.com/a/5/c/a5c2651e2b2ac3b4d95cb920a054864f73778592.png)
and 
.
and 
are cyclic quadrilaterals. Since 
,![\[\angle IFC = \angle IGB = 90^\circ,\]](http://latex.artofproblemsolving.com/5/2/3/5238437ed0fc6b8325e9818d1bcc20a5a566d8d2.png)
by AA similarity. Dropping an altitude from 
gives half the radius while dropping and altitude from 
and 
.