AoPS Academy
be an acute triangle with orthocenter 
, and let 
be a point on the side 
, lying strictly between 
and 
. The points 
and 
are the feet of the altitudes from and , respectively. Denote by 
is the circumcircle of 
, and let 
be the point on such that 
is a diameter of . Analogously, denote by 
the circumcircle of triangle 
, and let 
be the point such that 
is a diameter of . Prove that 
and are collinear.
such that for every 
, 
, 
equals the number of times appears in the sequence.
be the sequence 
so that 
for each 
and so that the decimal number 
is divisible by 
for each 
. Show that the decimal 
is irrational.
points on the plane, no three of which are collinear, and some set of line segments between them. We say that a subset of these line segments is called a "pairing" if every one of these () points is the endpoint of exactly one of the chosen line segments (in other words, a pairing is a perfect matching).
, there exists such an arrangement of points and line segments (for some value of ) such that there are exactly 
distinct (but not necessarily disjoint) pairings.
be a polynomial with real coefficients such that 
for all 
. Prove that 
for some real constant 
and 
.
be polygons, no two of which are similar. Show that there are polygons 
where 
is similar to 
so that for no 
does contain a polygon that's congruent to 
. so that is wholly contained in , and so it does not intersect the edges of at all.
![\[ 2^n -2, \;\; 3^n -3, \;\; 4^n -4, \;\; 5^n-5, \, \ldots \ldots \]](http://latex.artofproblemsolving.com/d/b/1/db17ef301bf3fd01f8b310f91a14b9dee045e11f.png)
is 
. Also, are there infinitely many n for which the greatest common divisor is 
?
, define:
as its circumcentre, as its orthocentre and 
as its nine-point centre.
, 
and 
to be the midpoint of the smaller arcs.
as the isogonal conjugate of the Nagel point (i.e. the exsimillicenter of the incircle and circumcircle)
as the ninja point of 
as the ninja point of the contact triangle
Points , and 
are collinear, that is is the Feuerbach point.
Points , and are collinear
and 
.Find the minimum of ![$S=\sqrt[3]{\frac{a^3+b^3+c^3}{3}}+\frac{a+b+c}{9}$](http://latex.artofproblemsolving.com/9/d/9/9d93ff9d9f7355e452f1007aeb87423e3491353a.png)
be an acute-angled triangle with altitudes 
and 
. Let be the orthocentre, that is, the point where the altitudes meet. Prove that ![\[\frac{AB\cdot AC+BC\cdot CA+CA\cdot CB}{AH\cdot AD+BH\cdot BE+CH\cdot CF}\leq 2.\]](http://latex.artofproblemsolving.com/a/1/1/a11018e8855be22de7459fb8e63488997cc8d097.png)
as sides of triangle and 
as its circumradius
etc.
etc.
etc.
is cyclic (from 
), and the sides 
and 
extended meet at 
.
and 
.
and 
is a cyclic quadrilateral. Then, 
so 
Hence, 
as desired. The others follow similarly. 
This follows by the Trivial Inequality 
is equivalent to 
= 
= 
which is trivial .
as sides of triangle and as its circumradius etc. etc. etc.." I've been looking at this for like 2 days and I don't see how this is true... Could someone please provide some insight or hints.
. Without loss of generality, assume 
. Observe that 
, and also 
so that ![\[\frac{AF}{AH} = \frac{AD}{AB} \implies \frac{AB}{AH} = \frac{AD}{AF} = \frac{AD}{AC \cos \angle A} \implies AH \cdot AD = \frac{b^2 + c^2 - a^2}{2}.\]](http://latex.artofproblemsolving.com/f/2/4/f241eb5a9c3276f70a5a7a16fb122b540c6a843f.png)
Similarly, ![\[BH \cdot BE = \frac{c^2 + a^2 - b^2}{2}, \text{ and } \frac{a^2 + b^2 - c^2}{2}.\]](http://latex.artofproblemsolving.com/f/6/3/f633512597939511f9b34fe2d92dd5d1ac50b648.png)
Therefore our initial inequality is equivalent to ![\[\sum_{cyc}^{3} ab \leq \sum_{cyc}^{3} a^2, \]](http://latex.artofproblemsolving.com/0/4/d/04de9a9fd6d25e2365d76707e9b59d6d17490cd9.png)
which is trivial by Chebyshev's rearrangement inequality. 
where 
.
be a function such that 
.
.
.
.
.
,
.
and so the problem translates to 
,
,
,
,
,
,
,
,
,
since 
.
,
. 
+ 
+ 
+ 
= 
. 
. 
. 
. 
. 
+ 
. 
+ 
. 
and 
.
AM-GM completes the proof.
