[/quote]
,
be non-negative real numbers such that ![\[b+c\leqslant a+1,\quad c+a\leqslant b+1,\quad a+b\leqslant c+1.\]](http://latex.artofproblemsolving.com/b/9/e/b9e86e898c0536b7323a03611d5bdbf679caa710.png)
Prove that 
be a function satisfying
for all integers 
and 
.
and 
such that 
for all integers 
such that ![\[f(xy-1)+f(x)f(y)=2xy-1,\]](http://latex.artofproblemsolving.com/8/8/8/888ca39f2b7f8cec6d6426bee28d40eade40a66e.png)
for any real numbers 
and 
![\[
f : \mathbb{R}^+ \to \mathbb{R}
\]](http://latex.artofproblemsolving.com/e/7/b/e7bfc5b236fb6f00ef328f850b1b14632cbf8416.png)
satisfying the condition that for all 
,![\[
f(x + y^2) \geq f(x) + y.
\]](http://latex.artofproblemsolving.com/3/a/a/3aa20083835682dbd81be692eba65cab19e923e5.png)
of positive integers with 
prime, 
,
,
is a multiple of 
.
is the number of integers not more than 
a point in the space and 
the centroid of a tetrahedron 
.
be a 
makes a fixed angle with the tangent to 
be a rectangular hyperbola centered at the origin, scaled such that it is tangent to the logarithmic spiral at some point.
be a monic polynomial so that there exists another real coefficients 
that satisfy![\[P(x^2-2)=P(x)Q(x)\]](http://latex.artofproblemsolving.com/6/9/7/697faa929e4fda7a6e9b1cd97849bd42ffc14306.png)
Determine all complex roots that are possible from 
with exactly 9 elements that is subset of 
.
and 
that satisfy the following:
is an empty subset.
is 
and 
.
to 
is 
and 
.
and 
is 
is 
is tangent to the 
and 
.
and 
be the midpoints of sides 
,
be points on segment 
and 
,
is the angle bisector of 
and 
is the angle bisector of 
.
is parallel to 
if and only if 
for which, for all 
,![\[
f(x) f(yf(x)) + y f(xy) = \frac{f\left(\frac{x}{y}\right)}{y} + \frac{f\left(\frac{y}{x}\right)}{x}
\]](http://latex.artofproblemsolving.com/9/0/c/90c180110402e1a32b70edb2b0a03a28727457d1.png)
Y by mathmax12, Adventure10
Let 
be an acute triangle. Points 
and 
lie on the segments 
and 
, respectively, such that 
and the segment 
passes through the orthocenter of the triangle . Lines tangent to the circumcircle of the triangle 
at points and intersect at point 
. Prove that points 
are concyclic.
be an acute triangle. Points 
and 
lie on the segments 
and 
, respectively, such that 
and the segment 
passes through the orthocenter of the triangle . Lines tangent to the circumcircle of the triangle 
at points and intersect at point 
. Prove that points 
are concyclic.
be the 
-
,
be the orthocenter of 
and 
therefore
so 
which implies 
.
,
as the points symmetric to the sides 
and 
are tangent to 
.
and 
in 
.
.
be the orthic triangle 
bisects 
and 
.
be midpoint of arc 
be reflection of 
and similarly, 
.
is cyclic 
is the tangent throug
is also a line.
t
wrt 
and 
a
a
and 
so 
which means 
is the reflection of 
.
is the reflection 
and name the points 
analogously. These points lie on the circumcircle of 
,
are collinear, thus 
are collinear similarly.
are concyclic. This is trivial enough, as 
and similarly 
.
wrt 
as desired.
be the reflections of 
and 
respectively. Notice 
since 
Also, 
,
,
,
,
.
,
are collinear along the internal bisector of 
is cyclic with diameter 
,
Now, observe that the isosceles condition gives 
Thus, the Ratio Lemma yields 
so 
.
,
is the external bisector of 
,
.
