[/quote]
,
distinct positive integers and then writes in blue the 
sums of each two red numbers. At most how many of the blue numbers can be prime?
Prove that
Where 
Prove that
Let 
Prove that
be a positive integer. Find the number of reduced fractions 
for which 
,
denotes the fractional part of 
.
,
,
be the centers of the circles circumscribing triangles ADE, BDF, CEF. Prove that the orthocenter of triangle 
is tangent to sides 
and 
at 
and 
respectively, and 
is the circumcenter of 
.
.
denote the maximum number of 
-
board such that each 
such that![\[
f(N) \leq cN^2
\]](http://latex.artofproblemsolving.com/7/4/8/748109c3d63203f9a67a167434e10a909f38e83f.png)
for all positive integers 
.
into 
or 
can sometimes be quite complex. That's why I’ve written a program to assist with this process.
such that for all positive real numbers 
and 
,![\[
\frac{x + f(y)}{x f(y)} = f\left( \frac{1}{y} + f\left( \frac{1}{x} \right) \right)
\]](http://latex.artofproblemsolving.com/7/d/f/7df25755bc33bef970fffb392fb5b6b789390881.png)
such that for all 
and 
,
such that 
or 
,![\[
1 \leq a_1 < a_2 < \cdots < a_{(p-1)/2} < p
\]](http://latex.artofproblemsolving.com/6/8/e/68e3d73edec43b2cc480832adf23bbbb5f4c7bee.png)
such that![\[
\prod_{\substack{1 \leq i < j \leq (p-1)/2}} (a_i + a_j)^2 \equiv 1 \pmod{p}.
\]](http://latex.artofproblemsolving.com/2/9/e/29edf6ab8cee2a526e7753a140c5a291e987856a.png)
be distinct real numbers such that![\[
a + b + c + \frac{1}{abc} = \frac{19}{2}
\]](http://latex.artofproblemsolving.com/5/1/e/51eef77358db2b6c98f0fb0fb4e30db837a03642.png)
Find the maximum possible value of 
.
.The solution to 
is only 
for some 
Y by gold46, math_lover81, Arshia.esl, Adventure10, Mango247, and 15 other users
Hi mathlinkers!
I want to organise a Geometry marathon. This is to improve our skills in geometry. Whoever answers a pending problem can post a new problem. Let us start with easy ones and then go to complicated ones.
Different approaches are most welcome
.
Here is an easy one
1. Inradius of a triangle is equal to
.find the sides of the triangle and P.T one of its angle is 
[moderator edit: topic capitalized.]
I want to organise a Geometry marathon. This is to improve our skills in geometry. Whoever answers a pending problem can post a new problem. Let us start with easy ones and then go to complicated ones.
Different approaches are most welcome
.Here is an easy one
1. Inradius of a triangle is equal to
.find the sides of the triangle and P.T one of its angle is 
[moderator edit: topic capitalized.
]
.
where 
.
,
.
are positive, 
.
.
,
are similar.
is equilateral if and only if so is 
.
be the circumcenter of an acute triangle 
and let 
be the circle with center 
that is tangent to 
and tangent to side 
at 
.
and 
again at 
and 
respectively. The lines 
and 
meet 
and 
.
and 
intersect at 
.
.
.
.
,
.
.
and 
are parallel too.
is a parallelogram. We can know that 
.
and 
at 
and AC at 
.
.
intersect 
,
and 
.
and 
at 
(Obviously, line 
,
, Let 
,
.find the relation between 
.
is equilateral.
, 
.
be a cyclic quadrilateral inscribe in circle 
with diameter 
and suppose that 
is the circumcircle of triangle 
and 
it's diameter. Consider 
.
is cyclic.
intersect 
at 
.
.
concur, say at 
.
,
and we have 
.
implying 
is cyclic and 
.
intersects 
other than 
is a diameter. But 
is also a diameter. Follows that 
.
,
and 
are outwardly constructed. 
.
is a symmedian.
is cyclic so :
so we are done
and 
are regular pentagons outside the triangle . Prove that 
,
,
,
.
and the others will follow analogously.
,
are collinear.Hence 
all go through 
is the unique center of spiral similarity sending the triangles 
to each other. So those four lines concur at the second intersection of the circumcircles of the pentagons.
such that 
.
are concurrent.
,
,
,
cut 
at 
be the common tangent of circumcircle of 
.
.
.
then we want to prove that 
are collinear
be the line through 
and let ![$\ell_{3]}$](http://latex.artofproblemsolving.com/2/0/a/20a6399dc043da87254cbd87e644b609d4a67545.png)
intersect the circles at 
then we have that 
is bisector of 
in one of minor arcs 
and the other one of 
and 
is outside of minor arc of 
so 
so we must have 
easily by a little calculation
and is tangent to 
and 
.
be the point on arc 
of parabola 
is foot of 
lies on 
is tangent to 
.
to be 
,
are also symmetric about its axis. Let their parameters be 
and the parameter of 
be 
.
.
,
.
.
.
,
are the respective exradii and 
are the inradii of 
respectively then show that:
where 
is the semiperimeter of 
