denote the set of positive integers. Find all functions 
satisfying the condition
for all 
be a scalene triangle, 
its medial triangle, and 
a point on the Thomson cubic (= 
)
).
be the circumcevian triangle of 
be the pedal triangle of 
be the reflection in 
of 
.
,
cyclically.
be the reflection in 
of 
,
cyclically.
be the reflection in 
of 
,
cyclically.
,
,
and the orthocentroidal circle of 
lines are drawn on the plane such that no three of them are concurrent and no two are parallel. At least 
of the bounded regions formed are colored black. A triangle formed by three lines is called a \textit{good triangle} if it lies entirely within a black region. Prove that there are at least 
good triangles. (A good triangle is a bounded region with finite area.)
cities, including a capital city. Some pairs of cities are connected by two-way flights. Given a city 
,
is called an antitour from 
,
by a sequence of exactly 
(not necessarily distinct) flights.
,
.
satisfies the quadratic equation 
and deduce that the value of 
.
.
,
is 
.
![$f \in C^1([0,1]), g \in C^2(f([0;1]))$](http://latex.artofproblemsolving.com/f/5/2/f52369171ef8571734950120c43aacac10154fcd.png)
.
?
,
are positive integers and 
is odd.
and 
.
with multiplication of matrixes operation is making an isomorphic-group structure with 
.
.
p
a
and the line 
.
by 
and 
for all 
,
Show that 
is an integer for all 
.
,
by the inequality
,
and 
.
from 
to 
and 
from 
to 
but in vain.
and integers 
such that 
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
be a triangle and let 
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 
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 
