[/quote]
,
be a triangle and 
an interior point of 
is less than or equal to 
.
be a positive integer and consider the set 
.
of numbers not previously chosen such that for every distinct 
,
divides 
nor 
,
.
.
be the midpoint of the arc 
not containing 
and define 
similarly. Show that the orthocenter of 
is the incenter 
of 
.
table filled with natural numbers, we say it is a divisor table if:
-
,
-
,
for every 
.
is given. Determine the smallest natural number 
,
such that 
,![\[
f(x^2+f(y))=f(xf(x))+y
\]](http://latex.artofproblemsolving.com/d/e/d/dedd2338779fcc6723077ce586713fb4a68b9931.png)
be an arbitrary point on the side 
.
intersect 
at 
.
.
lies on the line 
be the reflection of point 
and 
be the circumcenter and the orthocenter of triangle 
.
be the points where the reflection of line 
with respect to line 
intersects the circumcircle of triangle 
is a point on the line 
such that 
,
.
![$p(x)\in\mathbb{R}[x]$](http://latex.artofproblemsolving.com/9/a/e/9aecb18edc17f7391a44db6e4f9d1502ab1d92f7.png)
such that for all 
:
such that:
passes through 
and 
again at points 
such that 
and 
at 
at 
.
is parallel to 
bisects arc 
and line 
meets 
.
at point 
on the circumcircle of 
is such that 
.
and 
meet on 
be an inscribed circle of 
,
and 
be diameters of 
be the pole of 
and 
with respect to 
cuts 
.
cuts 
at 
at 
.
and 
be midpoint of 
and 
,
,
and perpendicular bisector of 
are concurrent.
grid. The rabbit's starting square is 
(unknown to the hunter), and after 
rounds, the rabbit is at square 
.
which shares a point with 
squares of the hunter's choosing. If the rabbit is in one of these squares, the rabbit is captured and the game ends.
is a parallelogram. Let 
such that 
.
at 
and 
.
.
be a non-cyclic quadrilateral. Let 
and 
be the circumcentre and the circumradius of the triangle 
.
and 
in a similar way. Prove that![\[\frac{1}{O_1A_1^2-r_1^2}+\frac{1}{O_2A_2^2-r_2^2}+\frac{1}{O_3A_3^2-r_3^2}+\frac{1}{O_4A_4^2-r_4^2}=0.\]](http://latex.artofproblemsolving.com/f/f/5/ff575c3012e0b86c5aeeec32aae5268a4dad74d2.png)
Y by
A non-degenerate triangle 
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of 
in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is 
injective?
Note: The function is called injective if for any 
, 
is given in the plane, let 
be the set of points which lie strictly inside it. Also let 
be the set of circles in the plane. For a point 
, let 
be the reflection of 
in sides 
respectively. Define a function 
such that 
is the circumcircle of 
. Is 
injective?Note: The function
is called injective if for any 
, 
This post has been edited 1 time. Last edited by 827681, Apr 20, 2025, 6:24 PM
