be the set of integers, and let 
be a function. Prove that there are infinitely many integers 
such that the function 
defined by 
is not bijective.
,
such that 
.
and 
be positive integers, and let 
be a 
grid of unit squares.
or 
rectangle. A subset 
of grid squares in 
edges of the grid squares.
such that
whenever 
is defined. What is 
Y by Davi-8191, jhu08, megarnie, mathematicsy, mathlearner2357, rayfish, Adventure10, Mango247, and 6 other users
Let 
be a point in the plane of 
, and 
a line passing through . Let 
be the points where the reflections of lines 
with respect to intersect lines 
respectively. Prove that are collinear.
be a point in the plane of 
, and 
a line passing through . Let 
be the points where the reflections of lines 
with respect to intersect lines 
respectively. Prove that are collinear.This post has been edited 1 time. Last edited by tc1729, Apr 25, 2012, 9:56 PM
.
and 
in the expression left it unchanged, and since I was running out of time, said that you could show the same for switching 
.
is the same, and they are collinear.
are cyclic. Darn.
are isogonal with respect t
.
are isogonal with respect t
.
are isogonal with respect t
.
.
.
and 
.
we find,
Okay that wasn't too bad. Now similarly we can compute,
Note that 
,
and 
are just reflections over the real axis of 
,
Then it suffices to show that,
Now we expand by minors to find,
and we're done.
![\[
\begin{vmatrix}
a_{11} & a_{12} & a_{13} \\
a_{21} & a_{22} & a_{23} \\
a_{31} & a_{32} & a_{33}
\end{vmatrix} = a_{11} \begin{vmatrix} a_{22} & a_{23} \\ a_{32} & a_{33}\end{vmatrix}- a_{12} \begin{vmatrix} a_{21} & a_{23} \\ a_{31} & a_{33}\end{vmatrix} + a_{13} \begin{vmatrix} a_{21} & a_{22} \\ a_{31} & a_{32}\end{vmatrix}
\]](http://latex.artofproblemsolving.com/4/3/0/4303e1e80bfa2af32ac91a5488ea0584d1d33141.png)
matrices, by considering the alternate sum and difference of the smaller 
matrices. If you have EGMO, it's mentioned in the addendum on linear algebra.
by reverse reconstruction as the intersection of 
and 
there is an involution swapping 
,
and 
.
,
is the reflection of ![\[a_1=\overline{a} , b_1=\overline{b} \text{ and } c_1 = \overline{c}\]](http://latex.artofproblemsolving.com/e/6/d/e6d2fd41b1b34208b22eef98a66cdbb0d209cf39.png)
Now, note that since 
,
Similarly, we obtain that,![\[b' = \frac{\overline{b}(\overline{c}a-c\overline{a})}{\overline{b}(\overline{a}-\overline{c})+b(c-a)}\]](http://latex.artofproblemsolving.com/b/a/c/bac638de120d8e65baa8fa567f028cd0d413d2e3.png)
and![\[c' = \frac{\overline{c}(\overline{b}a-b\overline{a})}{\overline{c}(\overline{b}-\overline{a})+c(a-b)}\]](http://latex.artofproblemsolving.com/6/1/0/6102ed5282407b52ab9b31e68250a888b8bf333f.png)
Now, let 
.
We then let, 
, 
and 
.
,
and 
.
one can do a quick check and verify that 
,
,
,
and 
.![\[\overline{E}=\frac{R(aAQ-bBP)}{Q(aAR-cCP)}\]](http://latex.artofproblemsolving.com/c/7/c/c7c01765226e0b556ddfdde912b4918d26af110d.png)
we wish to have 
.
So, it suffices to show that 
which is quite clear since,
as desired. So, it is clear that 
.
thus 
in quadrilateral 
thus 
Similarly 
tangent as well and now it is easy to check these three must be the same line.
.
,
are symmetric wrt 
.
,
,
,
and 
and 
are similar
and thus 
are cyclic
is the simson line of 
We want to prove that 
.
by reflection, we have that 
by reflection, we have that 
.
.
,
are isogonal conjugates in 
and quadrilateral 
,
are equal iff. 
are isogonal conjugates. This can also be approached with the theory of perfect isogonal six-point sets; 
is a perfect isogonal six-point set).
,
.
make up a complete quadrilateral with intersections in 
must be the same!
move P on 
Then our 
desired points each have 
so we need to check 
cases so that they are collinear.
t
in the original problem we have the lines through 
that are parralel to 
so we need to have 4 cases again. These cases are where 
for 
goes now theire reflection about a point in infinity just lies on the line at infinity , doing a back transformation we het the concurrency changes to collinearity and we are done
,
,
.
is defined to be 
,
with respect to 
and we are done.
