AoPS Academy
, the angle bisector at 
meet the opposite sides at 
, respectively. Prove that if the quadrilateral 
is cyclic, then
table filled with natural numbers, we say it is a divisor table if:
-th row are exactly all the divisors of some natural number 
,
-th column are exactly all the divisors of some natural number 
,
for every 
.
is given. Determine the smallest natural number 
, divisible by , such that there exists an divisor table, or prove that such does not exist. be a right-angled triangle where ∠
=90°.Let 
be an altitude of that triangle and points 
and 
be the midpoints of and 
, respectively.If 
is the circumcenter of the triangle 
, prove that 
and are paralel.
, let 
be a point (
). Let 
be the anticomplement of , and let 
be the 
-anticomplementary conjugate of . Prove that the line 
is parallel to the real asymptote of the circular pivotal isocubic with pivot .
and 
. Prove that there always exists a positive integer 
such that 
.
is a power of a prime number, but for arbitrary then well.....
be a positive integer. Define a sequence by setting 
and, for each 
, letting 
be the unique integer in the range 
for which 
is divisible by 
. For instance, when 
the obtained sequence is 
. Prove that for any the sequence 
eventually becomes constant.
, points 
lie on sides 
, respectively, such that 
. Altitudes 
meet at orthocenter 
. Points and lie on segment 
such that 
and 
. Lines 
and 
intersect at point 
. Compute 
. be a right angled triangle with 
. Point is the midpoint of side And be an arbitrary point on 
. The reflection of 
over 
intersects lines 
and at 
and , respectively. The circumcircles of 
and intersect again at 
. Prove that the center of the circumcircle of 
lies on 
.
be a triangle with orthocenter and circumcircle 
. Let 
be the reflection of 
in 
and let 
the reflection of in 
. Let be the midpoint of segment 
. Show that the tangent to in is perpendicular to 
.
be the midpoint of 
Reflect over to 
and Reflecting over sends to . Hence 
is a parallelogram.
is the nine-point circle of 
then showing 
be midpoint of . Take as the origin. 
. From 
. From 
. Take altitudes from and to and respectively and get equations of lines and their intersection which is . So, 
. We need to prove 
is parallel to . Which means their slopes should be equal.
.
,
,
;
,
,
.Then 
and 
.Note that the displacement vectors can be written as 
and 
.Then,by EFFT,if we have that ![$$a^2[c^2\frac{S_{AC}-S^2}{S^2}-b^2\frac{S_{AB}-S^2}{S^2}]+b^2[(b^2-c^2)\frac{S_{AB}-S^2}{S^2}+c^2\frac{S^2+S_{BC}}{S^2}]+c^2[(b^2-c^2)\frac{S_{AC}-S^2}{S^2}-b^2\frac{S^2+S_{BC}}{S^2}]=0,$$](http://latex.artofproblemsolving.com/c/b/0/cb0c8dd0d07f62dd2103e507d98a1d32b1e19c67.png)
we are done. This can be shown by some algebraic manipulations.
such that 
on unit circle, then 
.
and 
be the circumcentre and centroid of 
respectively.
maps 
so 
.