,
.
and 
is non-decelerating if 
holds for all positive integers 
.
is convergence-inducing, if the following statement is true for all real sequences 
:
is convergent and tends to 
for all positive integers 
,
,
,
is bounded from above.
light bulbs are arranged around a circle, and are consecutively numbered with
.
days we change the current on/off configuration as
,
let 
and 
denote the feet of the altitudes from 
and 
,
intersect circumcircle 
.
intersect circumcircle 
.
and 
passes through the orthocenter of triangle 
,
,![\[
a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1},
\]](http://latex.artofproblemsolving.com/4/e/c/4ec892df19b4ec97af9bb1e5d1954e6c26be3172.png)
for all integers 
.
such that![\[
\frac{a_n}{n}
\]](http://latex.artofproblemsolving.com/3/d/1/3d1d012e7c41e1e09abd14626a79b1fb1cbab34d.png)
is an integer.
,
to the equation![\[
\frac{1}{x_1} + \frac{1}{x_2} + \cdots + \frac{1}{x_n} + \frac{1}{x_1 x_2 \cdots x_n} = 1
\]](http://latex.artofproblemsolving.com/5/c/4/5c40a3f0eaa58e438c8dcaad395d7ab9e8f4f439.png)
be the incenter of triangle 
.
,
,
be the orthocenters of triangles 
,
,
,
,
,
,
,
terms in this sequence, say 
and 
,
is odd, then switch their placement, while the other terms stay in place. This creates a new sequence.
to 
,
be a fixed integer. A game is played by 
cards numbered 
.
denote the configuration after 
turns (so 
is the initial configuration). Show that 
.
be a positive integer. The integers 
are written on a board. In a move, Alice can pick two integers written on the board 
such that 
is an even number, erase both 
and 
from the board and write the number 
on the board instead. Find all 
,
to 
and can't make any further moves.
Y by
Let 
be a convex quadrilateral and let 
be the centroid, circumcentre, orthocenter & nine-point centre of 
respectively. We define the points 
analogously. Now, we define the area centroid 
of as the intersection of 
the qausicircumcentre 
as the intersection of 
quasiorthocentre 
as the intersection of 
and the quasinine-point centre 
as the intersection of 
Prove that the points 
are collinear with 
and 
be a convex quadrilateral and let 
be the centroid, circumcentre, orthocenter & nine-point centre of 
respectively. We define the points 
analogously. Now, we define the area centroid 
of as the intersection of 
the qausicircumcentre 
as the intersection of 
quasiorthocentre 
as the intersection of 
and the quasinine-point centre 
as the intersection of 
Prove that the points 
are collinear with 
and 
This post has been edited 1 time. Last edited by Entrepreneur, Nov 15, 2024, 8:23 AM
