[/quote]
,
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
be a positive integer. Dominoes are placed on a 
board in such a way that every cell of the board is adjacent to exactly one cell covered by a domino. For each 
or 
.
of positive real numbers that satisfy the system:![\[
\begin{aligned}
11bc - 36b - 15c &= abc \\
12ca - 10c - 28a &= abc \\
13ab - 21a - 6b &= abc.
\end{aligned}
\]](http://latex.artofproblemsolving.com/1/7/0/170e1cfaee90f5c37428fa75192ac889261a8f83.png)
be positive reals. Prove that there is a cyclic permutation 
of 
such that the inequality:![\[
\frac{a}{xa + yb + zc} + \frac{b}{xb + yc + za} + \frac{c}{xc + ya + zb} \geq \frac{3}{x + y + z}
\]](http://latex.artofproblemsolving.com/5/5/3/553527fc40a1b7d29dbdfb4b5e63635308f46c85.png)
and 
.
and circumcenter 
.
,
,
.
with diameter 
,
at a point 
.
meets line 
at 
and 
intersect at a point 
on 
.
Prove that: 
is the sum of 
square numbers
Let 
and 
is the sum of 
is the sum of 
and incenter 
and let 
,
,
are selected on sides 
,
such that 
,
,
.
intersects 
other than 
.
and 
meet on 
,
,
,
,
,
initially contains one coin. The following operations are allowed
,
,
;
,
,
and 
.
coins.
Prove that
,
.
.
be a real number and let 
be arbitrary real numbers in 
.
and 
are then defined recursively by
.
for all sufficiently large 
is a prime number
Y by
In the acute triangle 
, 
. The points 
and 
are the feet of the altitudes from 
and 
respectively. The line 
intersects the line 
at 
. Show that the bisectors of the angles 
and 
are perpendicular.
, 
. The points 
and 
are the feet of the altitudes from 
and 
respectively. The line 
intersects the line 
at 
. Show that the bisectors of the angles 
and 
are perpendicular.This post has been edited 2 times. Last edited by parmenides51, Aug 8, 2020, 10:20 AM
be the circle with diameter 
.
be defined as intersection point of angle bisectors of 
be intersection point of 
and 
.
are concyclic. Thus,
If we managed to show that 
,
are concyclic, then 
.
,
,
.
with 
and 
and using this we have,
and hence we are done.
is cyclic. Hence 
,
