such that:
with all 
is real number
be the set of integers. Determine all functions 
such that, for all integers 
and 
,
Proposed by Liam Baker, South Africa
be a positive integer. Find all real solutions 
to the system:![\[a_1^2 + a_1 - 1 = a_2\]](http://latex.artofproblemsolving.com/a/4/e/a4e26b83c929b82800576bac11e12de0c3aa1cf6.png)
![\[ a_2^2 + a_2 - 1 = a_3\]](http://latex.artofproblemsolving.com/9/9/5/995e8eec93828b82901b0639463f907f5ff95d08.png)
![\[\hspace*{3.3em} \vdots \]](http://latex.artofproblemsolving.com/e/e/1/ee15e1fefe2ad550d3c9b1c51207eaf66fbfa41f.png)
![\[a_{n}^2 + a_n - 1 = a_1\]](http://latex.artofproblemsolving.com/7/c/7/7c727d1f2044c1bf7c80981e1074bb3cde807b8f.png)
of positive integers is called central if for every positive integer 
terms of the sequence is equal to 
of positive integers such that for every central sequence 
there are infinitely many positive integers 
.
Y by samrocksnature
In an acute-angled triangle
, the altitudes 
and 
meet at the point 
. Points 
and 
are the midpoints of altitudes and . It turned out that the center 
of the circle inscribed in the triangle 
lies on the bisector of the angle 
. Prove that the triangle 
isosceles.
, the altitudes 
and 
meet at the point 
. Points 
and 
are the midpoints of altitudes and . It turned out that the center 
of the circle inscribed in the triangle 
lies on the bisector of the angle 
. Prove that the triangle 
isosceles.This post has been edited 2 times. Last edited by parmenides51, May 3, 2021, 8:37 AM
(exterior angles are equal from above)
(cyclic quad)
are the midpoints of 
respectively(cant we 
?)
be 
,
,
and 
.
,
.
are collinear thus 
and from here its easy to notice that 
