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 
.
be the circumcircle of triangle 
,
is the midpoint of 
and 
be the second intersection of 
and 
,
be a transversal of 
be intersections of 
with 
be a tangent line of 
is cyclic
Y by amatysten, narutomath96, BEHZOD_UZ, Tawan, Jalil_Huseynov, Adventure10, Mango247, and 3 other users
Four segments divide a convex quadrilateral into nine quadrilaterals. The points of intersections of these segments lie on the diagonals of the quadrilateral (see figure). It is known that the quadrilaterals 1, 2, 3, 4 admit inscribed circles. Prove that the quadrilateral 5 also has an inscribed circle.
![[asy]
pair A,B,C,D,E,F,G,H,I,J,K,L;
A=(-4.0,4.0);B=(-1.06,4.34);C=(-0.02,4.46);D=(4.14,4.93);E=(3.81,0.85);F=(3.7,-0.42);
G=(3.49,-3.05);H=(1.37,-2.88);I=(-1.46,-2.65);J=(-2.91,-2.52);K=(-3.14,-1.03);L=(-3.61,1.64);
draw(A--D);draw(D--G);draw(G--J);draw(J--A);
draw(A--G);draw(D--J);
draw(B--I);draw(C--H);draw(E--L);draw(F--K);
pair R,S,T,U,V;
R=(-2.52,2.56);S=(1.91,2.58);T=(-0.63,-0.11);U=(-2.37,-1.94);V=(2.38,-2.06);
label("1",R,N);label("2",S,N);label("3",T,N);label("4",U,N);label("5",V,N);
[/asy]](//latex.artofproblemsolving.com/4/6/4/46427c44da0a2576460300fa6c5f84472442b1e1.png)
Proposed by Nairi M. Sedrakyan, Armenia
![[asy]
pair A,B,C,D,E,F,G,H,I,J,K,L;
A=(-4.0,4.0);B=(-1.06,4.34);C=(-0.02,4.46);D=(4.14,4.93);E=(3.81,0.85);F=(3.7,-0.42);
G=(3.49,-3.05);H=(1.37,-2.88);I=(-1.46,-2.65);J=(-2.91,-2.52);K=(-3.14,-1.03);L=(-3.61,1.64);
draw(A--D);draw(D--G);draw(G--J);draw(J--A);
draw(A--G);draw(D--J);
draw(B--I);draw(C--H);draw(E--L);draw(F--K);
pair R,S,T,U,V;
R=(-2.52,2.56);S=(1.91,2.58);T=(-0.63,-0.11);U=(-2.37,-1.94);V=(2.38,-2.06);
label("1",R,N);label("2",S,N);label("3",T,N);label("4",U,N);label("5",V,N);
[/asy]](http://latex.artofproblemsolving.com/4/6/4/46427c44da0a2576460300fa6c5f84472442b1e1.png)
Proposed by Nairi M. Sedrakyan, Armenia
touches the sides 
and intersects the side 
are the same.
and 
are homotetic with center 
and WLOG 
and both 
lie closer to 
.
.
lie outside of a triangle and 
.
.
are tangent to the sides 
and 
correspondingly. 
have equal angle measures.
homotetic to 
and tangent to 
homotetic to 
and tangent to 
(it is always possible). If 
have equal arcs inside triangle 
(see figure) is homotetic to 
(with center 
.
(with respect to 
)
.
.
quadrilateral 5 is tangential.
has an incircle if and only if
"
and 
have inscribed circles, then we obtain quadrilateral 
also has an inscribed circle!
are concurrent. By applying the Monge d'Alembert theorem one can prove that if 
is the intersection of two diagonals then 
concurrent on 
.
and the circle with the circle lies on 
that touches 
is actually 
.
Let 
and 
are the radii of the incircles of 
and 
respectively.
. It is very sad story, that I did not solve it during contest 
and 1 other side of quadrilateral 5. Then Monge's theorem on circles 1,6,7 and circles 1,3,7 finishes the problem.
