[/quote]
,
,
is the point of tangency of the incircle with the side 
.
and 
are the intersections of the angle bisectors of 
and 
with the circumcircle of 
be the antipodal point of 
in the circumcircle of 
,
be the antipodal point of 
.
and 
are similar.
be a positive integer. Prove that there exist three permutations 
,
,
of 
such that ![\[\left|\sqrt{a_k}+\sqrt{b_k}+\sqrt{c_k}-2\sqrt{N}\right|<2023\]](http://latex.artofproblemsolving.com/9/8/0/98097fb721722b9103ef65d72322956877e262cb.png)
for every 
.
such that
are all powers of 
.
passes through 
and 
such that 
and 
at 
at 
.
is parallel to 
with real coefficients such that 
.
polynomial:![\[\underbrace{P(P(\ldots P(P}_{n}(x))\ldots))- \underbrace{Q(Q(\ldots Q(Q}_{n}(x))\ldots))\]](http://latex.artofproblemsolving.com/5/e/1/5e17b712fca5c08dc9004701f8fc4c12fb81b204.png)
.
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
.
be the midpoint of 
be an arbitrary point on 
.
over 
,
and 
lies on 
.
define 
if 
or 
,
where 
are all prime factors of 
with nonnegative integer coefficients such that 
divides 
for every nonnegative integer 
satisfy the equation 
.
such that
be a given integer. Prove that infinitely many terms of the sequence 
,![\[a_k=\left\lfloor\frac{n^k}{k}\right\rfloor,\]](http://latex.artofproblemsolving.com/8/3/2/832113b1ef1fe211e1eaead25129873aead1dab4.png)
are odd. (For a real number 
,
denotes the largest integer not exceeding 
all of whose digits are odd.
,
,
,
be distinct positive integers, 
.
and 
such that 
does not divide any of the numbers 
,
,
.
Y by Adventure10, Mango247
Quadrilateral 
inscribed in a circle 
. Diagonals 
and 
intersect at a point 
. On segments 
and 
points 
and 
are selected respectively. Straight 
crosses in points 
and 
. The circumscribed circles of triangles 
and 
cross the segment in points 
and 
respectively (assume that all points , , , , , are different). Prove that 
.
inscribed in a circle 
. Diagonals 
and 
intersect at a point 
. On segments 
and 
points 
and 
are selected respectively. Straight 
crosses in points 
and 
. The circumscribed circles of triangles 
and 
cross the segment in points 
and 
respectively (assume that all points , , , , , are different). Prove that 
.
.
so 
so 
as desired.
