[/quote]
,
,
,
,
.
and 
.
bisects the line segment 
.
,
.
and the external angle bisector of 
at points 
and 
,
,
.
![$P \in \mathbb Q[x,y]$](http://latex.artofproblemsolving.com/a/3/8/a384cbf38fd3a7f812b22baaeb0860e7f6eb26c0.png)
with 
and ![$\forall (a,b) \in \mathbb Z^2 \cap [0,d]^2, P(a,b) \in \mathbb Z$](http://latex.artofproblemsolving.com/a/6/5/a65ecb4bb2d9b5e6096f322c2de8bd11b945ed6a.png)
.
?
,
,
are positive real numbers, prove that![\[ \left(x + y + z\right)^2 \left(yz + zx + xy\right)^2 \leq 3\left(y^2 + yz + z^2\right)\left(z^2 + zx + x^2\right)\left(x^2 + xy + y^2\right) .\]](http://latex.artofproblemsolving.com/4/b/9/4b9b7fdbc13c51277b4f0b7f2083b3b385b9c8f9.png)
and 
.
and 
intersect at point 
,
is the point on segment 
such that 
.
is isosceles.
and let 
be nonnegative reals with sum at most 
.
and 
such that![\[\sum_{i=1}^k n_ia_{n_{i-1}}<2005.\]](http://latex.artofproblemsolving.com/d/b/b/dbb91fe3e036496f59530c371eafd32eae191e55.png)
every next number (from the fifth number) is the unit number of the sum of the four numbers preceding it. Is there any cases that we get the numbers 
and 
in this series?
such that 
is unbounded and![\[2f(m)f(n)-f(n-m)-1\]](http://latex.artofproblemsolving.com/0/f/7/0f77b5addbf8f5e603eb1b6923cdac795cd132e9.png)
is a perfect square for all integer 
and 
be the circumcircle and the circumcentre of an acute-angled triangle 
.
intersects 
.
be the circle with diameter 
.
and 
intersect 
and 
respectively. The point 
is chosen on the line 
so that 
.
.
,
be the midpoint of 
on 
is such that 
is an altitude of the triangle 
be the reflection of 
,
and 
be the point such that the circumcentre of triangle 
is the midpoint of the segment 
.
.
and 
.
Y by
Let 
be the center of the circle inscribed in a triangle 
. Let 
be a circle passing through point 
and touching 
at point . Let 
be a circle passing through point 
and touching 
at point . Prove that one of the points of intersection of the circles 
lies on the circle circumscribed around triangle .
be the center of the circle inscribed in a triangle 
. Let 
be a circle passing through point 
and touching 
at point . Let 
be a circle passing through point 
and touching 
at point . Prove that one of the points of intersection of the circles 
lies on the circle circumscribed around triangle .This post has been edited 1 time. Last edited by parmenides51, Mar 13, 2021, 7:04 AM
