[/quote]
,
,
,
is the point on 
such that 
.
be the circle through 
at 
the circle through 
be the second intersection of 
lies on 
.
Prove that
is given. It is known that triangles 
,
,
,
are all acute. Prove that the orthocenters of these triangles, together with the centroid of triangle 
be reals such that 
and 
Prove that
and 
with integer coefficients is called 
if from the divisibility of both differences 
and 
by 
,
and 
are divisible by 100. Does there exist such an important pair of polynomials 
,
,
and 
is also important?
satisfy:
![$[(x-2)^2+(y-4)^2][ (x-1)^2 + (y-2)^2] = 36$](http://latex.artofproblemsolving.com/b/1/0/b108dc2cac5b7491fb83418eb36c7184f6d5d30c.png)
be real numbers such that 
.
such that: 
.
and 
.
grid, taking turns. Petya goes first. On his turn, a player writes an uppercase Russian letter in an empty cell (each cell can contain only one letter). When all cells are filled, Petya is declared the winner if there are four consecutive cells horizontally spelling the word ``ПЕТЯ'' (PETYA) from left to right, or four consecutive cells vertically spelling ``ПЕТЯ'' from top to bottom. Can Petya guarantee a win regardless of Vasya's moves?
satisfy 
.
Also, find the values of 
and 
.
Y by Adventure10
Given a circle 
, choose a diameter 
and mark an arbitrary point 
other than 
and 
. In one of the two arcs determined by the diameter , consider two points 
and 
such that 
. Draw the segments 
and 
to obtain three curvilinear triangles 
and 
. (The sides of the curvilinear triangle 
are the segments 
and 
and the arc 
). In each triangle inscribe a circle. Show that the sum of the radii of the three constructed circles is less than or equal to the radius of .
, choose a diameter 
and mark an arbitrary point 
other than 
and 
. In one of the two arcs determined by the diameter , consider two points 
and 
such that 
. Draw the segments 
and 
to obtain three curvilinear triangles 
and 
. (The sides of the curvilinear triangle 
are the segments 
and 
and the arc 
). In each triangle inscribe a circle. Show that the sum of the radii of the three constructed circles is less than or equal to the radius of .This post has been edited 1 time. Last edited by parmenides51, Sep 20, 2022, 4:54 AM
