[/quote]
,
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
,
,
is the point on 
such that 
.
be the circle through 
tangent to 
at 
the circle through 
at 
be the second intersection of 
lies on 
.
Prove that
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?
.
Also, find the values of 
and 
.
Y by Adventure10, Mango247
Circles 
and 
are tangent internally to circle 
at points 
and 
, respectively, as seen in the figure. The inner tangent common to and touches these circles in 
and 
, respectively. Show that the 
and 
lines intersect the circle at diametrically opposite points.
and 
are tangent internally to circle 
at points 
and 
, respectively, as seen in the figure. The inner tangent common to and touches these circles in 
and 
, respectively. Show that the 
and 
lines intersect the circle at diametrically opposite points.This post has been edited 2 times. Last edited by parmenides51, Dec 11, 2022, 12:18 AM
be the circle centers and let the common tangent line shown be 
This takes 
to a parallel line 
which is tangent to 
then clearly 
so 
must be the midpoint of the arc not containing 
But this tangent point is the image of 
and 
and 
)
which completes the solution
