[/quote]
,
and 
be positive integers.
and 
.
.
and 
be the two foci of ellipse 
.
is a point on 
be the center of the excircle of 
.
,
elements with 
of 
if the product of the elements of 
be positive reals with sum 1. Show that for any 
,
satisfying 
for all 
,
.
,
,
are pairwise distinct real numbers satisfying 
Prove that
Let 
Prove that
chessboard such that no two pawns attack each other?
and 
be two positive integers. Prove that the integer![\[a^2+\left\lceil\frac{4a^2}b\right\rceil\]](http://latex.artofproblemsolving.com/1/4/5/14509b500811314c0c363da994e6de875ba468ba.png)
is not a square. (Here 
denotes the least integer greater than or equal to 
,
be a point on line 
satisfying 
.
be tangent to 
at 
.
on 
and 
on 
analogously, using the excircles opposite 
and 
,
is cyclic, then 
is tangent to the circumcircle of 
.
and 
Prove that
Let 
Prove that
and incenter 
,
are selected on sides 
,
,
.
intersects 
other than 
and 
meet on 
Y by Mango247, Mango247
Around the acute angle triangle 
circumscribe a circle. The chord 
is the bisector of the angle of the triangle and intersects the side BC at the point 
, the chord 
is perpendicular to its side 
and intersects it at the point 
. Find the ratio 
if 
(Tooth Vladimir)
circumscribe a circle. The chord 
is the bisector of the angle of the triangle and intersects the side BC at the point 
, the chord 
is perpendicular to its side 
and intersects it at the point 
. Find the ratio 
if 
(Tooth Vladimir)
This post has been edited 2 times. Last edited by parmenides51, Jul 14, 2020, 9:57 PM
and 
so 
is angle bisector. But 
