[/quote]
,
and 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
is the incentre and point 
is defined such that 
and 
is the perpendicular bisector of 
and 
be the circles passing through 
and 
respectively such that 
is tangent to 
to be a point such that it lies on both the circles 
and 
are perpendicular.
be an acute triangle. Let 
be a point on side 
be a point on side 
such that lines 
are parallel. Let 
be an interior point of 
.
and 
meet side 
and 
,
and 
.
and 
intersect at a point 
.
,
are collinear.
.
and 
,
as the greatest exponent 
such that 
divides 
.
and infinitely many 
such that 
.
and 
.
is given. Let 
to 
lies on the arc 
.
.
with diameter 
. The circle with center 
meets 
at 
.
is perpendicular to 
.
![\[ x + y^2 = p^m \]](http://latex.artofproblemsolving.com/e/c/f/ecf553036f22327dd457dc06611b6bc1b0ef072c.png)
![\[ x^2 + y = p^n \]](http://latex.artofproblemsolving.com/5/b/d/5bd672934a395aabd6865ca4b50d8da75c0fe07e.png)
For 
positive integers and 
being a prime.
and 
.
Equality holds when 
or 
.
intersect again at 
.
Y by
On the sides 
and 
of the rectangle 
, mark points 
and 
respectively, that the area of the triangles 
, 
, 
is exactly the same. Find the length of the segment 
, if the length of the segment 
is equal to 
.
and 
of the rectangle 
, mark points 
and 
respectively, that the area of the triangles 
, 
, 
is exactly the same. Find the length of the segment 
, if the length of the segment 
is equal to 
.This post has been edited 1 time. Last edited by parmenides51, Dec 19, 2022, 12:26 PM
