Prove that
Let 
Prove that
has rational sides and equal angles. Show that it is regular.
such that the equation
and 
its altitude. We draw parallel lines from 
to the vertical sides of the triangle and we call 
their points of intersection with 
respectively. The parallel line from 
to 
intersects the line 
be the symmetric of 
with respect to the line 
the projections of 
onto 
is the point of intersection of the lines 
and 
,
.
be defined by 
(a) Show that 
has exactly one point of discontinuity.
times differentiable real function 
has at least 
distinct zeros in the closed interval ![$[a,b]$](http://latex.artofproblemsolving.com/8/e/c/8ecbd1ba3da8f2adef66a63f2ab32c47e63fa734.png)
and that the polynomial 
has only real zeroes. Show that
has at least one zero in 
denotes the 
be a closed map with the property that the inverse image of each point in 
is a compact subset of 
.
is compact whenever 
is compact in 
denote the sum of the alternating harmonic series. Rearrange this series as follows
Assume as known that this series converges as well and denote its sum by 
.
respectively the 
-
has always integral solutions for 
and 
whenever 
is a positive integer.
is a sequence of real numbers such that
Prove that if 
converges, then 
for all 
where 
,
the tangents at the points of intersection are concurrent.
Let 
symmetric matrix such that each row and column consists of some permutation of the integers 
Show that each of the integers 
must appear in the main diagonal of 
convergent?
Y by NO_SQUARES, ehuseyinyigit
Let 
be a triangle with incircle 
, which touches sides 
and 
at points 
and 
, respectively. Choose points 
and 
on the line 
such that 
and 
. Let 
be the intersection of lines 
and 
. Define 
and 
as the intersections of 
and 
with lines 
and 
, respectively. Assume that 
is the intersection of 
and 
. Prove that 
.
be a triangle with incircle 
, which touches sides 
and 
at points 
and 
, respectively. Choose points 
and 
on the line 
such that 
and 
. Let 
be the intersection of lines 
and 
. Define 
and 
as the intersections of 
and 
with lines 
and 
, respectively. Assume that 
is the intersection of 
and 
. Prove that 
.Attachments:
at 
.
ia a parallelogram. 
intersect 
at 
at 
bisect 
.
bisects segment 
,
