[/quote]
,
of positive integers is called central if for every positive integer 
,
terms of the sequence is equal to 
,
,
,
of positive integers such that for every central sequence 
,
,
,
.
be an acute triangle. Points 
,
lie on a line in this order and satisfy 
.
and 
be the midpoints of 
and 
,
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
,
and 
and 
and 
be all the positive integers smaller than 
such that
for all 
.
be non-negative real variables with 
Prove that
Equality holds iff 
or 
where 
?
be a convex quadrilateral such that diagonals 
and 
intersect at right angles, and let 
be their intersection. Prove that the reflections of 
are concyclic.
and 
Prove that
Let 
Prove that
we are given :
and one side length of the triangle equals 
,
be a trapezoid with parallel sides 
and 
,
.
intersects 
again at 
respectively. Prove that 
are concurrent.
and 
Prove that
in it. Prove that there are an infinite number of perfect cubes in this progression. (A perfect cube is an integer of the form 
,
is an integer. For example, 
,
and 
there are elements 
such that 
.
days later (but perhaps earlier, as well). Hedvig received the trophy today. Determine all integers 
for which we can be absolutely certain that she cannot receive the trophy again in 
Y by Adventure10
The cyclic quadrilateral 
is given. Let the extensions of the sides 
and 
beyond points 
and 
, respectively, intersect at the point 
. We denote the feet of the perpendiculars from the point on the diagonals 
and 
by 
and 
, respectively. Prove that 
where, 
is the midpoint of the side 
.
is given. Let the extensions of the sides 
and 
beyond points 
and 
, respectively, intersect at the point 
. We denote the feet of the perpendiculars from the point on the diagonals 
and 
by 
and 
, respectively. Prove that 
where, 
is the midpoint of the side 
.This post has been edited 1 time. Last edited by parmenides51, Jun 21, 2022, 2:28 AM
and Let 
be the midpoints of 
then :
are coaxial and their radical axes is line 
where 
are the orthocenters of 
and we have : 
we only need to show 
