and 
pebbles, respectively. In the first round Barry chooses a positive integer 
and Andrew takes away 
pebbles from one of the two heaps (if 
,
,
,
.
at 
.
similarly.Prove that 
are collinear.
grid with 
rows and 
columns, and each square of the grid is filled with a non-negative integer. Let 
the there exist two identical rows (meaning same numbers in the same order).
and natural number 
are there such that 
and 
?
and 
.
be real numbers such that 
Prove that
Let 
Prove that
and 
, then we can deduce 
\le 2 $ ?
Y by
Let 
and 
be points on a circumference 
with center 
, in that order, such that 
is perpendicular to 
; 
is their intersection. 
is the convex quadrilateral such that and are the tangency points of with segments 
and 
respectively. The perpendicular lines to 
and 
through 
and 
, respectively, intersect at 
; the perpendicular lines to and 
through and 
, respectively, intersect at 
, and the perpendicular lines to and 
through and 
, respectively, intersect at 
. 
is the circumcenter of triangle 
. Prove that 
and are collinear.
Proposed by CDMX
and 
be points on a circumference 
with center 
, in that order, such that 
is perpendicular to 
; 
is their intersection. 
is the convex quadrilateral such that and are the tangency points of with segments 
and 
respectively. The perpendicular lines to 
and 
through 
and 
, respectively, intersect at 
; the perpendicular lines to and 
through and 
, respectively, intersect at 
, and the perpendicular lines to and 
through and 
, respectively, intersect at 
. 
is the circumcenter of triangle 
. Prove that 
and are collinear.Proposed by CDMX
This post has been edited 2 times. Last edited by Justpassingby, Jan 20, 2022, 1:31 AM
Reason: Added proposer
Reason: Added proposer
be the intersection of the lines perpendicular to 
.
say)
.
are colinear (and so are 
)
.
,
and 
-
.
are colinear: 
so 
and 
so 
.
to be parallel to 
.
