,
,
,
as seen in the following picture.
.
until 
divides 
.
meters. From what height is the ball was dropped?
is obtained by pasting two unit cubes 
and 
.
,
is the midpoint of the edge 
.
?
,
when divided by 
,
,
,
?
and 
be real numbers. Let 
be the matrix with entries ![\[a_{ij} = \begin{cases}1,&\text{if }x_i + y_j\geq 0;\\0,&\text{if }x_i + y_j < 0.\end{cases}\]](http://latex.artofproblemsolving.com/c/0/8/c0856fe7b2b113180659952c4d216a01d8849f64.png)
Suppose that 
is an 
matrix with entries 
,
.
.
.
.
are collinear, and 
are arbitrary points
be the sum of the squares of the digits of 
.
such that 
.
with centroid 
and circumcircle 
centred at 
,
meets 
;
intersect at 
;
and 
intersect at 
.
of the triangle 
lies on 
are collinear. Prove that 
.
,
and 
be points on sides 
is cyclic. Let lines 
and 
intersect at point 
be the midpoints of 
and 
,
is the foot of the perpendicular from 
,
and 
intersect at second point 
different from 
and 
,
for all integers 
.
of positive integers, with 
is a perfect square.
.
lie on 
are isogonal in 
.
.
lies on 
such that 
.
intersects 
passes through the Euler center of triangle 
Y by
Let 
be a convex quadrilateral and let 
be the centroid, circumcentre, orthocenter & nine-point centre of 
respectively. We define the points 
analogously. Now, we define the area centroid 
of as the intersection of 
the qausicircumcentre 
as the intersection of 
quasiorthocentre 
as the intersection of 
and the quasinine-point centre 
as the intersection of 
Prove that the points 
are collinear with 
and 
be a convex quadrilateral and let 
be the centroid, circumcentre, orthocenter & nine-point centre of 
respectively. We define the points 
analogously. Now, we define the area centroid 
of as the intersection of 
the qausicircumcentre 
as the intersection of 
quasiorthocentre 
as the intersection of 
and the quasinine-point centre 
as the intersection of 
Prove that the points 
are collinear with 
and 
This post has been edited 1 time. Last edited by Entrepreneur, Nov 15, 2024, 8:23 AM
