[/quote]
,
be a continuous function. A chord is defined as a segment of integer length, parallel to the x-axis, whose endpoints lie on the graph of 
.
chords, one of which has length 2025. Find the minimum possible value of 
has been partitioned into disjoint pairs 
)
,
equals 
or 
.![\[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \]](http://latex.artofproblemsolving.com/0/4/4/04422a7d2266b92a6b3f215b698f078cda85091f.png)
ends in the digit 
.
be an acute-angled triangle with circumcircle 
.
is internally tangent to 
and also tangent to 
at 
.
and 
intersect 
and 
respectively. Let 
and 
be points on line 
is the midpoint of 
and 
is the midpoint of 
.
and 
meet at 
and intersect 
and 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
.
its circumcentre. A line tangent to the circumcircle of the triangle 
intersects sides 
.
be the image of 
.
is tangent to the circumcircle of triangle 
be a semicircle with diameter 
touches 
in a point 
touches 
and 
and 
are perpendicular.
with no parallel sides is inscribed in a circle 
.
are inscribed in triangles 
,
and 
,
,
,
.
and 
Prove that
,
,
,
are integers such that 
.
and 
of 
such that for each integer 
with 
,![\[ n\mid a_i - b_i - c_i
\]](http://latex.artofproblemsolving.com/1/a/e/1ae8d54f068b646ac7a1e43d75e7f26510cfdc95.png)
be an odd prime and 
be integers so that the integers 
are divisible by 
Prove that 
be real numbers such that 
Prove that 
Where 
.
and 
respectively denote the midpoints of the arcs 
denote the midpoint of arc 
(the arc 
be the circle that is tangent to 
be the circle that is tangent to 
,
friends, 
people are seated around a circular table in a fixed order. Ali places 
Y by Adventure10, Mango247
Let 
be a cube. On the sides 
, 
and 
we consider the points 
, 
and 
respectively. On the sides 
, 
and 
we consider the points 
, 
and 
respectively. Let 
be the distance between the lines 
and 
, 
be the distance between the lines 
and 
, and 
be the distance between the lines 
and 
. Suppose that the distances , and are pairwise distinct. Prove that the lines 
, 
and 
are concurrent.
be a cube. On the sides 
, 
and 
we consider the points 
, 
and 
respectively. On the sides 
, 
and 
we consider the points 
, 
and 
respectively. Let 
be the distance between the lines 
and 
, 
be the distance between the lines 
and 
, and 
be the distance between the lines 
and 
. Suppose that the distances , and are pairwise distinct. Prove that the lines 
, 
and 
are concurrent.
