AoPS Academy
[/quote]
, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
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 
. It is known that the graph of contains exactly 
chords, one of which has length 2025. Find the minimum possible value of .
has been partitioned into disjoint pairs 
(
) so that for all 
, 
equals 
or 
. Prove that the sum ![\[ |a_1-b_1|+|a_2-b_2|+\cdots +|a_{999}-b_{999}| \]](http://latex.artofproblemsolving.com/0/4/4/04422a7d2266b92a6b3f215b698f078cda85091f.png)
ends in the digit 
.
be an acute-angled triangle with circumcircle 
. A circle 
is internally tangent to at 
and also tangent to 
at 
. Let 
and 
intersect at 
and 
respectively. Let 
and 
be points on line such that 
is the midpoint of 
and 
is the midpoint of 
. Lines 
and 
meet at 
and intersect again at 
and 
respectively. The ray 
meets the circumcircle of triangle 
again at 
.
.
be a triangle and 
its circumcentre. A line tangent to the circumcircle of the triangle 
intersects sides at and at 
. Let 
be the image of under 
. Prove that the circumcircle of the triangle 
is tangent to the circumcircle of triangle .
be a semicircle with diameter and midpoint . Let be a point on different from and .
touches in a point , the segment 
in a point , and additionally the segment . The circle 
touches in a point and additionally the segments 
and .
and 
are perpendicular.
with no parallel sides is inscribed in a circle 
. Circles 
are inscribed in triangles 
, respectively. Common external tangents are drawn between 
and 
, and 
, and 
, and and , not containing any sides of quadrilateral . A quadrilateral whose consecutive sides lie on these four lines is inscribed in a circle 
. Prove that the lines joining the centers of and , and , and the centers of and all intersect at one point.
, 
, 
, 
are integers such that 
.
and 
of 
such that for each integer 
with 
, we have![\[ n\mid a_i - b_i - c_i
\]](http://latex.artofproblemsolving.com/1/a/e/1ae8d54f068b646ac7a1e43d75e7f26510cfdc95.png)
be an odd prime and 
be integers so that the integers 
are divisible by . divides each of .
be real numbers such that 
Prove that 
Where 
be a triangle with circumcircle 
. Let 
and 
respectively denote the midpoints of the arcs and that do not contain the third vertex. Let 
denote the midpoint of arc 
(the arc including ). Let be the incenter of . Let 
be the circle that is tangent to and internally tangent to at , and let 
be the circle that is tangent to and internally tangent to at . Show that the line 
, and the lines through the intersections of and , meet on .
friends, 
people are seated around a circular table in a fixed order. Ali places apples for serving directly in front of himself and wants to distribute them among everyone. Since Ali and his friends dislike eating alone and won't start unless everyone receives an apple at the same time, in each step, each person who has at least one apple passes one apple to the first person to their right who doesn't have an apple (in the clockwise direction). such that after some number of steps, the situation reaches a point where each person has exactly one apple.
with 
and the circle of radius centered at 
Let be the midpoint of 
The line 
intersects a second time at 
Let be a point on such that 
Let be the intersection of and Prove that 
,so if N is the reflection of A in C,we are done.
and A is the center of the circle 
,we have that 
which means that B,A,D,N are concyclic.
which holds iff 
,but because it holds iff 
which is true if 
.But 
so we are done.
and 
is isosceles.
and 
Prove that
Prove that 
is the reflection of ![\[-1=(C',C;B,E)\overset{D}=(C',C;M,N)\]](http://latex.artofproblemsolving.com/0/1/3/013f2653790e677e1891bb0495fd8e13785abd81.png)
Which is equivalent to 
.