AoPS Academy
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 .
be real numbers such that 
Find the maximum value of the expression 
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.
moves each. In each move, Petya chooses 9 consecutive numbers and decreases each by 2. Vasya chooses 10 consecutive numbers and increases each by 1. They alternate turns, starting with Petya. Prove that Vasya can act in such a way that after each of his moves, there are always at least five positive numbers, regardless of how Petya plays.
be the point of tangency of the excircle of a triangle 
(corrsponding to 
) with the side 
. The line 
through is parallel to the bisector of 
. Lines 
and 
are analogously defined. Prove that 
have a common point.
.
denote the incentre of 
.
denote the second intersections of corresponding external bisectors of the angles of the triangle with the circumcircle 
of 
denote the centre of spiral sim sending 
.
we have 
with 
.
.
holds (as do the other analogous equalities). From here, we remark that the perpendiculars from 
,
and 
to lines 
concur. But now remark that 
and analogous perpendicularities imply that these perpendiculars are in fact just the lines 