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be a convex quadrilateral with 
and 
. Let 
and 
be points on segments 
and 
, respectively, such that line 
intersects lines 
and 
at points 
and 
, respectively. It is given that 
.Let the midpoint of 
be 
and the midpoint of be 
.Prove that the points 
and 
lie on a circle.
points are marked on it, dividing it into equal arcs. Petya and Vasya are playing the following game. Petya chooses a positive integer 
and announces this number to Vasya. To win the game, Vasya needs to color all marked points using 
colors, such that each color is assigned to exactly two points, and for each pair of same-colored points, one of the arcs between them contains exactly 
marked points. Find all for which Petya will be able to prevent Vasya from winning.
board has some of its cells coloured red, blue, or green. Each cell is coloured with at most one colour, and some cells may remain uncoloured. Additionally, there is at least one cell of each colour. Two coloured cells are said to be friends if they have different colours and lie in the same row or in the same column. The following conditions are satisfied:
, , and 
be positive integers. Prove that ![\[ \sum_{i = 1}^n \min\left(\left\lfloor \frac{x}{i} \right\rfloor, m \right) = \sum_{i = 1}^m \min\left(\left\lfloor \frac{x}{i} \right\rfloor, n \right). \]](http://latex.artofproblemsolving.com/b/7/a/b7aebfe11f646d604d2e8cb9d5bd70ec796949e8.png)
o 
cyclic we need that 
and 
are both beyond (with respect to 
).
and is cyclic 
.
.
and 
and 
.
.
is cyclic.
such that 
.we know that 
is cyclic and 
is cyclic . If 
then applying Reim to ![$ (KLB'C'),(ALB') [=ABC]$](http://latex.artofproblemsolving.com/1/2/d/12df27435c81cfb359ee1ea4022793f4ad28f064.png)
yields 
which is the tangent to 
at 
. let 
point of 
in the same semi-plane bounded by 
as .
thus 
then 
are colinear . cyclic 
from 
cyclic we deduce 
thus
a triangle, let 
be the foot of the bisector relative to and 
be the foot of the trisectrix relative to closer to the side 
(
). Let and be two point on the line 
on the side of with respect to , such that 
and 
. Prove that is cyclic if and only if lines 
and 
are parallel.
