[/quote]
,
around a circle so that the sum of every 
be a scalene triangle, and let 
be a point on side 
satisfying 
.
and 
are points inside 
and 
are similar and quadrilaterals 
and 
are cyclic. Let lines 
and 
meet at 
and lines 
and 
meet at 
.
and 
are parallel.
be an interior point of a triangle 
meet the sides 
in 
respectively. Show that ![\[ \frac{AP}{PD} = \frac{AF}{FB} + \frac{AE}{EC}. \]](http://latex.artofproblemsolving.com/7/5/9/759482bace109eb619aaea692625f7854faa9fec.png)
.
represented by tokens to players 
to 
so that they all have the same number of tokens in their urns.
beats 
when, when 
.
What is 
satisfy 
.
and the incircle 
of 
at 
,
meets 
at 
and intersects 
at 
meets 
at 
.
(different from 
and 
.
.
is situated inside the parallelogram 
such that 
.
.
of 
be the projection of 
.
that is not on 
intersects 
at 
.
and 
intersect at 
.
is an isosceles triangle.
.
on 
respectively. Circumcircle 
of triangle 
.
intersects 
at 
.
cuts 
respectively. Construct parallelogram 
.
are collinear.
a grid of 9*9, we choose a little square in 
is a strictly increasing sequence of positive integers such that the sub-sequences ![\[s_{s_1},\, s_{s_2},\, s_{s_3},\, \ldots\qquad\text{and}\qquad s_{s_1+1},\, s_{s_2+1},\, s_{s_3+1},\, \ldots\]](http://latex.artofproblemsolving.com/2/6/8/2686c6e4c1110bce322dd876aba37d5273bffe05.png)
are both arithmetic progressions. Prove that the sequence 
is itself an arithmetic progression.
:
and 
on the points 
meet the segment 
at its midpoint if and only if 
.
Prove that
Let 
Prove that
Let 
Prove that
Y by Funcshun840
A computer science teacher has asked his students to write a program that, given a list of 
numbers 
, calculates the list 
where 
is the number of times the number 
appears in the list. So, for example, for the list 
, the program returns the list 
.
Next, the teacher asked Alexandre to run the program for a list of
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to 
appears in the list. Find the largest value of for which, whatever the initial list of positive integers 
, it is possible for Alexander to do what the teacher asked him to do.
numbers 
, calculates the list 
where 
is the number of times the number 
appears in the list. So, for example, for the list 
, the program returns the list 
.Next, the teacher asked Alexandre to run the program for a list of
numbers. Then he asked him to apply the program to the resulting list, and so on, until a number greater than or equal to 
appears in the list. Find the largest value of for which, whatever the initial list of positive integers 
, it is possible for Alexander to do what the teacher asked him to do.
and we are done.
