[/quote]
,
be a prime and 
a nonempty set. Suppose that for each element 
there exist elements 
,
such that 
is divisible by 
,
denotes the cardinality of 
different point-like cities were founded one by one; from the second onwards, each newly founded city and the nearest existing city (the older one, if there were two) were declared sister cities. The surviving map of the empire shows the cities and the distances between them, but not the order in which they were founded. Historians have tried to deduce from the map that each city had at most 41 sister cities.
,
,
is divided into cyclic quadrilaterals with pairwise disjoint interiors. None of the vertices of the cyclic quadrilaterals in the decomposition is an interior point of a side of any cyclic quadrilateral in the decomposition or of a side of the quadrilateral 
.
is an integer. Is it true that for each perfect polynomial 
so that for each integer 
for which 
with the property that 
is divisible by 
.
be a positive integer, with divisors 
.
is always less than 
,
and 
be sequences of real numbers for which 
and
for all positive integers 
for all 
)
be an integer. Let 
,
,
be positive real numbers such that ![\[n^2 + 1 > \left( t_1 + t_2 + \cdots + t_n \right) \left( \frac{1}{t_1} + \frac{1}{t_2} + \cdots + \frac{1}{t_n} \right).\]](http://latex.artofproblemsolving.com/1/a/b/1ab410ebc17d9f6c38a3e7ae0f6efa955ffddfd3.png)
Show that 
,
,
are side lengths of a triangle for all 
,
,
.
be an acute triangle with incenter 
and circumcenter 
.
and 
at 
and 
respectively, and 
is the reflection of 
over 
meet at 
,
and 
,
.
and 
,
.
and 
.![\[P=\frac{1}{5-2a}+\frac{1}{5-2b}+\frac{1}{5-2c}.\]](http://latex.artofproblemsolving.com/c/3/7/c3708768a84c2bbb1e4fae502a9bb1e92cb13d02.png)
and 
Prove that
Y by Adventure10, Mango247
Let 
be the orthocenter of the acute triangle 
. Let 
and 
be altitudes of the triangle (
, 
). A variable line 
passing through intersects the segments ![$[BC']$](//latex.artofproblemsolving.com/b/6/0/b6013fe9fad33e3e8959c2e509ffa2beee29bc26.png)
and ![$[CB']$](//latex.artofproblemsolving.com/b/5/2/b5284ba241ca4c71e590f2d95cd40045852e6ff9.png)
in 
and 
. The perpendicular lines of from and intersect and in 
and 
. Determine the locus of the midpoint of the segment ![$[ PQ]$](//latex.artofproblemsolving.com/5/c/0/5c076b044ce2bda4259906c297b4488dc105383e.png)
.
Gheorghe Szolosy
be the orthocenter of the acute triangle 
. Let 
and 
be altitudes of the triangle (
, 
). A variable line 
passing through intersects the segments ![$[BC']$](http://latex.artofproblemsolving.com/b/6/0/b6013fe9fad33e3e8959c2e509ffa2beee29bc26.png)
and ![$[CB']$](http://latex.artofproblemsolving.com/b/5/2/b5284ba241ca4c71e590f2d95cd40045852e6ff9.png)
in 
and 
. The perpendicular lines of from and intersect and in 
and 
. Determine the locus of the midpoint of the segment ![$[ PQ]$](http://latex.artofproblemsolving.com/5/c/0/5c076b044ce2bda4259906c297b4488dc105383e.png)
.Gheorghe Szolosy
This post has been edited 9 times. Last edited by perfect_radio, Mar 16, 2006, 11:09 PM
,
,
,
.
.
and sorry for the typos 
.
of the initial problem!
-
(the page 51 with the number 5.17). Am I right ?![\[ \boxed {\ 2005\ -\ 2002\ =\ 3\ >\ 0\ } \]](http://latex.artofproblemsolving.com/d/8/0/d808289c087e4a719de80a28e484fad4384281dd.png)
Question: Who is the author of this problem from R.M.O., 2005 ?
I understood)
