[/quote]
,
be a positive integer. Find the number of permutations 
,
,
of the
,
,
, 
.
positive integer the equation: 
has positive integer solutions.
be an equilateral triangle. Let 
be a point inside of it such that the square root of the distance of 
of positive integers is called central if for every positive integer 
terms of the sequence is equal to 
of positive integers such that for every central sequence 
there are infinitely many positive integers 
.
,
be two sequences of positive real numbers. Suppose 
is a sequence of positive real numbers such that 
for all 
.
.![\[
\left( \frac{M+z_2+\dots+z_{2n}}{2n} \right)^2
\ge
\left( \frac{x_1+\dots+x_n}{n} \right)
\left( \frac{y_1+\dots+y_n}{n} \right). \]](http://latex.artofproblemsolving.com/8/c/4/8c4a72f57363eb1ef114825c8d36191bd5f42cb7.png)
be a triangle. On side 
take points 
and 
such that 
take points 
and 
such that 
and on side 
take points 
such that 
Let 
and 
are concurrent.
that verify the following equation :
lie the following points: 
and 
on 
,
on 
,
on 
.![\[
P = AM\cap BN,\quad
R = KM\cap LN,\quad
S = KN\cap LM,
\]](http://latex.artofproblemsolving.com/7/e/a/7ea90a0f1261ed7445bbac58fb0cdd76810259f3.png)
and let the line 
meet 
.
,
are collinear.
.
intersects 
and 
and 
,
of triangle 
lies on the circle 
and 
coincide.
be a triangle and 
and 
be two points on 
,
such that 
and let 
.
to the interior angle bisector of 
intersect 
in 
.
.
,
be its incenter. Incircle touches sides 
at 
,
to 
, respectively. Rays 
intersect 
at 
,
touches incircle at 
.
Y by anantmudgal09
Let 
be a positiv integer. Ana chooses the positive integers 
and for any non-empty subset 
she computes the sum ![\[s_A=\sum_{k
\in A}a_k.\]](//latex.artofproblemsolving.com/7/8/a/78afe6d1036a006b2252023d8f9619285fe0c6d5.png)
She orders these sums 
Prove that there exists a subset 
with 
elements such that, regardless of the integers chosen by Ana, these can be determined by only knowing the sums 
with 
be a positiv integer. Ana chooses the positive integers 
and for any non-empty subset 
she computes the sum ![\[s_A=\sum_{k
\in A}a_k.\]](http://latex.artofproblemsolving.com/7/8/a/78afe6d1036a006b2252023d8f9619285fe0c6d5.png)
She orders these sums 
Prove that there exists a subset 
with 
elements such that, regardless of the integers chosen by Ana, these can be determined by only knowing the sums 
with 
This post has been edited 1 time. Last edited by oVlad, Apr 12, 2025, 9:45 AM
and 
contains 
does it solve the problem?Since it contains the single element?
denotes the 
-
.
and 
are rather small, but all other 
'
not 
so it's useless information.
