Sequence inequality
by BR1F1SZ, May 10, 2025, 11:32 PM
Let 
be a sequence of positive integers satisfying the following property: for all positive integers 
, for all distinct integers 
and for all distinct integers 
,
![\[
a_{m_1} + a_{m_2} + \cdots + a_{m_k} \leqslant a_{n_1} + a_{n_2} + \cdots + a_{n_\ell}.
\]](//latex.artofproblemsolving.com/d/1/6/d1632ff7e5e55319cdbf56f45dca9ed84343aa47.png)
Prove that there exist two integers 
and 
such that 
for all 
.
be a sequence of positive integers satisfying the following property: for all positive integers 
, for all distinct integers 
and for all distinct integers 
,![\[
a_{m_1} + a_{m_2} + \cdots + a_{m_k} \leqslant a_{n_1} + a_{n_2} + \cdots + a_{n_\ell}.
\]](http://latex.artofproblemsolving.com/d/1/6/d1632ff7e5e55319cdbf56f45dca9ed84343aa47.png)
Prove that there exist two integers 
and 
such that 
for all 
.
on the board. Charlotte has two operations available: the GCD operation and the LCM operation.
and 
.
.
and, for each of them, determine the minimum number of GCD operations Charlotte must use.
be an integer. We consider a square grid of size 
divided into 
unit squares. The grid is called balanced if:
,
or 
.
.
,
such that any balanced grid always contains an 
square whose absolute sum of the 
cells is less than or equal to 
.
of distinct positive real numbers is called radiant if it satisfies the following property: if 
is also an element of 
?
or 
.
.
such that 
.
into two regions, two parallel planes divide it into three regions, and two intersecting planes divide space into four regions. Consider the six planes which the faces of the cube 
lie on, and the four planes that the tetrahedron 
lie on. How many regions do these ten planes split the space into?
be an acute-angled triangle in which 
and 
.
lie on segment 
and point 
lie on segment 
such that 
,
and 
.
be the circumcenter of triangle 
,
the orthocenter of triangle 
the point of intersection of the lines 
and 
.
to the system:![\[a_1^2 + a_1 - 1 = a_2\]](http://latex.artofproblemsolving.com/a/4/e/a4e26b83c929b82800576bac11e12de0c3aa1cf6.png)
![\[ a_2^2 + a_2 - 1 = a_3\]](http://latex.artofproblemsolving.com/9/9/5/995e8eec93828b82901b0639463f907f5ff95d08.png)
![\[\hspace*{3.3em} \vdots \]](http://latex.artofproblemsolving.com/e/e/1/ee15e1fefe2ad550d3c9b1c51207eaf66fbfa41f.png)
![\[a_{n}^2 + a_n - 1 = a_1\]](http://latex.artofproblemsolving.com/7/c/7/7c727d1f2044c1bf7c80981e1074bb3cde807b8f.png)
November 2016