[/quote]
,
be a function such that 
The fractional part of 
can be expressed in the form 
,
and 
are relatively prime positive integers. Find the remainder when 
is divided by 
.
is a function satisfying 
and 
,
.
then minimum possible number of values of 
satisfying 
,![$x\in[0,170]$](http://latex.artofproblemsolving.com/3/d/2/3d2c6ead3059f96f853bba6cfc7440e394227ab9.png)
is?
$
be real numbers such that 
.
a function satifying 
for all 
.
,
,
such that for all 
,![\[
mn + f(n!) = f(f(n))! + n \cdot \gcd(f(m), m!).
\]](http://latex.artofproblemsolving.com/c/0/c/c0cdf1bcb8053f5dc26e9df6355207b5e479f861.png)
and the recursive relation
for all natural number 
.
.
and 
Prove that
of the set 
,
such that:
,
,
,
.
?
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
Prove that
Let 
Prove that
.
measures 40 units. Let 
be the orthocenter of triangle 
its circumcenter. Let 
be the foot of the altitude from 
and 
the foot of the altitude from 
.
.
passes through point 
.
Y by
Problem Statement
Given a positive integer n \geq 3 , consider a permutation \pi = (a_1, a_2, \dots, a_n) of \{1, 2, \dots, n\} . For each i ( 1 \leq i \leq n-1 ), define d_i as the minimum absolute difference between a_i and any subsequent element a_j ( j > i ), i.e.,
d_i = \min \{ |a_i - a_j| \mid j > i \}.
Let S_n denote the maximum possible sum of d_i over all permutations of \{1, \dots, n\} , i.e.,
S_n = \max_{\pi} \sum_{i=1}^{n-1} d_i.
Proposed Construction
I found a method to construct a permutation that seems to maximize \sum d_i :
1. Fix a_{n-1} = 1 and a_n = n .
2. For each i (from n-2 down to 1 ):
- Sort a_{i+1}, a_{i+2}, \dots, a_n in increasing order.
- Compute the gaps between consecutive elements.
- Place a_i in the middle of the largest gap (if the gap has even length, choose the smaller midpoint).
Partial Results
1. I can prove that 1 and n must occupy the last two positions. Otherwise, moving either 1 or n further right does not decrease \sum d_i .
2. The construction greedily maximizes each d_i locally, but I’m unsure if this ensures global optimality.
Request for Help
- Does this construction always yield the maximum S_n ?
- If yes, how can we rigorously prove it? (Induction? Exchange arguments?)
- If no, what is the correct approach?
Observations:
- The construction works for small n (e.g., n=3,4,5,...,12 ).
- The problem resembles optimizing "minimum gaps" in permutations.
Any insights or references would be greatly appreciated!
Given a positive integer n \geq 3 , consider a permutation \pi = (a_1, a_2, \dots, a_n) of \{1, 2, \dots, n\} . For each i ( 1 \leq i \leq n-1 ), define d_i as the minimum absolute difference between a_i and any subsequent element a_j ( j > i ), i.e.,
d_i = \min \{ |a_i - a_j| \mid j > i \}.
Let S_n denote the maximum possible sum of d_i over all permutations of \{1, \dots, n\} , i.e.,
S_n = \max_{\pi} \sum_{i=1}^{n-1} d_i.
Proposed Construction
I found a method to construct a permutation that seems to maximize \sum d_i :
1. Fix a_{n-1} = 1 and a_n = n .
2. For each i (from n-2 down to 1 ):
- Sort a_{i+1}, a_{i+2}, \dots, a_n in increasing order.
- Compute the gaps between consecutive elements.
- Place a_i in the middle of the largest gap (if the gap has even length, choose the smaller midpoint).
Partial Results
1. I can prove that 1 and n must occupy the last two positions. Otherwise, moving either 1 or n further right does not decrease \sum d_i .
2. The construction greedily maximizes each d_i locally, but I’m unsure if this ensures global optimality.
Request for Help
- Does this construction always yield the maximum S_n ?
- If yes, how can we rigorously prove it? (Induction? Exchange arguments?)
- If no, what is the correct approach?
Observations:
- The construction works for small n (e.g., n=3,4,5,...,12 ).
- The problem resembles optimizing "minimum gaps" in permutations.
Any insights or references would be greatly appreciated!
