such that
Proposed by Ngo Van Trang, Vietnam
be the roots of the equation 
.
?
such that 
.
and 
.
and 
such that 
is 
,
.
be the set of integers. Determine all functions 
such that for all integer 
of a triangle 
are positive integers. Let
for any positive integer 
.
and 
,
has two roots, where one is twice of the other, find all possible values of 
Find the value of 
such that 
is an integer.
is 
in triangle 
to the side length 
.
.
which verify relation![\[
\left( f'(x) \right)^2 + f''(x) \leq 0, \forall x \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/3/a/2/3a235e8be4c61c32f376999e61d64973db21dc75.png)
is a finite set, let 
of subsets of 
if 
for each 
,
for each 
.
are there? Prove your answer.
be a continuous function such that 
for all 
.
for 
.
be a function differentiable at 0 with 
.
Show that the following statements are equivalent:
there exist 
such that 
such that 
be a positive sequence and let 
be a constant such that ![\[ \sum_{k=1}^{n-1} \lambda^2_k < K \lambda^2_n \;(n=1,2,...).\]](http://latex.artofproblemsolving.com/5/3/c/53ccb27390a368e1a74afae1da4f7566ad86af8a.png)
Prove that there exists a constant 
such that ![\[ \sum_{k=1}^{n-1} \lambda_k < K' \lambda_n \;(n=1,2,...).\]](http://latex.artofproblemsolving.com/e/7/4/e744b1b1179f879dbea84a53622afc5ca8e45fe2.png)
such that 
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!
