Page 25 - MATINF Nr. 3

An unusual configuration

Leonard Mihai Giugiuc 1

Our goal in this paper is to find the admissible domain of k such that the inequality
2
2
2 2
2
2
a + a + · · · + a + ka a . . . a ≥ n + k
1 2 n 1 2 n
X n (n − 1)
does hold for any real numbers a 1 , a 2 , . . . , a n satisfying a i a j = , where n ≥ 4 is
2
1≤i<j≤n
a fixed integer.
Preliminaries

X 2
1. a 1 + a 2 + · · · + a n ≥ n or a 1 + a 2 + · · · + a n ≤ −n. Indeed, as (a i − a j ) ≥ 0, the
1≤i<j≤n
conclusion follows immediately.
n (n − 1)
X
2. If a 1 , a 2 , . . . , a n are real numbers satisfying a i a j = and n ≤ a 1 + a 2 +
2
1≤i<j≤n
»
· · · + a n < (n − 1) n , then all of them are strictly positive.
n−2
2
2
Indeed, set a 1 +a 2 +· · ·+a n = ns, with 1 ≤ s < √ n−1 so a +a +· · ·+a = n (ns − n + 1) ,
2
2
n(n−2) 1 2 n
2
2
2
2
from which a 1 + a 2 + · · · + a n−1 = ns − a n and a + a + · · · + a 2 n−1 = n (ns − n + 1) − a .
n
1
2
2
2
By Jensen’s inequality, (a 1 + a 2 + · · · + a n−1 ) 2 ≤ (n − 1) a + a + · · · + a 2 . Thus,
1 2 n−1
2
2
2
2
2
2
(ns − a n ) ≤ (n − 1) [n (ns − n + 1) − a ], hence a − 2sa n − n (n − 2) s + (n − 1) ≤ 0,
√
√
n
n
then s − (n − 1) s − 1 ≤ a n ≤ s + (n − 1) s − 1 and as 1 ≤ s < √ n−1 , we have
2
2
√ n(n−2)
2
s − (n − 1) s − 1 > 0, from which a n > 0. Analogously a 1 , a 2 , . . . , a n−1 > 0.
X n (n − 1)
3. If a 1 , a 2 , . . . , a n are real numbers satisfying a i a j = and a 1 +a 2 +· · ·+a n ≥
2
1≤i<j≤n
Ä 2 ä
n, then there exists t ≥ 1 such that a 1 + a 2 + · · · + a n = n t +1 .
2t
4. We’ll prove that k ≥ 0. Indeed, let k < 0. For every t ≥ 1, set a 1 = a 2 = · · · = a n−1 = t
2
ã
Å
and a n = n−(n−2)t 2 . Hence, we get X a i a j = n (n − 1) , a 1 + a 2 + · · · + a n = n t + 1
2t 2 2t
i<j
2
t n−2 [n−(n−2)t ] Ä 2 ä 2
2
2
2
and a 1 a 2 . . . a n = . We have: a + a + · · · + a = n 2 t +1 − n (n − 1) and
2 1 2 n 2t
2
2
t 2n−4 [n−(n−2)t ]
2 2
2
a a . . . a = 4 . But
n
1 2
Ç ã 2 å
t + 1 t [n − (n − 2) t ]
Å 2 2n−4 2 2
lim n 2 − n (n − 1) + k · = −∞,
t→∞ 2t 4
1
Profesor, Colegiul Nat , ional ,,Traian”, Drobeta Turnu Severin, leonardgiugiuc@yahoo.com
25