44 2 Algebra
is^1 n( 2 −^1 n)−^1 ·n= 2 nn− 1. Since during the process the value of the expression kept
decreasing, initially it must have been greater than or equal to 2 nn− 1. This proves the
inequality.
Let us summarize the last idea. We want to maximize (or minimize) ann-variable
function, and we have a candidate for the extremum. If we can move the variables one
by one toward the maximum without decreasing (respectively, increasing) the value of
the function, than the candidate is indeed the desired extremum. You can find more
applications of Sturm’s principle below.
130.Leta, b, cbe nonnegative real numbers such thata+b+c=1. Prove that
4 (ab+bc+ac)− 9 abc≤ 1.131.Letx 1 ,x 2 ,...,xn,n≥2, be positive numbers such thatx 1 +x 2 +···+xn=1.
Prove that
(
1 +
1
x 1)(
1 +
1
x 2)
···
(
1 +
1
xn)
≥(n+ 1 )n.132.Prove that a necessary and sufficient condition that a triangle inscribed in an ellipse
have maximum area is that its centroid coincide with the center of the ellipse.
133.Leta, b, c >0,a+b+c=1. Prove that
0 ≤ab+bc+ac− 2 abc≤7
27
.
134.Letx 1 ,x 2 ,...,xnbenreal numbers such that 0<xj≤^12 , for 1≤j≤n. Prove
the inequality
∏n
( j=^1 xj
∑n
j= 1 xj
)n≤∏n
( j=^1 (^1 −xj)
∑n
j= 1 (^1 −xj))n.135.Leta,b,c, anddbe nonnegative numbers such thata≤1,a+b≤5,a+b+c≤14,
a+b+c+d≤30. Prove that
√
a+
√
b+√
c+√
d≤ 10.136.What is the maximal value of the expression
∑
i<jxixjifx^1 ,x^2 ,...,xnare non-
negative integers whose sum is equal tom?137.Given then×narray(aij)ijwithaij=i+j−1, what is the smallest product of
nelements of the array provided that no two lie on the same row or column?
138.Given a positive integern, find the minimum value of
x^31 +x 23 +···+xn^3
x 1 +x 2 +···+xn
subject to the condition thatx 1 ,x 2 ,...,xnbe distinct positive integers.