42 2 Algebra
126.On a sphere of radius 1 are given four pointsA, B, C, Dsuch that
AB·AC·AD·BC·BD·CD=
29
33
Prove that the tetrahedronABCDis regular.127.Prove that
y^2 −x^2
2 x^2 + 1+
z^2 −y^2
2 y^2 + 1+
x^2 −z^2
2 z^2 + 1≥ 0 ,
for all real numbersx, y, z.128.Leta 1 ,a 2 ,...,anbe positive real numbers such thata 1 +a 2 +···+an<1. Prove
that
a 1 a 2 ···an( 1 −(a 1 +a 2 +···+an))
(a 1 +a 2 +···+an)( 1 −a 1 )( 1 −a 2 )···( 1 −an)
≤
1
nn+^1129.Consider the positive real numbersx 1 ,x 2 ,...,xnwithx 1 x 2 ···xn=1. Prove that
1
n− 1 +x 1+
1
n− 1 +x 2+···+
1
n− 1 +xn≤ 1.
2.1.6 Sturm’s Principle.........................................
In this section we present a method for proving inequalities that has the flavor of real
analysis. It is based on a principle attributed to R. Sturm, phrased as follows.
Sturm’s principle.Given a functionfdefined on a setMand a pointx 0 ∈M,if
(i)fhas a maximum(minimum)onM, and
(ii)if no other pointxinMis a maximum(minimum)off,
thenx 0 is the maximum(minimum)off.
But how to decide whether the functionf has a maximum or a minimum? Two
results from real analysis come in handy.
Theorem.A continuous function on a compact set always attains its extrema.
Theorem.A closed and bounded subset ofRnis compact.
Let us see how Sturm’s principle can be applied to a problem from the first Balkan
Mathematical Olympiad in 1984.