Advanced book on Mathematics Olympiad

478 Real Analysis

computations, that the arithmetic–geometric mean is related to elliptic integrals. The

relation that he discovered is

M(a,b)=

π

4

·

a+b

K

(a−b

a+b

),

where

K(k)=

∫ 1

0

1

√

( 1 −t^2 )( 1 −k^2 t^2 )

dt

is the elliptic integral of first kind. It is interesting to note that this elliptic integral is

used to compute the period of the spherical pendulum. More precisely, for a pendulum

described by the differential equation

d^2 θ

dt^2

+ω^2 sinθ= 0 ,

with maximal angleθmax, the period is given by the formula

P=

2

√

2

ω

K

(

sin

(

1

2

θmax

))

.

336.The functionfn(x)=xn+x−1 has positive derivative on[ 0 , 1 ], so it is increasing

on this interval. Fromfn( 0 )·fn( 1 )<0 it follows that there exists a uniquexn∈( 0 , 1 )

such thatf(xn)=0.

Since 0<xn<1, we havexnn+^1 +xn− 1 <xnn+xn− 1 =0. Rephrasing, this means

thatfn+ 1 (xn)<0, and soxn+ 1 >xn. The sequence(xn)nis increasing and bounded,

thus it is convergent. LetLbe its limit. There are two possibilities, eitherL=1, or

L<1. ButLcannot be less than 1, for when passing to the limit inxnn+xn− 1 =0,

we obtainL− 1 =0, orL=1, a contradiction. ThusL=1, and we are done.

(Gazeta Matematica ̆(Mathematics Gazette, Bucharest), proposed by A. Leonte)

337.Let

xn=

√

√√

√

1 + 2

√

1 + 2

√

1 +···+ 2

√

1 + 2

√

1969

with the expression containingnsquare root signs. Note that

x 1 −( 1 +

√

2 )=

√

1969 −( 1 +

√

2 )< 50.

Also, since

√

1 + 2 ( 1 +

√

2 )= 1 +

√

2, we have