. . . (4- . — ! !), 2002

9. | | 9.

Ϡ.

1.
:



ab
(a+b)/2

, a b- . ,
a2


ab
+b=(ab)2 0

; , a=b.
,
x(ax) ((x+ax)/2)2=a2/4;

a+ 1

a
2


a(1/a)
=2蠠a > 0.

2.
: (a1a2an)1/n (a1++an)/n, , a1= = an.
n=2m m. m=1 . , m, m+1. , akak+2m ((ak+ak+2m)/2)2.
(a1a2 a2m+1)1/2m+1 (b1b2 b2m)1/2m,
bk=(ak+ak+2m)/2,
(b1 b2m)1/2m 1

2m
(b1++b2m) = 1

2m+1
(a1++a2m+1).
n . n < 2m m. an+1==a2m=(a1++an)/n=A. , (a1++an)+(an+1++a2m) = nA+(2mn)A=2mA a1 a2m=a1 anA2mn. a1 anA2mn (2mA/2m)2m=A2m, ..a1 an An; , a1= = an.

3.
a1,,an (a1++an)2 n(a12++an2). ,
(a1++an)2=
ai2+2

i < j
aiaj
ai2+

i < j
(ai2+aj2)=n
ai2.
4.
0acostdt=sina 0asintdt=1cosa, , cost 1, sina a, 1cosa a2/2

.. cosa 1 a2

2


,
sina a a3

6

,
cosa 1 a2

2
+ a4

24

.. ( a 0).
5.
, tga a 0 a < p/2. AB- 1 O, B- ; C- OA , S- BOC. a =2S < 2SAOB=tga.
6.
0 p/2 f(x)=x/sinx , f(x)=(tgxx)/(cosxsin2x) > 0. , f(a) f(p/2), ..a/sina p/2 0 < a < p/2.
7.
f(x)=acosx+bsinx,
f(x)


a2+b2

. , j,
cosj =a/


a2+b2


sinj =b/


a2+b2

,
f(x)=


a2+b2
cos(jx)


a2+b2

. , j =x+2kp, ..
cosx=a/


a2+b2


sinx=b/


a2+b2

.

9APP. | | 9.

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.
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