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# 2012二试(2)

2012().(40)

:PE2=PH·PQ

.

.(40)

a,b,ca<b<c,

(a,b)+(b,c)+(c,a)

2

(a,b,c).,(x,y)xy.

.(50)

1{aan

n}:a1=1,a2=1+√2,

2.:

(1){a

(2)liman}n;

n→+∞H,P2a=n?1

.(40)

:PE2=PH·PQ

.

∠PEB=∠ACH.∠ACH=90??∠A.PAB,AP=BP=EP,,∠ABE=∠PEB.,

,CQ⊥PQ,BE⊥CA,

∠QPE=∠EPH,

.(Q,C,H,E,,,∠EQH=∠ACH.,∠EQH=∠BEP=∠PEH.,△EPH～△QPE,EPEPPE2=PH·PQ.

40)

a<b<c,a,b,c

(a,b)+(b,c)+(c,a)

2

(a,b,c).,(x,y)xy.

gcd(a,b,c)=d,a=da1,b=db1,c=dc1,gcd(a1,b1,c1)=1.

(1)a1=1,

1+(b1,c1)+1

2?3+2(b1,c1)=b1+c1?(b1,c1)|3,

,(b1,c1)=13.(b1,c1)=1,b1=2,c1=3;(b1,c1)=3,b1=3,c1=6.

.,(a,b,c)=(d,2d,3d)a1??2,(d,3d,6d).(2)(a1,b1)=p,(b1,c1)=q,(c1,a1)=r,p,q,r

a1=pra2,b1=qpb2,c1=qrc2,

(a,b)+(b,c)+(c,a)

pra2+pqb2+qrc2=1

2,

min{a2,b2,c2}=a2??2,a1,b1,c1

p+q+r??pr+pq+qr.

,p,q,r

,

(a,b)+(b,c)+(c,a)

a1+b1+c1

??.

3

2

.

,

a1

p??2,

p+q+r??2r+p+q,

,r??0,

,a1??2

1.

,

(a,b,c)=(d,2d,3d)

(d,3d,6d)(

,d

)..(

50

)

{an}

1

2.

:

:a1=1,a2=1+

√2

,

an

an?1

2

=

(1)

(2)lim

{an}

an

;

n→+∞

2

,bn=

n

an

,

1

2

,

,b2=

2

4

,···.

k+1cn=

,

bk+1=

1

k+2

.

,bn=

n

an+1.

a2.n+1n

an??1

,an+1=

n+1

n+1,

cn=cn?1+

c1=1.

n+k.

1n√

,an=

nk=1

n

k>

k=1

2k√k+

2

k+1?

√k+n

k>

an

n+1>

2

n,

3

n

.

,lim

an

3

n→+∞

.

n

,

k=1

k

=2

k(

√3

k?1)=2(n+1)

k.

k=1

n

k=1

3

3

1+lim

n→+∞

√1

3.

n

<

2

n

an

,lim

n→+∞

.

,lim3.

an

n→+∞

.(M

50

)

{1,2,···,25},

,

M

.

M0={25,24,23,21,18,12}123.

,

6

123,

A={1,2,···,25}.S

,A

7

S

.

,

4

234

C17+C7+C7+C7=98,

22+23+24+25=94

.

....

,

,

,{25,24,···,19}

5

,,

53

5

,,

10

,

6

,

,

4.

4

2

,

M

6,

,

5

25+

24+23+21+18=111.

,

123,

.

5

,

12

.

,

5

,

1

,

.

5

2

,

123,

15.

,{25,24,···,15}

,

6

.

,6

15

,

10

.

,

3

,

6

15

1,2,3,4,5

,

2

6,7,8,9,1024,

1

.

,6

T

25

15,

24

16

,

,16∈T.

5,

,20∈T..

25?24=1

1

,

,23∈T.

123.

,22,21∈T.24?20=4,

4,

16∈T,

,

4