bồi-dưỡng-hsg-tỉnh-môn-toán.pdf

TI LIU BI DNG HC SINH GII

Phm Kim Chung www.k2pi.net T : 0984.333.030 Mail : [email protected] Tr.

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S GD&T NGH AN

TRNG THPT NG THC HA

www.k2pi.net

MT S BI TON CHN LC BI DNG HC SINH GII MN TON

VIT BI : PHM KIM CHUNG THNG 12 NM 2010

PHN MC LC Trang

I PHNG TRNH BPT HPT CC BI TON LIN QUAN N O HM

II PHNG TRNH HM V A THC

III BT NG THC V CC TR

IV GII HN CA DY S

V HNH HC KHNG GIAN

VI T LUYN V LI GII

DANH MC CC TI LIU THAM KHO

1. Cc din n : www.dangthuchua.com , www.math.vn , www.mathscope.org , www.maths.vn ,www.laisac.page.tl, www.diendantoanhoc.net , www.k2pi.violet.vn , www.nguyentatthu.violet.vn ,

2. thi HSG Quc Gia, thi HSG cc Tnh Thnh Ph trong nc, thi Olympic 30-4

3. B sch : Mt s chuyn bi dng hc sinh gii ( Nguyn Vn Mu Nguyn Vn Tin )

4. Tp ch Ton Hc v Tui Tr

5. B sch : CC PHNG PHP GII ( Trn Phng - L Hng c )

6. B sch : 10.000 BI TON S CP (Phan Huy Khi )

7. B sch : Ton nng cao ( Phan Huy Khi )

8. Gii TON HNH HC 11 ( Trn Thnh Minh )

9. Sng to Bt ng thc ( Phm Kim Hng )

10. Bt ng thc Suy lun v khm ph ( Phm Vn Thun )

11. Nhng vin kim cng trong Bt ng thc Ton hc ( Trn Phng )

12. 340 bi ton hnh hc khng gian ( I.F . Sharygin )

13. Tuyn tp 200 Bi thi V ch Ton ( o Tam )

14. B sch : CHUYN CHN LC ( Nguyn Vn Mu, Trn Nam Dng, Nguyn Minh Tun )

15. B sch : CC DNG TON LUYN THI I HC ( Phan Huy Khi )

16. v mt s ti liu tham kho khc .

TI LIU BI DNG HC SINH GII


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17. Ch : Nhng dng ch mu xanh cha cc ng link n cc chuyn mc hoc cc website.

Phn I : PHNG TRNH BPT HPT CC BI TON LIN QUAN N O HM


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PHN I : PHNG TRNH BPT - H PT V CC BI TON LIN QUAN N O HM

1. Tm c|c gi| tr ca tham s m h{m s : 2y 2x 2 m 4xx 5 c cc i . S : m < -2

2. Cho h{m s :

3 21 xsin 1, xf(x)

0 , x 0

x 0 . Tnh o h{m ca h{m s ti x = 0 v{ chng minh h{m s t cc tiu

ti x =0 .

3. Tm cc tr ca h{m s : y f(x) | x| x 3 . S : x =0 ; x=1 4. X|c nh c|c gi| tr ca tham s m c|c phng trnh sau c nghim thc :

a) x 3 3m 4 1 x3 m4 1m 0 . S : 7

9

9m

7

b) 4 2x 1 x m . S : 0 m 1

c) 2 2 4 2 2m 1 x 1 x 2 2 1 x 1 x 1 x

5. X|c nh s nghim ca h phng trnh :

2 3

3 2

y 2

xlog y 1

x

log S : 2

6. Gii h phng trnh :

2 22

y x

2

3 2

x 1

y 1

(x 2y 6) 2log (x y 2) 1

e

3log

. S : (x,y)=(7;7)


2 y 1

2 x 1

x 2x 2 3 1

y 2y 2 3 1

x

y


2x y y 2x 1 2x y 1

3 2

1 4 .5 2 1

y 4x ln y 2x 1 0

9. Gii phng trnh : 3 5(x 5) logx 3 log (x ) x3 2

10. Gii bt phng trnh : 4 (x 6)(2x(x 2) 1)(2x 1) 3 6 3 xx 2 . S : 1

2x 7

11. Gii bt phng trnh :

53 2x 2x 6

2x 13

12. Gii phng trnh : 2 23x 2 4x 29x 3 1 x x 1 0 13. Gii phng trnh : 33 2 24x 5x 6 7x 9x 4x

14. Tm m h phng trnh sau c nghim :

2 xy y x y 5

5 x 1 y m . S :

m 1; 5

15. X|c nh m phng trnh sau c nghim thc :

41

x x 1 m x x x 1 1x 1

.

16. Tm m h c nghim:

x 1 y 1 3

x y 1 y x 1 x 1 y 1 m

17. Gi s 3 2f(x) ax bx cx d (a 0) t cc i ti 1 2x ;x . CMR:

2

1 2

f '''(x) 1 f ''(x), x x ,x

f '(x) 2 f '(x)

18. Cho h{m s : 2 3f(x) cos 2x 2(sinx cosx) 3sin2x m . Tm m sao cho 2(x) 36,f m

19. Trong c|c nghim(x;y) ca BPT :

2 2x ylog x y 1 . Tm nghim P = x + 2y t GTLN

20. ( thi HSG Tnh Ngh An nm 2009 ) Gii phng trnh : x 22009 x +1- x =1 . S : x=0 21. ( thi HSG Tnh Ngh An nm 2009 ) . Tm m h phng trnh sau c ba nghim ph}n bit :

2

x y m

y 1 x xy m x 1 S :

3 3m

2



4

4

22. Gii h PT :

4 4

3 3 2 2

x y 240

x 2y 3 x 4y 4 x 8y


4 3 3 2 2

3 3

x x y 9y y x y x 9x

x y x 7 . S : (x,y)=(1;2)


2

2 2

4x 1 x y 3 5 2y 0

4x y 2 3 4x 7

25. Tm m h phng trnh sau c nghim :

2 xy y x y 5

5 x 1 y m . S :

m 1; 5

26. X|c nh m phng trnh sau c nghim thc :

41

x x 1 m x x x 1 1x 1

.

27. Tm m h phng trnh :

23 x 1 y m 0

x xy 1 c ba cp nghim ph}n bit .

28. Gii h PT :

2 y 1

2 x 1

x x 2x 2 3 1

y y 2y 2 3 1

29. ( thi HSG Tnh Ngh An nm 2008 ) .Gii h phng trnh :

x y sinxesiny

sin2x cos2y sinx cosy 1

x,y 0;4

30. Gii phng trnh : 3 2 316x 24x 12x 3 x


2x y y 2x 1 2x y 1

3 2

1 4 .5 2 1

y 4x ln y 2x 1 0

32. Gii phng trnh : x 33 1 x log 1 2x

33. Gii phng trnh : 33 2 2 32x 10x 17x 8 2x 5x x S


5 4 10 6

2

x xy y y

4x 5 y 8 6


2 2

2 2

x 2x 22 y y 2y 1

y 2y 22 x x 2x 1


y x

1x y

2

1 1x y

y x

37. ( thi HSG Tnh Qung Ninh nm 2010 ) . Gii phng trnh :

2 21 1

x5x 7

( x 6)x

51

Li gii : K : 7

x5

Cch 1 : PT

4x 6 36(4x 6)(x 1) 0 x

2(x 1)(5x 7). x 1 5x 7

Cch 2 : Vit li phng trnh di dng :

2 21 15x 6 x(5x 6) 1 x 1

V{ xt h{m s :

21 5

f(t) t , t7t 1



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5

38. ( thi HSG Tnh Qung Ninh nm 2010 ) X|c nh tt c c|c gi| tr ca tham s m BPT sau c nghim :

3 2 33x 1 m( x x 1)x

HD : Nh}n lin hp a v dng : 3

3 2x x 1 (x 3x 1) m

39. ( thi HSG Tnh Qung Bnh nm 2010 ) . Gii phng trnh :

3 2x 3x 4x 2 (3 2) 3xx 1

HD : PT 3

3(x 1) (x 1) 3x 1 3x 1 . Xt h{m s : 3 tf t) t ,t( 0

40. ( thi HSG Tnh Hi Phng nm 2010 ) . Gii phng trnh :

3 23 2x 1 27x 27x 13x2 2

HD : PT 33 32x 1 (3x 1) 2(2x 1) 2 (3x 1) f( 2x 1) f(3x 1)

41. Gii phng trnh :

42. ( thi Khi A nm 2010 ) Gii h phng trnh :

2

2 2

(4x 1)x (y 3) 5 2y 0

4x y 2 3 4x 7

HD : T pt (1) cho ta : 2

2 1].2x 5 2y 5 2y f([(2x 2x) f(1 5) 2y )

H{m s : 2 21).t f '(t) 3tf(t) (t 1 0

225 4x

2x 5 2y 4x 5 2y y2

Th v{o (2) ta c :

22

2 5 4x4x 2 3 4x 72

, vi 03

x4

( H{m n{y nghch bin trn khong ) v{ c

nghim duy nht : x1

2.

43. ( thi HSG Tnh Ngh An nm 2008 ) . Cho h:

x y 4

x 7 y 7 a(a l{ tham s).

Tm a h c nghim (x;y) tha m~n iu kin x 9. HD : ng trc b{i to|n cha tham s cn lu iu kin cht ca bin khi mun quy v 1 bin kho s|t :

x y 0 x4 16 . t x , t [t 3;4] v{ kho s|t tm Min . S : a 4 2 2


4 xy 2x 4

x 3 3 y

y 4x 2 5

2 x y 2

45. X|c nh m bt phng trnh sau nghim ng vi mi x : 2

sinx sinx sinxe 1 (e 1)sinx2e e 1e 1

46. ( thi HSG Tnh Tha Thin Hu nm 2003 ) . Gii PT :

2 22 5 2 2 5

log (x 2x 11) log (x 2x 12)

47. nh gi| tr ca m phng trnh sau c nghim: 4m 3 x 3 3m 4 1 x m 1 0

48. (Olympic 30-4 ln th VIII ) . Gii h phng trnh sau:

2 22

y x

2

3 2

x 1e

y 1

3log (x 2y 6) 2log (x y 2) 1

49. Cc bi ton lin quan n nh ngha o hm :

Cho

x

2

(x 1)e , x 0f(x)

x ax 1, x 0 . Tm a tn ti f(0) .

Cho

acosx bsinx, xF(x)

ax b 1, x 0

0 . Tm a,b tn ti f(0) .

2 2x xlnx , x 0

F(x) 2 4

0, , x 0

v{

xlnx, x 0f(x)

0, x 0 . CMR : F'(x) f(x)

Cho f(x) x|c nh trn R tha m~n iu kin : a 0bt ng thc sau lun ng x R : 2| f(x a) f(x) a| a . Chng minh f(x) l{ h{m hng .



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Tnh gii hn :

x

3

1 2

4

tanN lim

2sin

x 1

x 1 Tnh gii hn :

2 32x 2

2 2x 0

e 1N lim

ln(1 x

x

)

Tnh gii hn :

3

3x 0

3 32x x 1N

1m

xli

x Tnh gii hn :

sin2x

4

s

x

nx

0

ie eN lim

sinx

Tnh gii hn :

0

3

5x

x 8 2

siN lim

n10x Tnh gii hn :

2 32x 2

6 2x 0

e 1N lim

ln(1 x

x

)

Tnh gii hn :

sin2x sin3

7x

3x

0

eN lim

e

sin4x Tnh gii hn :

x 4

3x 0 38

4 xN

xim

2l

Tnh gii hn :

9

x 0

3x 2x.3 cos4x

1 sinx 1

2N lim

sinx

Cho P(x) l{ a thc bc n c n nghim ph}n bit 1 2 3 nx x x; ; ...x . Chng minh c|c ng thc sau :

a) 2 n

2 n

1

1

P''(x ) P''(x ) P''(x )... 0

P'(x P'( P'(x) )x)

b) 2 n1 ) )

1 1 1... 0

P'(x P'(x P'(x )

Tnh c|c tng sau :

a) nT osx 2cos2x ... nc(x) c osnx

b) n 2 2 n n1 x 1 x 1 x

(x) tan tan ... tan2 2 2 2 2 2

T

c) 2 3 n n 2n n nCMR : 2.1.C 3.2.C ... n(n 1)C n(n 1).2

d) 2nS inx 4sin2x 9sin3x ...(x) s sn innx

e)

n 2 2 2 2 2 2

2x 1 2x 3 2x (2n 1)(x) ...

x (x 1) (x 1) (x 2) x (n 1) (x n)S

50. Cc bi ton lin quan n cc tr ca hm s :

a) Cho R: a b 0 . Chng minh rng :

n na b a b

2 2

b) Chng minh rng vi a 3,n 2 ( n N,n chn ) th phng trnh sau v nghim : n 2 n 1 n 2(n 1)x 3(n 2)x a 0

c) Tm tham s m h{m s sau c duy nht mt cc tr :

22 2

2 2y (m 1) 3

x x

1 x 1 xm 4m

d) Cho n 3,n N ( n l ) . CMR : x 0 , ta c :

2 n 2 nx x x x1 x ... 1 x ... 1

2! n! 2! n!

e) Tm cc tr ca h{m s : 2 2x x 1 x xy 1

f) Tm a h{m s : 2y f(x) 2 xx a 1 c cc tiu .

g) Tm m h{m s :

msinx cosx 1

ymcosx

t cc tr ti 3 im ph}n bit thuc khong

90;

4

51. Cc bi ton chng minh phng trnh c nghim :

a) Cho c|c s thc a,b,c,d,e . Chng minh rng nu phng trnh : 2ax b c x d e 0 c nghim thc thuc

na khong [1; ) th phng trnh : 4 3 2bx cx dxax e 0 c nghim.

b) Cho phng trnh : 5 4 3 25x 15x xP( ) xx x 3 7 0 . Chng minh rng, phng trnh c mt nghim thc

duy nht.

Phn II : PHNG TRNH HM V A THC


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PHN II : PHNG TRNH HM-A THC

1. Tm h{m s : f :R R tho m~n ng thi c|c iu kin sau :

a)

x 0

f(x)lim 1

x

b) 2 2f x y f x f y 2x 3xy 2y , x,y R

2. Tm h{m s : f :R R tho m~n iu kin sau : 2008 2008f x f(y) f x y f f(y) y 1, x,y R

3. Tm h{m s : f :R R tho m~n iu kin sau : f x cos(2009y) f x 2009cos f y , x,y R 4. Tm h{m s : f :R R tho m~n ng thi c|c iu kin sau :

c) 2009xf x e

d) f x y f x .f y , x,y R

5. Tm h{m s : f :R R tho m~n iu kin sau : f y 1f x y f(x).e , x,y R

6. Tm h{m s : f :R R tho m~n iu kin sau : 2f x.f x y f(y.f x ) x 7. ( thi HSG Tnh Hi Phng nm 2010 ) Tm h{m f : tha m~n :

2(x) 2yf(x) f(y) f y f(x) , ,x,yf R

Phn III : BT NG THC V CC TR


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PHN III : BT NG THC V CC TR

1. Cho 2 2 2a,b,c R: a b c 3 . Chng minh rng : 2 2 2a b b c c a 3

2. Cho c|c s thc khng }m a,b,c . Chng minh rng :

2 2 2 2 2 22 2 2 2 2 2a b a b b c b c c a c a a b b c c a

3. Cho c|c s thc a,b,c . Chng minh rng :

2 2 2 2

2

a b c 81 a b 13a b c

b c a 4 42a b

4. Cho c|c s thc khng }m a,b,c tho m~n : a b c 36abc 2 . Tm Max ca : 7 8 9P a b c

5. Cho 3 s thc dng tu x,y,z . CMR :

a b c 3

a b b c c a 2

6. Cho a,b,c >0 . Tm GTNN ca :

6

2 3

a b cP

ab c

7. Cho c|c s thc dng x,y,z tha m~n : 2 2 2yx z 1

CMR :

2 2 22x (y z) 2y (z x) 2z (x y)

yz zx xy

8. Cho c|c s thc dng a,b,c . CMR :

bc ca ab a b c

a 3b 2c b 3c 2a c 3a 2b 6

9. Cho c|c s thc dng a,b,c . CMR : 3 3 3 3 3 3

1 1 1 1

abca b abc b c abc c a abc

10. Cho c|c s thc tha m~n iu kin : 2 2 2

1 1 11

a 2 b 2 c 2 . CMR : ab bc ca 3

11. Cho c|c s thc dng tha m~n iu kin : 2 2 2ba c 3 . CMR :

1 1 13

2 a 2 b 2 c

12. Cho x,y,z l{ 3 s thc dng ty . CMR :

x y z 3 2

x y y z z x 2

13. Cho c|c s thc dng a,b,c . CMR :

2 2 2 2a b c 4(a b)a b c

b c a a b c

14. Cho c|c s thc dng a,b,c tha m~n : abc=1 . CMR : 3 3 3

1 1 1 3

2a (b c) b (c a) c (a b)

15. Cho 3 s thc x,y,z tha m~n : xyz=1 v{ x 1 y 1 z 1 0 . CMR :

22 2x y z

1x 1 y 1 z 1

16. Cho a,b,c l{ c|c s thc dng bt k . CMR :

2 2 2

2 2 2 2 2 2

(3a b c) (3b c a) (3c a b) 9

22a (b c) 2b (c a) 2c (a b)

17. Cho c|c s thc dng a,b,c tha m~n : 2 2 2ba c 1 . CMR :

1 1 1 9

1 ab 1 bc 1 ca 2

18. Cho c|c s thc a,b,c tha m~n : 2 2 2ba c 9 . CMR : 2(a b c) 10 abc

19. Cho a,b,c l{ c|c s thc dng : a+b+c =1 . CMR :

3 3 3

2 2 2

a b c 1

4(1 a) (1 b) (1 c)

20. (Chn THSG QG Ngh An nm 2010 ) Cho c|c s thc dng a,b,c tha m~n :

4 4 4 2 2 2b c ) 25(9(a a b c ) 48 0 . Tm gi| tr nh nht ca biu thc :

2 2 2a b c

b 2c c 2a aF

2b



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Li gii 1 : T gi thit :

4 4 4 2 2 2 2 2 2 4 4 4 2 2 2 2

2 2 2 2 2 2 2 2 2 2

b c ) 25(a b c ) 48 0 25(a b c ) 48 9(a b c ) 48 3(a b c )

3(a b c ) b c ) 48 0

9

3 b c

(a

1625(a a

3

Ta li c :

4 4 42 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2

a b c a b c (a b c )

b 2c c 2a a 2b a (b 2c) b (c 2a) c (a 2b) (a b b c c a) 2(a c b a cF

b)

Li c :

2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 (a b c )b b c c a a(ab) b(bc) c(ca) (a b c ) b c ca [a b a ] a b c3

Tng t :

2 2 2

2 2 2 2 2 2 a b cc b a c b) a b c .(a3

T ta c :

2 2 2

Fa b c

13

. Du bng xy ra khi v{ ch khi : a=b=c=1.

Li gii 2 : ( P N CA S GD&T NGH AN ) p dng bt ng thc AM GM, ta c

2 2 2 2 2a (b 2c)a a (b 2c)a 2a2

b 2c 9 b 2c 9 3.

Tng t

2 2 2 2 2 2b (c 2a)b 2b c (a 2b)c 2c,

c 2a 9 3 a 2b 9 3.

Suy ra:

2 2 2a b cF

b 2c c 2a a 2b

2 2 2 2 2 22 1a b c a (b 2c) b (c 2a) c (a 2b) (*)

3 9.

Li |p dng AM GM, ta c

3 3 3 3 3 3 3 3 3

2 2 2 3 3 3a a c b b a c c ba c b a c b a b c (**)3 3 3

.

T (*) v{ (**) suy ra:

2 2 2 2 2 22 1

F a b c a b c (a b c )3 9

2 2 2 2 2 2 2 2 22 1

a b c a b c 3 a b c3 9

.

t 2 2 2t 3 a b c , t gi thit ta c:

2

2 2 2 4 4 4 2 2 225 a b c 48 9 a b c 3 a b c

2

2 2 2 2 2 2 2 2 2 163 a b c 25 a b c 48 0 3 a b c3

.

Do 2 32 1

F t t f(t)9 27

vi t 3; 4 (* * *) .

M{

t 3;4min f(t) f(3) 1 (* * **) . T (***) v{ (****) suy ra F 1.

Vy minF 1 xy ra khi a b c 1 .

21. ( thi HSG Tnh Ngh An nm 2009 ) Cho c|c s thc dng x,y,z . Chng minh rng :

2 2 2 2 2 2

1 1 1 36

x y z 9 x y y z z x

Li gii 1 :

BT ~ cho tng ng vi :

2 2 2 2 2 2 1 1 19 x y y z z x 36x y z

Ta c :

32 xy yz zx

xyz (xy)(yz)(zx)3

Do :

22 2

3

27 xy yz zx1 1 1 xy yz zx 27

x y z xyz xy yz zx(xy yz zx)



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Li c : 2 2 2 2 2 2 2 2 2 2 2 2y y z z x y 1 z 1) (z x 1) 29 x 6 x (y 3 (xy yz zx) Nn :

22 27 9VT 4 3 (xy yz zx) . 108 6 (xy yz zx)

xy yz zx xy yz zx

9108 6 2 (xy yz zx) 1296 VT 36

xy yz zx

Li gii 2 : ( P N CA S GD&T NGH AN )

Bt ng thc cn chng minh tng ng (xy + yz + zx)(9 + x2y2 + z2y2 +x2z2) 36xyz p dng bt ng thc Csi ta c :

xy + yz + zx 3 2 2 23 x y z (1)

V{ 9+ x2y2 + z2y2 +x2z2 12 4 4 412 x y z hay 9 + x2y2 + z2y2 +x2z2 12 3 xyz (2)

Do c|c v u dng, t (1), (2) suy ra: (xy + yz + zx)(9 + x2y2 + z2y2 +x2z2) 36xyz (pcm).

Du ng thc xy ra khi v{ ch khi x = y = z =1 22. ( thi HSG Tnh Qung Ninh nm 2010 ) Cho c|c s thc dng x,y tha m~n k : x y 1 3xy . Tm gi| tr

ln nht ca : 2 2

3x 3y 1M

y(x 1) x y 1) x

1

y(

Li gii :

Ta c : 3xy x y 1 2 xy 1 xy 1 xy 1 (*)

Ta c :

22

2 2 2 2 2 2 2 2 22

3xy 3xy 1 (1 3xy)1 1 1 3xy(x y) (x y)

y y (3

2xy3x 3y 1 2xyM

y (3x 1) x (3y 1) x 9xy 3x 1) x (x y(3y 1) x y 4x) y1

23. ( thi HSG Tnh Qung Bnh nm 2010 ) Cho c|c s thc dng a, b, c . CMR :

3 3

3 3

3

3

c a b c

b c aa

a b

b c

HD :

3 3

3 3

3 3 3

3 3 3

a a1

b b

a b c3

b c a

a3

b

24. ( thi HSG Tnh Vnh Phc nm 2010 ) . Cho x, y, z 0 tha m~n : 2 2 2yx z 1 . Tm gi| tr ln nht ca

biu thc : P 6(y z x) 27xyz

HD :

2 2 22 2 2y z 1 x6 2(y z ) x 27x. 6 2(1 x ) x 27x

2P

2 MaxP 10

25. ( thi HSG Tnh Hi Phng nm 2010 ) . Cho 2 2 20: a bb,c ca, 1 . Chng minh rng :

3 3 36

2b 3ca7

HD : C th dng c}n bng h s hoc Svacx 26. Cho x,y,z l{ c|c s thc dng tha m~n : xyz 1 . Chng minh rng :

4 4 3 4 4 3 4 4 3

6 6 6 6 6 6

(x (y (z

x y

y ) z ) x )12

y xzz

Li gii : t 2 2 2a;y b;z cx abc 1 . Bt ng thc ~ cho tr th{nh :

3 3 3

3 3

2 2 2 2 2

3 3 3

2

3

(a (b (c

a b

b ) c ) a )12

b acc

p dng Bt ng thc AM-GM cho 4 s ta c :

42 2 3 6 4 2 4 2 4 2 6 2 4 2 4 2 4 6 6 3 3(a ab ) b a b a b b b b a b ab a a a 4 ba



11

11

27. ( thi HSG Tnh ng Nai nm 2010 ) . Cho a,b,c > 0 . Chng minh rng :

2 2 2

1 1 1 3(a b c)

a b b c c a b2( ca )

HD :

BT

2 2 2 2 2 2(a 1b ) (b c ) 1 1 3(a b c)

2 a

(c a

b b c a

)

c 2

V{ ch :

2

2 2 (a b)a b2

28. ( thi HSG Tnh Ph Th nm 2010 ) . Cho x,y,z 0: x y z 9 . Chng minh rng :

3 3 3 3 3 3x y z

xy 9 yz 9 zx

z x

9

y9

29. ( thi chn T Ninh Bnh nm 2010 ) . Cho a,b,c l{ d{i ba cnh mt tam gi|c c chu vi bng 4. Chng minh

rng : 2 2 2272

a 2abcb c27

HD : B{i n{y th chn phn t ln nht m{ o h{m .

30. ( thi HSG Tnh Bnh nh nm 2010 ) . Cho a,b,c >0 . CMR : 3 3 3b c

aa

ca abcb

bc

HD :

4 2 2 2 2 4a (a b c ) (a b c)

a b cabc 3abc 27abc

VT

31. ( thi chn HSG QG Tnh Bnh nh nm 2010) . Cho x,y,z >0 tha m~n : 2 xy xz 1 . Tm gi| tr nh

nht ca : 3yz 4z

Sx 5xy

x y z

32. ( thi chn HSG Thi Nguyn nm 2010 ). Cho c|c s thc x,y,z tha m~n iu kin :

1 2 31

1 x 2 y 3 z .

Tm gi| tr nh nht ca : P xyz

33. ( thi chn HSG QG tnh Bn Tre nm 2010 ) . Cho 2 2 2ba,b c :a c, 0 3 . Chng minh bt ng thc :

1 1 11

4 ab 4 bc 4 ca

34. ( thi chn T trng HSP I H Ni 2010 ) . Cho c|c s thc dng x,y,z . Tm gi| tr nh nht ca :

2 2 2

3 3 3 2 2 2

y y z z x 1xP

3xyz

x y 3(xy yz zxz )

Li gii 1 :

t : x y z

a; b; c abc 1y z x

. Lc :

2 2 2

b c 13

3

aP

b (a b c)c a

Ta c :

2(ab bc ca)

(a b c) abc(a b c) (ab)(ac) (ab)(bc) (ac)(bc)3

Li c :

2

2 2 2 2

2

2b

b

1 a 1

a b

1 b 1 a

b

c 1 1 12 ab bc ca

a b cc acc b

1 c 12

c ca

Do : 2

13(ab bc ca)

(ab bc ca)P ( Vi ab bc ca 1 )

Li gii 2 :

t : z

a; b; c ay

z

x

ybc

x1 . Lc :

2 2 2

2

b c 13abc 13(a b c)

c a 3(ab b

aP

c ca) (a b c)b

35. Bi ton tng t : Cho x,y,z 0: xyz 1 . Chng minh rng : 2 2 2

x y z 34

x y zy z x



12

12

Li gii : t : 1 1 1

a; b; c abc 1x y z

.

BT ~ cho tr th{nh :

2 2 2 2

2

a b c 3abc (a b c) 9

c a b ab bc ca a b c (a b c) . Vi : 33a abcb c 3

36. ( thi chn i tuyn H Vinh nm 2010 ) . Cho a,b,c l{ c|c s thc thuc on [0;1] v{ a b c 1 . Tm gi|

tr ln nht v{ nh nht ca :

2 2 2

1 1 1P

a b c1 1 1

HD : Dng pp tip tuyn v{ Bt ng thc :

2 2 2

1 1 11 ,

x y (x yx,y 0;

)x y 1

1 1 1

37. ( thi chn HSG QG tnh Lm ng ) . Cho a,b,c l{ c|c s thc dng . Chng minh rng :

2 2 2

2 2 2 2 2 2b c a ab b b bc c c cac ab

aa

Li gii :

C1 : ( THTT) Ta c :

2 2 2 2 2 2b c b cc a 2(a b c) a b c

c b a

ab

b a c

a

Do :

2 2 2 2 2 2b c a a2.VT 2 b a b b 2VP

c a

a

b b

ab b

b

C2 : Ta c : 2 2a ab b a b c(Mincopxki)

M{ :

Sv

2 22 22

acx

2

o

aaVT a

b

ab bab bab

ab

b c

38. ( thi chn i tuyn trng Lng Th Vinh ng Nai nm 2010 ) . Cho a,b,c 0:abc 1 . Chng minh

rng : 2 2 2ab bc c a ba c

HD : BT a b c

a b cb c a

. Ch l{ :

2 2 ac 3a a cb

a ba

b c

Li gii 2 : Ta c : 2 2 2 2 2 2 33ab 3 (a )bab bc b c 3b

39. ( Chn T HSG QG tnh Ph Th nm 2010 ). Cho a,b,c 0 . Chng minh bt ng thc :

3

3 3

2

3

2 2a b c

b c c a b

3 2

a 2

HD :

2 2

3 33

b c b c b c a 1 a2 3 2

a a a 2(a b c) b c3 2

40. ( thi HSG Tnh Ngh An nm 2008 ) . Cho 3 s dng a,b,c thay i . Tm gi| tr ln nht ca :

bc ca abP .

a 3 bc b 3 ca c 3 ab

Li gii 1 : t b c

x; y; zc a

axyz

b1 . Lc :

z x y 1 xP 1

x 3z y 3x z 3y 3 x 3z . Li c :

2 2 2

2 2 22

x x (x y z) (x y z) 3

x 3z 4x 3zx (x y z) (xy yz zx) (x y z)(x y z)

3

Do : 1 3 3

P 13 4 4

. Du = xy ra khi v{ ch khi : x = y = z =1 .

Li gii 2 : ( P N CA S GD&T )

t x a ,y b,z c;x,y,z 0; .

Khi : 2 2 2yz zx xy

P .x 3yz y 3zx z 3xy



13

13

Ta c 2 2 2

3yz 3zx 3xy3P

x 3yz y 3zx z 3xy

2 2 2

2 2 2

x y z3 3 Q

x 3yz y 3zx z 3xy

|p dng bt BCS ta c

2

2 2 2

2 2 2

2 2 2

x y zx 3yz y 3zx z 3xy

x 3yz y 3zx z 3xy

Q. x y z 3xy 3yz 3zx

2

2

x y zQ

x y z xy yz zx. Mt kh|c

2x y z

xy yz zx3

Suy ra 3

Q4, do

9 33P P .

4 4

Du bng xy ra khi v{ ch khi a b c. Vy gi| tr nh nht ca P bng 3

.4

41. ( d b HSG Tnh Ngh An 2008 ) . Cho ba s dng a,b,c tho m~n : 2 2 2a b c 1 . Tm gi| tr nh nht

ca biu thc :

2 2 2a b cP .

b c c a a b

Li gii 1 : Gi s :

1 1 1b c

b c c a aa

b . p dng bt ng thc Chebysev ta c :

22

2 2

2 2

2

2

2a b c 1 1 1 1 1 1 1 1P . a

b c c a a b 3 b c c a a b 3 b c c a a b

3 3

2(a

b c

bb c) 2 (a c3 )

Li gii 2 : p dng BT Swcharz :

4 2 2 2 2

2 2 2 2 2 2 2 2 2

4 4a b c (aP .

a b c) b c a) c a b) b

b c )

( ( ( c ) a(b c ) c(a( ba )

Li c :

32 2 2 2 2 2 2

2 2 2a b c . b c 1 2aa(b2(b c )

c )32 2

42. ( chn i tuyn QG d thi IMO 2005 ) . Cho a,b,c >0 . CMR :

3 3 3

3 3 3

a b c

(a b) (b c) (c a)

3

8

Li gii : b c a

x; y; z ; xyz 1a b c

. Bt ng thc ~ cho tr th{nh : 3 3 31 1 1 3

8(1 x) (1 y) (1 z)

p dng AM-GM ta c :

363 3 2

11 1 13

81 x 1 x

3

8(1 x) 2 1 x

Ta cn CM bt ng thc : 2 2 21 1 1 3

4(1 x) (1 y) (1 z)

B :

2 2

1 1 1x,y 0

1 xy1 x 1 y

B n{y c CM bng c|ch bin i tng ng a v BT hin nhin : 2 2xy(x y) (1 xy) 0

Do :

2

2 2 2 2

1 1 z 1 z(z 1) 1 z z 1

1 xy z 1(VT

1 z) (1 z) (1 z) z 2z 1

Gi s : 3z Max{x,y,z} 1 yz z zx 1 . Xt h{m s :

2 2

2 4

z z 1 z 1; f '(z) 0, z 1

z 2z 1 (z 1)f(z)

Suy ra : 3

f(f ) 1)(z4

.



14

14

43. ( thi HSG Tnh H Tnh nm 2008 ) . Cho 0:x yx y,z z, 1 . Tm gi| tr nh nht ca :

1 x 1 y 1 z

1 x 1P

y 1 z

Li gii 1 :

22

2

1 x

1 x

x(1 x) 1 x 1 1 x 0 1 x 0

1 1 x ( lun ng )

Thit lp c|c BT tng t ta c : P 2

Ch : tm Max cn s dng BT ph :

1 x 1 y 1 x y1 , x y

1 x 1 y 1 x y

4

5 v MaxP 1

2

3

44. ( thi HSG lp 11 tnh H Tnh nm 2008 ) . Cho x,y,z 0: x y z 1 . Chng minh bt ng thc :

1 x 1 y 1 z x y z2

y z z x x y y z x

Gii : BT

x y z x y z 3 xz xy yz2

y z z x x y y z x 2 y(y z) z(z x) x3

(x2

y)

Ta li c :

22 2 2 xz yz zxxz xy yz (xz) (xy) (yz)

VPy(y z) z(z x) x(x y) xyz(y z) xyz(z x) xyz(x y) 2xyz(x y z)

M{ :

2(xy yz zx)

xyz(x y z) (xy)(yz) (xz)(zy) (zx)(xy) VP3

3

2

45. ( thi HSG Tnh Qung Bnh 2010 ) . Cho 0:a ba b,c c, 3 . Chng minh rng :

3 3 31 1a c ab c 1b 5

46. Cho a,b,c l{ d{i 3 cnh tam gi|c ABC . Tm GTNN ca :

2a 2b 2cP

2b 2c a 2a 2c b 2b 2a c

HD :

2a 6a 6a

2b 2c a (a b c)(3a)(2b 2c a)

47. Cho 0:a ba b,c c, 1 . Tm GTLN, GTNN ca : 2 2 2a 1 b 1P b c ca 1

HD . Tm GTNN : p dng BT Mincopxki ta c :

2 22 2

2 2 2 1 3 3 3a 1 b 1 c 1 a a b c2 2 2

3P a b c

2

Tm GTLN :

B : CM bt ng thc : 2 2 21 a a 1 b b 1 1 (a b) (a b)

Bnh phng 2 v ta c : 2 2 2 2(1 a a 1 a b (a b) 1 a b (a b))(1 (1 ab b ) b b)a 0

48. ( thi chn HSG QG tnh Hi Dng nm 2008 ) . Cho a,b,c 0:a b c 3 . Tm gi| tr nh nht ca biu

thc :

2 2 2

3 3 3

a b c

a 2b b 2c c 2aP

HD : AM-GM ngc du .

Ta c :

2 3 33 2

3 3 3 6

a 2ab 2ab 2 2 2 4a a a b a a b(a a 1) a b ab

3 9 9 9a 2b a 2b 3 ab

Do :

2a b c2 4 7 4

(a b c) (ab bc ca) 19 9

P (a b9

c3

)3

49. ( chn T trng chuyn Bn Tre ) . Cho x,y,z 0 . Tm GTLN ca :

1 1

x y z 1 (1 x)(1 y)(1M

z)

Gii : t x y z t 0 , ta c :

3x y z 3

(1 x)(1 y)(1 z)3

. Lc : 31 2

M7

t 1 (t 3)

Xt h{m s :

3

1 27, t 0

t 1 (t 3t

)f( )



15

15

50. Cho a,b,c 0 . Chng minh rng :

4 4 4 2 2 23a 1 3b 1 3c 1 a b c

b c c a a b 2

HD : Ta c : 44 4 4 4 1 31 a a a a3 4 2a 1 4a

Do :

3 4

Svacxo

4a 4a...

b c ab acVT

51. Cho a,b,c 0 . Chng minh rng :

1 1 1 9 4 4 4

a b c a b c a b a c b c

HD :

52. Cho a,b,c 0: a b c 1 . Chng minh rng :

b c c a 3 3

4a 3c ab b 3a b c 3b ac

b

c

a

53. Cho a,b,c 0 . CMR :

2 2 2 2 2 2 2 2 2

a 1 1 1 1

6 a b c3a 2b c 3b 2c a 3c 2a b

b c

54. Cho a,b,c 0:ab bc ca 3 . CMR : 2 2 2a b c

abc2a bc 2b ca 2c ab

55. Cho a,b,c 0 . CMR :

3 3 3

2 2 2

1 a 1 b 1 c3

1 a c 1 c b 1 b a

56. Cho a,b,c 0:abc 27 . CMR :

1 1 1 3

21 a 1 b 1 c

57. Cho a,b,c 0 . CMR : 2

1 1 1 27

b(a b) c(c b) a(a c) (a b c)

58. Cho a,b,c 0 . CMR :

b c c a a b

a b c 3a b c

59. Cho (a,b,c 1;2) . CMR :

c b

b c

b a a c1

4b c c a 4c ab a b4a

60. Cho a,b,c 0:abc 1 .CMR :

3 6

a b c ab bc1

ca

61. Cho x,y,z 0 . CMR :

2 2 2

3 3 3

x z 1 x y z

2 y z xxyz y xyz z xyz x

y x z y

62. Cho 1 1 1

1a

a,b,cb c

0: . CMR :

2 2 2a b c a b c

a bc b ac c ba 4

63. Cho x,y,z 0 . Tm Min ca :

3 3 3 3 3 33 3 3

2 2 2

x y zP 4(x y ) 4(y z ) 4(z x ) 2

y z x

64. Cho a,b,c 0: b ca 3 . CMR : a b c ab bc ca

65. Cho a,b,c 0:abc 1 . CMR:

1 1 11

a b 1 b c 1 c a 1

66. Cho x,y,z 0 . CMR :

x1

x (x y)(x z) y (x y)(y z) z (x z)(y z)

y z

67. ( thi HSG Tnh Bnh Phc nm 2008 ). Cho a,b,c 0 . CMR :

3 3 3

2 2 2 2 2 2

a b c a b c

2a b b c c a

68. ( thi HSG Tnh Thi Bnh nm 2009 ) .Cho c|c s thc x , y , z tha m~n 2 2 2x y z 3 . Tm gi| tr ln nht

ca biu thc: 2 2F 3x 7y 5y 5z 7z 3x

69. ( thi HSG TP H Ch Minh nm 2006 ) . Cho a,b,c l{ c|c s thc khng }m tha: a b c 3 . Chng minh:

2 2 2

2 2 2

a b c 3

2b 1 c 1 a 1.

70. Cho a,b,c > 0 . Chng minh rng :

2a 2b 2c3

a b b c c a



16

16

HD : t a

;y ;z xb c

xc a

yz 1b

. p dng B : 2 2

1 1xy 1

1 xy1 x 1

2

y

71. Chng minh cc Bt ng thc :

a) 2 2 2

b c c a a blog a log logb c 3 a,b,c 2

b)

b c alog c log a log 9 a,b,c 1b c c a a b a b c

2

c)

72. Cho 0: xyx,y yzz zx, 3 . Tm gi| tr nh nht ca : 22 3 2 3 23 22P x y (xy z z 1) (y 1) (z 1)x

Gii : 73.

Phn IV : GII HN DY S


17

17

PHN IV : GII HN DY S

1. Cho d~y s :

1

2

n 1 3 n

x 1

x 7 log x 11

. Chng minh d~y s c gii hn v{ tnh gii hn .

HD : Xt h{m s : 23f (x(x) 17 lo 1) 5g ,x (0; ) , ta c : 2 x (0;5)11)

2xf '(x) 0

n,

(x l 3

Do : 0 f(5) f(x) f(0) 5 . M{ n 1 nf )x (x , do bng quy np ta CM c rng : n , n0 x 5

Li xt h{m s : 23(g( x 11) x, x (0;5)x) 7 log . Ta c : 2 x (0;5)11)l

2xg'(x) 1 0

(x 3,

n

Suy ra phng trnh f(x)=x c nghim duy nht x = 4 .

Theo nh l Lagrage n(x 4)c ; sao cho : n n n

1f(x ) f(4) f '(c) x 4 x 4

11ln3

( V 2 211)ln

2c 2c 1f '(c)

(c 11ln32 11c ln33

). Do :

1

n 1 1

n1

x 4 x 011 ln3

4

2. Cho phng trnh : 2n 1x x 1 vi n nguyn dng . Chng minh phng trnh ~ cho c duy nht mt nghim

thc vi mi n nguyn dng cho trc. Gi nghim l{ nx . Tm nlimx

Gii : T phng trnh : 2n 2n2n 1x 1

1) 1 1x x 1 x(x ) 0 x(x 1) 0x(xx 0

t 2nn

1 x) 1f (x x .

+) Nu x 1 , ta c : 2nnf '(x) (2n 1).x 1 0 . Hn na

xf(1) 1; lim f(x)

, suy ra phng trnh

c nghim n (1x ); duy nht .

Xt hiu :

2n 2 2n 1 2n 1n 1 n n n n n n n n n n n 1 n n n) f (x ) x 1 x x 1 x xf (x x (x 1) 1 f (x ) f (0, x ) Hay :

n 1 n n n n 1 n 1 n n 1f (x ) f (x ) 0 f (x ) x x . (Do h{m f(x) tng ) .

Vy d~y n{x } l{ d~y gim v{ b chn di bi 1 nn c gii hn . Gi s : n a(lim 1x a )

Ta s chng minh a=1 . Tht vy, gi s a > 1 .

3. ( thi HSG Tnh Qung Bnh nm 2010 ) Cho d~y s 1

2n n

n 1 n

u

{u } :u u

1

u

2010

. t : 1 2 nn2 3 n 1

u u...

u u

uS

u

.

Tm : nlimS

Li gii :

Ta c : 2

k 1 k 1k 1

k 1 k

k k k k k

1 k 1 k 1

kk

k k k

u u u u u u u u 1u u 2010 (*)

2010 u 2010 u .u 2010.u u u

1

u

T h thc (*) cho k = 1,2,n ta c : nn 1

1S 2010 1

u

Li c : 2

nn 1 n n

uu u

2010u

D~y {un} tng .

Gi s {un} b chn trn . Suy ra tn ti gii hn hu hn : nlimu a(a 1) . Do , t : 2 2 2n n

n 1 n n 1 n

u u au lim u a a a 0

2010 2010 201u li

0mu

( V l )

Suy ra d~y {un} tng v{ khng b chn trn, nn : n nn 1

1limlimu 20100 limS

u



18

18

4. ( thi HSG Tnh Bnh nh nm 2010 ) . Cho d~y s 1

2n n

n 1 n

1 2

x1

x

{x , n 1

2

x } :x

. Chng minh d~y s {xn}

c gii hn v{ tm gii hn . Li gii :

Xt h{m s : 2x

f(x) 1 x , x2

(1;2) . Ta c : f '(x) 1 x 0 1 2), x ( ; . Do :

3

1 f(2) f(x) f(1) 22

. T thay x bi : 1 2 nx x ,...,x; ta c : 1 2 n,x ,...,1 x x 2

Suy ra d~y n{x } b chn .

Gi s d~y s c gii hn l{ a, lc a tha m~n pt : 2a

a 1 a a 22

Ta s CM gii hn n{y bng nh l kp :

Xt hiu :

2

2

nn 1 n n n

2x2 1 2

1x 1 x x x

22 2 2

2 2

Li c : n n n2 1 x 21 x 2 2 x 22 2 2

Do : n 1 n2

2 2x x2

(*) . T (*) cho n = 1,2, v{ nh}n li vi nhau ta c :

n

n

1

1 1

2x2x

22

. M{ n 1

1 nlim x2

2 0 limx2

2

5. ( Bi ton tng t ) . Cho d~y s 1

n 2

nn 1

1

3

u1, n 1

2

u

{u } :

u

. Tm nlimu .

6. ( thi HSG Tnh Bn Tre nm 2010 ) . Cho d~y s 1

n 2 2

n 1 n n n n

1

x x

x{x } :

1 x xx 1

. Chng minh rng

d~y s trn c gii hn v{ tm gii hn . Li gii :

Ta c : 2 2 nn 1 n n n n

2 2

n n n n

2xx x 1 x x 1

x x 1 xx

x 1

Bng quy np ta chng minh c rng : nx 0, n 1,2,...

Li c :

2 2 2 2

2 2

n n n n n nMincopxki

22

n nMincopxki

1 3 1 3x x 1 x x 1 x x

2 2 2 2

1 1 3 3x x 2

2 2 2 2

T suy ra : n 1 nx x

Vy d~y n{x } gim v{ b chn di bi 0 nn tn ti gii hn hu hn. Gi s

2 2

n a 1limx a a a a a 0a 1

7. ( thi HSG Tnh Ngh An nm 2009 ) . Cho d~y s : 1

n 1 2 n 1n 2

2

2x ... (n

x

{x } : xx , n 1

n

1)x

1n )(

Tnh nlimU vi 3

n nU (n 1) .x

Li gii : Ta c :



19

19

+) 2x

1

3

+) Vi n 3 ta c : 2 31 2 n 1 n n n nx nx n(n nx2x ... (n 1)x 1)x xn

2 31 2 n 2 n 1 n 1 n 1 n 1x (n 1)x (n 1) (n 1) (n 1)x (n 1)2x ... (n 2)x 1 x x

T suy ra : 3

3 3 nn n n 1 3

n 1

2x (n 1) n 1 n

n nx (n 1) xx nn n

x1n

(*)

T (*) cho n = 3,4ta c : 2 2 2

n n n 1n

3

22 2

n 1 n 22

xx x x n 1 n 2 2 n n 1 3 12 4. ... . ...

(n. . ... x

x x x 1)x n n 1 3 n 1 n 4 n (n 1) n

Do : 3

n 2

4(n 1)limU lim 4

n (n 1)

.

9. ( thi HSG Tnh H Tnh nm 2010 ) . Cho d~y 2n nn 1

n

0

n

2

x 0

{x } : x (x 3), n

1x

3x0

. Chng minh d~y c gii hn v{

tm gii hn . Li gii :

Bng quy np ta chng minh c n 0, nx 0

+) TH1 : Nu 0x 1 , quy np ta c n 1, nx 0 . Hin nhin nlimx 1

+) TH1 : Nu 0x 1 ,

Xt h{m s : 2

2

x(xf(x)

3)

13x

trn khong (1; ) ta c :

2 2

2 2

xf '(x) 0

(x 1)x (1; ) f(x),

(3f( ) 1

x1

1)

Do : 2 1 ) 1x f ,x .( .. . quy np ta c : nx 1, n

Li c : k k k kk kk

2

2

k

2

k 1 2

(x 3) 1)x

1

x 2x (xx x 0

3x 3x 1

ng vi

kx 1

T ta c : 1 2 n n 1x ....x x x 1 . D~y s gim v{ b chn di nn tn ti gii hn hu hn .

Gi s : 2

n 2

a a 3

1limx a 0 a a 1

3a

+) TH3 : Nu 00 1x , Xt h{m s :

2

2

x(xf(x)

3)

13x

trn khong (0;1) ta c :

2 2

2 2

(x 1)x (0;1) 0 f(0) f(

xf '(x) 0,

(x) f( ) 1

1)3x1

Do : 2 1f(x ) (0;1x ),... quy np ta c : n (0; nx 1),

ta c : k k k kk kk

2

2

k

2

k 1 2

(x 3) 1)x

1

x 2x (xx x 0

3x 3x 1

ng vi k0 1x

Do : 1 2 n n 10 x x ... x x 1 . D~y s tng v{ b chn trn nn tn ti gii hn hu hn . Gi s :

2n 2

a a 3

1limx a 0 a a 1

3a

Kt lun : nlimx 1

10. ( Bi ton tng t ) . Cho 0; a 0 l{ hai s ty . D~y 0

2n n n

n 1 2

n

(u 3a)

a

u

{u } : uu ,n 0,1,...

3u

. Chng minh d~y

c gii hn v{ tm gii hn .

11. ( Chn i tuyn H Vinh nm 2010 ) . Cho d~y s

0

2n n n

n 1

n

1

1 2(u 1

u

){u } : uu , n 0,1..

u 1.

. Tm nlimu



20

20

12. ( thi chn T HSG QG KonTum nm 2010 ) . Cho d~y s thc n{a } x|c nh nh sau :

1

n 1 n

n

1

1(na a )

a

1a

.

Chng minh rng : nn

alim 2

n

13. ( thi HSG Tnh Hi Dng nm 2006 ) . Cho d~y s thc n1 n 1

2

n

x2006; x 3x

x 1

. Tm n

xlim x

14. ( thi HSG Tnh Ph Th nm 2008 ) . Cho d~y s n{x } tha m~n :

1

n 1 n n n n

1

x (x 1)(x 2)(x 3

x

1 , 0x ) n

. t ni

n

i 1

1

xy

2 . Tm nlimy .

HD : 2

2 2

n 1 n n n n n n n n

n n n 1

1 1 1x (x 1)(x 2)(x 3) 1 x 3x 1 x 3x 1

x 2 x 1 1x

x

Sau chng minh d~y tng v{ khng b chn trn .

15. Cho d~y 1n 2n 1 n n

x a 1):

2010x(

0 9xx

2 0 x

. Tm : 1 2 n

n2 3 n 11 1 1

x x xlim ...

x x x

HD : Xt h{m s : 2x 2009x

f(x) , x 12010 2010

. Ta c : f(x) > 0 , x 1 f(x) f(1) 1 . Bng quy np chng minh

c rng : nx 1, n . Xt hiu :

2

n n n nn 1 n n n 1 n

x x x (xx 0,

2010 2010 201

1)x

0x 1 x x

Gi s 2nlimx a a 1 201 2009a a 0;a 10a a ( Khng tha m~n ). Vy nlimx Li c :

2 n n 1 nn 1 n n n 1 n n n

n 1 n n 1 n n 1

x 1 12010x x ) x 1) 2010 2010

x 1

x x2009x 2

(x 1)(x 1) x 1 x 1010(x x (x

16. ( Bi tng t ) . Cho d~y s : 1

24n n

n 1 n

x 1

): xx x N *

2

(x, n

4

. Tm gii hn 23 23 23

1 2 n

2 3 n 1

x x xlim ...

x x x

17. ( thi HSG Tnh Bnh Phc nm 2008 ) . t 2 2f(n) (n n 1) 1 vi n l{ s nguyn dng . Xt d~y s

n n

f(1).f(3).f(5)...f(2n 1)(x

f(2).f(4).f(6)...f nx

)):

(2

. Tnh gii hn ca d~y s : 2n nu n .x

HD : Ch : 2

2

f(k 1) (k 1)

f(k) (k 1

1

1)

18. Cho d~y s n(a ) x|c nh bi :

n

i 1

1

2

i n

2a

a n

008

,n 1a

. Tnh 2n

nim al n

HD : Ta c 22 2

1 2 n n n 1 n n n 1

n 1a n n 1 a n a aa ... a a

11 a

n

(1)

Trong (1) cho n=1,2,3.v{ nh}n n li tm : an

19. Cho d~y s (nx ) tha : 1 n 1

n

2006x 1,x 1 (n 1)

1 x

. Chng minh d~y s (

nx ) c gii hn v{ tm gii hn y

20. ( thi HSG QG nm 2009 ) . Cho d~y s 1

n 2

n 1 n 1 n 1

n

1x

2):

x 4x xx

2

(

, n 2

x

. Chng minh rng d~y n(y ) vi

n 2

n

i 1 i

1y

x c gii hn hu hn khi n v{ tm gii hn .

Li gii : :



21

21

Xt h{m s : 2x x

f(x)2

4x , ta c :

2

2x 4 1f '(x) 0,

24x

4xx0

Li c : 2 1 1f(x ) 0,(do x 0)....x bng quy np ta chng minh c nx 0, n .

Xt hiu : 2 2

n 1 n 1 n 1 n 1 n 1 n 1 n 1n n 1 n 1 n

2

n 1 n 1 n 1

x 4x x x 4x x 4xx x 0,(do x

2x 0,

xn

x 4)

2 x

Suy ra d~y n{x } tng v{ nx 0, n . Gi s tn ti gii hn hu hn

nn 0im ( )a l x a

. Suy ra :

22a aa a a a 0

2

4a4a

(V l ) .

Vy d~y n{x } tng v{ khng b chn trn nn :

nnlimx

Li c :

2

2n 1 n 1 n 1 2 n n n 1 n 1n n n 1 n 1 n 1 n n n 1 n 1 2 2 2

n 1 nn n 1 n n 1 n

x 4x x x (x x ) xx 2x 4x x (x x )

1 1 1x x

x xx .x xx

2 .x x

Do : 1n n2 2 2

1 2 n 1 n ni 1

n

ni 1 1

1 x1 1 1 1 1 1 1y ... lim y 6

x x x x xx x x

.

21. Xt d~y s thc n(x ),n N x|c nh bi :

0

3n n 1 n 1

2009

6x 6sin(x

x

), n 1x

. Chng minh d~y c gii hn hu hn

v{ tm gii hn .

HD : S dng bt ng thc : 3x

x ins x,x6

x 0

Xt h{m s : 3f(x) 6x 6sinx ,x 0 . Ta c : 3 2

1 6(1 cosx)f '(x) 0, x>0

3 (6x 6sinx)

Do : f(x) 0 x, 0 . M{ 2 1 1 n n 1f(x ) 0(do x 0) ...x f(x ) 0, nx

Xt hiu : 33

3

n 1 n 1 n 1n n 1 n 1 n 1 n 1

2 2

n 1 n 1 n 1 n 1 n 13

n 1

)x 6sin(x

6x x 6sin(xx 6x x 0)

6sin(x ) 6sin6x x 6x x(x )

(S dng Bt ng thc : 3

3xx inx 6x xs 6sinx 0, x 06

)

Do d~y n{x } gim v{ b chn di, nn tn ti gii hn hu hn . Gi s : nlimx a 0(a ) , ta c pt :

3 3a 6a 6sina a 6a 6sina . Xt h{m s : 3g(t) t 6sint 6t , ta c : 2g'(t) 3t 6cost 6, g''(t) 6t 6sint 0, t 0 g'(t) g(0) 0 g(t) g(0) 0 . Do pt c nghim duy

nht a 0 .

22. Cho d~y (xn) c x|c nh bi: x1 = 5; xn + 1 = 2

nx - 2 n = 1, 2, . Tm n 1

n1 2 n

xlim

x .x ...x

23. Cho d~y

1 n

n

1

2

n+ n

x = 3(x

x = 9x +11x + 3; n 1, :

N)

n .

. Tm n 1

nn

xlim

x

HD : Chng minh d~y nx tng v{ khng b chn :

D thy nx 0, n , xt : 2

n n2

n

2

n nn 1 n

n

n n

n

8x 11x 3x x 9x +11x + 3 x 0,

9x +11x + 3x 0

x

Gi s n

2

n

a 1

lim x a a 0 a 9a 1a 3 3a

8

1

( Khng tha m~n ) n

nlim x

Do : n n

n 1

2n n n

x 11 3lim lim 9 3

x x x



22

22

24. Cho d~y s n(u ) x|c nh bi cng thc

1

2 2

n+1 n n

u = 2008

u = u - 4013u + 2007 ; n 1, n N.

a) Chng minh: nu n + 2007; n 1, n N .

b) D~y s (xn) c x|c nh nh sau:

n1 2 n

1 1 1x = + + ... + ; n 1, n N.

u - 2006 u - 2006 u - 2006

Tm nlimx ?

25. ( thi HSG Tnh Tr Vinh-2009)Cho d~y s (nU ) x|c nh bi:

1

33n 1 3 n

U 1

4U log U 1 , n 1

3

Tm n

n

limU

26. Cho d~y s nn

0

xn n

n 1 x

x 1

): 2 l(x x 1

ln2 1

n2 1x

2

. Chng minh d~y (xn) c gii hn v{ tm gii hn .

HD : Chng minh d~y gim v{ b chn di .

27. Cho phng trnh : n n 1x x .... x 1 0 . Chng t rng vi n nguyn dng th phng trnh c nghim duy

nht dng nx v{ tm nx

lim x

.

28. Cho d~y s n{u } x|c nh bi

1

n n

n 2n

1

C n

u

u .4

. . Tm nlimu

29. ( thi HSG Tnh Ngh An nm 2008 ) . Cho phng trnh:x

1x n 0

2008 (1). Chng minh rng: vi mi n

N* phng trnh (1) c nghim duy nht, gi nghim l{ xn. Xt d~y (xn), tm lim (xn + 1 - xn). Li gii : ( P N S GD&T )

Vi n N*, xt f (x) = x

1x n

2008 ; x R.

f/(x) = - x

ln2008

2008 - 1 < 0 x R.

=> f(x) nghch bin trn R (1).

Ta c: n

n 1

1f(n) 0

2008

1f(n 1) 1 0

2008

=> f(x) =0 c nghim xn (n; n + 1) (2). T (1) v{ (2) => pcm.

Ta c: xn - n = nx

1

2008 > 0 => xn > n.

=> 0 < xn - n < n

1

2008.

Mt kh|c: limn

10

2008 => lim(xn - n) = 0.

Khi lim (xn - 1 - xn) = lim{[xn + 1- (n + 1)] - (xn - n) + 1} = 1 MT S BI TON TM GII HN KHI BIT CNG THC TNG QUT CA DY S .

30. Cho d~y s 1

n n 1n

n 1

2

24

u

u : 9uu , n 2

5u 13

. Tm nlimu ?

Gii :



23

23

31. Cho d~y s 1n2

n n 1

1

2

2

uu :

u u 1 , n 2

. Tm nn

ulim

n

HD : Tm c : n 1

n

2u cos

3

v{ ch : x

n nu u10 lim 0n n n

32. Cho d~y s 1

n 2

n 1

n

u

u :2 2 1 u

u2

1

2

, n 2

. Tm nn

nlim 2 .u

HD : Tm c n n 1

u sin2 .6

suy ra : nn

n n

n

n

sin3.2

lim 2 .u lim3 3

3.2

33. Cho d~y s 1

n 1nn

2

n 1

u

u :u

1 1

3

u, n

u2

. Tm n

nnlim 2 .u

HD : Tm c n n 1

u tan3.2

34. Cho d~y s 1

nn 1

n

n 1

2

3

u, n 2

2(2n 1

u :

u)u 1

u

. Tm i

n

ni 1

lim u

35. Cho d~y s : 1

2

n 2 n n 1

1

2

u 2u N *

u

u

u , n

. Tm n 1

nn

ulim

u

HD : Tm c n

n

n2u 1 1

42 2

. Suy ra :

n 1

n 1

n 1

n 1

x nn nn

1

22 2 2

12 1

1 1 1u 4lim

u 21 1 11 1

41 11

2 1

22 2

2 22

36. Cho d~y s 1

n n 1n

n 1

u

u :u

1 3

3

3 u, n 2

u

. Tnh nn

ulim

n

HD : nn

u tan3

37. Cho d~y s n(u )x|c nh nh sau : nu 2 2 2 .... 2 ( n du cn ) . Tnh

1 n

nn

2u .u ...ulim2

HD : t : nn

n n 1

ux x cos

2 2

v{ ch : 1 2 n 1 2 nn n

n 1

sinu .u ...u 1 2x ...x2 2

si

.

2

x

n



24

24

38. Cho d~y s

1

n2

n 1 n n n

1b

2):

1 1b b

(b

b (n 1)2 4

. Chng minh d~y hi t v{ tm n

nlim b

HD : Chng minh : n n n 1

1b .cot

2 2

Phn V : HNH HC KHNG GIAN


25

25

PHN V : HNH HC KHNG GIAN

1. Cho hnh chp tam gi|c u c th tch l{ 1. Tm gi| tr ln nht ca b|n knh mt cu ni tip hnh chp. 2. Cho t din ABCD c : AB=a; CD=b ; gc gia AB v{ CD bng . Khong c|ch gia AB v{ CD bng d. Tnh th tch

khi t din ABCD theo a,b,d v{ . 3. Trong c|c t din OABC c OA, OB, OC i mt vung gc vi nhau v{ th tch bng 36. H~y x|c nh t din sao

cho din tch tam gi|c ABC nh nht.

4. Cho hnh hp ABCD.A1B1C1D1 . C|c im M, N di ng trn c|c cnh AD v{ BB1 sao cho 1

MA NB

MD NB . Gi I, J ln lt

l{ trung im c|c cnh AB, C1D1 . Chng minh rng ng thng MN lun ct ng thng IJ. 5. Gi O l{ t}m ca mt hnh t din u . T mt im M bt k trn mt mt ca t din , ta h c|c ng vung gc

ti ba mt cn li. Gi s K, L v{ N l{ ch}n c|c ng vung gc ni trn. Chng minh rng ng thng OM i qua trng t}m tam gi|c KLN.

6. Cho hnh chp S.ABC . T im O nm trong tam gi|c ABC ta v c|c ng thng ln lt song song vi c|c cnh SA, SB, SC tng ng ct c|c mt (SBC), (SCA), (SAB) ti c|c im D,E,F .

a) Chng minh rng : OD DE DF

1SA SB SC

b) Tm v tr ca im O trong tam gi|c ABC th tch ca hnh chp ODEF t gi| tr ln nht. 7. Cho hnh hp ABCD.A1B1C1D1 . H~y x|c nh M thuc ng cho AC1 v{ im N thuc ng cho B1D1 ca mt

phng A1B1C1D1 sao cho MN song song vi A1D. 8. C|c im M, N ln lt l{ trung im ca c|c cnh AC, SB ca t din u S.ABC . Trn c|c AS v{ CN ta chn c|c

im P, Q sao cho PQ // BM . Tnh d{i PQ bit rng cnh ca t din bng 1.

9. Gi O l{ t}m mt cu ni tip t din ABCD. Chng minh rng nu 0ODC 90 th c|c mt phng (OBD) v{ (OAD) vung gc vi nhau .

10. Trong hnh chp tam gi|c u S.ABC (nh S ) d{i c|c cnh |y bng 6 . d{i ng cao SH = 15 . Qua B v mt phng vung gc vi AS, mt phng n{y ct SH ti O . C|c im P, Q tng ng thuc c|c cnh AS v{ BC sao

cho PQ tip xc vi mt cu t}m O b|n knh bng 2

5 . H~y tnh d{i b nht ca on PQ.

11. Cho hnh lp phng ABCD.A1B1C1D1 cnh bng a . ng thng (d) i qua D1 v{ t}m O ca mt phng BCC1B1 . on thng MN c trung im K thuc ng thng (d) ; M thuc mt phng (BCC1B1) ; N thuc mt |y (ABCD) . Tnh gi| tr b nht ca d{i on thng MN .

12. Cho t din ABPM tho m~n c|c iu kin : 0 2AM BP; MAB ABP 90 ; 2AM.BP AB . Chng minh rng mt

cu ng knh AB tip xc vi PM. 13. ( thi HSG Tnh Qung Ninh nm 2010 ) Cho im O c nh v{ mt s thc a khng i . Mt hnh chp

S.ABC thay i tha m~n : OA OB OC a; SA OA;SB OB;SC OC ; 0 0 0ASB 90 BSC 60 CSA; ; 120 . Chng

minh rng : a. ABC vung . b. Khong c|ch SO khng thay i .

Gii : a) t : SO = x .

Ta c : C|c tam gi|c OAS, OBS, OCS vung nn : 2 2SA SB SC ax .

Do : 2 2 2 2 2AB S SB a )A 2(x ; 2 2 2 0 2 2SC 2SA.SCAC SA os120 3(x. a )c ;

2 2 2 0 2 2SB SC 2SB.SBC os6C.c a )0 (x 2 2 2AB BCAC hay tam gi|c ABC vung ti B.

b) Gi M l{ trung im AC , do c|c tam gi|c SAC, OAC l{ c|c tam gi|c c}n nn :

SM ACAC (SOM) AC OS

OM AC

Tng t, gi N l{ trung im AB, ta CM c : AB SO

Suy ra : SO (ABC) .

Do mi im nm trn ng thng SO u c|ch u A, B, C . Suy ra SO i qua t}m ng trn ngoi tip M ca tam gi|c ABC . Trong c|c tam gi|c vung ABC v{ SBO ta c h



26

26

thc :2 2 2

2 2 2

1 1 1

BM AB BC

1

BM

1 1

OB BS

2 2 2 2

1 1

OB BS

1 1

AB BC 2 2

2 2 2 2 2 2 2

1 1 1 1 33a x a

22(x x a2x

) axa a

14. ( thi HSG Tnh Vnh Phc nm 2010 ) . Cho hnh chp S.ABCD c |y ABCD l{ hnh ch nht , AB = a ;

BC 2a . Cnh bn SA vung gc vi |y v{ SA=b . Gi M l{ trung im SD, N l{ trung im AD . a) Chng minh AC vung gc vi mt phng (BMN) b) Gi (P) l{ mt phng i qua B, M v{ ct mt phng (SAC) theo mt ng thng vung gc vi BM .

Tnh theo a, b khong c|ch t S n mt phng (P) . Li gii :

t AS x;AB y;AD z x.y y.z z.x 0;| x| b;| y | a;|z| a 2

Ta c : AC AD AB y z v{ 1

BN AN AB z y2

Do : 2

2 2 2(ay a 0 A1 2)

AC. C BNN z2 2

B

Li do : 1

MN SA MN AC2

Hay : AC (BMN) AC BM

Gi s (P) ct (SAC) theo giao tuyn (d) BM M{ do (d) v{ AC ng phng (d)/ /(AC)

Gi O (AC) (BD)

Trong mt phng (SDB) : SO ct BM ti I. Qua I k ng thng (d) // (AC) ct SA, SC ln lt ti H, K . Mt phng (MHBK) l{ mt phng (P) cn dng . Li v : I l{ trng t}m tam gi|c SDC v{ HK//AC nn :

SH SK SI 2

SC SA SO 3 (1)

Theo cng thc tnh t s th tch ta c :

SMBK SMHB

SDBA SDCB

V VSM SB SK 1 SM SH SB 1. . ; . .

V SD SB SA 3 V SD SC SB 3

2SABCD

SKMHB SKMB SMHB SDBA

V2 b 2V

aV V V

3 3 9 (2)

Ta li c : KMHB MKH BKH1 1 1

S MI.HK BI.HK BM.HKS2

S2 2

(3)

M{ : 2 22 2 2

HK AC a3.a

(a 2)3 3 3

; 1 1BM AM AB AS AD AB x z y2 2

2 2

2 2 2 2 2 2 6aBM) z1 1 3 b

( x y b a BM4 4 2 2

(4)

T (3), (4) suy ra : 2 22 2

KMHB

a 3(b1 b 2 6a )6S .

2

a 3a

2 3 6

(5)

T (2), (5) suy ra : 2

SKMHB

2 2 2 2KMHB

b 23V 18a 2d(S,(P))

S 9a. 3(

2ab

6a 6ab ) 3(b )

15. ( thi HSG Tnh Bnh Phc nm 2010 ) . Cho hnh lp phng ABCD.ABCD c cnh bng a . Trn AB ly im M, trn CC ly im N , trn DA ly im P sao cho : AM CN D'P x x a)(0 .

a) CMR tam gi|c MNP l{ tam gi|c u, tm x din tch tam gi|c n{y nh nht .

b) Khi a

x2

h~y tnh th tch khi t din BMNP v{ b|n knh mt cu ngoi tip t din .



27

27

16. ( thi HSG Tnh B Ra Vng Tu nm 2008 ) . Cho t din ABCD c c|c cnh AB=BC=CD=DA=a , AC x; BD y . Gi s a khng i, x|c nh t din c th tch ln nht.

17. ( thi HSG Tnh B Ra Vng Tu nm 2009 ) Cho khi t din ABCD c th tch V . im M thuc min trong tam gi|c ABC . C|c ng thng qua M song song vi DA, DB, DC theo th t ct c|c mt phng (DBC), (DCA), (DAB) tng ng ti A1 ; B1 ; C1 .

a) Chng minh rng : 1 1 1MA MB MC

1DA DB DC

b) Tnh gi| tr ln nht ca khi t din 1 1 1MA B C khi M thay i .

18. ( thi HSG Tnh Hi Phng nm 2010 ) . Cho t din OABC c OA, OB, OC i mt vung gc . Gi ; ; ln

lt l{ gc to bi c|c mt phng OBC, OAC, OAB vi mt phng (ABC ).

a) Chng minh rng : 2 2 2 2 2 2tan tan tan 2 tan .tan .tan

b) Gi s OC=OA+OB . Chng minh rng : 0OCA OCB ACB 90 19. ( thi HSG Tnh Ngh An nm 2008 ) . Cho t din ABCD c AB = CD, AC = BD, AD = BC v{ mt phng (CAB)

vung gc vi mt phng (DAB). Chng minh rng: 1

CotBCD.CotBDC = .2

Li gii 1 : t : BCD ; BDC

Ta c :

BAC BDCABC DCB

ABC BCD

BAD BCDCBD ADB

ABD CDB

Gi H l{ hnh chiu ca C ln AB . t HC x .

Do CBA DAB

CH DH(CBA) (BDA)

Trong tam gi|c vung BHC : HC HC x

sin BC ADBC sin sin

HC x xtan BH

BH BH tan

.

Trong tam gi|c vung AHC : HC HC x

sin AC BDAC sin sin

.

HC x xtan AH

AH AH tan

Trong tam gi|c BCD : 2 2

2 2 2

2 2

x x x xCD BC os 2 . cos

sin sinsiBD 2BC.BD.c

n sin

(1)

Li c :

2 2 2H AD 2AHD .AH oc sAD.

2 22

2 2

x x x xHD 2 . .cos

tan sintan sin

(2)

M{ tam gi|c CHD vung nn :

2 2(1) (

22)

CH HDCD

2 2 2 2

2

2 2 2 2

x x x x x x x x2 . cos x 2 . .cos

sin sin tan sinsin sin tan sin

2 22 2 1(1 cot ) (1 cot ) 2(cot .cot 1) 1 cot (1 cot ) 2cot .cot cot .cot2

Li gii 2 : ( P N CA S GD&T )

t AD BC a,AC BD b,AB CD c,BAC A,ABC B,ACB C.

Ta c ABC nhn v{ ABC = DCB = CDA = BAD.

Suy ra BCD ABC B;ABD BDC CAB A, 1



28

28

H CM AB , v CAB DAB nn 2 2 2CM DAB CM MD CM DM CD , 2 .

|p dng nh l cosin cho tam gi|c BMD ta c 2 2 2MD BM BD 2BM.BD.cosMBD, 3

T (1), (2), (3) ta c 2 2 2 2CM BM BD 2BM.BD.cosA CD 2 2 2 2 2 2BC BD 2BM.BD.cosA CD a b 2abcosA.cosB c

1cosC cosA.cosB sinA.sinB 2cosA.cosB cot A.cot B .

2

20. ( thi HSG Tnh Ngh An nm 2008 ) .Cho khi chp S. ABCD c |y ABCD l{ hnh bnh h{nh. Gi M, N, P ln lt l{ trung im ca c|c cnh AB, AD, SC. Chng minh rng mt phng (MNP) chia khi chp S.ABCD th{nh hai phn c th tch bng nhau.

21. ( thi HSG Tnh Ngh An nm 2009 ) . Cho tam gi|c ABC , M l{ mt im trong tam gi|c ABC. C|c ng thng qua M song song vi AD, BD, CD tng ng ct c|c mt phng (BCD), (ACD) , (ABD) ln lt ti A, B, C . Tm M sao cho MA'.MB'.MC' t gi| tr ln nht.

Li gii 1 : t DABC MABD MAC BDC MBC AA B CV ; ;V; V V V V V V V VV V v{ :

DA a; BD b; DC c; MA' x;MB' y;MC' z

Ta c : CV d(C,(ADB)) MC' z

V d(M,(ADB)) CD c ; tng t : A B

V Vx y x y z; 1

V a V b a b c

p dng bt ng thc AM-GM : 3x y z xyz abc

1 3 xyza b c abc 27

. Du = xy ra

x y z 1

a b c 3

Do : MA'.MB'.MC' t gi| tr ln nht khi v{ ch khi M l{ trng T}m tam gi|c ABC . Li gii 2 : t : DA a; BD b; DC c; MA' x;MB' y;MC' z

Ta c : A'M x

A M .DA DADA a

; B M y

B M .DB .DBDB b

;

Li gii 2: ( P N S GD&T ) Trong mt phng (ABC) : AM BC = {A1}; BM AC = {B1}, CM AB = {C1} Trong (DAA1) : K ng thng qua M song song vi AD ct DA1 ti A

Xt tam gi|c DAA1 c MA // AD nn MBC1

1 ABC

SMAMA'

DA AA S

Tng t ta c MAC1

1 ABC

SMBMB'

DB BB S

, 1 MAB

1 ABC

MC SMC'

DC CC S

Suy ra MBC MAC MAB ABCMA' MB' MC'

1 doS S S SDA DB DC



29

29

Ta c 3MA' MB' MC' MA' MB' MC'

3 . .DA DB DC DA DB DC

Suy ra MA.MB.MC 1

27DA.DB.DC (khng i)

Vy gi| tr ln nht MA.MB.MC l{ 1

27DA.DB.DC, t c khi 1 1 1

1 1 1

MA MB MCMA' MB' MC' 1 1

DA DB DC 3 AA BB CC 3

Hay M l{ trng t}m tam gi|c ABC 22. ( Tp ch THTT : T10/278 ; T10/288 ) . Cho t din S.ABC vi SA=a; SB =b ; SC = c . Mt mt phng ( ) thay i

i qua trng t}m ca t din ct c|c cnh SA, SB, SC ti c|c im SA, SB, SC ti c|c im D, E, F tng ng .

a) Tm gi| tr nh nht ca c|c biu thc : 2 2 2

1 1 1

SD SE SF

b) Vi k : a=b=c=1, tm gi| tr ln nht ca : 1 1 1

SD.SE SE.SF SF.SD

Li gii : t : SD x; SE y ; SF z

G l{ trng t}m t din nn : 1 1 SA 1 aSG SA SB SC .SD .SD4 4 SD 4 x

Do D,E,F, G ng phng nn : a b c

4x y z . T ta c :

2

2 2 2

2 2 2 2 2 2 2 2 2

1 1 1 a b c 1 1 1 16a b c (1)

x y zx y z x y z a b c16

Du bng xy ra

2 2 2

2 2 2

2 2 2

a b c

4a

a b c

4

x

y

z

b

a b c

4c

23. ( thi HSG Tnh Ngh An nm 2009 ) . Cho t din ABCD c d{i c|c cnh bng 1 . Gi M, N ln lt l{ trung im ca BD, AC . Trn ng thng AB ly im P , trn DN ly im Q sao cho PQ song song vi CM . Tnh d{i PQ v{ th tch khi AMNP . Li gii 1 :

Gi s : AB x;AC y;AD z v{ : AP

m;AQ n.AC (1 n)ADAB

Ta c : 1

x.y y.z z.x2

Lc :

1 n

AC y;AM x z ;AP m.x;AQ n.AN (1 n)zAD .y (1 n)z2 2

Suy ra : 1

CM AM AC x 2y z2

nPQ AQ AP mx y (1 n)z

2

Do CM // PQ nn :

km

2

n 2PQ kCM k k

2 3

k1 n

2

Vy : 221PQ 2y x z |PQ

1 1| 2y x z

3

3PQ

9 3 3

Li gii 2 : ( P N CA S GD&T ) Trong mt phng (BCD) k BK // CM (K CD)



30

30

Trong (ABD) DI ct AB ti P Trong (AKD) DN ct AK ti Q PQ l{ giao tuyn ca (DNI) v{ (ABK) , do NI // CM, BK // CM nn PQ // CM Gi E l{ trung im PB, ME l{ ng trung bnh tam gi|c BPD nn ME // PD hay ME // PI Mt kh|c t c|ch dng ta c I l{ trung im AM nn P l{ trung im AE. Vy AP = PE = EB

Suy ra AP 1

AB 3

MC l{ ng trung bnh tam gi|c DBK nn BK = 2CM = 3

Suy ra PQ AP 1

BK AB 3 PQ =

1

3BK =

3

3

AMNP

AMCB

V AM AN AP 1 1 1. . .

V AM AC AB 2 3 6

VAMCB =1

2 VABCD (Do M l{ trung im BD)

ABCD l{ t din u c d{i cnh bng 1 nn VABCD = 2

12(vtt)

Suy ra VAMCB = 1 2 2

.2 12 24

. Vy VAMNP = 1

6 V AMCB =

2

144(vtt)

24. ( d b khi D 2008 ) . Cho t din ABCD v{ c|c im M, N, P ln lt thuc c|c cnh BC, BD, AC sao cho

BC 4BM; AC 3AP; BD 2BN . Mt phng (MNP) ct AD ti Q . Tnh t s AQ

AD v{ t s th tch hai phn ca khi

t din ABCD c ph}n chia bi (MNP). Li gii :

t : AB b;AC c; AD d

Ta c :

1AN b d (2)2

1AC 3AP AP c (3)

3

Do C,D,I v{ M, N, I thng h{ng nn :

AI mAC (1 m)AD 3 1 1 1c (1 m)d n b c (1 n) b d

4 4 2 2AI nAM (1m

n)AN

n

ID 14 2AI 3AD AC2DI CD1 n IC 3

AI 2AM 3AN2 IN 2NI 2M

m

n 2

1 m 1m

2c d

N1 33n 1 n AI IM 30 2 24 2

Gi s : AQ kAD . Do P, Q, I thng h{ng nn :

3p 1 pp

p 1 3 53 2AQ pAP (1 p)AI kd c (1 p) c d 5AQ 3AP 2AI 3PQ 2QI3 2 2 33(1 p)

kk52

Suy ra : QI 3

PI 5

Ta li c : IQND QPMCDN

IPMC IPMC

V VIQ IN ID 3 2 1 2 13. . . .

V IP IM IC 5 3 3 15 V 15 (4)

3 1BC 4BM AC AB 4 AM AB AM b c (1)4 4



31

31

M{ :

BCDABCD

PMCI MIC

d A,(BCD) .SV AC CB.CD.sinC 3 4 2 4. . .

V PC MC.CI.sinC 2 3 3 3d P,(MIC) .S (5)

T (4) v (5) suy ra : PQDNMC PQDNMC

ABCD ABMPQN

V V13 3 13 13

V 15 4 20 V 7

25. ( thi HSG Tnh H Tnh nm 2008) . Cho hnh chp t gi|c u S.ABCD c gc gia mt bn v{ |y l{ . V ng cao SH ca hnh chp, gi E l{ im thuc SH v{ c khong c|ch ti hai mt phng (ABCD) v{ (SCD) bng nhau . Mt phng (P) i qua E, C, D ct SA, SB ti M, N .

a) Thit din l{ hnh g ? b) Gi th tch c|c khi t din S.NMCD v{ ABCDNM ln lt l{ V1 , V2 . Tm 3V2 =5V1.

26. ( thi chn T HSG QG tnh Qung Bnh nm 2010 ) . Cho t din ABCD . Gi trung im ca AB, CD ln lt l{ K , L . Chng minh rng bt k mt phng n{o i qua KL u chia khi t din n{y th{nh 2 phn c th tch bng nhau.

27. ( thi HSG Thnh Ph Cn Th nm 2008 ) . Trong khng gian cho hnh chp S.ABC , trng t}m ABC l{ G . Trung im ca SG l{ I . Mt phng ( ) i qua I ct c|c tia SA, SB, SC ln lt ti M, N, P ( Khng trng vi S ) . X|c

nh v tr ca mt phng ( ) th tch khi chp S.PMN l{ nh nht .

28. ( thi HSG Tnh Hi Dng nm 2008 ) . Cho hnh lp phng 1 1 1 1BAB .A CD DC cnh bng 1 . Ly c|c im M,

N, P, Q, R , S ln lt thuc c|c cnh AD, AB, BB1, B1C1, C1D1, DD1 . Tm gi| tr nh nht ca d{i ng gp khc khp kn MNPQRSM .

29. Cho hnh chp t gi|c S.ABCD, c |y ABCD l{ mt hnh bnh h{nh. Gi G l{ trng t}m ca tam gi|c SAC. M l{ mt im thay i trong min hnh bnh h{nh ABCD .Tia MG ct mt bn ca hnh chp S.ABCD ti im N .

t: Q = MG NG

NG MG

a) Tm tt c c|c v tr ca im M sao cho Q t gi| tr nh nht . b) Tm gi| tr ln nht ca Q.

30. Trong mt phng (P) cho tam gi|c ABC . Ly im S khng thuc (P) . Ni SA, SB, SC . I l{ mt im bt k trong tam gi|c , gi AI ct BC ti A1 , CI ct AB ti C1 , BI ct AC ti B1 . K IA2//SA, IB2//SB, IC2//SC

2 2 2(SBC);B (SAC);A C (SAB) . CMR : 2 2 2

1 2 1 2 1 2

SA SB SC6

A A B B C C

31. ( thi HSG Tnh ng Thp nm 2009 ) .Cho hnh chp S. ABCD c |y ABCD l{ na lc gi|c u ni tip

ng trn ng knh AD = 2a. SA vung gc vi mp ( ABCD ) v{ SA = a 6 . a) Tnh khong c|ch t A v{ B n mp ( SCD ). b) Tnh din tch ca thit din ca hnh chp S.ABCD vi mp( ) song song vi mp( SAD) v{ c|ch

mp(SAD) mt khong bng a 3

4.

32. Cho t din OABC vi OA = a, OB = b, OC = c v{ OA, OB, OC i mt vung gc vi nhau. Tnh din tch tam gi|c ABC theo a, b, c. Gi , , l{ gc gia OA, OB, OC vi mt phng ( ABC). Chng minh rng:

2 2 2sin sin sin 1 .

33. Cho hai na ng thng Ax, By cho nhau v{ nhn AB l{m on vung gc chung . C|c im M, N ln lt chuyn ng trn Ax, By sao cho AM+BN = MN . Gi O l{ trung im AB, H l{ hnh chiu ca O xung MN .

a) Chng minh rng H nm trn mt ng trn c nh. 35. Khi M kh|c A, N kh|c B 36. Cho hnh lp phng ABCD.ABCD c c|c cnh bng a. Vi M l{ mt im thuc cnh AB, chn im N thuc cnh

DC sao cho AM+DN=a a). Chng minh ng thng MN lun i qua 1 im c nh khi M thay i.

b) . Tnh th tch ca khi chp B.AMCN theo a. X|c nh v tr ca M khong c|ch t B ti (AMCN) t gi| tr

ln nht. Tnh khong c|ch ln nht theo a.

37. Cho h nh t die n OABC a) Go i M la mo t ie m ba t ky thuo c mie n trong cu a h nh t die n OABC va x1; x2; x3; x4; la n l t la khoa ng ca ch

t M e n bo n ma t (ABC), (OBC), (OAC) va (OAB). Go i h1; h2; h3; h4 la n l t la chie u cao cu a ca c h nh cho p tam gia c O.ABC; A.OBC; B.OAC va C.OAB.

Ch ng minh to ng 31 2 4

1 2 3 4

xx x x

h h h h la mo t ha ng so .



32

32

b) Ca c tia OA, OB, OC o i mo t h p v i nhau m 1V

Vo t go c 600. OA = a. Go c BAC ba ng 900.

a t OB+OC = m. (m >0, a > 0). Ch ng minh m > 2a. T nh the t ch kho i t die n OABC theo m va a 45. Cho t din ABCD c d{i c|c cnh AB, CD ln hn 1 v{ d{i c|c cnh cn li nh hn hoc bng 1. Gi H l{

hnh chiu ca A trn mt phng (BCD); F, K ln lt l{ hnh chiu ca A, B trn ng thng CD.

a) Chng minh: 2CD

AF 1 - 4

.

b) Tnh d{i c|c cnh ca t din ABCD khi tch P = AH.BK.CD t gi| tr ln nht.

46. a) Cho hnh chp S.ABC c |y ABC vung ti A , bit AB = a , AC = a 3 ; ng cao hnh chp l{ SA = a 3 ; M l{

im trn on BC sao cho BM = 1

BC3

. Tnh khong c|ch gia hai ng thng AM v{ BS

b) Cho hai na ng thng Ax, By cho nhau. Hai im C, D thay i ln lt trn Ax v{ By sao cho:

1 2 3

AC BD AB .Chng minh rng: mt phng (P) cha CD v{ song song vi AB lun lun i qua mt im c nh I

trong mt phng (Q) cha Ax v{ (Q) song song By. 47. ( thi HSG Tnh Tr Vinh nm 2009 ) .Cho hnh chp tam gi|c u S.ABC c cnh |y AB=a, cnh bn SA=b.

Gi M,N ln lt l{ trung im AB v{ SC. Mt mt phng ( ) thay i quay xung quanh MN ct c|c cnh SA v{ BC theo th t P v{ Q khng trng vi S.

1) Chng minh rng AP b

BQ a

2) X|c nh t s AP

AS sao cho din tch MPNQ nh nht

48. Cho t din ABCD c b|n knh ng trn ngoi tip c|c mt u bng nhau . Chng minh rng c|c cnh i din ca t din u bng nhau .

49. Cho t din ABCD c c|c ng cao ;BB';CC'AA' ;DD' ng quy ti mt im thuc min trong ca t din . C|c

ng thng ;BB';CC'AA' ;DD' li ct mt cu ngoi tip t din ABCD theo th t l{ 1 1 1 1;B ;C ;DA .

1 1 1 1

AA' BB' CC' DD' 8

AA BB CC DD 3 .

50. Cho t din ABCD c AB vung gc vi AC v{ ch}n ng vung gc h t A n mt phng (BCD) l{ trc t}m tam

gi|c BCD . Chng minh rng : 2 2 2 26 AB AD AB CD DB CC

51. ( thi HSG TP H Ni nm 2004 ) .Cho t din ABCD DA=a, DB=b, DC=c i mt vung gc vi nhau.Mt im M tu thuc khi t din.

a) .Gi c|c gc to bi tia DM vi DA, DB, DC l{ , ., .CMR : 2 2 2sin sin sin 2

b) .Gi A B C DS ,S ,S ,S ln lt l{ din tch c|c mt i din vi nh A, B, C, D ca khi t din. Tm gi| tr

nh nht ca biu thc: A B C DQ MA.S MB.S MC.S MD.S

52. ( thi HSG TP H Ni nm 2005 ) .Hnh chp S.ABC c c|c cnh bn i mt vung gc v{ SA =a, SB=b, SC=c. Gi A, B, C l{ c|c im di ng ln lt thuc c|c cnh SA, SB, SC nhng lun tha m~n SA.SA =SB.SB=SC.SC. Gi H l{ trc t}m ca tam gi|c ABC v{ I l{ giao im ca SH vi mt phng (ABC). a) Chng minh mt phng (ABC) song song vi mt mt phng c nh v{ H thuc mt ng thng c nh. b) Tnh IA2+IB2+IC2 theo a, b, c.

53. ( thi HSG TP H Ni nm 2006 ) .Cho t din u ABCD c cnh bng 1. C|c in M, N ln lt chuyn ng trn c|c on AB, AC sao cho mt phng (DMN) lun vung gc vi mt phng (ABC). t AM=x, AN=y.

a) . Cmr: mt phng (DMN) lun cha mt ng phng c nh v{ : x + y = 3xy. b) . X|c nh v tr ca M, N din tch to{n phn t din ADMN t gi| tr nh nht v{ ln nht.Tnh c|c gi| tr .

54. ( thi HSG TP H Ni nm 2008 ) . Cho hnh chp S.ABCD c SA l{ ng cao v{ |y l{ hnh ch nht ABCD, bit SA = a, AB = b, AD = c.

a) Trong mt phng (SBD), v qua trng t}m G ca tam gi|c SBD mt ng thng ct cnh SB ti M v{ ct cnh SD ti N. Mt phng (AMN) ct cnh SC ca hnh chp S.ABCD ti K. X|c nh v tr ca M trn cnh SB sao cho th tch ca hnh chp S.AMKN t gi| tr ln nht, nh nht. Tnh c|c gi| tr theo a, b, c.

b) Trong mt phng (ABD), trn tia At l{ ph}n gi|c trong ca gc BAD ta chn mt im E sao cho gc BED

bng 450. Cmr: 2 22 b c 2 b c

AE2



33

33

55. Cho hnh chp S.ABCD, |y l{ hnh bnh h{nh t}m O. Hai mt bn SAB v{ SCD vung gc ti A v{ C cng hp vi

|y gc . Bit ABC . Chng minh SBC v{ SAD cng hp vi |y ABCD mt gc tha m~n h thc :

cotcot os.c .

56. Cho hnh chp S.ABC, |y ABC l{ tam gi|c vung ti B vi AB=a, SA vung gc vi mt phng (ABC) ; mt (SAC) hp vi mt phng (SAB) mt gc v{ hp vi mt phng (SBC) mt gc . Chng minh rng :

acos

cos[ ( )].SA

cos( )

57. Cho hnh chp S.ABC . M v{ P ln lt l{ trung im ca SA v{ BC, N l{ im ty trn cnh AB. Chng minh rng thit din to bi (MNP) chia hnh chp th{nh hai phn c th tch bng nhau .

58. Cho hnh chp t gi|c u S.ABCD . M, N, P ln lt l{ c|c im trn AB, AD, SC sao cho : AM 1 AN 1 SP 2

; ;AB 3 AD 2 SC 5

.

Mt phng (MNP) chia hnh chp th{nh hai phn c th tch 1 2V ; V . Tm t s :

1

2

V

V

59. Cho hnh chp S.ABC . Ly M trn SA v{ N trn SB sao cho SM 1 SN

; 2MA 2 SB

. Thit din qua MN v{ song song vi AC

chia hnh chp th{nh hai phn. Tnh t s th tch hai phn ni trn . 60. Cho hnh lp phng ABCD.A'B'C'D' . Gi M v{ N l{ t}m ca |y ABCD v{ mt bn DCCD . Thit din to bi mt

phng (AMN) chia hnh lp phng th{nh hai phn. Tm t s th tch hai phn .

61. Cho hnh lp phng ABCD.ABCD cnh bng a. Ko d{i BA, BC, BB c|c on tng ng AM=CN=BP = 3

a2

. Thit

din to bi mt phng (MNP) chia hnh lp phng th{nh hai phn. Tnh t s th tch hai phn . 62. Cho lng tr tam gi|c u ABC.ABC . Gi OO l{ trc ca lng tr ( ng thng ni 2 t}m ca hai |y ) . P l{ mt

im trn OO sao cho : O'P 1

O'O 6 . Gi M v{ N tng ng l{ trung im ca AB v{ BC . Tit din to bi mt phng

(MNP) chia lng tr th{nh hai phn. Tnh t s th tch hai phn . 63. Cho hnh chp t gi|c u c c|c mt bn to vi |y gc . Thit din qua AC v{ vung gc vi mt phng (SAD)

chia hnh chp th{nh hai phn. Tm t s th tch hai phn . 64. Cho hnh chp S.ABCD c |y l{ hnh vung cnh bng a . on SA=a vung gc vi |y. M l{ im trn AC v{ t

AM=x, 0 x a 2 . Dng thit din qua M song song vi BD v{ vung gc vi (ABCD) . X|c nh v tr ca M thit din c din tch ln nht. Khi din tch ca thit din l{ ln nht, h~y tnh t s th tch ca hai phn m{ thit din n{y chia hnh chp.

65. Cho hnh chp t gi|c u S.ABCD c |y bng a, chiu cao h . Dng thit din qua A v{ vung gc vi (SAC) sao

cho n ct SB, SC, SD tng ng ti B, C, D . X|c nh v tr im C trn SC sao cho : SAB C D SABCD

1V V

3 .

66. Cho hnh chp t gi|c u S.ABCD . Gi M, N tng ng l{ trung im ca AD v{ DC . H~y x|c nh v tr im P nm trn phn ko d{i ca SD v pha D sao cho thit din to bi (MNP) chia hnh chp th{nh hai phn c th tch bng nhau.

67. Cho t din vung O.ABC ( OA, OB, OC i mt vung gc ) , P l{ im nm trong |y ABC . t AP BP

u ; v ;AO BO

CP

wCO

. Gi l{ gc to bi ng thng OP vi (ABC) . Chng minh rng : 2 2 2 2u v w 2 cot .

68. Cho hnh chp t gi|c S.ABCD, |y l{ t gi|c li ABCD. Mt mt phng ct c|c cnh SA, SB, SC, SD ln lt ti K, L,

P, N . Chng minh rng : BCD ADB ABC ACDSA SC SD SB

S S S. . SSK S

.N

.P S SL

.

69. ng cho ca hnh hp ch nht , to vi ba kch thc a, b, c c|c gc ; ; . Chng minh rng :

3 3 32

12 12 12

a b c

co2

s cos co178V

s ( }y V l{ th tch khi hp ) .

70. Cho hnh chp O.ABC, trong OA, OB,OC i mt vung gc vi nhau. K ng cao OH ca hnh chp . t

HOA ; HOB ; HOC . Chng minh rng : 2

.cot .cotcot4

.

71. Cho ABCD l{ t din u c cnh bng 1 . M v{ N l{ hai im di ng trn AB, AC sao cho (DMN) lun vung gc vi (ABC) . X|c nh v tr ca M v{ N t din ADMN c th tch ln nht v{ b nht.



34

34

72. Cho hnh chp S.ABCD c |y l{ hnh bnh h{nh . Gi K l{ trung im ca SC. Mt phng qua AK ct SB, SD ti M, N .

t 1 S.AMNK S.ABCDV V ; V V . Chng minh rng : 1V1

V3

3

8 .

73. Cho t din S.ABC v{ G l{ trng t}m ca t din . Mt phng quay quanh AG ct c|c cnh SB, SC ti M, N . t

S.ABC S.AMN 1V; V VV . Chng minh rng : 1V4 1

9 V 2

Phn VI : MT S KIM TRA I TUYN


35

35

PHN VI : MT S KIM TRA I TUYN

S GD&T NGH AN TRNG THPT NG THC HA

Gio vin ra : Phm Kim Chung

BI KIM TRA CHT LNG I TUYN THAM GIA K THI HSG TNH NM HC 2010 2011

( Ln th 1 ) Thi gian lm bi : 180 pht

_____________________________________

Cu 1 . Gii phng trnh : 2(x 21)l x xn 1 2x

Cu 2 . X|c nh tt c c|c gi| tr ca tham s m h phng trnh sau c nghim duy nht :

22

22

m2x

y

m2y

y

xx

Cu 3 . Cho a,b,c 0 . Tm gi| tr nh nht ca biu thc :

4a b 3c 8c

Pa b 2c 2a b c a b 3c

Cu 4 . Cho d~y s nx ,n N* , c x|c nh nh sau : 12

x3

v{ nn 1n

xx ,

2(2n 1n

)xN *

1

. t

n 1 2 nx ..y xx . . Tm nnlim y

.

Cu 5 . Cho hnh chp S.ABCD c SA l{ ng cao v{ |y l{ hnh ch nht ABCD, bit SA = a, AB = b, AD = c. Trong mt phng (SBD), v qua trng t}m G ca tam gi|c SBD mt ng thng ct cnh SB ti M v{ ct cnh SD ti N. Mt phng (AMN) ct cnh SC ca hnh chp S.ABCD ti K. X|c nh v tr ca M trn cnh SB sao cho th tch ca hnh chp S.AMKN t gi| tr ln nht, nh nht. Tnh c|c gi| tr theo a, b, c.

Cu 6 . Cho hnh lp phng 1 1 1 1BAB .A CD DC c d{i bng 1 . Ly im 1E AA sao cho 1

AE3

. Ly

im F BC sao cho 1

BF4

. Tm khong c|ch t 1B n mt phng FEO ( O l{ t}m ca hnh lp

phng ). Cu 7 . Tm h{m s 0: 0; ;f tho m~n :

xf xf(y) f f(y) x, 0, y ; )(

__________________________Ht__________________________

Thanh Chng ,ngy 03 thng 12 nm 2010



36

36

HNG DN GII V P S

Cu 1 . Gii phng trnh : 2(x 21)l x xn 1 2x

(1)

Li gii : iu kin : x 1

Lc : PT 2 22(x 1)ln(x 1) x 2(x 1)ln(2x 21 0x xx )

Xt h{m s : 2f(x) 2 x 1 ln( 2x, x 1x 1) x Ta c : f '(x) 2ln(x 1) 2x ;

2 2x

f ''(x) 2x 1 x 1

;

2

2f '''(x) 0,

(x 1)x 1

Li c : f ''(0) 0, f '''(0) 0 nn h{m s g(x) f '(x) t cc i ti x 0

Do : f (0f ) 0, x'(x) 1

Vy h{m s 2f(x) 2 x 1 ln(x 1) 2xx nghch bin trn khong 1; . Nhn thy x 0 l{ mt nghim ca phng trnh (1), suy ra phng trnh c nghim duy nht x 0 .

Cu 2 . X|c nh tt c c|c gi| tr ca tham s m h phng trnh sau :

22

22

m2x

y

m2y

y

xx

c nghim duy nht .

Li gii : iu kin : 0;y 0x

H ~ cho tng ng vi : 2 2 2

2 2 2

2x y

2y

y m

x x m

(*)

T h (*) nhn thy v tr|i ca c|c phng trnh khng }m, nn nu h c nghim (x,y) th : x 0;y 0

Do : 2 2 23 2 2

x 0,y 0y x 0

(*) 2x y2x (1)

(x y)(2xy x )

y

y 0

mx m

Do b{i to|n tr th{nh tm tham s m phng trnh (1) c nghim dng duy nht.

Xt h{m s : 3 2f(x) 2x x , x 0

Ta c : 2x 0

f '(x) 6x 2x; f '(x) 0 1x

3



37

37

Nhn v{o bng bin thin ta thy, phng trnh (1) c nghim dng duy nht khi v{ ch khi : 2m 0 . Vy vi mi m R h phng trnh ~ cho c nghim duy nht.

Cu 3 . Cho a,b,c 0 . Tm gi| tr nh nht ca biu thc :

4a b 3c 8c

Pa b 2c 2a b c a b 3c

Li gii :

t :

x a b 2c a y z 2x

y 2a b c b 5x y 3z(x,y,z 0)

z a b 3c c z x

Lc :

4 y z 2x 2x y 8(z x) 4y 2x 4z 8xP 17

x y z x y x z

2 8 2 32 17 12 2 17

Du = xy ra khi v{ ch khi :

4 3 2a t

2

2y 10 7 2b t t

2

2xR,t 0

2x2z 2

c 2 1 t

Cu 4 . Cho d~y s nx ,n N * c x|c nh nh sau : 12

x3

v{ nn 1n

xx ,

2(2n 1n

)xN *

1

. t

n 1 2 nx ..y xx . . Tm nnlim y

Li gii :

T : nn 1n n 1 n

x 1 1x 2(2n 1)

2(2n 1)x 1 x x

. t : nn

1v

u , ta c : 1

n 1 n

3v

2

v 2(2n 1) v

D d{ng tm c cng thc tng qu|t ca d~y : n 1(2n 1)(2n 3)

v2

Do : n 1n 1

1 1 1 1 1x

v 2n 1 2n 3 2n 1 2(n 1) 1

suy ra :

n 1 2 n 1

1 1 1 1 1 1 1y x x ... 1

2 1 2.2 1 2.2 1 2.3 1 2(n 1) 1 2n 1 2n 1x ... x

Do : nn n

1lim y lim 1 1

2n 1

Cu 5 . Cho hnh chp S.ABCD c SA l{ ng cao v{ |y l{ hnh ch nht ABCD bit SA = a, AB = b, AD = c. Trong mt phng (SBD) v qua trng t}m G ca tam gi|c SBD mt ng thng ct cnh SB ti M v{ ct cnh SD ti N. Mt phng (AMN) ct cnh SC ca hnh chp S.ABCD ti K. X|c nh v tr ca M trn cnh SB sao cho th tch ca hnh chp S.AMKN t gi| tr ln nht nh nht. Tnh c|c gi| tr theo a, b, c.

Li gii : Do G l{ trng t}m tam gi|c SDB, suy ra G cng l{ trng t}m tam gi|c SAC. Do AG ct SC ti trung im K ca SC.



38

38

t : 1S 1

x ; y2

M SN x, y 1 1

SB 2SD

Theo cng thc tnh t s th tch ta c : SANK SAKM

SADC SACB

V VSA SN SK y SA SK SM x. . ; . .

V SA SD SC 2 V SA SC SB 2 Li c

SADC SACD SABC

1 1V V abc

2V

6 v{ : SANK SAKM SANKMV VV . Nn ta c :

SANK SAKM SANKMSANKM

SADC SACB SABCD

V V 2V x y abc(x y)V

V V V 2 12

(*)

Ta li c :

SM 2SN SD ySD; SM SB xSB; SG SO

S

S

SD B 3

N

V O l{ trung im ca BD nn : 1 1

SO SD SB SG SN SM3y 3x

2 (1)

M{ : M, N, G thng h{ng nn t (1) ta c :

1 1 y 11 x

3y 3x 3yy

21

1

Thay v{o (*) suy ra : 2

SANKM

yabc y

3y 1 abc yV

24 8 3y 1

Xt h{m s : 2y 1

f(y) y 13y 1 2

Ta c :

2

2

3y 2y 2f '(y) ; f '(y) 0 y

33y 1

.

Bng bin thin :

Nhn v{o bng bin thin ta thy :

1y4 2 1

Minf(y) y ; Maxf(y) 29 3 2

y 1

T ta c :

SANKMabc

Max V MN / /BD9

SANKMabc

Min V8

M l{ trung im SB, hoc N l{ trung im SD.

Cu 6 . Cho hnh lp phng 1 1 1 1BAB .A CD DC c d{i bng 1 . Ly im 1E AA sao cho 1

AE3

. Ly im

F BC sao cho 1

BF4

. Tm khong c|ch t 1B n mt phng FEO ( O l{ t}m ca hnh lp phng ).

Li gii : Chn h trc ta Ixyz sao cho 1I A(0;0;0);A (0;0;1);D(1;0;0);B(0;1;0)



39

39

Lc : O l{ trung im AC nn 1 1 1

O ; ;2 2 2

; 11 1

E 0;0; ;F ;1;0 ; B 0;1;13 4

Mt phng (OEF) i qua O v{ nhn vct 1 5 3

; ;3 24

OE,OF8

l{m vct ph|p tuyn nn c phng trnh :

1 1 5 1 3 1x y z 0

3 2 24 2 8 2

hay : 8x 5y 9z 3 0

Vy : 12 2 2

5 9 3 11d B (OEF)

1708;

5 9

Cu 7 . Tm h{m s 0: 0; ;f tho m~n : xf xf(y) f f(y) x, 0, y ; )(

Li gii :

Cho y = 1, suy ra : xf xf(1) f f(1) . t f(1) a , ta c : xf(ax) f(a) (1)

T (1) cho 1

xa

, suy ra : 1

f(1)=f(a) f(a)=1a

Cng t (1) cho ta : 1

f(ax)x

(2)

T (2) cho a

ax y f(y)y

Th li ta thy a

f(y) (a 0)y

l{ h{m s cn tm .



40

40

S GD&T NGH AN TRNG THPT NG THC HA

Gio vin ra : Phm Kim Chung (S&GT) - Nguyn Th Tha (HH)

BI KIM TRA CHT LNG I TUYN THAM GIA K THI HSG TNH NM HC 2010 2011

( Ln th 2 ) Thi gian lm bi : 180 pht

_____________________________________

Cu 1. Gia i phng tr nh: 3 23x 3 5 2x x 3x 10x 26 0 Cu 2. T m ta t ca ca c gia tri cu a tham so m e he phng tr nh sau co nghie m :

2 2

3 3

2 2

3 3

log y

log

x lo

y

g 1 2m 3

log x 1 2m 3

Cu 3. Cho a, b, c dng thoa ma n ab bc ca abc . Ch ng minh ra ng:

2 2 2 2 2 2

b c a 1 1 1

a b c a c3

b

Cu 4. Cho h{m s 4 3 2f(x) x ax bx cx d . Ta k hiu o h{m bc n ( n nguyn dng ) ca

f(x) l{ (n)f x . Chng minh rng nu f(x) > 0, x R th : (1) (2) (3) (4)F(x) (x) f f (x) f (x) 0,( x Rf x) f

Cu 5. Cho t die n ABCD co DA vuo ng go c v i ma t pha ng (ABC), tam gia c DAB ca n va a y ABC la

tam gia c vuo ng ta i B co BAC . Go i la go c ta o b i hai ma t pha ng DAC va DBC . Ch ng minh

ra ng: 21 cos

tan .tancos

.

Cu 6. Cho h nh la p phng ABCD.A'B'C'D' ca nh a . V i M la mo t ie m thuo c ca nh AB, cho n ie m N thuo c ca nh D'C' sao cho AM D'N a . T nh the t ch kho i cho p B'.A'MCN theo a va xa c i nh vi tr cu a ie m Me khoa ng ca ch t ie m B' e n ma t pha ng A'MCN a t gia tri l n nha t.

Cu 7 . Cho h{m s f :R R tha m~n h iu kin :

4

x,y R

f(1) 1

f(x y) f(x) f(y) 2xy,

1 f(x)0f ,

x xx

.

Tnh gii hn :

32f(x)

x 0

1 f1

e(x

L limln 1 f )

)

(x

---------------------------------He t------------------------------

Thanh Chng, ngy 10 thng 12 nm 2010



41

41

HNG DN GII V P S

Cu 1. Gia i phng tr nh: 3 23x 3 5 2x x 3x 10x 26 0

Li gii : K : 2

15

x

2

2

2

3x 3 3 5 2x 1

3x 3 3 5 2x 1

x 12)3x 3 3 5 2x

PT (x 2)(x x 12) 0

3(x 2) 2(x 2)(x 2)(x x 12) 0

3 2( (

1x 2) x 0

Xt h{m s : 2 x 12,5

f(x) x 1;2

x

Ta c : 1

f '(x) 2x 1, f '(x) 0 x2

. Suy ra : 5

12

;

1Minf(x) Min f( 1);f ;f f 0

2

5 5

2 2

Do : 23 2 5

(x x 12) 0, x 1;23x 3 3 5 2x 1

Vy phng trnh c nghim duy nht : x = 2 .

Cu 2. T m ta t ca ca c gia tri cu a tham so m e he phng tr nh sau co nghie m :

2 2

3 3

2 2

3 3

log y

log

x lo

y

g 1 2m 3

log x 1 2m 3

Li gii : K : x,y 0

t : 2

3

2

3

uu

v y

log x 11,v 1

log 1

. Lc h PT tr th{nh :

2

2

u v 2m 2 (1

u 2m 2 (2)

)

v

.

Ly (1)-(2), ta c : u v u v 1 0 u v ( Do u+v+1 > 0 u,v 1 )

Lc b{i to|n tr th{nh tm m phng trnh : 2 u m 2u 2 c nghim u 1 .

Xt h{m s : 2f(u) u u 2 , ta c : f '(u) 2 uu 1 , 10 . V{ ulim f(u)

.

Do , PT trn c nghim u 1 khi v{ ch khi f(12m u) 2 1f( ) m

Li gii : Ta c : ab bc ca abc1 1 1

1a b c . t :

1 1 1x y z 1 x,y,x, y,

a cz

b0z . Bt ng thc

cn chng minh tr th{nh : 2 2 2

2 2 2x y z xy z x

3 y z .

p dng BT Svac-x ta c : 4 4 4 2 2 2 2

2 2 2 2 2 2

x y z ( y z )

y

xVT

x y z xz x y y z z x

.

Ta s chng minh : 2 2 22 2 2

2 2 2 2 2 2

(x y yz) xx

x

zy z

x3 3

y y z z x y y z z x

( do x y z 1 )

3 2 3 2 2 3 2 2 2xy y yz zxx 2(x y yz z z x) (*)

Theo bt ng thc AM-GM ta c : 3 2 4 2 2 3 2 4 2 3 2 4 2xy 2 x y 2x y; y yz 2 y z ; z z 2 z xx x . Cng c|c BT trn

ta chng minh c (*). Vy : 2 2 2V (xT 3 y z ) . pcm

Cu 3. Cho a, b, c dng thoa ma n ab bc ca abc . Ch ng minh ra ng:2 2 2 2 2 2

b c a 1 1 1

a b c a c3

b

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