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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 .
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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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17. Ch : Nhng dng ch mu xanh cha cc ng link n cc chuyn mc hoc cc website.
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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)
7. Gii h phng trnh :
2 y 1
2 x 1
x 2x 2 3 1
y 2y 2 3 1
x
y
8. Gii h phng trnh :
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
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Phn I : PHNG TRNH BPT HPT CC BI TON LIN QUAN N O HM
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22. Gii h PT :
4 4
3 3 2 2
x y 240
x 2y 3 x 4y 4 x 8y
23. Gii h phng trnh :
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)
24. Gii h phng trnh :
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
31. Gii h phng trnh :
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
34. Gii h phng trnh :
5 4 10 6
2
x xy y y
4x 5 y 8 6
35. Gii h phng trnh :
2 2
2 2
x 2x 22 y y 2y 1
y 2y 22 x x 2x 1
36. Gii h phng trnh :
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
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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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Phn I : PHNG TRNH BPT HPT CC BI TON LIN QUAN N O HM
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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
44. Gii h phng trnh :
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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Phn I : PHNG TRNH BPT HPT CC BI TON LIN QUAN N O HM
Phm Kim Chung www.k2pi.net T : 0984.333.030 Mail : [email protected] Tr.
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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.
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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
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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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Phn III : BT NG THC V CC TR
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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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Phn III : BT NG THC V CC TR
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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
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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
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Phn III : BT NG THC V CC TR
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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
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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
.
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Phn III : BT NG THC V CC TR
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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( )
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Phn III : BT NG THC V CC TR
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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
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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.
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Phn IV : GII HN DY S
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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
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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 :
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Phn IV : GII HN DY S
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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
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Phn IV : GII HN DY S
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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 : :
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Phn IV : GII HN DY S
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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
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Phn IV : GII HN DY S
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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 :
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Phn IV : GII HN DY S
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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
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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
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Phn V : HNH HC KHNG GIAN
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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
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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 .
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Phn V : HNH HC KHNG GIAN
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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
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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
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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)
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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
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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 .
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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
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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.
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Phn V : HNH HC KHNG GIAN
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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
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Phn VI : MT S KIM TRA I TUYN
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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
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Phn VI : MT S KIM TRA I TUYN
Phm Kim Chung www.k2pi.net T : 0984.333.030 Mail : [email protected] Tr.
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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
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Phn VI : MT S KIM TRA I TUYN
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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.
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Phn VI : MT S KIM TRA I TUYN
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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)
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Phn VI : MT S KIM TRA I TUYN
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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 .
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Phn VI : MT S KIM TRA I TUYN
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S GD&T NGH AN TRNG THPT NG THC HA
Gio vin ra : Phm Kim Chung (S>) - 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
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Phn VI : MT S KIM TRA I TUYN
Phm Kim Chung www.k2pi.net T : 0984.333.030 Mail : [email protected] Tr.
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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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Phn VI : MT S KIM TRA I TUYN