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TI LIU BI DNG HC SINH GII
Phm Kim Chung THPT NG THC HA T : 0984.333.030 Mail : [email protected] Tr.
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S GD&T NGH AN
TRNG THPT NG THC HA
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 GI I ( 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. v mt s ti liu tham kho khc . 15. 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. = + + +2y 2x 2 m 4xx 5Tm cc gi tr ca tham s m hm s : c cc i . S : m < -2
2. + =/=
=
3 21 xsin 1, xf(x)0 , x 0
x 0Cho hm s : . Tnh o hm ca hm s ti x = 0 v chng minh hm s t cc tiu
ti x =0 . 3. ( )= = y f(x) | x | x 3Tm cc tr ca hm s : . S : x =0 ; x=1 4. Xc nh cc gi tr ca tham s m cc phng trnh sau c nghim thc :
( ) ( )+ + + =x 3 3m 4 1 x3 m4 1m 0 a) . S : 79
9m7
+ =4 2x 1 x mb) . S :

3
3
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. Xc nh m phng trnh sau c nghim thc : ( ) ( ) + + + =
41x x 1 m x x x 1 1
x 1 .
27. Tm m h phng trnh : ( ) + + =
+ =
23 x 1 y m 0
x xy 1 c ba cp nghim phn 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 y2
1 1x yy x
37. ( thi HSG Tnh Qung Ninh nm 2010 ) . Gii phng trnh : =
2 21 1x
5x 7( x 6)
x5
1
Li gii : K : > 7x5
Cch 1 : PT + = = +
4x 6 36(4x 6)(x 1) 0 x2(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 hm s : = >
21 5f(t) t , t
7t 1
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38. ( thi HSG Tnh Qung Ninh nm 2010 ) Xc nh tt c cc gi tr ca tham s m BPT sau c nghim : + 3 2 33x 1 m( x x 1)x
HD : Nhn 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 ( ) + + ++ = + +33(x 1) (x 1) 3x 1 3x 1 . Xt hm 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. ( 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 )
Hm s : + == + > 2 21).t f '(t) 3tf(t) (t 1 0 = = =225 4x2x 5 2y 4x 5 2y y
2
Th vo (2) ta c :
+ + =
222 5 4x4x 2 3 4x 7
2 , vi 0 3x
4 ( Hm ny nghch bin trn khong ) v c
nghim duy nht : =x 12
.
42. ( 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 mn iu kin x 9. HD : ng trc bi ton cha tham s cn lu iu kin cht ca bin khi mun quy v 1 bin kho st :
= x y 0 x4 16 . t = x , t [t 3;4] v kho st tm Min . S : +a 4 2 2
43. Gii h phng trnh : + + =
+ = +
4 xy 2x 4
x 3 3 y
y 4x 2 5
2 x y 2
44. Xc nh m bt phng trnh sau nghim ng vi mi x : ( ) + 2sinx sinx sinxe 1 (e 1)sinx2e e 1e 1
45. ( thi HSG Tnh Tha Thin Hu nm 2003 ) . Gii PT : + +
= 2 22 5 2 2 5
log (x 2x 11) log (x 2x 12)
46. nh gi tr ca m phng trnh sau c nghim: ( ) ( ) + + + =4m 3 x 3 3m 4 1 x m 1 0
47. (Olympic 30-4 ln th VIII ) . Gii h phng trnh sau: +=
+ + + = + + +
2 22
y x2
3 2
x 1ey 1
3log (x 2y 6) 2log (x y 2) 1
48. 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 += + + = =
2 2x xlnx , x 0F(x) 2 40, , x 0
v >
= =
xlnx, x 0f(x)
0, x 0 . CMR : =F'(x) f(x)
Cho f(x) xc nh trn R tha mn iu kin : >a 0 bt ng thc sau lun ng x R : + < 2| f(x a) f(x) a | a . Chng minh f(x) l hm hng .
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Tnh gii hn :
=
x
3
1 2
4
tanN lim2sin
x 1x 1
Tnh gii hn :
+=
+
2 32x 2
2 2x 0
e 1N limln(1 x
x)
Tnh gii hn :
+ + =
+ 33 x 0
3 32x x 1N 1m xlix
Tnh gii hn :
=
sin2x
4
s
x
nx
0
ie eN limsinx
Tnh gii hn :
+=
0
3
5 x
x 8 2si
N limn10x
Tnh gii hn :
+=
+
2 32x 2
6 2x 0
e 1N limln(1 x
x)
Tnh gii hn :
=
sin2x sin3
7 x
3x
0
eN lim esin4x
Tnh gii hn :
=x 4
3x 0 384 xNx
im2
l
Tnh gii hn :
=
+ 9 x 0
3x 2x.3 cos4x1 sinx 1
2N limsinx
Cho P(x) l a thc bc n c n nghim phn bit 1 2 3 nx x x; ; ...x . Chng minh cc ng thc sau :
a) + + + =2 n2 n
1
1
P''(x ) P''(x ) P''(x )... 0
P'(x P'( P'(x) )x)
b) + + + =2 n1 ) )
1 1 1... 0P'(x P'(x P'(x )
Tnh cc 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
49. Cc bi ton lin quan n cc tr ca hm s :
a) Cho + R: a b 0 . Chng minh rng :
+ +
n na b a b2 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 hm s sau c duy nht mt cc tr : + +
= + +
22 2
2 2y (m 1) 3x x
1 x 1 xm 4m
d) Cho n 3,n N ( n l ) . CMR : =/x 0 , ta c : + + + + +

Phn II : PHNG TRNH HM V A THC
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PHN II : PHNG TRNH HM-A THC
1. Tm hm s : f : R R tho mn ng thi cc iu kin sau :
a)
=x 0
f(x)lim 1x
b) ( ) ( ) ( )+ = + + + + 2 2f x y f x f y 2x 3xy 2y , x,y R 2. Tm hm s : f : R R tho mn iu kin sau : ( ) ( ) ( ) = + + + + 2008 2008f x f(y) f x y f f(y) y 1, x,y R 3. Tm hm s : f : R R tho mn iu kin sau : ( ) ( ) ( )( )+ = + f x cos(2009y) f x 2009cos

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