so crAo 9UC vA qAo rAo rci rru cHeN Hec srNH crOr Lop tzrHpr cAr rixnru:ir xcuyEx iviu rrgc zots _zpt6 HIIOr'{G bAN CHAM oT rrrr CHiI\H rrrffc naON ToAN ruron7 ddn chiim vd"bidu cli€m gd
Trang 1so crAo 9UC vA qAo rAo rci rru cHeN Hec srNH crOr Lop tzrHpr cAr rixn
ru:ir xcuyEx iviu rrgc zots _zpt6
HIIOr'{G bAN CHAM oT rrrr CHiI\H rrrffc naON ToAN
(ruron7 ddn chiim vd"bidu cli€m gdm c6 04 trang)
f Hufng dfln chung
1) N.1 bqc sinh l4rn bai k*r6ng theo cZch n6p tronf dap hnnhrn g dring thi cho
tdn quy dinh:, n\\r: ^ ^LrtAtLt' r X ,.,r-
?-2) ViQc chi ti(5t h6a (ntiu c6) thang di6m tiorg hufng d5n ch6m phai bio d6m
hgre ldm sai lQch huong dan ch6m vd phii duoc thdng nh6t thgc hiQn trong t0
hgre ldm sai lQch hucrng da" ch6ir vd phii OIq"
chdm
II Drlp rln vi thang tli6m
1) Giei b6t phucmg trinh: 32xa -1 6x' -97s -9J2* -1 + 2> 0
D Giei he phucrng trinh:
1) Di6u kiQn: *
= 1.
(l) <+ 32xa -16x2 -9x -7 29rl21s a -g
<+ (, *l)(zz*' +32x2 +t6x +?)(^12,* i* 1)
= e(zx - 4 1 il f
e(, -\l!zx3 +32x2+16x +?)(Jz.-t*1)-,r]= o
I 4, o
2) He phuong trinh -=
{;[; :l', : ;' i' :;r)r;r,
o[*@+vx, +v+w)=30
lry(*+y)+ ry+x+y-ll
Ddt x* y=tt; W=v
[*("+ v) = 3e
L-* u+v=ll o[*0r- uv):3s
lu+v=ll-uv
f luv -5
-lt'*v=6
I [*=a
LIr* v=5
HQ trd thdnh:
@
Trang 2Yoi uv = 5;u * y = 6 Giei trucmg hgrp ndy ta dugc nghiQm
(*;v)
Y6i uv = 6;u + y = 5 Giei trudrng hqp niy ta dugc nghiQm
(*;y) H (t;2) ho6c (z;r)
{.r )f
BAi 2 Tir cdc chfr s6 1,2,3 c6 th6 lgp duo c bao nhiou s5 tF nhi6n c6 2015 cht
sO sao cho m6i'cht s6 1,2,3 xu6t hiQn dirng 16 6n.
Ldi eif,i:
Gqi r, ld sd c6c s6 tunhi6n c6 n chl sO du-ry c lflp thanh tu c6c cht s6
I,2,3 , n 16 vi c6c cht s6 tr6n xu6t hiQn 16 Hn (*)
X6t mQt sO U6t ki c6 n chtr sO 1n -Zk +I)
TII1 : SO aO th6a mdn diAu kiQn (*), c6 3 c6ch th6m hai chfi s6 giiing
nhau viro cu6i sd d6, dii duoc m6t s6 c6 n+2 chir s6 th6a m6n di6u
kiQn bdi to6rr
TH2 : SO AO kh6ng th6a mdn (*) Vi m 16 n6n trong trudmg hqrp nay c6
dring 2 chir s6 *u6t hiQn ch8n 6n @9i ld a,b) vd 1 cht'sd xu6t hiQn ld
tan Nen trong truong hgrp ndy c6 dirng hai c6ch thdm hai chfi sO AO
(ab hogc ba) v}o cuoi ai5 dugc mQt sd c6 n+ 2 chfi so thoa mdn diAu
VAy c6: xn*2 =3x, + 2(3' - *,) = *, * 7
T,i d6 tifn duo c: x2ors =3(3'?011 - 1)
.
,.4 ^ztlt -3 4 ^\ot'l c ,Uo| 2a)
- \.ao{r J,- ' + (i*, l- t L*r' Ll + ' .l- Crr, I
d
1
0,5 d
1
0,5 d
d
Bni 3 Cho tam gi6c nhgn LABC nQi ti6p ducnrg trdn t6rn O , AB < BC , D
li ch6n ducrrrg cao
"u6t ph6t tu B vi H ldtryc tOm cta tam giic ABC .
Eudrng th6ng qura D vu6ng g6c v6i OD chtc4nh lB t1i f .
Chimg rrlinh ,iog ffiP -fu .
Liri sii!:
Ggi K li glao rtiiSm oioa BD va (O) , K =t B Q li giao di6m oiua CK
vit PD Theo dinh l)t con bucnn, SUy ra D ld tmng di€m crta doqn PQ 1,5 d
Trang 3M[t kh6c D le,trung di6m oiua HK, do d6tt
hanh Suy ra DHP - HKQ.MiL HKQ=-g,lC .
Dovfly: ffi*fu
tu= ' 1^ l_
-Ut
n-Lz
1d
1,5 d
YL
1
Bii 4 Cho ddy (r, ) bitit x,
xo*t =
*(0.- +t + 4
-.)
J4.".)(t=o*N) Tfnh Lim(r, * 1)
Ldi rriii:
Ta c6: *,., =*(4 x,+l++rl44q)<+ 16x,*, = 4xn+l+ +Jffi
e 4(4x,*,+r) = (&;fr +2)'
2,,1
p
c)
4d
-t
Y Y1
?,C
D-i V
{t tvf,t ) ^
t,
G,
le
t0e
\.r tk.')t+
Trang 4!',* =b' + -!' g.
2a'+bc 2b'+ac 2c'+ab Lli eiii:
ndt dang thrlc dd cho tucmg ducmg v6i:]r* 1; ,* I , < t
o-A -# ,*#
n6t Alng thirc cAn chimg minh trd
MAt kh6c, theo U6t aang thtrc Cauch
Cho a,b,c ld c6c s6 thgc duong Chtmg minh
(x+y+z)z 2+ x2 ' 2+y' ' 2+ z'- 6+ x2 +y' + z' '
Do d6, ta cAn chi ra -@ ! Y +?'
, >t.
b+-r-+i"+Z-ThAt v+y, b6i it&ng thric tren tuong duong v6'i n, -i yz -t-z.x ) -1.
B6t dang cu6i lu6n dring theo b6t d6rg thtrc AM-GM.
Tt d6, ta c6 di6u phii chimg minh
2
x
Bii 5