Đề học sinh giỏi huyện Toán 8 năm 2018 - 2019 phòng GD&ĐT Nậm Nhùn - Lai Châu - THCS.TOANMATH.com

PHONG GD&DT NAM NHUN

^DE

THI

^{CHQN HQC SINH}

GIOI

^{CAP HUYEN}

Nim

hqc 2018-2019

Dt THI CHiNH THI,IC

(Di rhi

gin

0l trang)

Mdn thi: TOAN

⁸

Ngdy

thi: l3l0ll20l9

Thoi

^gian.ldm

bdi:

150

phtit

(Kh6ng k€ thoi gian

phft

de)

ui-uai

CAu 1: (4.0

tli5m):

Cho bi6u thric

( t 2 5-x \ t-2x A=l

-+----

- li---

Il-x x+l l-x') x''l

a.

Tim DKXD,

rut ggn bi6u thric

A

b.

Tim

x ^A6

e

^co

gi6tr!nguyCn

CAu 2: (2.0

tli6m):

^PhAn tich da thric sau thdnh nh6n

tri

a. x'-6x+8

b.

^xt

-3x'-l6x+48

CAu 3. (4.0

tli6m)

a. Chimg

minh r' -

n chia h6t cho 6

(n

e

z)

b.

Tim

^a, b ae

e(x) :

A(x)=2x3 +7x2+ax+bchia h6t

cho B(x\=f ar-1 Ciu

^{4. (4.0}

tli6m)

a. Cho x

-

^{y =}

7.Tinh

^gi6

trl

^{cria birlu}

thuc

^x'?

-2*v

⁺

v' -

^{5x + 5 v + 6}

b. Chrmg

minh:

^{a' + b2 + c' > ab + bc +} cav1i mgi a,b,c

CAu 5. (6.0

tli6m):

Cho tam gi6c

ABC

^{vu6ng t?i}

A.

M,N,P lAn

lugt li

trung di6m

cia AB,AC,BC.

a.

^{Chrmg minh}

ring:

^{Tri gi6c}

BMNP li

^{hinh binh hdnh.}

b.

^{Chtmg minh}

ring:

^{Tu giSc}

AMIN

^{ld hinh}

cht

^nhqt'

c.

_{VE Q} ^{ai5i xung}

v6i

^{P qua N, R d6i ximg}

vdi

^{P qua}

M.

^{Chtmg minh}

ring

R,A,Q

thing

hdng.

---n6t---

thi

sinh kh6ng

duqc

^sic dqng tdi ti€u, Gidm thi coi thi kh6ng

gidi

thich

ei

^thAm)

h.t-'

Y---

PHONG

GD&DT NAM

^NHUN

soc of rsI cnixH rntlc

xV rm cHeN

HQC

stxn cr0r cAr uuvflx Nim

hgc 2018-2019

M6n

thi: To6n

8

Than

o

em CAU

Y .Ii

D6p 6n

0 5

a

DKXD:

|

-2x

2x

-l

(l+r)(1-r)

-r

.' (l +x)(1

-x)

l+x+2-2x-5+x). 2x-l

(l-rxl-r) I (t-rxt-r)

) (l+x)0-x)

) ,, -, 1-x*0

x+l+0

^<+

l-2x*0

'l-x2

2x

-l

^=(

^=(

x+l

x*-l x*-

I

2

A=-

a 2x

-l

( t 2 5-x)

A=l

-+-

- ---= l:

ll-x -r+l l-x')

^=(

^=(

1 2 5-x

-+--- 1-x l+x (l+x)(1-x)

+x+2(l-x)-(5-x)

(l +.r)(l

-

^x)

0,5d 0,5

0 5

0 5

A nhin

^gi6

tri

nguy€n

thi: -2i(2x-l)

=2x-t eU(-Z): {lt;tz}

+)2x-1=l 2x-l=-l 2x-1:2 a2x:2 e2x:0 e2x:3

<+

, <+.r

³

:l(loai) -0(t/m) o':,

(t/m) Vay dC

A

nh4n gi6

tr!

nguy€n

thi

x e ⁰

2

D

3 2

2x-l=-2

e2x:-l

-l

2

(t/m) b

I

(4d)

a ^-r2

-6x+8

= x2 -6x ⁺⁹

-l

2 (2d)

Trang

ll4

0,25 0,25

0,5 0,25 0,25

I

(x'-6x+9)

^I

= (x

-:')- t'

= (.r-r+t)(x-3-t)

= (x-z)(x-+)

0,25 0,25 0,25 0,25

b

.rr -3.r2 -16.r+48

:

^(.rr

-

^3x'?) _- (16r - ⁴⁸⁾

= x'(x

-

³⁾

-

^16(x

-

³⁾

=

(x-:)(x'-to)

= (.r

- l)(x

_{- +)(x} ⁺ +)

0,25 0,25 0,2s 0,25

3 (4d)

a

Ta

c6: ,'-n:

ⁿ⁽ⁿ²

-

l) _{= (n}

-

l).n.(n + l)

ld tich

cta

3 s6 nguyEn li6n ti€p n6n chia htit cho 2 vd chia hi5t

cho 3

md 2 vit 3 nguy€n t6 cirng ^i nhau

=(n-l).n.(n+1)

chia h6t cho 6 (tlpcm)

1d 0,5 0,25 0,25

b

A(x):

^2*; +7x2+ax+bchia h6t cho

B(x): ,: *, *,

D4t ph6p

tinh

chia

2x3 +7 x2 +ax _{+ b}

2x3 +2x2

-2x

^{2x +5}

5x2

+(a+2)x+b ).x'+5x-5 (a-3)x+b-5

E6

A(x) chiah6tchoB(x)thi a-3 :

₀

vdb-5

_{= 0}

Suy ra

a:3

^vd

b:5

Vfy

de

A(x)

chia tr6t e1x; thi

a:

^{3 vd}

b:

⁵

1d

0,25 0,25 0,5

4 (4d)

a

Cho: x

-

^y

-

^{7. Tinh gi6}

tr!

^{cria bi}

x' -2xy ⁺

y' -5x+5y+6

Ta

c6:

_{x' -2xy+} ^y2

-5x-5y+6

=(x'-zxy+ y')-5@- y)+6

uc

= (*

- y)'-

5(x

-

^{y) + 6(*)}

Thay

x-

y

:

_{7 vdo bir3u}

thric(*)

_{ta duoc:}

t2 -5.7+6:

_ZO

o5

0,5 I

0,25 0,25 0,25 0,5 0,5 0,25 b

Ta c6'. a2 + b' ^>-zab

b1 +ct >2bc

c' +

a'

^>- 2ac

CQng v6

vdi

vi5 ba b6t dang thric tr€n, ta dugc:

2(a2 + b'z + _{c2 1} = ^21o6 * 6, *

"o,

)

^{a2 +b2 +c2 2.ab+bc+ca}

>ab +bc

*ca voi

mei a,b,c.

Chimg minh: ^a

+b +c

DAu

":" ra e a=b=c

0 5

VE hinh

R

N

a

0,25 0,25

I

N

ld trung di6m

cta

AC

=MN

ld dudng trung binh ^cria tam gi6c

ABC

:>MBNP

ld hinh binh hdnh MN BC

2 I

Ta c6

M

ld trung diem cira

AB

MN I IBC

MN I /BP MN ^__ BP a

0,25 0,s 0,25

0,5 0,5

+MP

lir dudng trung binh

cta

tam gi6c

ABC

-> MP//AC ,a eCreS

^=>

MPl.AB

^hay

zM:900

Tuong ^{tr,r :} Ta c6 PN ld duong trung binh ^crha tam _S16c

ABC

=>

pNr/AB

_md

AB rAC=> iN rAc

^hay

zN -

⁹⁰⁰

o AMPN

^{ld hinh}

cht

nhat (c6 3 g6c vu6ng) c[ra

AB

cta

BC di

di

M

a

T

aco trung

em

P a trung 5

(6d)

b

Trang3l4

\

c Ta c6

PM=MR

=R

^{d6i ximg}

voi

^P qua AB=>

lA,

=

lA, lM

=900

NP _-- NQ ZN _=90o

Ta c6

=Q

^{d6i xrlmg}

voi

^{P qua}

AC= ZA. -

^2tr,121

Ta c6

IRAQ ₌ ZA, + ZAr+

Z4+

lA4 =2ZAz+2ZAt=2LBAC = ^2.900 = ¹⁸⁰⁰

Vfy

^R,A,Q

thing

^hdng

0,5 0,5

0,75 0,2s