Đề thi HSG Toán 8 năm 2015 - 2016 phòng GDĐT thị xã Giá Rai - Bạc Liêu

Ho vir tOn thi sinh:. Chfr ky gi6m th! 1:

Sii nao danh:.

PHONG GD&DT THI XA GIA RAI KV THI CHON HSG LOP 8 VONG THI a.

OT CTTINH THTIC

(Gdm 0l trang)

NAM HgC 20ts - 2016

* MOn thi: TOAN * Ngiry thit 241412016 * Thoi gian: 150 phrlt (Kh6ng k€ thdt gian giao di)

DE

Ciu 1: (5 tli6m)

a) Chtmg minh rirg' n3 + 20n chia h6t cho 48, v6i mgi n ch8n. b)Tinh,S = 1' -2'+3'-4' +...+ 2015'? -20162 +20112

Cdu 2z (5 tli6m)

a) Tim n,a,,Z thoa mdn: r' + 4y' + 22 = 2r +l2y + 4z -14

b) Chtmg minh: r' + y' 2 *oa + ry4 vli r + y;r +a > O Ciu 3: (5 tli6m)

chobiSuthirc: x= - ro !rt +r+l :L -a +2r'-r+l . a) Rirt ggn bi6u thric l.

b) Chung t6 ring I kh6ng 6m vdi moi gi6 tri c[ra x.. c) Tim gi6tnnho nh6t cua A. Cffu 4: (5 tli6m)

Cho tam giSc ABC c6 ba goc nhgn; ba ducrng cao BD, CE vit AF clt nhau tqi H. Chirng minh ring: a) HD.HB = HE.HC b) BH.BD + CH.CE = BC2 .AE BE DF n\

-

-

-

- I ,CFDEBF

--- HBT ---

PHONG GD&DT THI XA GIA RAI Ti'TUI CHQN HSG LOP 8 VONG THI r---= :ii;;;-rn'.-, r frra rrfrri.rrr .nrrr,r,r. I NAM HQC 2015 - 2016 ON CUINH THUC

(GOm 04 trang) * M6n thi: TOAN

HTIONG oAN CHAM

C6u NQi dung

Ao S tIi6m I .- -.7 (5 drem)

a)

Ta c6: n eZ, nchdn = n - 2k,k eZ Khi d6: n3 +20n = 8k3 +40k

= sr(r'+ s)

= sr[(r'-r)+o]

= 8k(k -,)(* + r) + a8k

Md: k(* -,)(r + r)i6,yk e z

= IT(f -,)(r + r)i48 vr, .z - ) +ttt, +t l. Vay: n3 +20ni48;Vn eZ vd n chin

0,,25d 0,25d

0,,25d

0,5d

0,25d

0,25d,

0,5d

0,25d

b)

Tac6: S =l'-2'+3'-4' +...+ 2015'z -20162 +20172 = l'+ (l' - ,')*(s' - a')*...n(zot72 -20t6,)

= t*(: *z)(t-z)*(s++)(r-o)+...+ (zon+zoio) (zon -zoro)

=l+2+3+ 4+5+..'+2017

= (, * zon).20t7 :2 = 2035t53

,S = 2035153

0,5d

0,5d

0,5d

0,25d

0,5d

0,25d ) (5 tli6m)

a)

Ta c6: r' + 4y' + 22 = 2r +t2y e r'-2r+l+4y2 -l2y +9+ o (*-,)' * (r, -r)' * (, -r)' [e-r=o l'=t. olrr-3-o=ja:) lz_2=o l,=;

+42-14 z'-42+4=0 -0

1d

0,75d

0,5d

I

f* =,

VAv: ),, = 1 'J l" ') t4l- -'t l'- '

0,25d

b)

r s s- 4 4 laco: r'+a- zx:y+ry ert-roy+yt-ryo>0 e ra ("-r)-ao ("-y) > o o(*-r)(,^-y')=o o (, - r)' (" * y)(r' * y') > o (1u6n dring)

Vi: r +y=(*-r)' ro,(, *r).0,(r' *s')=o

VQy: "t * yt > roy * *yo vbi r,a * 0;r +Y > 0

0,25d

0,5d

0,5d

0,75d

0,25d

0,25d

3 (5 tli6m)

a)

Ta c6: ro l, Md: l'"t, Lr 'j

> (r' +l

YQy A lu

Ta lpi co:

(, =,4=f(,

YQy A =

o - tr' +2r' - r +l = (, +f)(r' - z +1)

z r2 -r+r=[r-!l *1ro;vr [ 2) 4 +l>O;Vr ,)(r'-n+t) >o;vr

u6n x6c dinh v6i mgi gi6 tri cua x

): ro + r' + r*t = (r+t)' (r, -r+1)

',* + t)' (*' - r +t)_ (r + r)' 'r' + 1) (r' -r + 1) 12 +r

, \2 (r+t)

=+ 12 +1

0,75d

0,25d,

0,25d

0,25d

0,75d

0,5d

0,25d

b)

ra co:l(' . t)' > o;v'

[.r'*l>0;Vz

_ (, * l)' ., .,.. tw.vI ' r'+l Vfly: A2liYr

0,5d

0,25d

0,25d

c) Ta c6: A >- 0;Y r V0y: Mi,nA- 0 e r+1 = 0 e r = -l

0,25d 0,75d

4 (5 tli6m) Hinh vE dring 0,25d

2

a)

x6t ^HEB(frE = eoo) vn ^HD r(*, = ro') \/ frE = 6D (2 g6caoi ainn; ) LHEB a LHDC (1 goc nhgn) HE HB HD HC * HD.HB = HE.HC

co:

ld

0,25d

0,25d

b)

Xet ^BHF(BFH = eoo) vir rcCo(BDC = e00) .o,

DBC chung + LBHF cn LBCD (1 e6c nhqn) BH BF < BC BD > BH.BD = BF.BC (r) 0,5d

Chimg minh tuong tU: ) ACHF a ACBE (1 g6c nhqn) CH CF CB CE = CH.CE = cr.ca(z) 0,25d

Tri (1) vd (2) ta duoc: BH,BD + CH ,CE = BF,BC + CF.CB

= BC (ar + cF) = sg' 0,5d

VOy: BH.BD + CH.CE = BC' 0,25d

c)

xdt ^ADr(^* = eoo)vit n+nr(u, - eo ) "0, 6ic chung = AADB a A,AEC (1 e6c nhon) AD AE - AB AC 0,,25d

x6t LADE vit LABC co: