De thi HSG lop 9 quan hong bang hai phong

PHoNG GrAo DVC vA DAo r4or,{AM HQC ZO1Z.. Ggi E, F lAn luot lir trung di6m cua AB vd CD.. chirng minh rdng BI : cI... chirng minh ring BI : cI.

PHoNG GrAo DVC vA DAo r4o

r,{AM HQC ZO1Z ZOt:

DE THI MON: TOAN _ LOP 9

Thdi gian." I50 philr (kh6rtg ki giao di1

EE CHINH THTTC

(Ei rhi gim 0 t rrang)

a

-Bii 1:(2.0 cli6m)

a Chirng minh ,irrg -l - * =1, - + +

V l + V : 4 5 + 4 7 ,19997 +.,19999> 2 4

b Cho ba s6 a, b, c + 0 th6a mdn:

I I I I r r - t r

_ t _ _ r

t [ - t l , - , z f - - ; V a V a + V b + { 6 =

v a ' v D v c J

chrmg minh ring tron g cdc s6 a, b, c c6 it nh6t mot s6 bing 2l

Bii 2: (2.5 cli€m)

a Giii phucrng trinh y3 = 2qf 2y -1-1

b chrmg minh khdng t6n tai cFry giittrf nguyCn (x; y) thoa mdn x, - 2y, = 2013

Biri 3: (2.0 cti6m) Cho tri gi6c ABCD o e = 400;D : g00 vir AD : BC Ggi E, F lAn luot lir trung di6m cua AB vd CD Tfnh eFD f

Itdi 4: (2.5 eli6m) Tir di€m A nim ngodi duong tron tAm 0, ke hai ti6p tuy6n AB vd

AC (B vd c ld titip diem) Ducrng thing qua A cet (o) tqi D vd E (D nim gifra A vd E), k6 day cung EN song song v6i BC, DN cit BC tai I chirng minh rdng BI : cI

Bii 5: (1.0 tti6m) cho a,b, c la dg ddi ba canh ctra m6t tam gi6c thoa m6n:

* ;- - A2012 * 62012 + Qt"'

a + b - c a - b + c - a + b + c Hdy x5c dinh dang.cfia tam gidc do.

6 + 2 J i

-Gidm thi khorcg giai th{ch gi th1m

PHoNG crAo uUC, vA DAo rAo

UON roAru - Lop e

tt+Atu Hec 2orz-2ar3

> 2 4

Bii 1: (2.0 eli6m)

'19997 + 19999

b Cho ba s6 a, ;e 0 thoa mdn- - L - I1 1 1 1

^ r _ | - v a

V a V b V c 3

4 l i * V u * i 6 - J o * z J i

-Chring minh rSng tron g citc

l z e -r 2 4 s .

s6 a, b, c c6 it nh6t m6t sd bing 27

D6p rln

a ) E d t { - - f L +

B : - r l , j

-Js + -Js Jt + Jt ' Jeeee + Jr ooor

Jz"n -Jz"n

-^ l 2 n + 1 + r / 2 n + 3 2

n 6 n A * B : Jrooor - r 1 0 0 - r 9 9

Ta c6

V i A > B n 6 n A t A * B r 2 r z o

b) Ta c6

1 1 1 1 , ^ ' ; + ^ r

e : ( V " u *V u *V;)-Vabc = o (1)

0,5d

0,5d

0o25aI

-"rir (1) vd (z) ta co: 3(V"b + Vui * VA)- V"b + e($ + Vu fiF

YQy itnhAt m6t trong ba so a, b, c co mQt s6 bing27

2 7 : 0

0,541

a Gi6i phuorrg trinh y3 =2ll2a-y -1

b chrmg minh kh6ng tdn tai cFry gi|tri nguydn (x; y) th6a mdn x2 - 2y2 = 2013

Biri 3: 12,$ai6m) cho ttt giac ABCD c6 e :400;fr=g00 ve AD: BC Goi E F

Dat y :?112x= ta c6 h6:

[ " ' * l = 2 y [ * ' * 1 = 2 y

j - < + {

L y ' * 1 : 2 x L t ' - y 3 = 2 ( y -

" )

:

'0',25d

0,25d

0,5d 2b (1.0 ili6m)

x' =2y' + 2013 + x16, d4t x = 2k+ i;(k e Z)thayvdo ta c6:

2y' - 4k' + 4k - 2012 e y2 = 2k2 + 2k-1006 (l)+ y chfin, y = 2t; (t e z)

Thay x;y viro (1) vd bi6n d6i: 2t2 :k(k + t) 503 (2)

X6t thay vr cria (2) ludn chEn; vp cfia (2) td t6 16 vi k(k+t) ch84 (Tfch 2 s6

nguydn li6n ti6p) vpy d6u ":" crta(2) kh6ng th6 xay ra<+ rhbng ta" * ,ap ,o

nguy6n (x; y) th6a mdn: xt - 2y' - 2013

0"25a1 02sd

l6n lucrt ld trung diOm cria AB vd CD Tinh gm Z

- VE hinh binh hinh ACKD ta c6:

D C K = A D C = 8 0 ' '

:+ BCK : BCD + DCK =1200

C6 CK: AD : BC n€n tam ei6c CKB cAn

^

tai C :> CKB = 30'

Lai co F la trung di€m cria cD ncn F cfrng ld trung di€m cua AK

n6n EF ld duong trung binh ctra tam giitc ABK

:> EF song song v6i BK

:> EFD = BID = CIK .z (+r")"f )

Tt d6 x6t tam gi6c KIC co EFD = 6lR =70a

1 , 0 d

1,0t1

Bii 4: (2.5 tti6m) Tt di6m A nim ngodi dudng trdn tAm O, ke hai ti6p tuy6n AB vd

AC (B vd c ld ti6p dicm) Dulng thing qua A cat 1o; t4i D vd E (D nim gifra A vd E), ke d6y cung EN song song v6i BC, DN cdt BC tai I chirng minh ring BI : cI.

Bii 5: (1.0 iti6m) Cho a,b, c le d0 ddi ba canh crja m6t tam gi6c th6a m6n:

C 2 0 1 3 6 2 0 1 3 a 2 0 1 3

+

-

+ - - a 2 0 1 2 + b 2 0 t 2 + C ' o t '

a * b - c a - b + c - a + b + c Hdy x6c dinh d4ng cua tam giSc do

- Ta c6 DBC = DEC;EN / /BC + BN : cE =+ BDN : enp = ABID.- AECD

BI EC

_ > - = _

ID CD

Tucrng tu ta c6:

Ta c6 AAEC * AACD n€n

CD = AC ; tu'ng tu ta cung.o gp = AB;

M e A C : A B n c n S = = ( 3 )

CD BD

Tt (1), (2) vd (3) ta c6 BI : CI ./

(1)

CI EB

- = _ l L l

I I ) B D

0o5d

0,5d

0,5d

l,Oil

_ - ' r <*_

Edp 6n

- T a c 6 :

{ u ' o t t - b 2 0 r ' ) ( u - b ) > 0 6 uzotz + b 2 0 i 2 } u 2 0 t ' b + a b t o , ' (t )

Tuong tu ta co:

b 2 0 t 2 + c 2 0 1 2 > 6 2 0 r r " + b c r o , , ( z ) ;

c2()12 + a20t2 >c2'lra + caron (:);

6p dr,rng b6t ding thric CoSi cho hai s6 ducrng ta c6:

+ ( - a + b +

" ) u r r , ' > 2 r l _ _ " , _ ( _ u * b + c ) a r o "

a20t3

= - a + b + c_ + a t o t t b + a ' o " c > 3 a t o t , (4 )

Tuong tU ta c6:

; 2 0 1 3

+ abr'r, + br.,rc > 36zotz (5)

a - b + c

^ 2 0 1 3

" , + a c r o ' r * 6 a 2 o r t > 3 c 2 o t z ( 6 )

a + b - c

Tt (1), (2), (3), (4), (5), (6) ta c6

C2013 I b2ot: 42013

+ - + -_: - ) u20rz * 62012 + crorr p6, rc_': xhy rakhi vd chi

a + b - c a - b + c - a + b + c

khi a: b : c tric ld ta c6 tam si6c dAu

0,25d

0,5d

0.25d