Đề thi chọn HSG lớp 12 môn Toán năm 2019 2020 trường chuyên Lê Quý Đôn Quảng Trị

so cD&Dr euANc rnl

TRUcTNG TIrPT cuuv0n

Ln eu.f BON

(DA thi gim cd 01 trang) Thdi gian ldm bdi: 150 philt

Cflu 1(6 tli6m)

a) Gi6i phucrng trinh (x -2)' + J.. 6 = 67 +Ji 1 -n

b) Tir c6c chfi s5 0, 3, 4, 5, 6,7, 8, glpp dugc bao nhi6u s5 chin, c6 ba cht s5 kh6c nhau

Chn 2(a ili6m). Trong mflt phhng tqa d0 Oxy, cho hinh vu6ng ABCD v]r cic di6m M, N thoa

mdn: fu+zMe --d ; zfii+ wD =6

a) Chimg minh tam grttc BMN vu6ng chn.

b) Tim tqa dQ di€m A,UietNp;21, dvdngthingBM cd phuong trinh x-2y-3:0 vit

di6m A c6 hodnh d0 nh6 hon2.

Cflu 3 (a tli6m).

a) cho c6c s5 thqc a,b,c th6a man ili6u kign a+b+c>1. chimg minh ring

4 14 o-at+b'+c' A +D rC 1 aJ

b) Tim GTLN, GTNN ctra him s(i: /(x) = x(l0 *Jtz-71

Cflu 4 (a tli6m).

Cho hinh ch6p S.ABC, c6 SA : SB : SC vd d6y ld tam gi6cvu6ng cdn v6i c4nh huyAn

AB : oJi.tvtqtb6n (SBC) hqp v6i mflt d6y mQt g6c (p saocho cosp =+. Tinh theo a th6 ' J13

tich kh6i ch6p S.ABC vi khoing cdch gifia hai duong thFngAB vd SC.

cffu 5 (2rli6m). Tim t6t cil cilchim si5 /:(0,+m) + (0;+oo) th6a mftn ding thirc

.f(x+ y)+ f(xy) : x * ! * xY, Yx,Y e (0;+m)

Hq vi t6n SBD

rci rnr cHeN EQr ruYEN DU rHr HSG LdP

t2 cLP rixn NAnn HQC: 2019 - 2o2o

Mdn thi: To6n

urldNc nAN cuAilr uoN roAx DE THI CH D ruvnx gsc THr Hsc cAp rixu

CAU D an

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a) Gini phuong trinh (x-2)' +J*+e =67 +Jir-, 1t), x e [-o;t t]

(r) o (J, + 6 -4) + (r - Ji r-,)+ *' - +* -60 = 0

-0 + x+6 e (x-10)

Qx:10 ( Vi +x+6>0,xe[-6;11] )

11 _: Jx+6 +4 t+.,/t 1-x 11 _:Jx+6+4 l+Jl1-x

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6d) I (

b) Gqi ot" As6 cAn t\m: a,b,c d6i mQt kh6c nhau, a+0 ' c lir

{a ,b , c\ c { o, 3, 4, 5, 6,7 ,8,9\ . . c=0,m6ic6chchon cs6c6 7 cdchchon a (kh6c c:0),m6ic6chchqn

c,a sdc6 6 c6ch chon b (kh6c c,a), n€nc6 7.6:42 s6loai ndy'

. c * 0, c6 3 c6ch chon c chian, mdi crich chon c sE c6 6 c6ch chgn a

(a * 0,a+ c), m6i c6ch chon c,a c66 c6ch chgn b (kh6cc,a ) ndn c6:

3.6.6=108 sd loai ndy.

V{y, tong cong c6: 42+lO8:150 s6 thoa m6n dO bei'

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a) E[t canh hinh vu6ng bing 3m. Qua M k6 dunng vu6ng g6c v6i BC cEt BC, AD

frnp:Tr,Ttr-- mn tuqt t4i E, F. Khi 416, tt de bdi ta c6: ] :: ,^: ^' =ABEM:AMFN' [BE:MF:2m

= fr: fr,us:vnt +6ME+ fr =9oo " vay ABMN wdng cdn t4i M'

N F A

C B E

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4d)

7 (

D @M) :x -2y -3 = O+(MN) :2x + y - 6 = 0 .

rqa d0 di6m M rd nghi€m cua h6 {;;1; _', =or= ,(g; o) .

IvF2 +FN2 :MN2 e4m2 + m' = 5 e m- 1 > AN : l,AM :2J,

)

Gqi A (a;b) , v6i a 12 , ta c6 h€

Giai hQ, voi a <2 , ta duqc A (1;2

(o-2)'*(b-2)' =r

(o-3)'+b2 =8

Di0m

sti chin.

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3 (4d) a) Vx,-y € R, ta c6 xo + yo ) x3 y+rlr' ( * ) o (, * y)' (r' + xy + y')> o, dirng Vx, y e tR.

Ap dgng ( * ), ta duoc

o,*( !)'=1r,*( !\'o \3, 3 (3.r

o' *( !l' = la,*(1)'u \3, 3 (31

.' *[1)' ,1.,'*(1)'. \3, 3 \3/

r r,r = (r) /r\3 (o +b+.)>l ; I . cQng ve theo vC ta co: \J/ -.3 13 3 J 14 4_ A +D +C ^r a' rb-+c- ) +c' ( dpcm ). Ddubangxay ra e o:6- s=! 33

0,5

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b) TXD: D =

/(r)>otr€n [-.E,Jir], r{-,):-/(x),vx e D

[0,.D], /(,) < o tron [-Jo,o] max J (x) = ,{,1fr,f (,) = X.T / (,) : - ,{,?fr,f (*)

Theo BDT C-B, ta c6

,r (r) < rJ0.r f0.(r2-r') = *.Ji.Jzz- *' . ^1n.x: +4- x2 :r rJil

=,r(r)<ttJiT, dau " : " xay ra <+, = JtT

YQy mal f (,)=1lJ[, khi -r="'fi. r ;.f (*)=-1lJ1, khi x=-fi

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4 (4d) ^s

A B

K

C

Isa t(ABCI I = H ld tam clucrng tritn (ABC). }dd, LABC vu6ng can tai c = H ld trung lsr: sB:.rc

di€m AB. Goi K ld trung iti6m cira BC. suy ra HK ld dudng trung binh cua a,4BC. lruxtttc n ri 1"" "'- -fuxtac uo oo \rr :#:;tr" ea : ali)=t* L BC - BC )' (sHK)=> SKH =Q'

__1 Cosrp :i + tanp : 2Ji + SH = HK.tane = aJI. ./l 3

so..,a. = !,qc.ac 1 a')Vr.uu. I s^,{BC'sF1 .6 a 2 J 6

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