Đề thi HSG toán 12 THPT chuyên Lê Quý Đôn-Quảng Trị năm 2019-2020

so cD&Dr euANc rnl rci rnr cHeN t2 cLP rixn cuuv0n TRUcTNG TIrPT rHr HSG LdP NAnn HQC: 2019 - 2o2o Mdn thi: To6n Ln eu.f BON (DA thi EQr ruYEN DU 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 2(a ili6m). Trong mflt phhng tqa d0 Oxy, Chn fu+zMe mdn: --d ; zfii+ cho hinh vu6ng ABCD v]r cic di6m M, N thoa 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 14rC o-at+b'+c' 1 A4+D a J b) Tim GTLN, GTNN ctra him s(i: /(x) = x(l0 *Jtz-71 Cflu 4 (a tli6m). Cho hinh ch6p S.ABC, c6 SA AB : : SB : SC vd d6y ld tam gi6cvu6ng cdn v6i c4nh huyAn oJi.tvtqtb6n (SBC) hqp v6i mflt d6y tich kh6i ch6p S.ABC vi khoing si5 /:(0,+m) + .f(x+ y)+ f(xy) vi t6n (p saocho cosp' =+. Tinh theo a th6 J13 cdch gifia hai duong thFngAB vd SC. cffu 5 (2rli6m). Tim t6t cil cilchim Hq mQt g6c : x (0;+oo) th6a mftn ding thirc * ! * xY, Yx,Y e (0;+m) SBD urldNc nAN cuAilr uoN roAx THr Hsc cAp ruvnx gsc D DE THI CH D CAU I ( 6d) a) Gini phuong trinh (x-2)' +J*+e rixu Di0m an +Jir-, 1t), x e [-o;t t] =67 o (J, + 6 -4) + (r - Ji r-,)+ *' - +* -60 = 0 11 + x+6 -0 e (x-10) _: Jx+6 +4 t+.,/t 1-x 1 (r) 11 +x+6>0,xe[-6;11] Jx+6+4 l+Jl1-x 1 Qx:10 ( Vi _: b) Gqi ) As6 cAn t\m: a,b,c d6i mQt kh6c nhau, a+0 ' c lir sti chin. 3, 4, 5, 6,7 ,8,9\ . {a ,b , c\ c { o, . c=0,m6ic6chchon cs6c6 7 cdchchon a (kh6c c:0),m6ic6chchqn ot" c,a . sdc6 6 c6ch chon b (kh6c c,a), n€nc6 7.6:42 s6loai 1 ndy' * 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: c 1 3.6.6=108 sd loai ndy. s6 thoa m6n dO bei' V{y, tong cong c6: 42+lO8:150 7 F N A 1 (4d) B C E bing 3m. Qua M k6 dunng vu6ng g6c v6i BC cEt BC, AD frnp:Tr,Ttr-ta c6: bdi tt de 416, F. Khi ] :: ,^: ^' =ABEM:AMFN' mn tuqt t4i E, a) E[t canh hinh vu6ng 1 [BE:MF:2m = fr: fr,us:vnt +6ME+ fr D @M) :x -2y -3 = O+(MN) rqa d0 di6m M IvF2 +FN2 rd nghi€m cua h6 :MN2 e4m2 + =9oo - 6=0 _', =or= cdn t4i M' 1 . ,(g; o) . 1 > AN : l,AM :2J, (o-2)'*(b-2)' =r m' = 5 e m- Giai hQ, voi a <2 , ta duqc A (1;2 ) vay ABMN wdng :2x + y {;;1; Gqi A (a;b) , v6i a 12 , ta c6 h€ " 1 (o-3)'+b2 =8 1 3 (4d) a) Vx,-y € R, ta c6 xo + yo ) x3 y+rlr' ( * ) o (, * y)' (r' + xy + y')> o, dirng Vx, y e tR. 0,5 Ap dgng ( * ), ta duoc !)'=1r,*( !\'o o,*( \3, 3 (3.r o' *( !l' la,*(1)'u \3, = 3 (31 .' *[1)' \3, r ,1.,'*(1)'. 3 r,r = (r) a'J I \3/ (o /r\3 +b+.)>l ; I . cQng ve theo vC ta co: \J/ 13+C3 4_) A-.3+D +c' 14 rb-+c- 33 b) TXD: D = ( dpcm ). ^r Ddubangxay ra [-.E,Jir], r{-,):-/(x),vx e e o:6- s=! 0,5 D /(r)>otr€n [0,.D], /(,) < o tron [-Jo,o] : max J (x) = ,{,1fr,f (,) = X.T / (,) - ,{,?fr,f (*) 0,5 Theo BDT C-B, ta c6 ,r (r) < rJ0.r f0.(r2-r') = *.Ji.Jzz- *' . ^1n.x: +4- =,r(r) SKH =Q' I aJI. .6 a 6 1 LAB > AH r (SCH). SH LAB Trong LSCH kdtludng cao HI-d(SC,AB)=nt CH Gt= ,t I a CO a r 5 (2d) = I -----'---:- T I ---------= I = 11 T d( AB:SCl= HI, SH' CH" -;3A" o3a f(x+y)+"f(xl,)- x+y+ry,Yx,y >0 (1) . x: !=)a f (4)=4. LAn luot thay (x;y) . {{t;t);{z;1);(3;1)} vdo (1), ta c6: I f tzl+ / rD:: [/1:) =: tl i/t:lo f Q)=5-\l \2)=2[l r+i * /(3):; [7'trl : Th6 x : !.! =l1r > O) vio ( l) ta thu dugc: - 1 .t l Hl: n'' a. 1 7 0.5 r I t' ' 1(r+-1+J(l):t+-t '',=Jtt+!)=t+1 " t. t =J'(x)=x.vx>z(1oo,*1=z''l. / ) Ti6p tuc th€, y = 2. Tt (1) ta suy ra (2x): x+2+2x f Ilf lr*2)+ 2)= **2.Vx> 0 [,ftr* Thu 0.5 0.5 = - f12x):2x.Yx >0 haY./(x)=x.Vx>0 lai,f(*) :x,Vx > 0 th6a mdn di6u kiQn bdi to6n.V{y "f(*) :x,Vx e (0;+"o). 0.5