Đề toán thi vào 10 chuyên các tỉnh, thành năm 2023

DA] HOC QUOC GIA HA N◊l DE THI TUYEN SINH VAO LOP 1o TRUONG D~ HQC KBOA HQC Tlf NHIEN TRUONG THPT CHUYEN KHTN NAM 2023 D~ chinh t!htrc MON TID: TOAN (Vong II) Tboi gian lam bai: 150 phut (khong k~ thoi gian phat cl~) Cau I (3,5 di~m) 1) Giai phtrcrng trinh 2x+ I+ 2✓4x + 6x = 4°J5x - x 2) Giai h~·phucrng lrinh xy(x+ y ) =30, { x +y3 Cau II (2,5 di~m) l) Tim tAt ca cac c~p s6 nguyen ducrng = 30+ 1x +y+ l 20 (x; y) th6a man 4.r + ( 1+ 3>') ( l + 7" ) = 2x ( Y + 7>' + 2)2) V 6i x, y, z la nhung s6 thvc ducrng, tim gia tri nh6 nhfit cua bi€u thuc: xt4 - x6 + y14 - y + + z'4 - z6 + M = x-~ y i + zx + zy + y-~ z i + xy + xz z i x i + yz + yx• Cau ITI (3 diim) ~ t1.ep : ducmg tron ,( 0) c6 ti~p Cho tam g,iac ABC nhQn v6-1 AB< AC n91 -.: BC o T cho TB > BC G91 P va E ,an ~ tucrt tuyen t(.li A cua (0) cM · la trung (hem cua TA va T C /) Chting minh r~ng tfr g iac APEB n9i ti8p 2) GQi giuo dit m tl u'r hai cua AE v&i (0) la F Liy G tlrn9c (0) cho FG song song vai AC Chtrng minh r~ng 3) Gi;,i If ATC =TAF la live {am Ctla lam giac ABC, D la giao diSm Clla AH va BC M la trung d6i xung v&i A qua BC N thu9c dtrcmg th~ng A}vf cho KN song song v&i HM L~y S tbu9c BC cho NS J_ NK D1Jng R thu(k ti a AK cho AR· AH = AD • Q la diBm cho PQ J_ AS va SQ J_ AO Chfrng minh r~ng di€m d6i x(rng ct:1a A qua QR tJrn9c dm:mg tr6n ducmg kinh DN diem BC K Cau IV (1 di~m) Vi~t m9t tram s6 nguyen ducrng dftu tien 1, 2, 3, , 100 vao mn · • , · • h J x ta1 ha1 o ke (2 c · h 10 Cl;lll C lllng) ma hai viet (J hai c6 bi~u tern crn 10~,c ,;ang , ? ° ~ HOC QU6C GIA HA NOi Bt THI TUYEN SINH VAo LOP 10 TRUONG BH KHOA HQC T\f NHliN TRUONG THPT CHUYtN KBTN NAM 2023 Dl cbinb thm: M6N THI: TOAN (cho tit ci cac thf sinh) Thai gian lam bai: 120 phut (kb6ng kj thoi gian phat cti) Cau I (3,S 4i@m) · 1) Giai phucmg trinh ✓x2 +6x+2023 +.Jx+3 = ✓x2 +5x+2025 + ✓5 2) Giai he phucmg trlnh (x+6y)(3x+ 2y) = 12, { 2x3 + 6y3 + 15x2 y+ 19y2 x+ x+ 6y = 12 Ciu Il (2,5 diim) 1) Gia su n la sb nguyen cho 3n3 -1011 chia bet cho 1008 Chung minh ring n-1 chia bet cho 48 2) V6i a,b,c la cac s6 duong thoa mm di~u ki~n ab+bc+ca = I Chung minh ring ( l+ 1 ) > )(1+2 )(1+l+a l+b l + c2 Ciu fil (3 ctiim) Cho hai ducmg trcm (0) va (O') ca di.nh cit t~ A va B cho nlm ngoai (0? va O' nim ngoai (0) Tren ducmg tron ( 0) lAy diam P di chuy~n cho P n~ ducmg tron (O? Duong thing AP cit (0') ~ C khac A 1) Chung minh ring hai tam giac OBP va O' BC d6ng d\lllg 2) GQi Q I~ giao diem ctia hai duong thing OP va O'C Chung minh ting QBC + ABP = 90" 3) Liy di!m D thuqc (0) cho AD vu6ng g6c O'C Chung minh ting trung diSm cua do'1} thing DQ luon nAm tren mqt duong tron cO dinh P thay d6i Ciu IV (1 ctiim) Gia SU A la ijp hqp cua ijp hqp gAm 30 th! s6 t\l nhien diu tien {0,1,2,3, ,29} a = b ) thl a + b + 30k khOng la tich cu.a hai s6 nguyen li!n tiep Chung minh ring s6 phin tu cua ttp hqp A nh6 h GIAO ovc v A aAo T~o c()NG u6A XA uor culi NGUIA VJtr NAM TRlfCJNG D~I HQC Sil PH~M HA N()I D9c lf p - T y- - Htnb pbiic Dt CH INH THIJ'C (Di th/ g6m co 0/ trang) Dt THI TUYEN S lNH LOP 10 TRUONG T HPT CHUYEN D~I IIQC su· PH~M NAM 2013 Mlln tbl: TOAN (Du11g c/ra 1119i thi si11I, 1l1i vao T111Ung THPT c/111yen D1;1i l,9c Sir phqm) Tl,iri gian lam bill: 90 p/rut (khong lei 1/rln g/011 philt d€) Bil I (2.5 diam) a) Rill gQn bi~u thm:: ,., = T +8E ix +.[; 16-4x > O r + 7" + r vmx x-2vx+4 vx vx+2 b) MQI khay mroc co nhie1 d.Q 12S"F bit diu cho vlio ni dal tu dti, cu sau m6i gib nhi~I dQ cua khay nuoc l(li gidm di 20% l-loi sau bao nhicu gib, nhiQI dQ clia khay nuoc chi hi 64°F? Bil (3.0 diim) a) Cho phuang 1rlnh r'-(Zm-l)x -(111' +1) = (I) (m Iii tham s:6) Chung minh rang vcri m9i gia trj cuo m pbuang trlnh (I) luon co hoi nghi¢m n xi Tim h¢ tbuc lien h¢ giihl r, xi cho b~ thm: kh6ng ph11 thuQC viio m b) Cho parabol (P): y = Ill'' (a., 0) di qua di~m A( -1: ½} Tim 19a dQ cua dic1:m M tren parabol (P) cho khodng calch tir di~m M d&i lr\JC tung gftp hoi Jin khoang each tir di€m M d€n lr\JC hoanh Bil (2,5 di~m) Cho hlnlh binh hanh ABCD c6 ABC= 1200 vi BC= 2AB D\lllg dubng tr6n (0) c6 duc:mg kinh A:C G9i £ F lin luQt lo cac giao diSm thu hai cun AJJ AD voi dubng tr6n ( 0) Duong thbg EF Jin luQt CAI enc dubng thing BC, BD ~i H, S Chung minh a) Tam gidc ABD Id 1nm giac vu6ng b) Tu gidc OBEH lo tu ginc: nQi ti~p c) SC la tifp tuySn cun dulmg tron (0) Bil ( 1,0 diEm) C6 hay khong cac s6 nguyen a, b cho (a+ bJ2023 )' =2024 + 202JJ20231 Bil (1,0 di~m) Trio bang 10 vi1!t da thuc l'(.t) = at' + b:r+ c (a,.0) Ta vi~I Jen bong thuc mm P,(x) = P(x+l)+P(x - l) r6i xoa di da thuc P(x) Ta viet lcn bang da Ihm: mm P,(x) = P,(.t + I); P,(.t-l) r6i x6a di da lhuc f:(x) Ta cu ti~p I\IC lam nhu lh6 nhieu !An Chung minh rang n6u cu lam nhu vjy nhiEu Jin 1hi d~n mQI Ilic mlo d6 ta nh6n duqc mQI da thuc khong c6 nghi~m Hit G/1/ d1u: H9c si11l1 kh611g duvc sri dcmg tai liju can b? coi ,1,; /cl,ong giai tMch gi them H9 len thf sinh: S6 boo donh: so olAo ovc vAE>Ao T~o TRU"ONG D~l HQC SU' P~M HA NOi CONG IIOA XA HOI CHU NGHIA VltT NAM D0c IJp - T11 - Htnh ph6c ot CHINH THUC (Di thi glim ,·6 0I 1ra11g) ot THI TUYtN SINFI LOP 10 TRU"O) 2Ji Rlllgoabi!ulhuc A vichungminh AS2 ,_ sl, ChCm minh b) Cho phwag trinh: x'-2(o+l)x+o' - 2a+ l = (x 16 an, a 16 lham ) g nbi a Ii 16 chinh plnnmg thl phuung lrinh d4 llho c6 hai nghietn cOng 16 nhDng s6 chlnh pbllOllg Bll (2,0 llii111) 11) Oiii ptnnmg lrinh: (3:x' + 4x+6)✓3x' +4•:x+ S = 27:r' + Jx JY( Jx+ I +Ji}= I b) OW h~ pbuong trlnb: {y+4,/Y = x' +3x-3-2(:x+l).Ji Bll (J,O 411,,,J Cho U1111 gijc nhQII AJJC kh6ng c1n ~i ti!p dulmg ln'ln IAm Ve dubng ldnh A.T c:ua duong aon ( 0) vi lly di4m P trm dop thing OT ( P • T) 09i E vi F tuung ung IA blnh woog g6c cua P trm etc cllrlmg thing AC v6 AB 09I H 11 blnh chiEu vulmg g6c cua A 1m1 qnh BC a) Qmng m.inh DAB , HAC vi ha.i dwng lblng BC, EF song song v6i b) Cho ,fJf vi EFi;iJ lfl U; di4m Q di dOng IJ6! dofll lhing UE (Q;,,U, Q• E) chieu - - £)uupg thing w6ag goc v6i AQ tfi dl4m (J cit* dutmg thing PE, PF tuong img tfi M, N G9i K 16 tlm duoag ln'ln ngO(li ti~p tam glk ANN Cbilng mlnb b6n diem A, kl, N, P cung lhu~ 11$ dutmg ln'ln vi OAH = KAQ c) Kl KD vu6ng g6c v6i BC (De BC) Chung mlnh dwng thing di qua dihn D vA sons sons v6i AQ lu6n di q111 mOI dl4m c6 djnh 1W (J,O llim) Cho c4c s6 lh\lC a, b, c tho6 a+ I,+ c 11m gl6 tri nh6 nh!t cua bi~ thuc: Bl! (2,0 41im) la-1 lb-I + ; ;, 2c-1 p = +- ; a'+2 61 +2 C1 +2 :> Tim de s6 nguyan 16 a, b v6 16 ngvym duong m 1h06 a' +b +l Bab= 4.s•, :hn de J, ) ~ I _P~lei tr& mOI dllimg tron B6nh s6 de di~ d6 mOl deb nglu nbibl boi bai di~~-• ,(hai di ~ - ~ dll\7C dAnh s6 b6I hal s6 khAc nhau) Mgi dly cung n6i 111;&1 tJm duQCicy~ ; ' : , : : : ; : d6l ~~~ de s6 b bai d!u ml\t Chung minh ring 0ng ban dAy cung d6 bing l cm chung, cho t6ng cua cAc J, g6n v6i - v'- ' , I \ vA u rA o TH UA THI.EN Hu t KY Tm TU Yt N srN H LOP 10 - NA M TRUONG TR PT CHUYtN QU HQ C 2023-2024 OC HQC - ou t M6o thi: TOAN (CHUYEN TOAN ) Thm gian lam bai: 150 phut, kh6ng kJ rhoi gian phat tie DE THI CHiNH nn JC Ci u (1,5 lli2m) \ a) Chfrng minh ·a g1 tr1· cua b'1eu th'uc p = ( + ✓ a a_ 12 : a ✓a ✓a a+ ✓ a+ - ✓ a + a _ :J; _ khong pb1,1 th u~ vao gia tri cua a, vo i a > va a I b) Cho a b, c la ba s6 nguy~n du ong th6a miln + = ! .Chung minh Q = a + 4b2 + 16c' a b c la mc}t s6 chinh phuong Ca o (1,5 4iim) - J * \ a) Tren m~t ph!ng tQa de} 0x y, cho parabol (P ) : y = 2x va duc mg tbiing ( d) : y = ~ x + m • Tim tfit ca cac gia tri cua m de ( d) cit (P) t(li hai die m pban bi~t A, B cho tam giac OAB vuong t~i A '\ I b) Giai h~ phuong trlnh {.J~x -2y- + 2y +4 y = x· + t = xy Ca u (2, dii m) I a) Ti mm d~ ph uong trinh x - 2{ m - l) x- m +2 m -3 =0 (x x1• x, th6aman Jx ~+ l - x =Jx;+ l+ x2 • I la An s6) c6 hai nghj~m \ b) Giai plmcrng trlnh 2( Jx +9 -3)( ✓9- x +3) =9 Cii u (3, ilii m) Chl) Lam giac nh Qn ABC ( AB < AC ) nc}i tiAp ducmg tron (0 ), c6 ducmg cao AD va tr\IC tii111 11 GQi E la di6 m tren (0 ) cho hai day AE va BC son g so ng v6 i E>uang th ~g EH CUI (0) ll_l i di~m th(r hai la F va cit duon g trung tr\l 'C cua BC t{li M a) Chun g minh ~ tru ~i ~m c(1a EH va AMOF la tu giac n9i tii p \ bJ Ch(mg minh OFA +O DF = l 80° c) G9 i K la di~m df>i xung v6 i A qua TiAp €n cua (0 ) ~i A cit dumig th ~g FK tcJi T Chun g minh hai du on g thi ng TH va BC song son g v6i C:iu (2,0 ttii m) a) Tim t~t ca cac s6 thµ c a cho a+ ✓2023 va 999 + ✓2023 d~u la ca c s& nguyen a b) Cho hai s6 th1,1c duong a, b tJ16 a mru1 4a + b = Tun gia trj nh nh it cua bieu thu c T == ~ + -b-+ 20 24 + b l + a 2a +b -B E T Thi si11J1 kh ong dU'{Yc su dtmg tai lifu Ctin bp coi thi kllong git ii thic /1 gl them HQ va ten thi sinh: bao danh: Chii ky cua Can bQ coi thi I: Chu IN THJ: TOA ~ (~b~ ea) ; Th&, g1an lam ba,: J50 phw kh6ng ke thin gian phal de Ngay thi: 03/6/2023 ~ ( /,5d i€m)Chobie uthuc K=( Jx_ +1)(✓x-x) a) Tim di!u kien xac dinh va rut g~m bi~u thuc K b) Tim tAt ca cac s6 thi,c X Chung minh a+ b + c > 4[~ + _!_ + ~b] + be ca a C:iu {/ < ,11im) I ChLing minh n + 3n + la s6 le voi mQi s6 t\f nhien n nm !lit ca cac s6 nguyen duong a, b cho 4a1 + b + 4; 4b1 + a + t\ d~u \a s& chinh phllong Cau -1 (3.0 iliin,) Cho tam ginc ABC nhQn, AB< AC Kc cac ducmg cao AD,BE,CF c{l\ li!i H Tir ke hai ti~p tuy~n AP,AQ d~n duimg tron tim 0, ducmg kinh BC (P,Q la cac tiJp di~m va P, F nim cung ph[a so voi dubng thing AD) I Chung minh AP1 = AB.AF va di~m A,P,D,O,Q nlm tren mQt ducmg tron Chung minh H, P, Q thing hang Chung minh PF,QE,AD d6ng quy Cau (0,5 ilid111) r ,_1 Tren P I n1~~ I· l • phing co di~m • ::11 •• I m , !"I • 1n y,• d6 kho~g c6 ba di~m nao thing hani • •• •• ~ !'I I" so GIAO D\JC vA DAO T~o Dt Tm TUYt N SINB VAo LOP 10 THPT TiNH DONG NAI NAM BQC 2023 - %024 M0n: To4n(chuy!n) Thai g.ian 16m bAi: I SO phul (DI g6m I trang, co cau) it cu1Nu THirtj CAu l (2,0 b>c;ab+bc+co>O vl a+b+c a I I I S Timgiilrinb6nbMcuabi&ithuc P'= a-b + b-c +;:;◄ 2Jab+bc+co Cl■ (O,S 4l'1n) Cbo x, y, z la c6c s6 cblnh phuvng Chung miDh ring (x+l)(y+l)(1+l) lu6n vi& duQC du6I dq tbns cua hai s6 chlnh phuung SO Gl AO DVC V.~ DAO TAO PH(: THQ Kl' THI TUYtN SI NH V 1\0 LOP 10 TRUNG ll()C PHO Tl16NG CHll \'EN 11 \JNG Vl i'UNG N.-lM HQC 202J - 2024 o t Cl-liNH TH(l"c Mon: Toan (Danh rho thi sinh thi chu~~n rin) Thai gwn ltim b -2 a) Chung J(2x+1)(271+l) 2z+l - + 4xy + 2x + 2y + 2z ,rV/ = / · x + l + Y+ l + z + ' < + 4z + 2y 2x + 1,0 41/m) Cho duong tron tam O nOi tiSp hlnh thoi ABCD GQi E, F, G, H la cac An Ju~ thuoc c4c c~ AB, BC, CD, DA s~o cbo EF, GH cung ti~p xuc vm (0) Tim gia tri nh6 nhlt cua bieu th(rc p = a Chung minh CG.A H = A • b) Chung minh EH song song FG Bai (1,0 dilm) Xet c4c nguyen a < b < C th6a mAn n nguyen a) Chung minh a < ta TI· (ill) · v sa _ l \ = + b3 + c3 - lh tr,(lL v([ \ 3abc la sA ult ca c4c s6 nguyi!n a, b, c (a < b < c) cho n ll mQt ulrc cila 2023 -BET- ~ / SO GIAO UllC \'Ao,,o T AO l!lli~ r, I • ~c11/N11 11£ nu T UI £_\' /11'111 I.GI' 10 T lll'T CUL\'t_,; P,\C ca"'(· ,'IA.\ I IIQC l OlJ • 10JJ ~16'1 TIii: T0 ,(,'1' Spy 1J1i: O! /W.~!.' n,u:I N THJ: TOA ~ (~b~ ea) ; Th&, g1an lam ba,: J50 phw kh6ng ke thin gian phal de Ngay thi: 03/6/2023 ~ ( /,5d i€m)Chobie uthuc K=( Jx_ +1)(✓x-x) a) Tim di!u kien xac dinh va rut g~m bi~u thuc K b) Tim tAt ca cac s6 thi,c X 1) Cluing minh ring ab - ✓1 - albl +cr+bl =I C iu (6.00 dli m) Giai cac pblJO'l'l g ninh, h~ phuon g trinh sau: a) (x- Ji)' +(x+JsY +(Ji - Js- ixf b) {-· - · x + I y + - x·' + y y l + t - - H~t'- - Thi sioh khong d trQ'C SU' dl,ID J! rai lifU Giam thi kbong giii thich gi them I ~n th I s 1· n11· H(l\ 3c.: ·S6 bao danh: Chfr ki gi:im thi I: :Chfr ki giam thi 2: lvJ tx -y,.,., , v ," · -~ (\ -; rl C~ v Cau Ill (1,0 di€m) 202 + 2" la s6 chinh phu ang Tim tAt ca cac s6 t\l' nhien n dS • + 220?7 Ci u IV (4,0 dli1n) h va khOng di qua tAm GQi A la Cho duong tron ( o) co n 1'i BC lh c(U cit AH g thin ng duo K, t~ - _ , , Chung minh BA D = CAE binh ban h va HA.HD = HK HM inh lah CE BH c gia tir g rin h ung Ch K) , duong thi ng di qua / va vuo ng c kha (/ / t(li (0) tron ng dua cit KD Tia ng minh rin g cac ducmg thi ng AK , BC va g6c voi duong thi ng BC cit AM t~ J Chu • HJ cimg di qua m(>t diem AK l(li A va cit cac c(lnh AB, AC M(>t ducmg tron thay d6i luon ti~p xuc voi g diem cua do{ln thi ng PQ Ch ung mi nh rin g trun la N i GQ t bi~ n phA Q P, t~ t hrq IAn h duang thin g AN luon di qua mOt diem cf, djn =1 di~u ldt n ! + ! + _!_ n ma a tho ng dua c thv s6 ba la c b, a, o c2 Ch b2 m) a2 Ci u V (J,0 dii Tim gia trj Jon nhAt cua bicu thl'.rc: ✓ I + I + I P ::2 + 2ca + 2a 2 2 /Sh + 2hc + 2c /Sa + 2ah + 2h V Jsc -H ET • nting SO( ln Ihao van ban c chir co 11g lch6 llii bo linh y ma va khong g d(ln SU' ,c Thi sinh dur co ,,, nh thi p' -1 chia h!t cho n ll t,i ::'iu IV (J,Odiim) Cho tam giac nhQn ABC (AB