De thi HSG quoc gia THPT nam 2016 mon Toan

<span class='text_page_counter'>(1)</span>nQ. irAo. DUC vA DAo. pe rur cHiNH. rao. rY rm cHeN Hec. GrA. rHpr. NAM 2016. THUC. ?. cror eu6c. sJNH. M6n : TOAN. ,. fiAhI ilHIhiffi. Thdi gian : 180 phrit (kh6ng kO thdt gian giao di) Ngiy thi thil nh6t : 06t01t2016 I. Bni 1 6. aiAm1. GiAi he phuong. trinh. I a*-y+22 =3 1*' -y'. -22=-l. (x,y,zelR.).. I o*' - 3y' - y -zzz =o. Bii 2 (5 diam\. a) cho day so (o,,) xitcdinh boi a,,=rn(2nz +l)-ln(n2 +r+l), vdi n=1,2,... Chung minh chi c6 hffu hpn s5 n sao cho {o,}<]. ,tt 2 b) Cho ddy s6 (b,) xhc dinh bryi b, =ln(Zn'+ l). minh tdn t4i vO h4n si5 n sao cho. {4,}.#. + ln(n2 + n. +l),. v. 6i. n = 1,2,.... Chfng. i. Trong d6 {x} ln ky hiQu phAn 16 cria sti thpc x:{x} = r-[r]. Bii 3 (5 diad. Cho tam gi6c ABC c6 B, C co dinh, A thay d6i sao cho tam gi6c ABC nhgn. Gqi D li trung di6m cria BC vd E, F tuong ung li hinh chi6u vudng g6c cta D l€nAB, AC. a) Ggi O tqi. Mvdll.. li tdm cria duong trdn ngopi ti6p tam giirc ABC. nf. cht,qO vil BC lan tuqr. Chung minh dulng trdn ngoqi ti6p tam giirc AMN di qua mQt eli€m cO Ainn.. b) C6c ti6p tuy6n cria dubng trdn ngoai ti6p tam Chring minh Tnthugc mQt dulng thing cd Ointr.. gi6c. AEF t4i E, F cit nhau tai Z. Bii. 4 (5 diA@. Nguoi ta tr6ng hai loai c6y kh6c nhau tr€n mgt miting d6t hinh cht nhflt kich thudc mxn 6 vudng lmdi o tr6ng mQt cdy). MQt cdch tr6ng c6y dugc gli ld iin tuqng n6u nhu:. i) SO luqng c6y dugc trdng cria hai lopi cdy bing nhau; ii) SO lugng ch6nh l$ch cira hai loai c6y trdn mdi trang khdng nhd hon mQt nua si5 O pfia hing d6 vi si5 lugng chOnh lQch cria hai lopi cdy tr6n m6i c6t kh6ng nho hon mQt nta sti 6 cria cQt d6. a) HAy chi ra mQt c6ch tr6ng cdy 6n tugng l<hi m = n = 20t6.. b) Chfng minh n6u c6 mQt crich tr6ng c6ry iintugng thi. : .. ";;;;-;. ;;;;;;;; :; ;;;;;. Cdn bp coi thi khdng. ci. ;;;;ff ;; ;;; ;;;. gidi th{ch gi. th€m.. m vd n t16u lA bQi cria 4. -.

<span class='text_page_counter'>(2)</span> Be GrAo DUC vA EAo. oe rru. rAo. ri. rm. cHeN Hec srNH GrOr euoc GrA Trrpr NAM. csixu rHUc. 2016. Mdn : roAN. r}Ar{ ffiI[NFf,. '%:*ry-. .,.,,".,.,*.._. Thoi gian : 180 phrit (kh6ng k€ thdi gian giao di) Ngiy thi thir hai:0710112016. ^_l. Bni5 $diem). Tim t6t cri cric so thgc a d6 t6n tpi hdm s6 ,r:lR.+R th6a m6n:. i) /(t) = 20t6; ii) f y,+ f (y))= /(r) +ay v0i msi x,y e R .(x+. .. Bni 6 Q di1m'). Cho tam giitc ABC n$i titip du<rng trdn (O) (vdi tdm O) c6 cic g6c o dinh B vit C dAu nhqn. LAy diOm M trdn.rrng 6D khdng chira A sao cho AM\<hdngvuong g6c voi BC, AM chttrung truc cua BC tai f. Oucrng trdir ngo4i ti6p tam giac eOfZEt w (N * A).. fqlit. a) Chung minh EIit = CAN. b) Gqi / li tdm dulng tron nQi titip vd G li chdn duong ph6n gi6c trong g6c A cula tam gi6c ABC. AI, MI, NI cht(o) 6n luqr tai D, E, F. Ggi ia'gtuJng li.iiao di6m cin DF voi AM vi DE vdi AN. Eucrng tron di qua P vi ti6p xric vdi ,qO tqil @i H (H * D), duong trdn di qua evd ti6p xric v6i eb tqi t cit oa @i K ( K * D). "hlOf chung minh duongtrdn ngoai tiiSp tam gi6tc GHK ti6p xtic v6i BC. BAi 7 Q diam). 56 nguy6n ducrng n duoc ggi ld s6 hoan chinh n6u n bing t6ng c6c udc sd duong cria n6 (khdng k6 chinh n6).. *g. !. a) Chung minh ring ntiu n li s6 hoin chinh. l€: thi. n c6 dpng. n= ptmr, trong tl6 p ld s6 nguy6n t5 c6 dqng 4k+1,. li. s. li s6 nguy6n duong c6 dpng 4h+tvit m. sti nguy€n ducrng khdng chia h6t cho p.. b) Tim tdt hoan chinh.. cir cric s6 nguyen. duong. n>l. sao cho. n-l vd n(n-+l) dOu li c6c s6. .. Thi sinh khdng &rp'c s* dqng tdi tiQu vd mdy tinh cdm my. Cdn bp coi thi khdng gidi thich gi th€m.. 2.

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