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179 chuyãšn ä Ề toã-n tuá»”i thæ  2 - thcs

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  15. 15. · • Tinh chat sau day cUa trvc tam tam giae co nhieu Lmg dl,lngtrong vi~e giai cac bal tap hlnh hQC. Tinh chat. Cho AABC co H la tnrc tam, 0 la tam dlti1ng tron nqoal tiep va M la trung diem BC. The thl AH = 20M. Chung minh. Xet tntong hop AABC nhon (cac tntong hop khac ehLmg minh tuang tl/). Va duong kinh BD ella (0), VI DAB = 90° nan DA.l AB. Ma CH .1 AB nan DA II CH. Tuang tl,l DC II AH. Suy ra t(( giae DAHC la hlnh blnh hanh. Do d6 DC =AH. Ma DC =20M (vl OM la duong trung binh ella aDBC) nan AH = 20M (dpcm), Bili t~p ap dl,lng Bili toan 1. Cho AABC nhon n¢i tiep duong tron tam O. G9i M, N va Plan luqt la trung diem cac canh BC, CA va AB. Chltng minh rang cac duong th~ng qua M song song vai OA, qua N song song vai OB va qua P song song vai OC dong quy.· . _ Loi giai. G9i H la true tam AABC; I va Elan luqt la trung diem ella HA va HO. ViAH = 20M nan AI = IH = OM. Ma AH 110M (do eung vuong g6e vai BC) nan cac t(( giae AIMO va IHMO la hinh blnh hanh. . T~ MOT TINH CHAT VE TR~C TAM TAM 61Ac D';NG HAl GIANG (GV THCS Thl tran C~m Xuy€m, HiJ Tinh) Suy ra MI II AO va E la trung diem ella MI. T((e la E thuoc duong th~ng qua M song song vai OA. Tuang tl,l E cling thuoc cac duong th~ng qua IJ song song vai OB va qua P song song vai OC. Tlt d6 suy ra dpem. Bid toan 2. Cho AABC nhon e6 H la tnrc tam. Chltng minh rang 9 diem g6m chan ba duong cao, trung diem ba canh va trung diem cac doan HA, HB, HC dong viano LOi giai. sadunq hinh ve va ehltng minh ella biJiteen 1 ta co: 1 1 ED =EM =EI = -MI = -OA. 2 2 Sa dl,lng cac ket qua tuang tl,l va ket hop vai OA = OB = OC ta suy ra 9 diem da eho cling thuoc duong tron tam E. Bili toan 3. Cho AABC nhon n¢i tiep duong tron tam 0 co cac duong cao·AN va CK. Duong tron nqoal tiep aBKN cat duong tron (0) tal diem th(( hai M. G9i I la trung diem ella AC. Chltng minh 1M .1 MB. LOi giai. G9i H la tnrc, tam ella AABC, J la trung diem BH. Ta thay J la tam duong tron ngo~i tiep aBKN va BH la duong kinh ella (J). • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • t www.VNMATH.com 15
  16. 16. .............................................................................................. · VI IHJO la hlnh blnh hanh nan IH II OJ. M~t khac, VI M, B la giao di~m ella hai duong tron (0), (J) nan OJ 1- MB. Suy ra IH 1- MB. Ma BH la duong kinh ella (J) nan HM 1-MB. Tli d6 suy ra , H, M th~ng hang va 1M 1- MB. Bcii toan 4. Cho tCt giae ABCD n¢i tiep duong tron (0 ; R). GQi H1, H2 , H3, H4 thCt nr la tnrc tam ella cac tam giae ACD, BCD, ABD va ABC. ChUng minh rang: a) BH1,AH2 , CH3 va DH4 dong quy. b) Bon di~m H1, H2 , H3, H4 dong viano H3 Or ·· H. D Li:fi giai. a) Vi AH1= BH2 (= 20M) va AH1II BH2 (vl cling vuong g6e vai CD) nan tCt gia~ AH1H2B la hlnh blnh hanh, Suy ra AH2 va BH1 cat nhau tai trung di~m I ella moi duong. ChUng minh tuang nr vai cac cap di~m khac ta suy ra BH1,AH2 , CH3 va DH4 dong quy tal trung di~m I ella moi duong (dpem). Vi tCtgiaeDOH4011ahinh blnh hanh nan 01H4 : =OD =R. : ChUng minh tuang nr ta c6 : 01 H1 =01 H2 =01H3 =R. : V~y bOn di~m H3, H4 cilng thuQc :H1, H2, dLtOng tron tam 0 1 ban kinh R (dpcm). Bid toan 5. Cho dLtOng tron tam 0 ban kinh R : va d~m P c6dinh narn ngoai dl.tOng tron. Va M'p : tuyen PAva cat tuyen PBC (A, B, C nam trEm (0)). : Chang minh rang khi cat tuyen PBC thay d6i thi : trve tam H ella MBC chay trim m¢t dLtOng cOdinh. ·: ·M : ········································Li:fi gild. GQi I la hlnh ehieu ella 0 tren BC, K ···la trung di~m ella AH. Lay 0' doi xUng vai 0 qua ··trung di~m ella PA. ··Ta thay 0' la di~m cO djnh va tCt giae AOPO' la ···hlnh blnh hanh, ··Suy ra O'P II AO va O'P = AO. ···COng vi tlt giae AOIK la hlnh blnh hanh nan ··AO II KI va AO =KI. ··Suy ra O'P II KI va O'P = KI. Tli d6 tlt giae ···O'PIK la hlnh blnh hanh, Suy ra O'K II PI. ··Ma PI 1-AK nan O'K 1-AK. ···Suy ra ~O'AH can tal 0' ~ O'H = O'A. ··Tli d6 H thu¢e duong tron cO djnh tam 0' ban ··kfnh O'A (dpem). ····Bcii ~p til luy~n. ·· Bcii 1. ChUng minh rang tnrc tam, tronq tam va ···tam duong tron nqoal tiep ella m¢t tam giae cling ·· . nam tren m¢t duong th~ng (duong th~ng Ole). ···Bcii 2. Cho duong tron (0) va ba di~m A, B, C ··thay 95i tren (0) sao eho tnrc tam H ella MBC la ···m¢t di~m co dinh nam trong (0). Tim quy tieh ···b) Lay 0 1 doi xUng vai 0 qua I. chan cac duong cao ella MBC. ·...•...••.•••••.•...............•.•.•............•..•..••.........••.•.......•.•••..........· www.VNMATH.com 16
  17. 17. " Tim otto ut 00 ilieu bien • THAI HOU HU~ (Gv. THCS Quang U:,c, Can Lt?c, Ha Tfnh) Co rat nhieu phLlong phap giai mt?t bet toen bat dlmg thLic (BDT), nhLlng chUng ta can tim mt?tphLlong phap giai ng~n 99n nhat. D~c bi~t, chUng ta can t<;lokinang chLing minh BDT. Khi tim hi€uve BDT, chUng toi nh~n thay mt?ts6 BDT co bien s6xec d;nh trong esc doen thi ta can xem xet kTgia tr; cec bien s6 do d€I~p ra mQtBDT dung, roi ta BDT dung nay, ta bien d6i thanh BDT phai chLing minh. Bid toan 1. Cha cac so thuc x, y, Z E [-1 ; 2] thea man dieu ki~n x + y + Z = O. ChUng minh rang x2 + y2 + z2 s 6. Loi gild. VI x E [-1 ; 2] nen (x + 1)(x - 2) s 0 <=> x2 ::; x + 2. TL/dng nry2 s y + 2; z2 s z + 2. C¢ng ve thea ve cua ba BDT tren ta dL/<;lc x2 + y2 + z2 ::; X + Y + Z + 6 = 6. D~ng thltc xay ra khi va chi khi x, y, Z E {-1 ; 2} va x + y + Z = 0 <=> trang ba so x, y, Z co hai 56 bang -1, so con lal bang 2.. Bid toan 2. Cha cac so thirc x, y, Z E [0 ; 2] thea man dieu ki~n x + y + Z = 3. ChUng minh rang x2 + y2 + z2 s 5. Loi giai. VI x, y, Z E [0 ; 2] nen (x - 2)(y - 2)(z - 2) ::;O. Sa dunq gia thiet x + y + Z = 3 ta dL/<;lc xyz - 2(xy + yz + zx) + 4(x + Y + z) - 8 ::; 0 <=> xyz + (x2 + y2 + z2) - (x + Y + z)2 + 4 ::;0 <=> xyz + (x2 + y2 + z2) ::;5. Tll d6 ket hop voi xyz ~ 0 ta suy ra x2 + y2 + z2 s 5. D~ng thltc xay ra khi va chi khi (x - 2)(y - 2)(z - 2) = 0; xyz = 0 va x + y + Z = 3 <=> trong ba so x, y, z co mot so bang '2, mot so bang 0, so con lal bang 1. Bid toan 3. Cha cac so th,!c a, b, C E [-2 ; 5] thea man dieu kien.a + 2b + 3c ::;2. ChUng minh ra'ng a2 + 2b2 + 3c2 ::;66. (De thi tuydn sinh lop 10, nam h9C 2009-2010 Sa GD-DT Ha Tfnh) Loi giai. T,LJdng t,! nhu bai ioen 1, tll (a + 2)(a.- 5) s 0 ta suy ra a2 s 3a + 10 cling cac ket qua tL/dng t,! b2::; 3b + 10 va c2::; 3c + 10. Suy ra a2+ 2b2+ 3c2::;3(a + 2b + 3c) + 60 = 66 (dpcm). D~ng thltc xay ra khi va chi khi a = -2; b = 5; c = -2. Bid toan 4. Cha cac so tbuc a, b, c E [0 ; 1]. ChUng minh rang a2 + b2 + c2::; 1 + a2b + b2c + c2a. Loi gild. Vi a, b e [0 ; 1] nen 2(1 a(1 - b) ~ a - b). b2(1TLJOng t'!b(1 - c) ~ - c); c(1 - a);~ c2(1 - a). C¢ng thea ve ba BDT tren ta suy ra a + b + c - (ab + bc + cal 2) 2a)~ (a2 + b2 + c - (a2b + b2c + c <=> 1 - (1 - a)(1 - b)(1 - c) - abc 2) (a2b + b2c + c2a)~ (a2 + b2 + c ­ 2a)<=> 1 + (a2b + b2c + c - abc ~ (a2 + b2 + c2) + (1 - a)(1 - b)(1 - c). Ma(1 - a)(1 - b)(1 - c) ~ 0 va abc ~ 0 nen suy ra dpcm. Bid toan 5. Cha 2010 so thuc [0 ; 1].a1, a2,·,; , a20W E ChUng minh rang (1+a1+a2+...+a201i~4(af +a~ + ... +a~OlO)' Loi gild. Voi cac 56 th,!c x, y bat kJ ta cO (x - y)2 ~ 0 <=> (x + y)2 ~ 4xy. Ap dl:mg voi x = 1, y = a1+ a2+ ... + a2010 ta cO (1 + a1+ a2+... + a2010)2 ~ 4(a1+ a2+... + a2010). www.VNMATH.com 17
  18. 18. .:xL ~ Co ehinh phttonghhong? Cho so t'! nhien S c6 2011 chlt so, trong d6 c6 2010 chlt so 5 va m¢t chlt so a khac 5. Hei S c6 phai la so chfnh phuong hay khOng? CAO QUaC CUONG (GV THCS Vinh Tuong, Vinh Pnuc) ------------------------~---~,.- .... ""7/ ~T... ? [L'IO -/~t: IlRO~"• --A.EL qua _" Cach dt,tng. Dlfng mot duong tron tam .°cat Ox, Oy tuong ltng tal cac diem A1' A2. Tren (0) dlfng lien tiep cac diem cac diem A3, A4,·· , A20 phan bi$t, khac A1,A2 thea man A2A3 = A3A4 =... = A19A20 = A1A2· Tren cung A1A2 cua (0) d,!ng lien tiep cac diem cac diern B1, B2,·., B17 phan biet, khacA1, A20 thea man B2B3 =... = B16B17 =A1A20.A20B1 = B1B2 = Ta dU<;1c cac tia OA2, OB 1, OB2,.. , OB17 thea , , , ,L ! (TTT2 so84) man dieu ki$n bal toan. Cht1ng minh. Theo each dlfng ta c6 =... =A190A20 =ApA2 =19°.A20A3 =A30A4 . Ma 192 = 361 nen A10A20 = 361° - 360° = 1°. Suy ra BpB2 =...A200B1 ~ =B20B3 ... = B160B17 = B170A2 =1°. Bi~n lu~n. Bal toan luon dlfng dU<;1C va c6 m¢t nghi$m hinh. Nh~n xet. Day la mot bal toan quen thuoc, Toa soan nhan dU<;1c nhieu loi giai cua cac ban, Ngoai each dlfng quen thuoc tren, cac ban con c6 them m¢t so each dlfng nlta, vai y tl10ng la Ian lu¢ tao ra cac g6c 60°, 30°, 15°,4°, 2°, 1° hoac 95°,5°,20°, 1°. Cac ban sau dU<;1c thuong ki nay: Tran Phuong Nga, 9C, THCS Thanh Thuy, Phu ThQ; Nguyen f)(fc Th9, 9C, THCS Phan B¢i chau, Tlt Ky, Hai Duong; Nguyen f)ang Huy, 6A, THCS Hoang Xuan Han, Dltc ThQ, Hli TInh; oso H6ng Qulm, 80, THCS Cao Xuan Huy, Dien Chau, Ngh~ An; Nh6m ban Le ChI Hieu, f)ao Anh Tuyet, Nguyen f)inh Hien, 57B Quang Trung, TP. Quang Ngai, Quang Ngai. ANH COM PA ~-----------------------------------~ Ma vai moi i E {1 ; 2 ;... ; 2010} ta c6 aj :::>: af . h" a b cCh ' rang --+--+--~2.ung min (vi theo gia thiet aj E [0 ; 1]). be + 1 ca + 1 ab + 1 Tlt d6 suy ra dpcm. B" 3 Ch 1 < aj <.1 ,. D~ng thltc xay ra khi va chi khi trong 2010 so 31. 0 2010 - ~ - 2009' Val a1, a2,..· , da cho c6 2009 so b~lng 0, so con lal bang 1. Blii t~p luy~n tap. . . a2000 va b1,b2,... , b2010 ta cac so th,!c duong. Blii 1. Cho cacso thuc a, b, c E [0; n Chltng minh rang Chltng minh rang a2 + 2b2 + 3c2 s 60; 2(a 3 + b3 + c3 ) s 3 + a2b + b2c + c2a. 1 a1+a2+ ... +a2010 1 --< <-­ 2010 - b1+b2 + ... + b2010 - 2009Blii 2. Cho cac so thirc a, b, c E [0 ; 1]. 18
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