HNG DN NI DUNG BI DNG HC SINH THI CHN HC SINH GII TON QUC GIA LP 12 THPT

(Km theo cc Cng vn s 11636/THPT ngy 25/12/2000 v 1403/THPT ngy 25/02/2002 ca B Gio dc v o to) I. Yu cu ti thiu v kin thc Ngoi cc kin thc ton theo Chng trnh ph thng (t lp 1 n lp 12) hin hnh, cc hc sinh d thi cn c trang b thm ti thiu mt s kin thc sau: 1. Phn S hc: - Cc khi nim v kt qu l thuyt c trnh by trong Chng I; 1, 2, 4 Chng II; 1, 2, 3 Chng III; Chng IV v Chng V cun"Bi ging s hc" ca nhm Tc gi: ng Hng Thng(Ch bin), Nguyn Vn Ngc, V Kim Thu (NXB Gio dc, 1994). - nh l nh Phcma, nh l Uynsn. - nh l le v nh l Trung Quc v cc s d. 2. Phn i s - Gii tch: a. Bt ng thc (Bt): - Cc bt ng thc i s: Bt Csi cho n (n Z, n 2) s thc khng m; Bt Bunhiacpxki cho hai b n s thc (n Z, n 2); Bt Trbsep cho hai dy n s thc (n Z, n 2); Bt Nesbit cho ba s thc dng; Bt Becnuli m rng. - Bt ng thc hm li (Bt ng thc Jensen). - Cc bt ng thc tch phn c trnh by trong mc 3 ca 2 Chng III SGK Gii tch 12 (Sch chnh l hp nht nm 2000, NXB Gio dc). - Kt qu ca V d 1.4 trong 1 Chng V cun"Bt ng thc" ca Tc gi Phan c Chnh (NXB Gio dc, 1993). b. a thc: - Khi nim nghim bi ca a thc v mt s kt qu n gin lin quan n nghim ca mt a thc: # nh l 1. a thc bc n (n N*) c ti a n nghim thc, mi nghim c k s ln bng s bi ca n. # nh l 2. Nu x0 l nghim ca a thc P(x) th x0 + a l nghim ca a thc P(x - a), vi aR cho trc. # nh l 3. Nu x0 0 l nghim ca a thc: P(x) = a0xn + a1xn 1 + ...... + an 1x + an , a0 0 v n N*, th 1/x0 l nghim ca a thc: Q(x) = anxn + an 1xn 1 + ...... + a1x + a0 . # nh l 4. Nu x0 l nghim bi k (k Z, k 2) ca a thc P(x) th x0 l nghim bi k 1 ca a thc o hm P/(x). # nh l 5. Nu x0 l nghim hu t ca a thc vi h s nguyn: P(x) = a0xn + a1xn 1 + ...... + an 1x + an , a0 0 v n N*,

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th x0 phi c dng p/q; trong p, q tng ng l c ca an, a0. # nh l Viet thun v o cho a thc bc n (n Z, n 2). - Cng thc ni suy Lagrange. - Khi nim a thc kh quy, bt kh quy. - nh l Bdu v s d trong php chia mt a thc cho nh thc bc nht x a. - a thc Trbsep v cc tnh cht c trnh by trong phn 1 Ph lc 3 cun"Bt ng thc"ca Tc gi Phan c Chnh (NXB Gio dc, 1993). c. Dy s - Hm s: - Phng trnh c trng v cng thc tnh s hng tng qut ca dy s c cho bi h thc truy hi tuyn tnh. - Cc khi nim: dy con, dy s tun hon v chu k ca dy s tun hon. - Mi lin h gia tnh hi t ca mt dy s v tnh hi t ca cc dy con ca dy s . - Mt s kt qu n gin v tnh n iu ca hm s: # Kt qu 1: Nu f v g l cc hm s ng bin (nghch bin) trn tp X th f + g cng l hm s ng bin (nghch bin) trn tp X. # Kt qu 2: Gi s f v g l cc hm s ng bin (nghch bin) trn tp X. Khi : i) Nu f v g ch nhn gi tr khng m (khng dng) trn X th f.g s l hm s ng bin trn tp X. ii) Nu f v g ch nhn gi tr khng dng (khng m) trn X th f.g s l hm s nghch bin trn tp X. # Kt qu 3: Gi s f l hm s ng bin v g l hm s nghch bin trn tp X. Khi , nu f ch nhn gi tr khng m (khng dng) trn X v ng thi g ch nhn gi tr khng dng (khng m) trn tp th f.g s l hm s nghch bin (ng bin) trn X. # Kt qu 4: Gi s g l hm s ng bin (nghch bin) trn tp X. K hiu g(X) l tp gi tr ca hm g vi tp xc nh X. Khi : i) Nu f l hm s ng bin trn g(X) th f(g(x)) s l hm s ng bin (nghch bin) trn X. ii) Nu f l hm s nghch bin trn g(X) th f(g(x)) s l hm s nghch bin (ng bin) trn X. # Kt qu 5: Nu f l hm s ng bin trn R th hai phng trnh sau s tng ng vi nhau: f(f(.... (f(x))....)) = x v f(x) = x. - Khi nim chu k c s ca hm s tun hon v mt s kt qu lin quan n hm tun hon: # nh l 6. Nu hm s f(x) tun hon trn tp X vi chu k c s T v nu: f(x) = f(x + A) x X th phi c A = kT, vi k Z. # nh l 7. Nu hm s tun hon f(x) c chu k c s T th hm s f(ax) (a 0) l hm s tun hon v c chu k c s T/a .

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# nh l 8. Nu cc hm s f1(x) , f2(x) tun hon trn X v tng ng c cc chu k T1, T2 thng c vi nhau th cc hm s f1(x) + f2(x) , f1(x) f2(x) , f1(x)f2(x) cng tun hon trn X. - nh ngha hm s ngc. - nh l v gi tr trung gian ca hm s lin tc trn mt on. - Kt qu cc Bi ton 1-7 trong 1 Chng II cun"Phng trnh hm" ca Tc gi Nguyn Vn Mu (NXB Gio dc, 1997). 3. Phn Lng gic: - H thc Sal cho cc cung lng gic. - Bt phng trnh lng gic v tp nghim ca cc bt phng trnh lng gic c bn. - Cc cng thc n gin tnh di ng phn gic,bn knh ng trn ni tip,bn knh ng trn bng tip ca mt tam gic theo di cc cnh v gi tr lng gic ca cc gc ca tam gic y. - Mt s bt ng thc thng dng trong tam gic:3 3 2 3 cosA + cosB + cosC 2

sinA + sinB + sinC

ABC. ABC. nhn ABC.

tgA + tgB + tgC 3 3 4. Phn Hnh hc:

Du "=" trong cc bt ng thc trn xy ra khi v ch khi ABC l tam gic u. a. Hnh hc phng: - Khi nim trng tm, tm t c ca mt h im v to ca chng xt trong h to cac. - Tm ng phng ca ba ng trn. - Hng im iu ho v Chm iu ho: nh ngha v mt s tnh cht n gin: # H thc Niutn, H thc cac. # nh l 9. Hai cnh ca mt tam gic cng cc ng phn gic trong, ngoi xut pht t nh chung ca hai cnh y lp thnh mt chm iu ho. - nh l Ptlm , nh l Xva , nh l Mnlaut , nh l Thales thun v o. - nh ngha ng trn Apoloniut, ng trn le (ng trn 9 im). - Kt qu ca cc V d 1,2 trong phn 4 4 Chng II SGK Hnh hc 10 (Sch chnh l hp nht nm 2000, NXB Gio dc). - H thc le trong tam gic: d2 = R2 2Rr trong : d, R, r tng ng l khong cch gia tm ng trn ngoi tip v tm ng trn ni tip, bn knh ng trn ngoi tip, bn knh ng trn ni tip ca mt tam gic.

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- nh ngha tch cc php bin hnh v mt s kt qu lin quan, nh ngha v cc tnh cht ca php ng dng: nh c trnh by trong TLGKT Hnh hc lp 11 Ban KHTN-THCB (NXB Gio dc, 1997). - Cc kt qu l thuyt lin quan ti cc php bin hnh trong mt phng c trnh by trong cun "Cc bi ton v hnh hc phng" (T.1 v T.2) ca Tc gi Praxolov V.V. (NXB Hi Phng, 1994). - nh ngha v cc tnh cht ca php nghch o c trnh by trong phn"Cc kin thc c bn" Chng 28 cun"Cc bi ton v hnh hc phng. T.2 ca Tc gi V.V. Praxolov (NXB Hi Phng, 1994). b. Hnh hc khng gian: - nh l Thales thun v o. - nh ngha khi a din u, khi t din gn u, khi t din trc tm v mt s kt qu lin quan: # nh l 10. T din ABCD l t din gn u khi v ch khi xy ra t nht mt trong cc iu sau: i) Cc mt ca t din c din tch bng nhau. ii) Bn ng cao ca t din c di bng nhau. iii) C t nht hai trong ba im sau trng nhau: tm mt cu ni tip, tm mt cu ngoi tip v trng tm ca t din. # nh l 11. T din ABCD l t din trc tm khi v ch khi xy ra t nht mt trong cc iu sau: i) Cc cp cnh i ca t din vung gc vi nhau. ii) Chn ng vung gc h t mt nh xung mt i din l trc tm ca mt y. iii) Tng bnh phng di ca cc cp cnh i bng nhau. - nh l v s tn ti ca mt cu ngoi tip khi a din. - Kt qu ca V d 1 trong 1 Chng II SGK Hnh hc 12 (Sch chnh l hp nht nm 2000, NXB Gio dc). - Khi nim trng tm, tm t c ca mt h im v to ca chng xt trong h to cac. - nh ngha v tnh cht ca tch c hng ca hai vect, tch hn tp ca ba vect cng mt s kt qu lin quan: nh c trnh by trong 3 v 8 Chng II SGK Hnh hc 12 (Sch chnh l hp nht nm 2000, NXB Gio dc). 5. Phn T hp: - Nguyn l Dirichlet, Nguyn l cc hn (hay Nguyn l khi u cc tr). - nh ngha nh x, n nh, ton nh, song nh, nh x tch. - Cc khi nim v kt qu c trnh by trong 1, 2 v 3 ca ti liu "V mt s vn ca gii tch t hp trong chng trnh THPT "(Bin son: Nguyn Khc Minh. Ti liu bo co ti Hi ngh tp hun gio vin ging dy chuyn ton ton quc, H Ni-1997). - Kt qu ca cc Bi ton 1, 4, 5 trong 4 ca bi vit ni trn. - Cc khi nim c bn ca L thuyt Graf: Graf; nh, nh c lp, cnh v hng, cnh c hng ca graf; Graf c hng; Graf n v hng hu hn; Graf y4

; Graf b; Graf con; Bc ca nh trong graf n v hng hu hn; Graf thun nht; ng i, di ng i, ng i khp kn, xch(c ti liu gi l ng i n gin), xch n, chu trnh(c ti liu gi l chu trnh n gin), chu trnh n, ng i le, ng i Hamintn, chu trnh le, chu trnh Hamintn trong graf n v hng hu hn; Graf lin thng, cy, graf le, graf Hamintn; Thnh phn lin thng ca graf n v hng hu hn. - Mt s kt qu n gin ca L thuyt Graf: # nh l 12. S nh bc l trong mt graf n v hng hu hn l mt s chn. # nh l 13. Trong graf n v hng n nh (n Z, n 2) tn ti t nht hai nh c cng bc. # nh l 14. Nu graf G n v hng n nh (n Z, n 2) c ng hai nh cng bc th G phi c ng mt nh bc 0 hoc ng mt nh bc n 1. # nh l 15. Mi graf n v hng hu hn khng lin thng u b phn chia mt cch duy nht thnh cc thnh phn lin thng. # nh l 16. Nu mi nh ca graf G n v hng n nh (n Z, n 2) u c bc khng nh hn n/2 th G l graf lin thng. # nh l 17. Graf G n v hng hu hn l graf le khi v ch khi hai iu kin sau c ng thi tho mn: i) G l graf lin thng. ii) Mi nh ca G u c bc chn. # nh l 18. Nu tt c cc cnh ca mt graf n v hng y 6 nh c t bi hai mu th phi tn ti t nht mt chu trnh n di 3 c tt c cc cnh cng mu. - Khi nim "chin lc thng cuc" trong cc bi ton tr chi. II. Yu cu ti thiu v k nng 1. Bit vn dng linh hot cc kin thc l thuyt vo vic gii quyt cc bi ton c th. 2. Bit phn tch mt cch hp l cc gi thit t tm ra hng gii quyt bi ton. 3. c bit, i vi cc bi ton c th gii c nh m hnh graf, cn: - Bit cch chuyn bi ton ban u (hoc mt phn ca bi ton ban u) v bi ton tng ng trn m hnh graf; - Bit cch s dng biu din hnh hc ca graf nh mt cng c to ra cc gi trc gic trong qu trnh suy lun, tm ti li gii cho bi ton; - Bit s dng cc khi nim, thut ng ca l thuyt graf trnh by li gii mt cch ngn gn, sng sa, cht ch v chnh xc. III. Mt s im lu 1. Cng thc ni suy Lagrange, nh l le v nh l Trung Quc v cc s d, php bin hnh nghch o trong mt phng, cc khi nim, kt qu ca l thuyt graf v cc bi ton tr chi l ni dung kin thc khng bt buc i vi cc hc sinh d thi Bng B.5

2. V nh l le v nh l Trung Quc: Ch yu cu hc sinh hiu ng cc nh l ny v bit vn dng chng trong cc tnh hung khng phc tp. 3. Hc sinh d thi ( c hai bng A v B) c php s dng cc khi nim, kt qu nu trong mc I nh khi nim, kt qu SGK. 4. Trong k thi chn hc sinh gii quc gia mn ton lp 12 THPT cc th sinh c php s dng cc kin thc v s phc trong phm vi chng trnh mn ton Ban KHTN-THCB (c). 5. hc sinh t c cc yu cu v k nng, nh nu mc II, cc gio vin bi dng cn ch : - Gip hc sinh hiu r bn cht ton hc ca cc khi nim, cc kt qu. - Gip hc sinh nm c cc tng ton hc n cha trong li gii ca cc bi ton c th. - Luyn tp cho hc sinh cc bi ton m li gii ca chng th hin mi lin quan gia cc phn kin thc. - Phn tch cho hc sinh thy con ng i n li gii ca cc bi ton. iu ny c bit quan trng trong vic luyn tp cc bi ton t hp. 5. nng cao hiu qu ca vic bi dng, ngoi cc ti liu dn mc I, cc gio vin bi dng c th tham kho cc ti liu sau: [1]. thi v ch cc nc (19 nc) T.1, T.2, T.3, NXB Hi Phng. [2]. H.Chng, Graf v gii ton ph thng, NXB Gio dc, 1992. [3]. N.V.Mu, Phng php gii phng trnh v bt phng trnh, NXB Gio dc. [4]. Ti liu bi dng hc sinh gii bc THPT mn ton, V THPT n hnh, 1997. [5]. Tp ch Ton hc v Tui tr.

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