HI TON HC H NI S GD & T HNG YN

**************** *******************

NGUYN VN MU, NGUYN VN PH

(CH BIN)

HI THO KHOA HC

CC CHUYN CHN LCBI DNG HC SINH GII TON

HNG YN 25-26/02/2017

Hi tho khoa hc, Hng Yn 25-26/02/2017

TRNG THPT CHUYN HNG YN -NI THP SNG NHNG TI NNG TON HC

Nguyn Th Hng Thy

Hiu trng trng THPT Chuyn Hng Yn

Trng Trung hc ph thng Chuyn Hng Yn c thnh lp ngy 23 thng 5 nm1997. Ban u, trng c tn l trng Ph thng Nng khiu Hng Yn. T nm 2003 nnay, trng mang tn trng THPT Chuyn Hng Yn. Trng ta lc ti s 1, ng ChuVn An, phng An To, thnh ph Hng Yn. Mc tiu ca nh trng l xy dng mtngi trng cht lng cao, trong hc sinh c bi dng nng khiu, ng thi cpht trin ton din v tr tu, nng lc t duy, nghin cu v th cht.

Tri qua gn 20 nm k t khi thnh lp n nay, trng THPT Chuyn Hng Yn khngngng pht trin ln mnh v mi mt. Trong , cng tc bi dng hc sinh gii, hc sinhnng khiu c coi l nhim v trng tm hng u. Tnh n nay, trng c tng s714 hc sinh ot gii hc sinh gii Quc gia, trong c 12 gii nht, 85 gii nh, 328 gii ba,290 gii khuyn khch; c 1 hc sinh t Huy chng ng quc t, 4 hc sinh c thamgia d thi Olympic Quc t Chu -Thi Bnh Dng. i vi b mn Ton, nh trng c92 gii, trong c 1 gii nht, 6 gii Nh, 53 gii Ba, 32 gii khuyn khch; 1 hc sinh ctham gia d thi Olympic Quc t Chu -Thi Bnh Dng.

Trong khng kh ca nhng ngy u xun mi inh Du 2017, trng THPT ChuynHng Yn c vinh d chn l n v ng cai t chc Hi tho Cc chuyn chn lc bidng hc sinh gii Ton THPT ca ngnh GD& T Hng Yn. y l dp sinh hot chuynmn b ch gp phn nng cao cht lng i ng gio vin, nng cao cht lng bi dnghc sinh gii mn Ton, ng thi cng l hot ng ngha hng ti k nim 20 nmthnh lp trng THPT Chuyn Hng Yn (1997-2017).

Tham gia vit bi cho Hi tho c 25 tc gi, nhm tc gi l cc chuyn gia u ngnhb mn Ton; hi vin Hi Ton hc H Ni; Bo Ton hc v Tui tr; ging vin trngi hc Thng Long, hc vin An Ninh nhn dn; gio vin b mn Ton thuc cc trngTHPT ca thnh ph Hi Phng, tnh Bc Giang, Bnh nh; gio vin Ton cc trng THPTChuyn HSP H Ni, Vnh Phc, Tuyn Quang v THPT Chuyn Hng Yn. Cc chuyn tp trung vo cc lnh vc khc nhau ca Ton hc nh: S hc, Hnh hc, T hp, Dy s,Bt ng thc, Phng trnh, Bt phng trnh. . . . Nhiu chuyn c cht lng chuynmn su, th hin s u t ca tc gi trong nghin cu, tng hp. Nhiu chuyn ctnh ng dng cao v mi, c th p dng trong cng tc bi dng hc sinh gii quc gia.

Hi tho nhn c s quan tm su sc ca Lnh o S GD& T, Hi Ton hc H Ni,cc trng THPT, THPT Chuyn trong v ngoi tnh. Nhn dp ny, trng THPT ChuynHng Yn xin chn thnh cm n s quan tm ch o ca Ban Gim c S GD& T, cc

2

Hi tho khoa hc, Hng Yn 25-26/02/2017

phng chuyn mn thuc S GD& T; xin by t tnh cm bit n su sc n GS-TSKHNguyn Vn Mu v Hi Ton hc H Ni phi hp gip v chuyn mn v cng tct chc Hi tho; xin cm n cc cn b qun l, cc thy c gio tham gia v vit bi gpphn to nn thnh cng ca Hi tho.

CHNG TRNH HI THO

Sng ngy 25.02.201710h00-11h00 Hp Ban T chc v Ban Chng trnh11h30-12h30 n tra

Chiu ngy 25.02.201713h30-13h45 Vn ngh cho mng13h45-14h15 Khai mcPht biu cho mng: Nguyn Th Hng Thy, HT trng THPT Chuyn Hng YnPht biu khai mc: Nguyn Vn Ph, Gim c s GD & T Hng YnPht biu dn: GS.TSKH Nguyn Vn Mu, Ch tch hi THHN

14h15-15h15 CC BO CO MI PHIN HP TON THiu khin: PGS.TS Trn Huy H, PGS.TS Nguyn Thy Thanh1. PGS.TS. m Vn Nh, Mt vi kt qu v ng bc ba2. TS. Nguyn Vn Ngc, Mt s bt ng thc ca cc hm hyperbolic3. TS V Tin Vit, Xung quanh mt bi ton thi IMO4. TS Nguyn Vit Hi, S ly tha5. ThS Nguyn B ang, T gic iu ha

15h15-15h45 Ngh gii lao

15h45-17h45 CC BO CO CHUYN PHIN HP TON THiu khin: PGS.TS. Nguyn Minh Tun, PGS.TS. H Tin Ngon6. Hong Minh Qun, Mt s m rng v p dng ca bt ng thc Klamkin7. ng Th Mn, Mt s phng php gii phng trnh hm trn tp s nguyn8. Trn Th Lan Hng, Phng trnh v h phng trnh qua cc k thi hc sinh gii9. Hunh Duy Thy, Tip cn cc phng php tm cc tr ca mt biu thc a bin10. L Th Mai, Mt s dng ton v bt ng thc i vi hm phn thc11. V Vn Thng, S dng bt ng thc Karamata kho st bt ng thc trong tam gic12. Phan Ngc Ton, S dng tnh cht gii tch trong cc bi ton s hc13. Phm Vn Dng, Mt s ng dng ca nh l Lagrange14. Mng Thanh Hng, Phng php lng gic gii phng trnh a thc bc cao

17h45-18h00 Tng kt hi thoiu khin: GS.TSKH. Nguyn Vn Mu18h30-20h30 n ti v giao lu vn ngh

3

LI M U

Nguyn Vn Ph

Tnh y vin - Gim c S GD& T Hng Yn

Thc hin Ngh quyt 29-NQ/TW Hi ngh ln th 8, Ban Chp hnh Trung ng khaXI v i mi cn bn, ton din gio dc v o to; Ngh quyt s 09-NQ/TU ngy4/10/2016 ca Ban Chp hnh ng b tnh kha XVIII v Chng trnh pht trin GD& T tnh Hng Yn giai on 2016-2020, mt s nh hng n nm 2025; K hoch321/KH_UBND ngy 29/12/2016 ca UBND tnh Hng Yn v thc hin Ngh quyt s 09-NQ/TU ngy 4/10/2016 ca Ban Chp hnh ng b tnh kha XVIII, S GD & T HngYn phi hp vi Hi Ton hc H Ni t chc Hi tho Cc chuyn chn lc bi dng hcsinh gii Ton THPT tnh Hng Yn ln th Nht nm 2017.

Trong nhng nm qua, cng tc pht hin v bi dng hc sinh gii ca tnh lun cquan tm ch trng. Hng nm, S GD & DT t chc k thi hc sinh gii cp tnh cc mnvn ha cho hc sinh lp 9 v hc sinh THPT, lm tt cng tc tuyn chn v bi dng chohc sinh tham gia k thi Hc sinh gii cp quc gia THPT. Kt qu k thi chn hc sinh giiquc gia hng nm c gi n nh, trong c nhiu em t gii cao. Trong 2 nm tr liy, c 3 hc sinh c tham gia thi Olympic Quc t Chu - Thi Bnh Dng. Song songvi vic bi dng hc sinh, cng tc o to, bi dng i ng gio vin c quan tm,nh mt khu quan trng to nn cht lng cng tc bi dng hc sinh gii. Nhiu hitho, cc lp tp hun chuyn mn c t chc hng nm theo quy m cp tnh vi mctiu nng cao kin thc, k nng, nghip v s phm cho gio vin.

Hi tho Cc chuyn Ton hc bi dng hc sinh gii tnh Hng Yn ln th Nht nm2017 c t chc nhm nh gi khch quan thc trng cng tc o to, bi dng hcsinh gii, t ra nhng gii php hu hiu nhm o to hc sinh gii trong nhngnm tip theo.

Hi tho c vinh d n tip nhiu nh khoa hc, nh gio lo thnh, cc chuyn giaTon hc u ngnh, cc chuyn gia gio dc v vi t Hng Yn ngn nm Vn Hin. Bant chc Hi tho hy vng y s l c hi cc gio vin b mn Ton ca tnh Hng Ynv cc tnh bn c trao i, chia s, rt ra nhng bi hc kinh nghim qu bu trong cngtc bi dng hc sinh gii ni chung, b mn Ton cp THPT ni ring. Hi tho Ton hccng s l tin chng ta t chc nhng hi tho tip theo i vi cc b mn khoa hckhc.

Nhn dp ny, S GD& T Hng Yn chn thnh cm n Hi Ton hc H Ni, cc nhkhoa hc, nh gio lo thnh, cc chuyn gia Ton hc u ngnh. S GD & T tnh ghinhn s ng gp ca cc chuyn gia gio dc, lnh o, cn b ph trch chuyn mn caS GD & T, cc thy, c gio b mn Ton vit bi v tham gia Hi tho. S ng gpca cc ng ch gp phn lm nn thnh cng ca Hi tho.

Hi tho khoa hc, Hng Yn 25-26/02/2017

MC LCChng trnh 22. m Vn Nh, Mt vi kt qu v ng bc ba 53. Nguyn Vit Hi S ly tha 124. Nguyn Vn Ngc, Mt s bt ng thc ca cc hm hyperbolic 235. Nguyn B ang, T gic iu ha 396.V Tin Vit, Xung quanh mt bi ton thi IMO 497. Nguyn Duy Lin, nh l thng d Trung hoa v mt s ng dng 558. Hong Phng Anh, L thuyt th v ng dng 799. Phm Vn Dng - Hong Th Nhung,

Phng php o hm trong chng minh bt ng thc 9010. Phm Vn Dng - Hong Th Minh Thy, Mt s ng dng ca nh l Lagrange 10511. Mng Thanh Hng, Phng php lng gic gii phng trnh a thc bc cao 11612. Lng Th Hng, Phng php hm sinh xc nh dy s 13013. Trn Th Lan Hng,

Phng trnh, bt phng trnh, h phng trnh qua cc k thi hc sinh gii 14314. L Th Mai,

Mt s dng ton v bt ng thc i vi hm phn thc bc hai trn bc nht 16615. ng Th Mn, Mt s phng php gii phng trnh hm trn tp s nguyn 17216. ng Th Mn, Cp ca mt s nguyn v ng dng gii mt s bi ton s hc 18417. Hong Minh Qun - Ngy Phan Tin,

Mt s m rng v p dng ca bt ng thc Klamkin 19318. Nguyn Th Tm - Hong Th Nhung - Trn Thi Hng, nh l Helly v ng dng 20519. V Th Thun - Nguyn Th an Qu, S chnh phng theo modul bc ty 21420. V Th Thun, S dng i lng bt bin v n bin trong ton t hp 22621. Hunh Duy Thy, Tip cn cc phng php tm cc tr ca mt biu thc a bin 24122. Phan Ngc Ton, S dng tnh cht gii tch trong cc bi ton s hc 25423. ng Th Mn, Mt s ng dng ca s phc trong i s v ton t hp 25924. V Vn Thng, S dng bt ng thc Karamata kho st bt ng thc trong tam gic 27025. Quch Th Tuyt Nhung, Mt s bi ton v dy s truy hi tuyn tnh cp hai 27926. Hong Tun Doanh, Mt s tnh cht ca hm s hc c bn v p dng 290

4

Hi tho khoa hc, Hng Yn 25-26/02/2017

MT VI KT QU V NG BC BA

m Vn NhTrng THPT Chuyn HSP H Ni

Tm tt ni dung

Mc ch ca bo co ny l trnh by mt phn ni dung ca cun gio trnh (xem[2]) nhm l gii s hnh thnh, pht trin ca mt s kt qu trong Hnh hc S cp vxu chui cc vn lin quan li thnh mt th logic. tc gi mong mun a n chongi c mt ci nhn ton cc v mn hc ny v s ra i ca H tin Hilbert, Htin Wayne v H tin Pogorelov ca Hnh hc Euclid, gii thiu M hnh Carte kim tra h tin do Pogorelov a ra tha mn ba yu cu: c lp-Phi mu thun-y . Tc gi tp trung trnh by nh l Pascal cho ng cong bc hai v ch ra nhl Newton, Pappus, Brianchon v m rng bi ton con bm. Vic s dng phngphp a thc v m rng trng cho ta hnh dung c s pht trin ca mn Hnh hcS cp t a thc bc 0, a thc bc 1 ri n a thc bc 2 v tip tc phi xt a thcbc 3. Bc cao hn na ca hnh hc s c tip tc nghin cu trong Hnh hc i s.

Mc tiu ca chuyn nhm trang b mt cch su sc cc kin thc, rn luyn tduy v k nng gii ton hnh cho cc em hc sinh kh, gii; cho cc em sinh vin, cchc vin cao hc. Vi mt vi bi ton m cc em tp dt nghin cu, xy dng ktqu mi.

1 nh l Pascalnh l 1 (Pascal). Gi s 6 im A1, A2, . . . , A6 nm trn ng trn (C). Khi , im giaoA = (A1A5) (A2A4), B = (A3A4) (A6A1), C = (A2A6) (A3A5) thng hng.

Chng minh. Gi I = A1A5 A2A6, J = A1A5 A3A4 v K = A3A4 A2A6. Theonh l Menelaus, vi tam gic I JK v cc b ba im (A, A2, A4), (B, A1, A6), (C, A3, A5)thuc ba cnh tng ng thng hng ta nhn c ba h thc sau:

AI

AJ.

A4 J

A4K.A2K

A2I= 1,

A6K

A6 I.A1I

A1J.

BJ

BK= 1,

CK

CI.A5 I

A5 J.

A3 J

A3K= 1.

Nhn cc kt qu ny li v bin i theo phng tch

1 =AI

AJ.

A4J

A4K.A2K

A2 I.A6K

A6I.A1 I

A1 J.

BJ

BK.CK

CI.A5 I

A5 J.

A3 J

A3K

=AI

AJ.

BJ