Tạp chí Toán học và Tuổi trẻ: Số 232 (Tháng 10/1996)

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Tạp chí "Toán học và Tuổi trẻ - Số 232" tháng 10/1996 giới thiệu đến các bạn những bài viết chuyên ngành Toán học như: Khi đặc biệt hóa bài toán, hãy nhìn bằng con mắt lượng giác, điền số vào hình lục giác,...

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Nội dung Text: Tạp chí Toán học và Tuổi trẻ: Số 232 (Tháng 10/1996)

i-, lj 1' lL( L ee ctAo DUC vA DAo rAo *nor oK.hi dd. U.0t h6& lrAi tudn e nAr NHiN nAxc coN rrir LudNG ,' crAc O DIEN TICU A FIN TRONG MAT PHANG oTA ,t i.f Ail Uru fl; 01,/"rl;. 1d.12 ... o eoEern sO eoio za)otm pot@ qgdte i al .\^ l A oTHI HQC SINH GI0I IiHOI PTCT IRUONG DHSP IH NOI 1995 . l$96 EOi luVdn todn Hdi Phong. D(tng thl hoi lu ben phoi lo Ngo D1c Duy
roAN Hoc vA rubr rRE MATHEMATICS AND YOUTH MIJC LIJC Trang o Ddnh cho cdr ban Trung hoc co s0 For Lower Secondary School Leuel Friends Vu Httu Binh - Khi dac bi6t hria bdi torin 1 o Ddnh cho ctic ban chudn bi thi vdo ilai hoc Tdng biin tdp : For College and Uniuersity Entrance _ NGUYEN CANH ToAN Exam Prepares Phd tdng bidru fip : Pham Bdo - Hay nhin bing con m5t luong gi6c 2 NGO DAT TI.J o Gidi bdi ki trwdc HOANG CHUNG Solutions of Problems in Preuious issue Cric bii c:iu'a s6 228 3 o Db ra ki ndy r-r0r o6ruc arEru rde : Problems in This Issue Nguy6n CAnh Toin, Hodng TU232 ".. TLOILIZ, LUt32, L2fi32 9 Chring NgO Dat Trl, LO Kh6c o Nguydn Thilc Hdo - Di6n tich aphin trong BAo, Nguy6n Huy Doan, met phing i 1 Nguy6n Vi6t Hai, Dinh Quang o Tim hidu sdu th|m todn hoc phd thdng HAo, Nguy6n Xudn Huy, Phan To Help Young Friends Gain Better Huy Khii, Vri Thanh Khidt, Le Understanding iru School Maths. Hei Kh6i, Nguy6n Van Mau, Hodng L6 Minh, Nguy6n Kh6c Ngo Thd Phiet - Tit mdt bdi torin thi Olympic Minh, Trdn Vdn Nhung, Qu6c td ddn kh:ii ni6m d6i phrrong tich 14 Nguy6n DEng Phdt, Phan o Nguydn Huy Doan - Thi chon hoc sinh gi6i Thanh Quang, Ta Hdng kh6i PTCT. Bia 3 QuAng DangHtng Thing, Vr1 o Gidi tri todn hoc Duong Thuy, Trdn Thdnh Fun with Mathematics Trai, Ld 86 Kh6nh Trinh, NgO Vi6t Trung, D4ng Quan Vi6n. Binh Phuong - Gii.j d6p bni ?im crla vdo vd dudng di Bia 4 Tudn Dang - Di6n s6 vio hinh luc gi6c Try sd tba sog.n : 45B Hlrng Chudi, HA NOi DT: 8213786 Bian tQp ud. tri sU; VU KJM T$UY 231 Nguy6n Vtrn Cir, TP Hd Chi Minh DT: 8356111 Trinh bay; QUOC HoNG
I ba,,,/, il.,r" *c lc* 1u*A, k.*" r* aG $ffililil mffiG oo r BilGtr ,. !' ffi6m Bmt mCIfiN ,' . vo HOu siNFI Chring ta bit til bdi'to6n sau atdu Tt (1) : b2 + c2 = 82 = 64. Blri to6n t finn ' canh huybn cia m\t tam Tt (2) : bc = = 5.8 = 40. giac uu1ng bidt duirng cao ing udi canh huybn Suy ra : (b - c)2 - bz + c2 - 2bc :64 - 80 = bang h uit. bdn hinh duing trbn nOi tidp bd.ng r. -16 ( !) Giai. X6t Nhu vdy kh6ng t6n t4i tam gi6c vu6ng th6a LABC vu6ng m6n gi6 thidt ctra bii torin 2, t(rc li kh6ng t6n dA (h. 1). Gqi tai tam giric vu6ng nio,cri drrdng cao rlng v6i BC:x.Ta c4nh huy6n h': Scm vri b5n kinh drrdng trdn d5:t AC : b, nOitidpr=Zcm. AB = c. Cdn Bay gid chung ta di sAu th6m v6 quan hQ gitra tinh r theo h h vd r trong tam giSc vudng. Ta cci : vi r. Blri to6n 3. Ch,ilng minh rang ffong tam Ta cci h6 6 x. L, gid.c uuOng c6 dudrug cao tng udi cgnh huybn phrrong bd.ng h ud. bd.n kinh Htnh I trinh : duirng trbn nQi tidp b2 +c2 = x2 (1) bd,ng r, ta c6 bdt bc: hx (2) *c-N,= 2r (3) hi- 2
Ddnh cho cdc bqn chudn b! thi vdo dgi (l v HAYNITTNBANG PHAM BAO (Hd N0i) Ta d6 bi6t c6c him sd lugng g76c c6 nhtng tinh chdt ri6ng vi c6c ph6p bidn ddi ri6ng. " L-*2 Ching h4n hdm sd sin"tr, cosr cd t{p gi6 tr! 4 I sinrl < 1, I cosrl < 1 v6i mgi r ; hErm sd tgr - | -v2 crf tdp gi6 tr! ld tafj c5c sd th{c. Cric him sd d
B,diTllzz9 Tim rn, n. € tr/dd A = g3m2 + 6n - 61 + 4 k s6 nguy€n td. Ldi gi6i (Dua tr6n bdi gi6i cria bpn D6 Mai Lidh 8 Tt Li6m He N6i) ViA e N* n6n Bnr2 + 6n - 6l > O. . 3m2 *6n -61 chia B du 2. yey Bmz * 6n - 67:3k+2(keN) Ldp lu6n m6t e6ch hJdng tu ta cfing di d6n Tt d,6 A : g3k + 2 * 4 = ZZk . g + 4 '2 4 ylees (2) \h 27 = 1 (mod 78)---+2y'+ .9 : 9 (mod 18) *A ! 13. ft (1) ve (2) suy raly'ees = Ylees - I Vi A nguy6n td suy ra A = 18 +-> g3k+1 = 9 ---+ Sm2+6n-61 =2 + Nghia ld ta c
Nhdn x6t. C6c ban sau dAy cti ldi giAi tdt : KiAn, 8T, Nguydn Khd.nh An,'7"t Trdn Ddng Nguydn Than h Tud.n (9"t Chuy6n l-1ac Son Hda Ninh, Nam D!nh, Ld Thi Thu Huong, Van 7, Binh) ; Nguy'6n Thd Vinh (9 Chuy6n To6n, NK Hai Hdu, Thanh H6a : Miri Thi Thu Sanh, huy{n Chrrong My - HA Tay) ; Nguydn Dic 8A Nga HAi, Nga San, La.Hbng Minh, 8T Neng Uinn (8u Chuy6n C2 Tam DAo - finh Phn) ; khi6u Bim Son ; NghQ :An: La Thi Td.m,9A Trd.n Tud.n Anh (8Todnl,o Qui Ddn Nha Trang PTTH Hung Dnng, Vinh ; Hh Tinh : Vo Si - Kh6nh ltrda) ; Ldm Thd.nh Dqt (9A THCS Nam,9CT, NK Drlc Thq ; Quing Binh :.DQng Vo Thi Sdu, Bac Li6u - Minh Hei) ; Hod.ng Thi T6 Nhu, 9T NK HAi Dinh, Trd.n Duc Son, Thanh l,am (9T Thoai Ngoc H6n, Long Xuy6n 8T, Nguydn Hilu QuYbn 9T ChuY6n QuAng - An Giang\ ;Va Quang Md'n (9ATPTTH s6 2 TRach ; Quing Ng6i : Nguydn Tdn Tich,8"t Phf CAt - giol, Dinh) ; La Phuong Thd.o (8^r Chuy0n MQ Drlc, Luo.ng Htu ThuQn, 8T THCS H6ng Bnng - Hai Phdng) ; L€ Hbng Chuyen Nghia Hinh ; Binh Dinh : Trd.nMinh Linh (8CI Nang Khidu Tx Ninh Binh - Ninh Md.n, 9A, PTTH 2 Pht C6t, Lim Ddng : -Flb Binh) ; Nguydn Van Khiam (9B'THCS Pham viQt Dic) gAzLa Loi, Di Linh; Kh6!h Hda : Ven Hai, Binh ChSnh - fP HC-M). Dinh Thd.i Minh Td'm', 92THCS Cam L6c, Cam DANG VIEN ILanh, Vd Trung Hba, 8T Le QyV D6n, Nha Trang Vinh Long : Nguydn Ch; Thdruh 8Tt, .PidiT4l228 : Cho tam giac ABC (AB < AC) Nguydn Hoimg Qudn, 8T2, NguY6n Binh uit cd.c tam. gid,c cd.n BAD, CAE (BA = BD, CA Khi6m. = CE) sao cho D nd.nt khd.c phia udi p ddi udi 2. MOt ban d tnrdng Le QuY Ddn, TuY6n AB-,J ndmJthac phia uoi B ddi udi AC uit t\BD: ACE. Goi M lit. trung didm. crta'BC. Quangd6 pai khi cho ringtt DB < EC, DM < on sanh L-f;;tt so i/tD u6i rt,i; MP. Mn -O, n f DB + BM, ME < EC + MC, B'M = MC suY ra Hd.y ";-h MD DM
Khianr (8r2 Nguy6n Hoing QuAn, Vinh Long), Theo bd d6 tr6n R(r) le da thrlc cd h6 sd Trdru Tudn Anh (8 To6n,Le Quy D6n Nha nguy6n. Trang Kh6nh Hba), Hoitng Tharuh Lam Q"I : PTTH chuy6n Thoai Ngoc Hdu Long Xuy6n An Ta cci un = (p + l)n + R(n) Giang), L€ Hbng Linh (8CT NK thi xd Ninh n ttt-I m-l Binh), VO Thanh Viet QA Qudc Hoc Quy 2 q,p, -2 f,@)o' p"' + 2 f{n)b, Nhon), Luong ViAt Tuimg (8A, THCS H6ng i=0 i:l i:0 Bdng, Hii Phbng),VuVan Phong (9a Chuy6n @ m-l Van Toen Vinh Ttrdng Vinh thtr), phan = )1,1"1ei -Df,@lP' (modp-) Thanh Gidn (8^ Hda Th6ng Tuy Phridc Binh i=o i:l Dinh), Trdn Tdt Dqt (8Ar Chu Vnn An TAy Hd @ He NQi), LA Trung Ki€n (9^ Nguy6n Tri Phttong, Thila Thi6n - HuO, Nguydn ViAt i: trt. Cuitng (9T NangKhidu Nga Son Thanh Hria),' Dec bi6t u, : (p + 7) + R(1) : rytn Nguydn Ch; Thdnh (8T Chuyen Nguyi5n Binh Khi6nr Vinh Long), Hit. Ttti P.huong Thdo (8 Ta chrlng minh da thrlc Todn NK.thi xd Bim Son, Thanh H'6a), Mai Q(x) = R(x) pttt(l - e) ld da thtlc cdn tim + Hdn Giang (ST Lo Khidt Quing Ngsi), Bui QuA vQy a, = (p + 1)tt + Q@) = (p + l)tt + Manh Hilng (9H THCS .Trung Vriong, Hir NQi), Nguydn Ngqc Hbo (8 PTCS M! Hria Tx + R(.n) * p'"(1 - e) : u,, * pttt(l - e) t pnl Yn Bdn Tre, Tinh B6n Tre). mdo, = (p + 1) + QG) : p + 1 +R(1) + __ y _ . _- - _- 6 I)ANG VIEN 'B,ii P"(1 - e) = nP"' *p''(1 - e) -ptrt TGl228 Cho p ld. sd nguy€n. td. Ching minh rdng udi moi s6 m nguyAn hhOng d.m, bdt Do dcip"r Ia UCLN ci,a a, , a),... hy, tbn tai ntQt da thilc Q c6 hQ sd rtguyAn sao Nh{n x6t : Ddy ld m6t bdi to6n kh
i S, : rL * no, v6i no = const > 0 {? !)). Hdu hdt ex= t;{-2:z,t@ cdc ban giAi dfng bdi toSn ddu dtta ra nhfrng ldi ii) Giei :27(l - yz) + 6y - 4lg o gini qu6 cdng k6nh vi n{ng n6. Ngodi ban = Ir -V78) torin kh6ng nhi6u. Cd 3 ban cho ldi gi6i sai, vi w +++btb) oScban dd m6c ph6i nhr:ngsai ldm. (Chinghan lorl +larl+.,+loLl :coqpl *.o** cri ban dalflp luin nhrl sau : TrI Snz n suy ra 6
bb Ddu ding thrlc dlt drroc khi vd chi khi a = +... +-= cogPk < lz 1ur+bz+... +bft) = F=y - 60o, trlc Id khi chdp SABC li mdt ttl + diQn d6u. : i2 h,ll (a, * a, + ... + ap) Ldi giai 2 kiaDang Drtc Hgnh,l0T, Phan &"tt 1$aki hieu dO dai hinh chidu criaftro, z). B6i Chdu, NghQ An vd Tri)u Van Son, l2F, Do dd : k
Quing Binh : Trd.n Chi Hba, Triln Hitu Nansi-Amsterdam, Luu Hod.ng Minh l}CL, Lqc, Nguydn Hitu Th.n PTTH Ddo Duy Tt, D6ng Hdi, Quing Binh ; Quing Tri : Trdn Thd Anh Nguydn ThinhVuqng,9 Li; chuy6n Xudn Thriy, Thila Thi6n - Hu6 : Dinh Trung l.Iod,ng. Nam Hh. MAI ANTI (i0CT, DHTH Hu6l B,bj L2t228. I Quing NgSi : Dinh Hftu Khd.nh Cho mach diAn Phd YGn : Pham. Nggc Td.n (Luong Van nhu hinh uo. Chdnh), V6 Hodng Phi (Trrtitng Nguy6n Hu6) Rz = Rz;R, = 3R, Minh IJ.hi: LA Chi Nguydn (TH chuy6n Phan . U khong ddi. Ngoc Hidn) Bd qua gid, tr[ I 3o) DAc bi6t, ban Bili Manh Hirng,l6p 9H, diQn trd c&a ddry PTCS TrungVudng, Hi NOi dd d6 xudt vd giAi ndi, hh6a K ud dring bni to6n tdng qurit hon sau ddy : Iamite hd. I Hinh chcip SABC c
Bidi.Tglz}z: ChobdndidrnA, (i ='f7) cing nim tr6n m6t drrdng trdn Goi ?, ld tam gi6c xtic dinh bdi ba trong Mn didm tr€n trd didm A,. Chrlng rninh b5n drrbng trbn Euler cria ?,, btin drrdng thing Simson cria didmA, ddi v6i 7, cnng oua m6t didm. ' (Dtrang thing Simson cria didm A; d6i vdi cAc t 6p rHCS. tam giric Q ld drrdng th6ng di qua chdn dtrdng vu6ng gdc k6 trlA, ddn B,niTll2S2: mdn Cho a, 6 ln 2 sd tU nhi6n th6a "1,:t:fii13.. aa=199668*1 (Sdc'l-rdng). Chtlng minh ring tich ob li bQi cria 5. Bni T10/232 : Trong kh6ng gian cho ba tia lr.Ar'r eA sY ox, oy, oz vd A, B, C l?,' ba didm c6 dinh ldn luot (Thurth H6a.) nAm trOn ba tia dd B,di T21232 : Tim nghiQm nguY6n cria GiA stl + a,rlb' rnQt cdp sei cOng co a, > 0, phtrong trinh sau : c6ns sai d > O. Vdi m6i sd nguv6n drlong re, tr6n Y2 = -2('u -*3Y -32) ;;;Zi:;; ldn hrst tdy cdc didm Ar,, B,l. C,, sao NcuvEN o0c rAN cho --oA=arroArr, : 1'lP Hb ('ht Minh). oB=a,,, * roBrrrOC=an +2oCn BAi T3l?32 : Cho r, y le hai sd thtrc th6a Chrlng rninh rang : m6n: xz +yz = rr[TZf +y {-[=?z U C{e drrdng tli&ng A,,BnBnCu,C,y',, ldn Iudt di qua c6c didm /, J, K cd dinh. Chrlng minh ring, 3r_1I :=,.q.^ 2l I, J, K thing hdng. ^. - ]'I{INH BANG GIANG llb t"hi Minh). (TP B,di T41232: Cho tam st{'c ABC n6i tidp drrdns trdn (O). soi M Ii trirns didm ctia BC. Tr6n'drrdns thiln"E'BC cci hai didm 1. J di d6ns cAc ob vAr li lu6n lu6n d"6i xtnE vai nhau o:ua M . Goi E, F ldi B,diLllL3,z: Tr6n m6t rn{t brlp cdu, b:in Iudt Ie giao didm"thrl hai cuaAI, AJ i,6i.iludng kinh .R : i m6t, cd dat bi'nh6 B cd khdi ludng tr6n (O) vi F/ ld trung didm cta EF. Tint quf mB = 2 kg. MOt con l5c tich didh H. vr errfic n[lNc don cci chi6u dii I = o(l (Bdc Thdi). 1m. khdi luons quA cdu A lit mo'=-l kg. I BdiT5l232: Cho hinh vuOnsABCD n6i tiSp K6o A dd day treo hop A I dtrdns trbn t6m O brin kinh .8. Chrlne minh v6i ohuon"E thins I 6 rangl Vdi moi didm M e (o, l?) ta cdtdng : drlns ilr6t sdc a : 60h MA4 + MB4 +MC4 +L/ID4li m6t.hhng s6. (hintr v'6)"rdi- bu6ng v[r er.,fx'r.trdN(i kh6ns vAn t6c diu. $1d N6i) Sau v"a cham. B trudt ddn vi tri IttI $ = 30(') cAc rdp rncs thi rdi kh6i ibm cdu. Tim lrrc c6ns div treo L- khi vAtA ddfi vitri cao nhdt sau va cham. Bei T6/232 -: I^ Tim tdt chgrri tr! nz d6' hQ .- Ldy g..= ,lombz vir b6 qua luc c6n kh6ng khi va nla sat. /. )x5-nw-!:O NGUYEN otlc Prtt. lv3 - mY -lx: o (Qudng Ngdi). Cd 5 nghiOni (nt, x, y ld cric s6 thgc). Bldi L21232 : Cho mach di6n 1 chi6u nhrt hinh t-E vAN QUANG v6. Tlong d
PROBLEMS IN THIS ISSUE has five solutions (rn, x, y e R). 'N7 1232: Prove that for everyr € R, For lower secondary schools ,5 Tll232: Let a, b be two natural numbers lsinlsssr +cosri < f,. satisfying TSl232: Let be given a bijection / : N ---* N, ' a4 : 1996 b8,- 1. Prove that there exists an infinite number of Prove that the product ab is amultiple of 5. triples (a, b, c) with o, b, c e N satisfying : T21232: Find integer-solutions of the a < b .< c, 2f(b) : f(a) + fk). equation : Tgl2S2: Let be given four points A, ( i = gl y2 - -21xt, -x3y -82). on a circle and let 7, be the triangle with these TSl232 z Let x, y be two real numbers points, except Ai, ffi vertices. Prove that the satisfying - four Euler circles of !, the four Simson lines of *l *yr: *r{r -j +y,[T=? . A, with respect to { pass through a common Prove that Ax +4y < 5. 'T41232 point. z Let ABC be a triangle incribed in (The Simson line of Ai with respect to a circle (O) arrd M be the midpoint of BC. Two points d J, s;rmmetric through M, move on the triangle T, the line passing through the line BC. Let E and ,F' be respectively the second orthogonal projections of A, on the sides of T). points of intersection of /1 and AJ with the circle (O). Find the locus of the midpoint If of T1Ol232: Let be given three serni-lines Or, EF. Oy, Oz in space and theree points A, B, C respctively on these semi-lines. T51232: Let be given a square ABCD inscribed in the circle with center O and Let + an'be an arithmetic progression with radius R. Prove that the quantity or > 0 and common difference d > 0. For every MA1 + MB4 + MC4 does not depend on M e positive integer n,let Arr, Bn, Cn be the points (o, R). respectively on Ox, Oy, Oz such that For upper secondary sch.ools OA=anOAn, OB : an+tOBn, OC = an*rOCn. TGl232: Find all values of rz such that the Prove that : system of equations (^ I) the lines AnBn, Brrcn, CAn pass lx5 -mx -J = 0 respectively through fixed points I, J, K; )y'-*y *r:0 2) I, J,1( are collinear SAI SOT NH6 a.NHUNG HAU Q UA... NGTJYEN THI PHT,ONG THAO . (PTCS L€ Lqi Hh DOng - Hd TAy) MOt ldn xem vd ghi cira hoc sinh chuy6n ban nhrrngkh6ngchrlngminh, trongdci crim6t tfnh l6p 10 tdi thiiy cd mOt vdi ch6 sai, C6c ch6u bAo chdt: 'c6 gi6o ch6p tr) trong s6ch gi5.o khoa ra. T6i xem a -b > l"l - lbl (sai,thidub = 0; a : -3). trong cu6n srich D{.I SO 10 in ldn thrl 3 nnm Hgc sinh d5 srl dung tinh chdt niy dd giAi 1995 dring ld cri m6t sd sai scit nh6. Sai ndy cri to5n. thd do khdu in dn nhrrng tai hai lA mdy ndm nay c6 girio vA. hoc sinh d ddy d6u kh6ng bidt. Thim Trang 74 c6 mdtbdt ding thfc ttrdng trr : chi cdn 6p dung dd giei cric bdi to6n khdc. Sau la -bl < lo +61 (sai v6ia = L,b : -l). day la m6t vai thi du : Cirng trang niy cring ed m6t bdt d&ng thfc sai : Trong phdn cdc tinh chdt cia hdm azl(a + 7) < 112 (sai, thi du o = 2). y = lxl (trang 39) da dua ra mQt sd tinh chdt 10
I l^ NGI-'YEN-TIJUC HAO Trong hinh hgc phd thdng; kh6i niQm khod.ng Ph6pbidnf(x) goi ld. tuydn tinh khi th6a mdn cd,ch gqita hai didm (tray dQ diti c.ioa m6t do4n hai di6u kien. thing), crlng nhrr kh6i ni6m s d d.o g6c, cho ph6p f(ox) : af(x) Va' x ta nrii ddn hinh bd.ng nhau vd ph1p diti hinh. Ph6p ddi hinh (cbn goildph€p dd.ng cu) la ph6p f(x + y) = f(x) + fQ),Yx, y biSn hinh bAo todn khoing oich gita hai didm Quy v6tqadO theo m6t cd sd I e1, ez], tav46t (cring nhrr sd do m6t B
Cho rn6t cap vecto bdt ki (kh6ng ddng GSi ct, b,+ c, d ldn ltrot le cdc vectd phrrong) u, u. oA, oB,oc, oD u = )rre, * )re2, u : pre, * p.e, ldn luot la cdc vectd "Dat -nl,-9t, -g' :', oA" OB" OC"OD ta cd Ta cci a' :f(a)'+ u, b' :f(b) * u, c' =f(c) * u, fu,uf = Vre, +Lre.,pre, *prer)+ d':f(d)*U=+ fu, ul = (Arp, - irlt )[e 1, e27 (6) va I^i' = f(b1 -f@) = f(b - a) = rt&l : p ,f(e +tt + fC'D' = ftd) - f(c) = f(d - c) : f(CD) [f(u 1 ,f1u11 11 ,f(e ,) +1 2f@ 2) ,) 2f(e ,)7 ti h', ci't V@h), ft1bfi lf(u), f(u)) : (4,!2 - ),y )V@ ), 1@)1 (7) vAy-frfr1 =ffi:detf So s6nh (6) vd (7) ta duoc Thd la ffong phdp bidn afin, tich ngoiti cfta nr.6i cQp uecto dbu duoc nhdn bdi mOt h€ s6 tfqu 7, flu)1 l1"tc rl, fte,)J . . - '-;;i-: 44 =detf (8) chung ld det f. Det f cung drioc goi ld dinh thrlc cita phdp bidn afin. Ta giA thidt det / * 0. Khi Thd Ie vdi moi cip vecto z, r., thi ta cci det f :-1, tich ngoii li nfit bd.t bidn afin. N6i chung, nd ehi li mQt bd,t bidn tuong ddi. Vg.l!) . ftt,u) - cons!: detf Phttong trinh cira ph6p biSn afin (10) n6u viSt theo tga dQ sG cho ta h6 phrrong trinh bdc Dd chinh ln y nghia hinh hoc (bdt bi6n) ctra nhdt det f : qua phdp bidn tuydn tinh f, tich ngoii cta moi cap vecto d6u bi nhAn vdi m6t h€ s6 nhu {' = o16 + Frtl + yr (11) nhau vd h6 sd dci lit det /. Nhtr vAy, tich ngodi +yz 1' = qz€ + Fzrl cria hai vecto lA ]fndt bdt biAn tuorlg d.6i cua ph6p bi6n tuydn tinh. trong dd €, qld toa d0 cta r t' ; ,r1' ldtoa d0 cria r' ; cbn 7 ,, T 2ld toa d6 cria u . Ph6p bidn afin Il. Ph6p tri6n afin trong m5t phing cd cic tinh chdt sau ddy : Cho didrn cd dinh O, m6r didm bdt ki M drioc 1) Bidn dudng thing thdnh drrdng thing bidu di6n bang vecto - didm Oiut ,: 2) Bho toin tinh song song cta drrdng thing, Ndu m6t ph6p bi6n ndo dci bidn didur M 3) BAo toAn ti s6 don cria 3 didm thing hing. thinh didm M'bidu di6n bdi r' : O?, sao cho 4) NhAn vdi m6t hO so chung (det fl tich r' : f(x) (9) ngoii ctia mgi c4p vecto'. nri / ld mOt ph6p bidn tutdn tinh, thi ta ndi Ban doc cd thd d6 ding chrlng minh cric tinh rang ph6p bidn dy \dndt ptrcp bi€n o,fitfu,m,.R6 chat tr6n. rAng ld tiong ph6p nay, didm O khdng ddi vd ddoc goi la tdnt cua ph6p afin tAm. III. Di€n tich afin Cbn ndu ph6p bid.n afin tAm dtroc kdru theo Theo trtlc gi5c, ta hidu di6n tich lh sd do mOt rn6t ph6p tinh tidn u, tttc ln ta cci ph6p phdn ndo dd cria m[t phing gi6i han trong m6t (10) chu vi kh6p kin, sd do dci ndi l6n str rOng l6n mrlc dO ndo ctra phdn m5t ph6'ng dy. Nd c
D6 dnng thdy rang vi6c chon hu6ng le thf tU cta c6c vectd canh ld tty y, bdi vi vi6c dy chi 6nh hydnr ddn ddu cria.tich .Fgoai ma th0i. l\ He qui. DiQn tich cfiam6t tam gi6c bdngLlZ tich ngoai cria hai vecto canh bdt ki. ,' QuA vdy, tam giScABC vA tam g;d,c CDA c6 cirng di6n tich vi ddi xrlng tim v6i nhau. V6y di6n tich cria m6i tam gieic dy bing 1/2 diQn tich cria hinh binh hinh' trlc -t Ta dirng gqch ngang d tiin'dd chi dien 'l\ tich. ABC: CDA = |llu,ull r Trong mp! ph&rrg afin, ta,hey cOng nhdn hai tinh chdt sau dAy. ' t ' Ta hey d* A : w. Tathdy . * ( 1) Di6n tich khOng ddi qua m6t ph6p tinh u*u*w:O ti5n. : ' +++ Ldy tich pgoiri ldn ludt vdi u, u thi Ching ha!, ndu L4' - gEl = Ce', thi * fw, y7 = Q, fu, uf * fw, uf = d.+ , fu, uf : .------_ . fu, uf - fu, wf = l1t,u7 ': 2) Dien tich kh6ng ddi qua phdp hdi xrlng tAm Th6 le ---+. ,--) ---t - = Chins han. n6u OA' - OA, OB' = 'OB, + OC':-OC,thi ,Gc = IZ2Itu,rtl : *l tu, *11= *l tr,'rll't-r;r) ---t .+ ABC = A,B,C, Chrty. Mudn tinh diQn tich;td-;di,lrirn au giric, ta chia nd'thinh tam giric vi-binh hinh r6i Bay gid ta dlnh nghia. tinh ttng phdn md cQng lai. Didn tich cila hinh binh hdnh tri tuy6t liL giri ctalisLassai.-hai-yeeis-canh-cta-nd Ndu mudn ti;rr di6n tich cria mOt hinh cc d-6-i chuvil_1_drrdngggle,_ta j_U1r.e_p!,ep_tjrrhj jsL. 1 phAn :'- I Dd kdt ludn bii ndy, ta c
:: r 5V*A;l*o^a*r["ar*t-;r"r4fi8%V_ ^, _ r^. r.r ltgr Bil rofN t,1:olYM-fi!,og9! rE oEn KHfi NtE],t'afi PHtrona rict{" - NcO rn6 PntEr ' ( Qudng, Nam - Dd Ndnl bidt khSi ni6m phrrong tich cria Cdc ban de + D'6i phuong tieh ai,e- P d6i v6i (S,) lu6n ' m6t'didm P ddi v6i m6t dudng trbn C(O ; R) lir lu6n li mQt sd drrong. sd thrrc : \ +Dudng tibn tam P brin t
NA AB2 +b2 - a2 Dinh li 3 : Cho 57, Sp, S, c6 3 tdm th&ng _ -NB AB2 + a2 -b2 hd.ng. Ndu tbn tai thi tbn tai nhibu hon mQt didm P c6 cilng ddi phuong tich ddi udi Dinh ngh[a: Dttdng thd.ng (L,) uuilng g6c S,4, Sa, Sc' uoi AB tai N theo ti sd : Chting minh : Vi A1, Lz, L3vu6ng gric v6i NA ABz +b2 - a2 AB, BC, CAvQy L1, L2, A, hoac song song hodc NB trtng nhau. Ap2 +a2 -b2 + Npu \ ll L2// A3 thi khdng tdn tai didm d.uoc gsi td'TRUC DdI PHUANG TICH. nio cri ctng ddi phuong tfch ddi vdi 3 dudng cia hai dudng trdn td.m Abd.n hinh a uit dtdng trbn. trbn tdm B bd.n kinh b. * Ndu A, = A, : A3 , cri v6 s6 didm c
HAY NHiN BANG .... dd chrlng minh (6) dQt a : tga, b = tgar, c , (tidp theo trdng 2) = tsa3v6i a, e ( - ;, ;) *rd
++ Ggf SJa gdc ldm bdi zz vd u ; P : (u, w) oji rur cHgN EAC srNH GIor y : (u , ut) ta phai chrlng minh cos a * I ) > Z_Sq{cgy . Tt m6t didm O bdt ki k6 cdcv6c xu6l PTCT- TRUONG DHSP IIA NOI td u , u, ro ldm thdnh mQt g- (D6 sd tr 11 cdu IIIz. BO d6 thi tuydn sinh vdo b*c*d o*c*d a*b+d a*b*c' 3 dai hoc) Cdu 5. Gqi II, I vd O theo thf tu la truc tam, tAm drrdng trbn n6i tidp vd tAm dudng trdn ngoai 3. Tim giri tri I6n nhdt vd nh6 nhdt cira tiSp mOt tam giSc nio dci. Chfng minh rang : (r +y111 -ry) OH > IH 1[r. ,_':- ^ (1 +r2)(1 +y2) Cnu 6. Hai ngudiA vdB ldn luot di6n sd vio 4. Giai bdt phrrong trinh cic d trong mOt b6ng k6 d vu6ng 5 x 5 theo crich sau : m6i ngudi khi ddn hiot chi di6n 1 sd vio 1 6 1135 nio dci ndub dc; cdn trdng j A di truoc vd chi di6n x ,[1-1r -.12 -+->- s6 1 ; B chi di6n sd 0. Sau khi hdt 6 trdng, ngudi ta tinh tdng ctia c6c sd trong m6i hinh vuOng 3 x [r) 3 vi goi S ld sd ldn nhdt trong e5c tdng dy. l*2+Yz:4 1) Hey n6u mOt chidn thudt ctaB dd v6i moi 5.ChohQ112*u2:9 t" c5ch di6n ctraA d6u cd S < 6. lxu+yz>6 2) IJey n6u m6t chidn thudt c:iuaA dd vdi moi Tim.r, J, z, u dd x z c5 gii, tri l6n nhdt. c6ch di6n cria B d6u cd S > 6. - NGl]YE,N HUY DOAN
DIEN SO VAO HINI-I LUC GIAC Girii dzip bai riU CUA VAo VA DUONG DI Ta ddruh s6 cdc phdng ctta khu tridn ldm nhu sau: Ta thd.y khu tridn lam c6 tdi da 10 cua udo, d cdc Hd.y dibn cd.c sd tit 1 ddn 19 uito cd.c duitng phbng duqc tri;n cila hinh luc gidctr€n sao cho tdng cd,c s6 dAnh s6 : 1,3, cila cd.c duitng trdn ndm tr€n m.Qt canh hoQc 4, 5, 6, 7, B, g, nd.m, bAn m|t dudng thang sotlg sotlg udi cd.c 10, 14. canh dbu bang nhau. ThQt uQy ta c6 thd dua ra duitn g di th6a indn TUAN DANG ydu cd.u lit. qua td,t cd cdc phbng m6i phdng mQt ldru qua cd.c a2a ud.o nAu bAn nhu sau : 1) + / +g +B + Cfrng ben dgc - 1 -212 -373-4 + -5+g + 10 - 11 - - ll lg ---> . Ban dqc m6i gili bei ldn ddu cho fC 2) *3 -4 -5 -6 - 12 -7 - 11 -B -9 +10+14-13-2---+f+fg+ Todn hoc ve tu6itrd xin luu y : - Bdichigli 1 lhn 3) -4 -5 -6 - 72 -7 - 11 +8 +9 + 70 - Ghi ddy d0 he tdn, dla cni @o quan, nhit -14-13-3-2+f+fg+ ri6ng) sd di6n thoai (ndu c6) vdo cudi bei. Bei 4) -5 -4 -G + 12 + 17 -7 -B -9 - 10 vi6t trdnh d6p xoa. -14-73-3-2+f+fg+ r Ndu ld bAi giAi : 5) -G -5 *4-72 -7 -11+8 +9 +70 - Ghi he t6n, dia chi theo l6p, trudng, -74-13-3-2+f+fg+ huyQn, tinh vea goc ffAn b6n phai. 56 cua bdi ghi bdn trdi. 6) -7 -6 -5 -4 - 12 - 77 --->8 -9 - 70 -14-13-3-2+f+fg+ - Gdp bitidon gidn. Khong dan nhi6u hd. 7) -8 -9 - 10 - 14 - 71 -/ --->g +g + 4-12-3-13+2-7-75- - bdo ndo. Ngodi phong bi ghi 16 : Ldi giiii cua sd 8) -9 +S +*2 7 ---+6 -5 --->4 - 72 --->77 - 10 ---> f + lg ---> o C6c ban hgc sinh c6 thd tham gia cdc muc: 14 73 - - -3 9) - 10 -9 --->B +/ + 11 - 14 - 13 - 12 - Hqc sinh tim tdi -6-5n4-3+2--->7+75+ - Giei bdi ki tru1c. 10) + i4 n 10 +9 +8 +7 +g +g +l + - Gieifi bdn hoc. 12-11+13+3+2-7+75+ - Nu cubi todn hgc. (Theo LA Thi Bich Hanh, SCT, chuy€n Tit Tda so4n chua nhdn ddng d6 ra cia cdc LiAnr, Hit NQi). Rd.t nhibu ban c6 gid.i dap ding. b4n hgc sinh. Xin cem on ciic ben. BINH PHT,ONG THVTI ISSN: 0866 - 8035 Sip chrl tai 'ITVT Nhd xudt bin gi6o duc Chi sd :12884 In tai nhi m5y in Di6n Hiing. 57 Giing V6 Gi6 2.000d M5 sd: 8BT34MG In xong vA ndp lrlu chi6u th6ng lO/1996 Hai nghin ddng