Bồi dưỡng HSG Phương trình hàm P2

So s a n h h§ so c i i a l i i y t h f r a cao n h a t d h a i ve c u a ( ) , t a dxxac: " V a y p{x) ' ' a „ " = 8a„ " = 2^ o dUdc: 14p(8) = O.p(lO) = NhiT vay: p{x) Do n = l a d a thiltc bac b a T\t (1) l a y x = - , t a dUdc: p ( - ) lay X = - , t a diWc: - p ( ) = p ( - ) = G i a i G i a suf d e g ( P ) = n So sanh bac c i i a h a i ve c i i a (1) t a t h u d u ^ c ''> = T i t (1) p ( ) = T t r (1) l a y x = 4, t a , K h i n = 0, t a dUdc d a t h i i c h t o g P ( x ) = a{x - 4)(a: + 4)(a; - ) , V x e R p(x) = ( x - ) ( T + ) ( a ; - ) , V x € R (2) T h u : l a i : v d i p ( x ) l a d a t h i i c xac d i n h b d i (2), t a c6: Vay P{x) P(x) = Ova = l a cac d a t h i i c h a n g t h o a m a n bai r a , X e t n = G i a s i i P ( x ) '''' c = c = c = = 210 n e n : 105a = 210 Tii (3) t a thay r i n g h ={x - y)'+ {y - z)'+ {z - x)' phai l a so c h i n : h = 2k, vdi k > K h i (3) t r Tom = ( x ' ' + 2/" + z") - 4x^2/+ x V - 4xy3 fc = l k = 2.3*= + 12* = 2.9*^ k h i P ( x ) Tiep theo t a chiing minh dftng thi'lc •^Jiong dong nhat 0, nghia la: 3xo e R : P(xo) 9[(a-6)" + (6-c)" + (c-a)"] = (2a - - c)" + (26 - c - a)" + (2c - a - 6)" " X -I- z^ = - ( x v + yz + zx) Suy =4[.T2,y2 ^ , y ^ ^ ^2^2 ^ 2.x,/x(x + 7/ + z)] = 4(.TV + 2/'^' + z'x^) 3m [(a - bf + (6 - cf + {c- Dat (8) x'' + y" + 2'' + ( x V + y'^' + V ) Vx e R T h a y vao phiiong t r i n h ham da cho d dau bai t a diXdc =m[{2a-b-cf+ -V^ = a - c, 2/ = - a, z = c - K h i x + i/ + z = Ta c6 [(a - 6)" + (6 - c)" + (c - a)*] = ( x ' ' + 2/" + z") 402 ^ (G) *a dudc P(xo).P(0) = 0 Trong (1) cho x = y = y P(0) = Trong (1) lay y = 2x t a dudc • P(3x).P(-x) = p ( x ) - p ( x ) , V x € R (7) Gia su p(^.) ^ „^.,.n + „„_i:;:»-i + ^ (2) „^ ^ (do P(0) = 0) Thay (2) t a dUdc: «n(3x)" + a „ _ i ( x ) " - i - I - • • • + ai(3x)] [ a „ ( - x ) " + • • • + a i ( - x ) ] = ( a „ x " + a „ - i x " - i + • • • + a i x ) ' - [a„(2x)" + a „ - i ( x ) " - i + • • • + ai(2x) ^ C, vdi C la hang so t h i t i t (1) c6 C = 2C2 ^ (3) a " ( - l ) " = (1 - 4") * = ( - ) " + 4">-,)£,^ ^"^"^^ (do ham so m u y = a„(2x2)" + a„_i(2x2)""^ + • • • + ai(2a;^) + ao] " = + 3" = ( - = ( a „ x " + a „ _ i x " - ^ + • • • + a j x + ao)^ , Vx € R nghich bien k h i < a < 1) Vay bac cua da thiJc P{x) man Vay t a t ca cac da thufc c l n t i m la: So sanh he so ciia x^" d hai ve ta dUdc • >iiii>» jt;; a „ " = 2(a„)2 n ) ^ ^ ^^^j ^ ^ ' , Vay deg(Q) < 1, suy P ( x ) = ax + b, Vx e R Thay vao (1) t a dudc (x - l ) x T2 _ bi^zili H ri%.ii'V "v?^t); + 6x ^{^+b)+y(j + b^=x + y,\fxeR\{0} •^ay + bx + ax + by = X + y, Vx € K\{0} +oo ''^eo dinh l i Lagrange, vdi moi x e (m; + 0 ) , t6n t g i G ( P ( x ) ; P ( x ) + x) '^ao cho _ P(x + P(x))-P(P(x)) _ P(x) P(x)+x-P(x) ^ ^ ^ ^ - - l • Ui t , Vfc i« = -Q(^)>vMo L P(x) = P ( x + _ P ( x ) ) - P ( P ( x ) ) , (2) D a t P ( i t ) = C?(fc) + K h i do (2) nen da thiic Q thoa man = 1+ [P'(x) - x l = + 0 (vi n > 3, na„ > O) P"(x) = xP(fc) + A:xP(^) = X + fcx, Vx ^ 0, fc ^ + l + k " n§n ton t a i n i cho vdi moi x > n i t h i P " ( x ) > 0, suy r a P'(x) dong bien •^ren (74,; + 0 ) Chon m = max {0, no, n j va xet x e (m; + 0 ) T i f G i a i Trong (1) lay y = A:x, vdi fc ^ va x 7^ t a dUdc Q{k) Do nen ton t a i no cho vdi moi x > no t h i P'(x) > x Lgi c6 X—»+oo =>P(fc) + ifcP(^) = + fc, VA: 5^ (1) P ( x ) = a „ x " + a „ _ i x " - i + • • • + a i x + ao, an^ Do Q{x) = T2 + hx va P ( T ) = r^ + hx thoa (1) Ket hian: Co hai cap da thiic thoa man yen can de bai la P(x) = P ( x + P(x)) = P(x) + P ( P ( x ) ) , V x e R G i a i Gia sir o„ > Xet trudng hdp deg(P) = n > Gia siif = Q ( l + +••• + n), V n = , , (•, ± 0 , k h i (do da thilc Q la ham lien tuc), den day t a gap mau thuan -Q{0) ,^, Q{k) CO P(o) = nen ao = Do lim [P(x) - x] = +00 X — • + 0 (3) lim X - t + O O P'(X) - P(x) = lim x—*+oo vS, * J2 4- Jj.'^oo P{x) ^ P{x) dieu mau thuan vdi P'(cx) = PiQix)) W^fiQJx)) X P{x) V i > max {mo, m } N h u ^^"g - anQ^'ix) - a „ P " ( x ) = c - a „ ( g " ( x ) + P"(x)) + P{R{x)) - anQ"{x) P{R{x)) - a„R-{x) P"-i(x) c ^ ^ - a „ ( Q ( x ) + P(x))r khong R-~Ux) Q-Hx) ,auu: • =^g(x) + P ( x ) the xay r a trirdng h(?p deg(P) = n > X e t deg(P) = G i a siit _ P(g(x)) - anQ"(x) P{x) ('^ "^^y a„ la he s6 bac cao nhdt ^jia da thrrc P ( x ) ) T i r gia thiet t a c6: Vay, v6i niQi x > max {mo, m } , t a c6 P'{c,)>P'{P{x))>P'{x)> — a„x" X"-'" = ax^ + bx, Vx R (do P(0) = 0) a„TQ"-i(x) P(P(x)) - a„P"(x) P"-nx) a„TP"-'(x) •g"-i(x) ^ a„TQ"-Hx)- Thay van (1) t a durtc 26 a[ax2 + ?>(x + 1)1 V l l v gidi han hai ve k h i cho x -* +oo, t a dUdc l i m ( Q ( x ) + P ( x ) ) = Hav (?(•'•) + -^(3^) l a da tluic hang Vay t a c6 dieu phai chiing m i n h [ax^ + 6(x + 1)' =ax^ + bx + a(ax^ + 6x) + (ax^ + 6x), Vx e R (2) TCr (2) lay X = dUdc a/;^ + fr^ = Q DO dang xet a > nen suy r a = Thay vao (2) dudc: 3.3 Phu-dng t r i n h d a n g P{f)P{g) = P{h) Bai t o a n t n g q u a t Gid sic / ( x ) , ^(x) va h{x) la cac da thiic he so thiic cho frudc thoa m.dn dieu kie.n: deg(/) + deg(g) = deg(/i) Tim tat c.a cac da thiic he so thiic P{x) cho a^x" + ahx^ = ax^ + a^x"* + abx^, Vx e R (v6 h') Tudng tit, vdi a„ < t h i cung suy r a vo h' neu deg(P) > Vay xet deg(F) = Gia sU P{x) = ax, Vx g R, a la hang so, thay vao (1) thay thoa man Neu P ( x ) la da thiic hang t h i thay vao (1) diTdc P ( x ) = Tom lai cac da thiic thoa man yeu cau bai l a P{x) = ax, Vx € R, a la hang so t i i y y B a i t o a n 3.36 (International Zhautykov Olympiad 2012) Cho P, Q, R la P(/(x)).P(3(x)) = P(/i(x)),Vx€R (1) I^ghiem r u a (1) c6 nhieu t i n h chat dftc biet giup chung t a c6 the xay di^ng (luoc t a l c a cac nghiem ciia no t ^ cac nghiem bac nho D j u h l y Neu P,Q la, nghiem cua phiiOng trinh ham (1) thi P.Q cung la nghiem ciia vhuanq trinh ham (1) ba da thiic vdi he so thuc cho P(Q(x)) + P ( P ( x ) ) = c , V x R Chx'cng minh rhng ho&c P ( x ) la da thiic hhng hoac R{x) + Q{x) la da thiic C h i h i g m i n h Ta c6 {P.Q) (/i(x)) = P {h{x)).Q {h{x)) = P {f{x)).P hhng G i a i Neu deg P = t h i P ( x ) la da thiic hSLng, t a c6 dieu phai chiJng minhGia si'r d e g P > Ta c6 cac nhan xet sau: = {9ix)).Q {f{x)).Q {g{x)) {PQ)U{x)).{PQ){9[x)) q u a Neu P ( x ) la nghiem cua (1) thi [ P ( x ) ] " cUng la nghi$m cua (1) Neu deg Q = t h i d e g P = va ngU^c l ^ i , nen khong giam t i n h t6ng qu^t ta gia sir deg Q > va deg P > Suy l i m Q{x) = oo va l i m P ( x ) = Q{x) = -1 deg Q = deg P > va deg P = n, la so lo, d6ng thrri ^ hrn^ j , a t T - l + ^ + ^ + - + ^ ^ t h l ^°ng kha nhieu trudng hdp h§ qua tren cho phep t a mo t a het cac nghiem ^'^^ (1) Dg l a m dieu t a CO dinh If quan sau day i t '[^inh l y Neu f, g, h la cac da thiCc he so thiCc thoa man dieu kien deg(/) + ^{9) - deg(/i) va thoa man mot hai dieu kien sau day: l i m T = l 411 410 (1) d e g ( / ) deg(9) (2) d e g ( / ) = deg(p) va f 0, r,g* + g* Id he so cua luy thica cao nhat cua cdc da thiCc f va g tUdng ling c u a x"''^«(^)+'"''*=8('^ d a thiic t h i i nhat v a d a thiic th,'r hai l^n lUdt l a i?*.(/*)'•.P*.(5*)"- N h u the bac ciia x"^^^f)+rdee(9) ^Bgtig hai d a thvic bang Khi vdi moi so nguyen duong n ton tax nhiiu nhat mot da thiic he so thycc K P{x) CO bdc n vd thoa man (1) " ChuTng m i n h Gia sur P l a da t h i i c bac n thoa man (1) Goi P*,f*,g*,h* Ian lirot l a he so cua luy thCra cao nhat cua P,f,g,h So sanh h§ so cua luy thfta cao nhat hai ve ciia cac da thi'rc'trong (1) t a CO • p^irr-R^xgr + =p*R'{rn9T{{fT-' n'-ifr-p^-iaT + {9T-1 (do r + V o) "' ^ H h U v^ly b a c cua ve trai cua (2) van l a n d e g ( / ) + rdeg((/), k h i bac ^ M a ve phai l a 7-deg(/t) = r ( d e g ( / ) + deg(5)) < n d e g ( / ) + rdeg(j?) ( m a u t h u a n ) _ ' •> P*.(/*)".P*.(.7*)" - P* = ^ • | n h l i dUdc chiing m i n h hoaii toan N h u vay neu gia sii ngUdc l a i , ton t a i mot da thiic Q he so thuc bac n, khac P, thoa man (1) t h i Q* = P* va t a c6 Q{x) = P{x) + R{x), vdi < r = deg(i?) < n ' Hbhu y Sii dung dinh li (2) vd h$ qua (1), ta thdy rdng neu Pi{x) la mot ^ma thAc bac nhat thoa man (1) vdi f,g, h la cdc da thiic thoa man dieu kien ^ • t t a dinh li (2) thl tat cd cdc nghiem cua (1) la P{x)=0,P{x) (ta quy Udc bac ciia da thiic dong nhat khong bang - o o , do r > d5ng nghia R khong dong nhat khong) Thay vao (1) t a dUdc [P{f) + R{f)].{Pig) " + R{g)]=P{h) + R{h) ^P{f)P{g) + P{f)R{g) + Rif)P{g) + R{f)R{g) ^P{f)R{g) + R{f)Pig) + R{f)Ri!j) = ^(M- (2) n d e g ( / ) + r d e g ( ) , r d e g ( / ) + ndeg(5), r d e g ( / ) + r d e g ( ) , v, ( n - r ) d e g ( / ) > (n - r ) deg(p) Vay ve t r a i cua (2) c6 h^c la n d e g ( / ) + rdeg(5) Trong k h i ve phai c6 bac • ' J rdeg{h) = r ( d e g ( / ) + deg{g)) < n d e g ( / ) + r deg(5) (mau thuan) TrvCdng hdp d e g { / ) = deg(5) K h i hai da thiic dau tien ve trai cua (2) CO cimg bac la n d e g ( / ) + r deg(ff) va c6 the xay r a sU triet t i e u k h i tb^'^ hien phep cong T u y nhien, xet he sS cao nhat cua hai da thiic nay, t a c6 + l),\fxeR (1) ^ K x ) = t h o a m a n phuong trinh h a m (1) Xet trUdng hop P ( x ) c6 bac nhat, ^ • ( x ) = ax + b {a,b l a hang so a ^ 0) Thay v a c (1) t a dUOc B [ax + b] [ax + a + b] = a{x'^ + x + 1) + 6, Vx € R B r deg(/) + ndeg(9) > r d e g ( / ) + r d e g ( g ) + l) = P{x'^+x H&ch (Tvfdng t\i nhvf h a i t o a n t n g q u a t ) De thay P{x) = v a n d e g ( / ) + r deg(5) > r d e g ( / ) + ndeg(5)- Do la P{x)P{x H; = Pih) + R{h) cac da thiic ve trai ciia (2) la ^ = [P,{x)r {vdi n = 1,2, ) H i a i t o a n 3.37 Tim tat cd cdc da thdc he so thiCc P(x) thoa man TrtfcJng h d p d e g ( / ) 7^ deg(5) Gia sii deg(/) > deg(5) K h i bac ciia D e y rang = l,P{x) a2 - a = a ( a + - 1) = 6(a + ) - a - = eR •• ' ' V n? v6 nghiem a 7^ Vay khong ton t a i d a thiic bac n h a t t h o a m a n (!)• Tiep theo t a x e t trUdng hop P ( x ) c6 bac 2, P ( x ) = ax^ + 6x + c, vdi a 7^0 ^ h a y vao (1) v a d5ng nhat he so n h u tren t a duOc a = 1,6 = 0, c = V?ly thiic bac h a i t h o a man (1) l a P ( x ) s x^ + Xet c a c d a thiic / ( x ) = x , g ( x ) = x + l , / i ( x ) = x2 + x + l 413 : K h i deg(/) = deg(5) = 1, deg(/) + deg(g) = = deg{h) Tiep theo ta chiing m i n h vdi moi so nguyen duong n ton tai nhieu nhat mot da thiic he so thuc P{x) CO bac ri va thoa man (1) Gia sii P la da thi'rc bac n thoa man (1) Goi P* la he so cao nhat ciia P So sanh h§ so cao nhat hai ve cua cac da thiic (1) ta ro , , ^P*f = P'^ P'= {do nghiem thuc, suy P(x) la da thiic bac chSn, gia sir deg(P) = 2m K h i p{x) dUdc bie'u dign dudi dang • o P ( x ) = (x2 + l ) " * + G{x), •(^2 N h u vay neu gia sii ton t a i mot da thjic Q h^ so thi^c b§c n, kh&c P, thoa Q{x) = P{x) + R{x), vdi < r = deg(fl) P*.R' + R\P' = (2) 2P*.R" N h u vay bac ciia ve trai ciia (2) la n + r, bac ciia da thiic d ph.ii la 2r, nhung 2r < n + r , den day ta gSp mau thuan Vay vdi mpi nguyen diidng n ton t a i nhieu nhat mot da thilc he so thuc P{x) c6 bac n thoa man (1) Ket hdp vdi he qua (1) siiy tat ca cac da thiic thoa man bai la P ( x ) = 0, P{x) = 1, P{x) ve so va de = a {a la bang s6) t h i thay vao (1) ta dirdc = a a(a - 1) = p Vay G(x) = Vay p(x) = (x^ + 1)"" T h i l lai thay thoa man Do t i t ca cac da thiic thoa man de bai la = P{h) + R{h) Hai da tlnlc dan tien t'f ve t r a i cua (2) c6 cimg bac la n + r va c6 the xay sir trict tieu k h i thac hien plicp cong Tuy nhien, xet he so cao nhat ciia hai da thiic nay, ta c6 he so ciia x"+'' da thiic t h i i nhat va da thiic t h i i hai Ian ludt la P'.R' R\P* Nhir the b§c ciia tOng hai da thiic bang :v,, ^, r , + • TvJt (ii) suy vdi n G Z, n > 1, t a c6 / ( n x ) > n / ( x ) , Vx G Q>o, suy !C« (3) = f > q Vq G Q>() Gia sii ton tai so hiiu t i q G (0; i j cho fiq) finq) • fix) G i a i Trong (i), lay x = va y = a, ta dUdc > /(I) ^ /(