! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! "! H4"I)1%J%)>K)L!ML)N=O9)P%E)Q)L4RS()/T&1)L4U&4)VE6) DW&()L"I&X)/Y)Z[).\]\*) L4^%)1%E&)$U6)_U%()`a*)@4b#c)24W&1)2d)#4^%)1%E&)1%E")>K) e%f&)4g)>0&1)23)24"I)489)Q)!"#$%&'()*+,-) ).*.)Q)B4%)#%C#()hhhF6E#4$%&2:FG&)) Bi=)`)j.c*)>%d6kF!#$%!$&'!()! y = −2x−1 x +1 (1) *! "* +$,%!( !(/!0123!.$143!5&!56!78!.$9!$&'!()!:";*! <* =>'!.%?!7@!71A'!B!.$C@D!:";!DE!$%&3$!7@!3FCG43!D-D$!7HC!.1I'!DJ3!7K3F!5&!7LM3F!.$N3F! Δ : 3x− 4y−19 = 0 *! Bi=).)j`c*)>%d6kF) O; #$%!FED!O!.$%,!'P3! tana = 1 2 *!=Q3$!F1-!.R9!DSO!01AC!.$KD! A = 1− cosa + cos2a sin2a − sina *! 0; #$%!()!T$KD!U!.$%,!'P3! (2+ i).z = 2+11i *!=Q3$!'V7C3!DSO!U*!!!! Bi=)7)j*c\)>%d6kF)W1,1!T$LX3F!.R>3$! (x +1) log 3 (x+2)+1 = (x 2 + 2x +1) log 9 (x+2) 2 *!! Bi=)l)j`c*)>%d6kF!W1,1!0Y.!T$LX3F!.R>3$! 2 2 x +1 + x ≤ 2 2+ x (x +1) 2 *! Bi=)\)j`c*)>%d6kF!=Q3$!.QD$!T$Z3! I = (x −1) 2 + x x dx 1 2 ∫ *!! Bi=)-)j`c*)>%d6kF!#$%!$>3$!D$ET![*\]#^!DE!7-G!\]#^!_&!$>3$!5CV3F!D?3$!O!5&!D?3$!043![\! 5CV3F!FED!5`1!'a.!T$N3F!:\]#^;*!Ba.!T$N3F!:b;!71!cCO!\d!5CV3F!FED!5`1![#!De.!D?3$![]!.?1!f! .$%,!'P3! SE.SB = 2a 2 *!=Q3$!.$A!.QD$!g$)1!D$ET![*\]#^!5&!g$%,3F!D-D$!71A'!f!723!'a.!T$N3F! :[#^;*!!! Bi=),)j`c*)>%d6kF!=R%3F!'a.!T$N3F!.%?!7@!hiG!D$%!.O'!F1-D!\]#!DE! AC = 2AB 5&!7j3$!#:k"lmkn;*! =12T!.CG23!.?1!\!DSO!7LM3F!.Ro3!3F%?1!.12T!.O'!F1-D!\]#!De.!7LM3F!.$N3F!]#!.?1!71A'!p:lm";*! =>'!.%?!7@!D-D!7j3$!\d]!012.!\!DE!$%&3$!7@!Z'!5&!T$LX3F!.R>3$!7LM3F!.$N3F!\p! _& x + 2y− 7= 0 *!!!!!! Bi=)a)j`c*)>%d6kF!=R%3F!g$V3F!F1O3!5`1!$I!.RqD!.%?!7@!hiGU!D$%!$O1!71A'!\:"mlmr;!5&!]:smsmt;d! 7LM3F!.$N3F! d : x +1 2 = y−1 −1 = z 2 *!=Q3$!DV(13!FED!F1uO!7LM3F!.$N3F!\]!5&!7LM3F!.$N3F!v*!=>'! .%?!7@!71A'!B!.R43!v!7A!D$C!51!.O'!F1-D!\B]!3$w!3$Y.*!! Bi=)+)j*c\)>%d6kF!xO1!3FLM1!$y3!FaT!3$OC!z!.$L!51I3!.{!"r$!723!""$!(-3Fd!$|!.$)3F!3$Y.!5`1! 3$OC!32C!3FLM1!723!.RL`D!7}1!3FLM1!723!(OC!cC-!"r!T$~.!.$>!RM1!71*!=Q3$!i-D!(CY.!7A!$O1!3FLM1! 71!3F•C!3$143!'&!FaT!3$OC*! Bi=)`*)j`c*)>%d6kF!#$%!idGdU!_&!D-D!()!.$/D!.$C@D!3€O!g$%,3F! 1;+∞ ⎡ ⎣ ⎢ ) *!=>'!F1-!.R9!3$w!3$Y.!DSO! 01AC!.$KD! P = 1 (x 3 +1) 2 + 1 ( y 3 +1) 2 + 1 (z 3 +1) 2 − 3 2x 2 y 2 z 2 + 2 *! ! mmm!nLmmm) ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! <! ) M!oV)LpB!)qrV!)estV)/uM)uV) !!! Bi=)`)j.c*)>%d6kF!#$%!$&'!()! y = −2x−1 x +1 (1) *! "* +$,%!( !(/!0123!.$143!5&!56!78!.$9!$&'!()!:";*! <* =>'!.%?!7@!71A'!B!.$C@D!:";!DE!$%&3$!7@!3FCG43!D-D$!7HC!.1I'!DJ3!7K3F!5&!7LM3F!.$N3F! Δ : 3x− 4y−19 = 0 *! "* x|D!(13$!./!F1,1*! <* W|1! M(m; −2m−1 m +1 )∈ (1),m ≠ −1 *! •;!=1I'!DJ3!7K3F!_&! x +1= 0 ⇒ d(M;TCD)= m +1 *!! •;!=O!DE! d(M;Δ)= 3m − 4. −2m−1 m +1 −19 3 2 + (−4) 2 = 3m 2 −8m−15 5 m +1 *! •;!=$‚%!0&1!RO!.O!DEƒ!!! ! m +1 = 3m 2 −8m−15 5 m +1 ⇔ 5(m +1) 2 = 3m 2 −8m−15 ⇔ 5(m +1) 2 = 3m 2 −8m−15 5(m +1) 2 = −3m 2 + 8m +15 ⎡ ⎣ ⎢ ⎢ ⎢ ⇔ m =1(t / m) m = − 5 4 (l) m = −9± 41 2 (l) ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⇒ M(1;− 3 2 ) *! „JG!71A'!D…3!.>'! M(1;− 3 2 ) *!! Bi=).)j`c*)>%d6kF) O; #$%!FED!O!.$%,!'P3! tana = 1 2 *!=Q3$!F1-!.R9!DSO!01AC!.$KD! A = 1− cosa + cos2a sin2a − sina *! 0; #$%!()!T$KD!U!.$%,!'P3! (2+ i).z = 2+11i *!=Q3$!'V7C3!DSO!U*!!!! O; =O!DEƒ! A = 1− cosa + 2cos 2 a −1 sina(2cosa −1) = cosa(2cosa −1) sina(2cosa −1) = cota = 1 tana = 2 *!! 0; =O!DEƒ! z = 2+11i 2+ i = (2+11i)(2− i) 5 = 15+ 20i 5 = 3+ 4i ⇒ z = 3− 4i *! „>!5JG! z = 3 2 + (−4) 2 = 5 *!!! Bi=)7)j*c\)>%d6kF)W1,1!T$LX3F!.R>3$! (x +1) log 3 (x+2)+1 = (x 2 + 2x +1) log 9 (x+2) 2 *!! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! s! †1HC!g1I3ƒ! x >−2 *! b$LX3F!.R>3$!.LX3F!7LX3F!5`1ƒ!! ! (x +1) log 3 (x+2)+1 = (x +1) 2log 9 (x+2) 2 ⇔ x +1= 1 0< x +1≠ 1 log 3 (x + 2)+1= 2log 9 (x + 2) 2 ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎡ ⎣ ⎢ ⎢ ⎢ ⎢ ⎢ ⇔ x = 0 x = 1 ⎡ ⎣ ⎢ ⎢ ⎢ *! „JG!T$LX3F!.R>3$!DE!$O1!3F$1I'! x = 0;x = 1 *!! Bi=)l)j`c*)>%d6kF!W1,1!0Y.!T$LX3F!.R>3$! 2 2 x +1 + x ≤ 2 2+ x (x +1) 2 *! †1HC!g1I3ƒ! x ≥ 0 *! ]Y.!T$LX3F!.R>3$!.LX3F!7LX3F!5`1ƒ! 2 2(x +1)+(x +1) x ≤ 2 2x 2 + 5x + 2 ⇔ 4(2x 2 + 5x + 2)≥(2 2(x +1) + (x +1) x ) 2 ⇔ 4(x +1) 2(x 2 + x) + x 3 −6x 2 −11x ≤ 0 ⇔ x(x 2 −2x +1)+ 4(x +1) 2(x 2 + x) − 4x(x + 3)≤ 0 ⇔ x(x−1) 2 + 4 x (x +1) 2(x +1) −(x + 3) x ⎡ ⎣ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ≤ 0 ⇔ x(x−1) 2 + 4 x . 2(x +1) 3 − x(x + 3) 2 (x +1) 2(x +1) + (x + 3) x ≤ 0 ⇔ x(x−1) 2 + 4 x (x −1) 2 (x + 2) (x +1) 2(x +1) + (x + 3) x ≤ 0 ⇔ x (x −1) 2 x + 4(x + 2) (x +1) 2(x +1) + (x + 3) x ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ≤ 0 ⇔ x (x−1) 2 = 0 ⇔ x = 0 x = 1 ⎡ ⎣ ⎢ ⎢ ⎢ (do x + 4(x + 2) (x +1) 2(x +1) + (x + 3) x > 0,∀x ≥ 0) *! Kết$luận:!„JG!.JT!3F$1I'!DSO!0Y.!T$LX3F!.R>3$!_&! S = 0;1 { } *! Cách$2:!]Y.!T$LX3F!.R>3$!.LX3F!7LX3F!5`1ƒ! 8 x +1 + x + 4 2x x +1 ≤ 4(2+ x (x +1) 2 ) ⇔ 4 2x x +1 ≤ x(−x 2 + 6x +11) (x +1) 2 *! •;!‡2C! x = 0 d!0Y.!T$LX3F!.R>3$!_CV3!7~3F*! •;!!‡2C! x > 0 d!0Y.!T$LX3F!.R>3$!.LX3F!7LX3F!5`1ƒ! −x 2 + 6x +11 (x +1) 2 ≥ 4 2 x(x +1) (1) ⇔ −x 2 + 6x +11> 0 (−x 2 + 6x +11) 2 (x +1) 4 ≥ 32 x(x +1) ⎧ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎪ ⇔ −x 2 + 6x +11> 0 x(−x 2 + 6x +11) 2 −32(x +1) 3 ≥ 0 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⇔ −x 2 + 6x +11> 0 x 5 −12x 4 −18x 3 + 36x 2 + 25x −32≥ 0 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⇔ −x 2 + 6x +11> 0 (x −1) 2 (x 3 −10x 2 −39x−32)≥ 0 ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⇔ −x 2 + 6x +11> 0 (x −1) 2 . −x(−x 2 + 6x +11)−4x 2 −28x−32 ⎡ ⎣ ⎢ ⎤ ⎦ ⎥ ≥ 0 ⎧ ⎨ ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⇔ x = 1(do − x(−x 2 + 6x +11)−4x 2 −28x−32< 0,∀x ∈ (0;3+ 2 5)) *! Kết$luận:!„JG!.JT!3F$1I'!DSO!0Y.!T$LX3F!.R>3$!_&! S = 0;1 { } *! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! ˆ! qv&4)$=;&F!]Y.!T$LX3F!.R>3$!5`1!_M1!F1,1!.R43!.O!D$j!0123!7‰1!.LX3F!7LX3F!0Š3F!T$‹T!0>3$! T$LX3F!$O1!52!7LO!5H!0Y.!T$LX3F!.R>3$!DX!0,3! B ≥ A *!‡F%&1!.O!.O!DE!.$A!.$/D!$1I3!D-D!D-D$! g$-D!(OCƒ! Cách$2:![€!vq3F!0Y.!7N3F!.$KD!\B!ŒWB!.O!DE! ! 4 2 x(x +1) = 8 2x(x +1) ≥ 8 2x + x +1 2 = 16 3x +1 *! ‡43!.{!:";!(CG!ROƒ! ! −x 2 + 6x +11 (x +1) 2 ≥ 16 3x +1 ⇔ (3x +1)(−x 2 + 6x +11)−16(x +1) 2 ≥ 0 ⇔ −(x −1) 2 (x + 5 3 )≥ 0 ⇔ x = 1(do x > 0) *! =O!DE!g2.!cC,!.LX3F!./!.R43*! Cách$3:!=O!DE!:";!.LX3F!7LX3F!5`1ƒ! ( 8 2x(x +1) − 16 3x +1 )+ 3x +17 3x +1 − 4(2x + 3) (x +1) 2 ≤ 0 ⇔ 8(3x +1−2 2x(x +1)) (3x +1) 2x(x +1) + (3x +17)(x +1) 2 − 4(2x +3)(3x +1) (3x +1)(x +1) 2 ≤ 0 ⇔ 8( 2x − x +1) 2 (3x +1) 2x(x +1) + (x −1) 2 (3x + 5) (3x +1)(x +1) 2 ≤ 0 ⇔ 2x = x +1 x −1= 0 ⎡ ⎣ ⎢ ⎢ ⎢ ⇔ x = 1 *! Bi=)\)j`c*)>%d6kF!=Q3$!.QD$!T$Z3! I = (x −1) 2 + x x dx 1 2 ∫ *!! =O!DEƒ! I = x 2 −2x +1+ x x dx 1 2 ∫ = (x −2+ 1 x + 1 x )dx 1 2 ∫ = ( x 2 2 −2x + ln x + 2 x ) 2 1 = − 5 2 + ln2+ 2 2 *! Bi=)-)j`c*)>%d6kF!#$%!$>3$!D$ET![*\]#^!DE!7-G!\]#^!_&!$>3$!5CV3F!D?3$!O!5&!D?3$!043![\! 5CV3F!FED!5`1!'a.!T$N3F!:\]#^;*!Ba.!T$N3F!:b;!71!cCO!\d!5CV3F!FED!5`1![#!De.!D?3$![]!.?1!f! .$%,!'P3! SE.SB = 2a 2 *!=Q3$!.$A!.QD$!g$)1!D$ET![*\]#^!5&!g$%,3F!D-D$!71A'!f!723!'a.!T$N3F! :[#^;*!!! =O!DEƒ! BC ⊥ AB BC ⊥ SA ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⇒ BC ⊥ (SAB)⇒ BC ⊥ AE *! [CG!RO! AE ⊥ BC AE ⊥ SC(gt) ⎧ ⎨ ⎪ ⎪ ⎩ ⎪ ⎪ ⇒ AE ⊥ (SBC)⇒ AE ⊥ SB *!!! „JG!\f!_&!7LM3F!DO%!.R%3F!.O'!F1-D![\]d!5>!5JG! SE.SB = SA 2 = 2a 2 ⇒ SA = a 2 *! „&! V S.ABCD = 1 3 SA.S ABCD = 1 3 .a 2.a 2 = a 3 2 3 :75 ;*! •;!=Q3$!v:fm:[#^;;*! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! "! #$!%&! SB = SA 2 + AB 2 = 2a 2 + a 2 = a 3 ⇒ SE = SA 2 SB = 2a 2 a 3 = 2a 3 3 '! ()!*+,! SE SB = 2a 3 / 3 a 3 = 2 3 ⇒ d(E;(SCD))= 2 3 d(B;(SCD)) (1) '! /!%&!012234567!898! d(B;(SCD)) = d(A;(SCD)) (2) '! :;!0<!*=>8?!?&%!*@/!46!A./!<!AB)! AH ⊥ (SCD) ⇒ AH = d(A;(SCD)) (3) '! #C!3D7E3F7!*G!3H7!I=,!J$K! d(E;(SCD))= 2 3 AH '! #$L!?/M%!*=>8?!406!%&! 1 AH 2 = 1 SA 2 + 1 AD 2 = 1 2a 2 + 1 a 2 = 3 2a 2 ⇒ AH = a 6 3 '! (+,! d(E;(SCD))= 2a 6 9 '!!!!!!!!! HI%)#;@)#<J&1)#KL!5BN!B)8B!%B&O!4'0156!%&!PM,!0156!QG!B)8B!*=>8?!%.8B!$!*G!%.8B!R98!40! *=>8?!?&%!*@/!LSA!OBT8?!301567'!USA!OBT8?!3V7!P/!W=$!0E!*=>8?!?&%!*@/!45!%XA!%M%!%.8B! 41E45E46!QY8!QZ[A!A./!\E]E<!ABN^!L_8! S AEFH = a 2 2 3 '!#`8B!ABa!A`%B!bBc/!%B&O!4'0156!*G!bBN^8?! %M%B!P/aL!\!Pd8!LSA!OBT8?!34567'!!e2IK!! V S.ABCD = a 3 2 3 ;d(E;(SCD))= 2a 6 9 '! BM=),)NOP*)>%Q6RF!#JN8?!LSA!OBT8?!AN.!Pf!gh,!%BN!A$L!?/M%!015!%&! AC = 2AB *G!Pi8B!53jD"kjl7'! #/dO!A=,d8!A./!0!%m$!PZn8?!AJo8!8?N./!A/dO!A$L!?/M%!015!%XA!PZn8?!ABT8?!15!A./!P/aL!p3"kD7'! #)L!AN.!Pf!%M%!Pi8B!0E1!R/dA!0!%&!BNG8B!Pf!qL!*G!OBZr8?!AJ)8B!PZn8?!ABT8?!0p! QG x + 2y− 7= 0 '!!!!!! stA!B$/!A$L!?/M%!10p!*G!05p!%&K BAI ! = C " ,I # (chung) !898!B$/!A$L!?/M%!10p!*G!05p!Pu8?!v.8?'! ()!*+,! AB AC = IB IA ⇒ AB 2 AC 2 = IB 2 IA 2 (1) '! USA!bBM%E!vN!p0!QG!A/dO!A=,d8!898! IA 2 = IB.IC (2) '! #C!3D7E3F7!A$!%&K! IB IC = AB 2 AC 2 = 1 4 ⇒ IB !"! = 1 4 IC ! "! = 1 4 (−20;−10) = (−5;− 5 2 )⇒ B(0;− 3 2 ) '! w7!xy/! A(7− 2a;a)∈ AI E!A$!%&K!! AC 2 = 4AB 2 ⇔ (22−2a) 2 + (a + 9) 2 = 4 (7− 2a) 2 + (a + 3 2 ) 2 ⎡ ⎣ ⎢ ⎢ ⎢ ⎤ ⎦ ⎥ ⎥ ⎥ ⇔ 15(a 2 − 2a − 24)= 0 ⇔ a = 6(l) a = −4(t / m) ⎡ ⎣ ⎢ ⎢ ⎢ ⇒ A(15;−4) '! SC#)$=;&(!(+,!03D"kjz7E!13{kjH2F7'!!!!! HT&4)$=;&(!#$!%&!ABa!A)L!8B$8B!P/aL!0!%|8?!AC!B$/!A$L!?/M%!10p!*G!05p!Pu8?!v.8?!8BZ!I$=K! AB AC = IB IA = IA IC ⇒ IC = 2IA ⇒ A '! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! }! BM=)U)NOP*)>%Q6RF!#JN8?!bB>8?!?/$8!*@/!B~!AJ•%!AN.!Pf!gh,€!%BN!B$/!P/aL!03Dk"k{7!*G!13HkHk}7E! PZn8?!ABT8?! d : x +1 2 = y−1 −1 = z 2 '!#`8B!%>I/8!?&%!?/•$!PZn8?!ABT8?!01!*G!PZn8?!ABT8?!v'!#)L! AN.!Pf!P/aL!U!AJ98!v!Pa!%B=!*/!A$L!?/M%!0U1!8B‚!8BƒA'!! #$!%&K! AB ! "! = (2;−2;6) / /(1;−1;3) E!PZn8?!ABT8?!v!%&!*t%!Ar!%Bi!OBZr8?! u ! = (2;−1;2) '! ()!*+,! cos(AB;d) ! = 1.2+ (−1).(−1)+ 3.2 1 2 + (−1) 2 + 3 2 . 2 2 + (−1) 2 + 2 2 = 3 11 '! w7!xy/! M(−1+ 2t;1− t;2t)∈ d ⇒ MA = 9t 2 + 20,MB = 9t 2 − 36t + 56,AB = 2 11 '! 6N!01!bB>8?!P„/!898!%B=!*/!A$L!?/M%!0U1!8B‚!8BƒA!bB/!U0wU1!8B‚!8BƒA'! #$!%&K! MA + MB = 9t 2 + 20 + 9t 2 − 36t + 56 = (3t) 2 + (2 5) 2 + (6− 3t) 2 + (2 5) 2 '! stA!B$/!*t%!Ar! a ! = (3t;2 5),b ! = (6− 3t;2 5) E!I…!v•8?!RƒA!PT8?!AB†%K! a ! + b ! ≥ a ! + b ! !E!A$!%&!! ! MA + MB ≥ (3t + 6−3t) 2 + (2 5+ 2 5) 2 = 2 29 '! 6ƒ=!R‡8?!h^,!J$!bB/!*G!%Bi!bB/! a ! ↑↑ b ! ⇔ 3t 6−3t = 2 5 2 5 ⇔ t = 1⇒ M(1;0;2) '! (+,!P/aL!%Y8!A)L!U3Dk{kF7'!! BM=)+)N*PV)>%Q6RF!<$/!8?Zn/!Bˆ8!?SO!8B$=!‰!ABZ!*/~8!AC!ŠB!Pd8!lB!IM8?E!By!ABc8?!8BƒA!*@/!8B$=! 8d=!8?Zn/!Pd8!AJZ@%!P[/!8?Zn/!Pd8!I$=!W=M!D{!OB‹A!AB)!Jn/!P/'!#`8B!hM%!I=ƒA!Pa!B$/!8?Zn/!P/! 8?Œ=!8B/98!LG!?SO!8B$='! ! w7!x/^!I…!h!3OB‹A7!QG!ABn/!?/$8!LG!8?Zn/!AB†!8BƒA! %Bn!8?Zn/!AB†!B$/!‰!ABZ!*/~8'! xy/!,!3OB‹A7!QG!ABn/!?/$8!LG!8?Zn/!AB†!B$/!%Bn! 8?Zn/!AB†!8BƒA!‰!ABZ!*/~8'! xy/!0!QG!R/d8!%c!B$/!8?Zn/!?SO!8B$='! e/•=!b/~8K! 0 ≤ x,y ≤ 60 E!898!bB>8?!?/$8!LŒ=!QG! B)8B!*=>8?!%&!v/~8!A`%B!R‡8?! S = 60 2 = 3600 3Pr8! *Ž7'! w7!ea!B$/!8?Zn/!?SO!8B$=!A$!OB^/!%&K! x− y ≤10 ⇔ −x +10≤ y ≤ x +10 (*) '! 5M%!P/aL!U3hk,7!ABN^!L_8!P/•=!b/~8!3•7!QG!OBY8! ?.%B!Iy%!AJ98!B)8B!*•!3R98!vZ@/7!*G!QG!Ic!OBY8!A…! %m$!R/d8!%c!0'!VBY8!?.%B!Iy%!8G,!%&!v/~8!A`%B! S '= S −( 1 2 50 2 + 1 2 50 2 )= 3600−50 2 = 1100 3Pr8!*Ž7'! ()!*+,! P(A)= S ' S = 1100 3600 = 11 36 '! ! ! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! ‘! BM=)O*)NOP*)>%Q6RF!5BN!hE,E€!QG!%M%!Ic!AB’%!vZr8?!AB=f%! 1;+∞ ⎡ ⎣ ⎢ ) '!#)L!?/M!AJŽ!8B‚!8BƒA!%m$!R/a=! AB†%! P = 1 (x 3 +1) 2 + 1 ( y 3 +1) 2 + 1 (z 3 +1) 2 − 3 2x 2 y 2 z 2 + 2 '! W4;&)XY#(!(@/!$ER!QG!B$/!Ic!AB’%!vZr8?!A$!%&K! 1 (a +1) 2 + 1 (b +1) 2 ≥ 1 ab +1 '! Chứng$minh.!4…!v•8?!RƒA!PT8?!AB†%!5$=%B,!“4%B”$JA€!A$!%&K! ! (ab +1)( a b +1) ≥ (a +1) 2 ⇒ 1 (a +1) 2 ≥ b (a + b)(ab +1) , (ab +1)( b a +1) ≥ (b +1) 2 ⇒ 1 (b +1) 2 ≥ a (a + b)(ab +1) '! 5f8?!AB•N!*d!B$/!RƒA!PT8?!AB†%!AJ98!A$!PZ[%K! ! 1 (a +1) 2 + 1 (b +1) 2 ≥ a (a + b)(ab +1) + b (a + b)(ab +1) = 1 ab +1 '! 6ƒ=!R‡8?!h^,!J$!bB/!*G!%Bi!bB/! a = b = 1 '! –O!v•8?!A$!%&K!!!! 1 (x 3 +1) 2 + 1 ( y 3 +1) 2 ≥ 1 x 3 y 3 +1 , 1 ( y 3 +1) 2 + 1 (z 3 +1) 2 ≥ 1 y 3 z 3 +1 , 1 z 3 +1 + 1 x 3 +1 ≥ 1 z 3 x 3 +1 '! 5f8?!AB•N!*d!R$!RƒA!PT8?!AB†%!AJ98!A$!PZ[%K! ! 1 (x 3 +1) 2 ∑ ≥ 1 2 1 x 3 y 3 +1 ∑ ≥ 3 2(x 2 y 2 z 2 +1) '! B4Z)3(!(@/!Ly/! a,b,c ≥1 A$!%&K! ! 1 a 3 +1 + 1 b 3 +1 + 1 c 3 +1 ≥ 3 abc +1 3•7'! Chứng$minh.!#$!%&!RƒA!PT8?!AB†%!W=•8!AB=f%K!! ! 1 1+ m 2 + 1 1+ n 2 ≥ 2 1+ mn ,∀ mn ≥1 '! –O!v•8?!A$!%&K!! ! 1 1+ a 3 + 1 1+ b 3 ≥ 2 1+ a 3 b 3 , 1 1+ c 3 + 1 1+ abc ≥ 2 1+ abc 4 , 2 1 1+ a 3 b 3 + 1 abc 4 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ≥ 4 1+ a 3 b 3 . abc 4 = 4 1+ abc '! 5f8?!AB•N!*d!R$!RƒA!PT8?!AB†%!AJ98!A$!%&!3•7!PZ[%!%B†8?!L/8B'! —=$,!Q./!RG/!ANM8E!MO!v•8?!A$!%&K! 1 x 3 y 3 +1 ∑ ≥ 3 x 2 y 2 z 2 +1 '!!!! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! Š! 4=,!J$K P ≥ 3 2(x 2 y 2 z 2 +1) − 3 2(x 2 y 2 z 2 +1) = 3 2 1 x 2 y 2 z 2 +1 − 1 2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ 2 − 3 4 ≥− 3 4 !'! 6ƒ=!R‡8?!h^,!J$!bB/!*G!%Bi!bB/! x = y = z =1 '!! SC#)$=;&(!(+,!?/M!AJŽ!8B‚!8BƒA!%m$!V!R‡8?! − 3 4 P.A!A./! x = y = z =1 '!!! Cách$2:!–O!v•8?!RƒA!PT8?!AB†%K! a 2 + b 2 + c 2 ≥ 1 3 (a + b+ c) 2 '!#$!%&K! 1 (x 3 +1) 2 + 1 ( y 3 +1) 2 + 1 (z 3 +1) 2 ≥ 1 3 1 x 3 +1 + 1 y 3 +1 + 1 z 3 +1 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ 2 '! –O!v•8?!RƒA!PT8?!AB†%!3•7!A$!%&K! 1 x 3 +1 + 1 y 3 +1 + 1 z 3 +1 ≥ 3 1+ xyz '!! 6N!P&K! P ≥ 3 (1+ xyz) 2 − 3 2x 2 y 2 z 2 + 2 '!! USA!bBM%K! 2x 2 y 2 z 2 + 2 ≥1+ xyz E!vN!P&K! P ≥ 3 (1+ xyz) 2 − 3 1+ xyz = 3 1 1+ xyz − 1 2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ ⎟ ⎟ 2 − 3 4 ≥− 3 4 '! HT&4)$=;&(!5B‹!˜!AJN8?!Qn/!?/^/!AJ98!A$!I…!v•8?!%M%!RƒA!PT8?!AB†%!B$,!v™8?!I$=K! w7!5BN!8!Ic!AB’%!vZr8?!ABN^!L_8! x i ≥1,i = 1,n A$!%&K! ! 1 x 1 +1 + 1 x 2 +1 + + 1 x n +1 ≥ n x 1 x 2 x n n +1 '! w7!(@/!$ER!QG!B$/!Ic!AB’%!vZr8?!A$!%&K! ! 1 (a +1) 2 + 1 (b +1) 2 ≥ 1 ab +1 '! #„8?!W=MA!Br8!A$!%&K! ! 1 (a + k) 2 + 1 (b + k) 2 ≥ 1 ab + k 2 ,k > 0 '! Chứng$minh.$ 4…!v•8?!RƒA!PT8?!AB†%!5$=%B,!“4%B”$JA€!A$!%&K! ! ab + k 2 ( ) ( a b +1) ≥ (a +k) 2 ⇒ 1 (a + k) 2 ≥ b (a + b)(ab + k 2 ) , (ab +k 2 )( b a +1) ≥ (b + k) 2 ⇒ 1 (b +k) 2 ≥ a (a + b)(ab + k 2 ) '! 5f8?!AB•N!*d!B$/!RƒA!PT8?!AB†%!AJ98!A$!PZ[%K! ! 1 (a + k) 2 + 1 (b + k) 2 ≥ b (a + b)(ab + k 2 ) + a (a + b)(ab + k 2 ) = 1 ab + k 2 '! 6ƒ=!R‡8?!h^,!J$!bB/!*G!%Bi!bB/! a = b = k '! ! !"#$%&'()*+,-) ).*.)) /0&1)23)&456)7)489):%&4)&4;&)<=)>?%)489)@4A))) B4%)#%C#()DE#4$%&2:FG&) ! l! ea!%B/!A/dA!Br8!*•!%M%!RG/!ANM8!MO!v•8?!B$/!RƒA!PT8?!AB†%!OB•!8G,!1.8!Py%!AB$L!bB^N!5=c8! š:›!AB=+A!?/^/!1ƒA!PT8?!AB†%!1G/!ANM8!U/8!“!U$hœ!%m$!%™8?!AM%!?/^'! HI%)#;@)#<J&1)#K)L) HI%):[)*OF)5BN!hE,E€!QG!%M%!Ic!AB’%!vZr8?!ABN^!L_8! x, y,z ≥1 '!#)L!?/M!AJŽ!8B‚!8BƒA!%m$!R/a=!AB†%! 1 (x 3 +1) 2 + 1 ( y 3 +1) 2 + 1 (z 3 +1) 2 ≥ 3 2(x 2 y 2 z 2 +1) '! HI%):[)*.F!5BN!hE,E€!QG!%M%!Ic!AB’%!vZr8?!ABN^!L_8! x, y,z ≥ 2 '!#)L!?/M!AJŽ!8B‚!8BƒA!%m$!R/a=!AB†%! P = 1 (x 3 + 8) 2 + 1 ( y 3 + 8) 2 + 1 (z 3 + 8) 2 − 3 64 2(x 2 y 2 z 2 + 64) '! ! !! !! ! !! !! ! !! ! !!! ! !!!!! ! ! !!!!! !! !! . x > 0,∀x ≥ 0) *! Kết$luận:!„JG!.JT!3F$1I'!DSO!0Y.!T$LX3F!.R>3$!_&! S = 0;1 { } *! Cách$ 2:!]Y.!T$LX3F!.R>3$!.LX3F!7LX3F!5`1ƒ! 8 x +1 + x + 4 2x x +1 ≤ 4(2+ x (x +1) 2 ) ⇔ 4 2x x. ! ˆ! qv&4)$=;&F!]Y.!T$LX3F!.R>3$!5`1!_M1!F1,1!.R43!.O!D$j!0123!7‰1!.LX3F!7LX3F!0Š3F!T$‹T!0>3$! T$LX3F!$O1!52!7LO!5H!0Y.!T$LX3F!.R>3$!DX!0,3! B ≥ A *!‡F%&1!.O!.O!DE!.$A!.$/D!$1I3!D-D!D-D$! g$-D!(OCƒ! Cách$ 2:![€!vq3F!0Y.!7N3F!.$KD!B!ŒWB!.O!DE! ! 4 2 x(x +1) = 8 2x(x +1) ≥ 8 2x + x +1 2 = 16 3x. +11)−16(x +1) 2 ≥ 0 ⇔ −(x −1) 2 (x + 5 3 )≥ 0 ⇔ x = 1(do x > 0) *! =O!DE!g2.!cC,!.LX3F!./!.R43*! Cách$ 3:!=O!DE!:";!.LX3F!7LX3F!5`1ƒ! ( 8 2x(x +1) − 16 3x +1 )+ 3x +17 3x +1 − 4(2x + 3) (x