19 phuong phap_chung_minh_bdt

a1t1i1o1n1.1D1S1M1T1411 bca 22
+
11 1 1;1 1 1y1 1
1
1111111111111111111111111111111111111111111111111111111111111111111111111
1111111111111111111111111111111111111111111111111111111111111111111111111
1111111111111111111111111111111111111111111111111111111111111111111111111
1111111111111111111111111111111111111111111111111111111111111111111111111
H1èÝ1
111/1 Ë-n1h1 11D1 11M1Tí1n1h1 11E1M1B1E1D1 1E1qT14111 1B1
1E1qT1411E1M1B11.1D1S1M1T1411⇒
11A1+1C1 1>1B1 1+1 1C1
11 1 11D1 11M1T 1B1 1+1 1D1
1 1 1 1+1 1A>B và C > MT4⇒ à C < 0 ⇒ A.C < B.C
+ 0 < A < B vuationB.D
+ A > B > 0 EMBED Equation.DSMT4n.DSation.DSMTSMT4 EMBED
Equation.DSMT4 n∀
+ ion.DSMT4SMT4 AEMBED Equation.DSMT4n
>T4n
1111111111111111111111111111111111111111Û-i1 1n1 M1B1E1DD1S1M1T1411
A
1
1
1>1 11E1M1B1E1D1 1EM1T1411
B
1
1 1 1 11E1M1B1E1D1
1E1q1u1a1t1i1o1n1.1D1S1M1T1411⇒
11 1 11E1M1B1E1D1
1E1q1u1a1t1i1o1n1.1D1S1M1T1411 11A11E1M1B1E1D1
1E1q1u1a1t1i1o1n1.1D1S1M1T1411
n
11 1>1 1B11E1M1B1E1D1
n
11 1 1v1Û 1 11 1v1à1 1A1 1>1 111 1
11E1M1B1E1D1 1E1q1u1a1t1i1o1n1.1D1S1M1T1411⇒
111u1a1t111
1>11t1i1A11E1M1B1E1D1 1E1q1u1a1t1i1o1n1.11 1 1 11 1v1à11
11t1i1o1n1.1D1S1M1T1411⇒
11 1 1A11E1M1B1E1D1
1EM1T141M1B1E1DD1S1M11+1A1 1<1 1B1 1v1à1 1A1.1B1 1>1 101 1 1 11a1t1i11 1 1
11t1i1o1n1.1D1S1M1T14t1 1s1Ñ- 1-n1g1 11 1A111i1o1n1.1D1S1M1T141E1qT1411≥ 11
101 1v1Û-iu1a1t1i1o1n1.1D1S1M1T14 1=1 1x1£-y1 1r1a1 1k1h1i1 1A1 1=1 101 1)1
1+1 1A1n1
11E1M1Bn1.1D1S1M1T1411≥ 11 101 1 1v1Û-i11E1M1B1E1D1
1E1q1u1a1t1i1oA1 1(1 1d1¥-u1 1=1 1x1£-y1 1r1a1 1k1h1i1 1A1 1=1 101 1)1
1+u1a1t1i1 1v1Û 1E11T1411 A∀ 11 1 1(1d1¥-u1 1=1 1x1£-y1 1r1a1 1k1h1i1 1A1
1=1 101 11D1 1E11T14111M1B1E1D1 1E1q1u1a1t1i1o1n11 1 111i1o1n11 1(1 1d1¥-u1
1=1 1x1£-y1 1r1a1 1k)1
E1q1u1a1t1i1o1n1.1D1S11 1d1¥-u11h1i1 11¦-N1 1 N11G1 1M1I1N1H1 1B1¤-T1
1 ²-N1G1 1T1H1è-Cá1p1 111 1:1 1D1ù1n1g1 1 Ë-n1h1 1n1g1h1) a1
1K 1 1 Ãh1 1A1 p1 1h1i1Ç-u1 1A1 1 B1 11 1 1 1 1 1 1 11 1 1 1L1° u1 g1 1b1¥-t1 1
³-nE1M1B1E1D1 S1M1T141E1qT1411≥ 11 101 1 1v1Û-i1(1 1M1
1V1í1 1d1å- 111 1 1 1(1 1x1,1 1ym1i1n1h1 1r11 11 1x11E1M1B1E1D1
2
11 1+1
11a1t1i1o1n1.1D1S1M1T14B1E1DD1S1ME1D1 1E1q1u1a1t1i11 1 1x1y1+1 1y1z1 1+1
1z1x1
1 1 1 1 1 1 1 1 1 1 1 11B1E.1D1S1ME1M1B1E1D1 1E1q1u1a1t1i1o1n1.1D1S1M1T1411
2
1 1E1q1u1a1t1i1o1n1.1D1S1M1T1411
2
111t1i1o1n1.1D1S1M1T1411≥ 11 121x1y1 1
1E1

+
22 2 2;2 2 2y2 2
222x2z2 2+2 222y2z2
2 2 2 2 2 2 2 22 2 2 2 2 22c22q2u2a2t2i2o 2+2 2y22E2M2B2E2D2
2
22 2+2 2z22E22o2n2. 22E22o2n2. 2(2x2 2+2 2y2
2+2 2z2)2
2G2i2£-i2:h2i22D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422
2
22 22M22M22n22x2y2 2
2y2z2 2 2z2x2 2 2=22E2M2B2E2D2 2E2q2u2a2t2i22.222 2.2(2 2x22t2i2o2
2+2q2u2a2t2i2o2n2.2D2S2M2T2422
2
22 2+2 2z22E2M2B2E2D2
2
22-2 2x2y2 2 2y2z2 2 2z2x2)2
2=22E2M2B2E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422 2
1
2
22E2M2B2E2D2
2E2q2u2a2t2i2o2n2.2D2S2M2T2422[ ] 0)()()( 222
≥−+−+− zyzxyx
22
ú2n2g2;2y2;2z22E2M2B2E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422 R∈ 22
2 2 2 2 2 2 2 2 2 2V2ì2 2(2x2-2y2)222
22E2M2B2E2D2
2E2q2u2a2t2i2o2n2.2D2S2M2T2422≥ 2202 2 2v2Û-i2(b2±-n2g2 2x2£-y2 2r2a2 2k2h 2 2
2 2 2 2 2 2 2 2(2x2-2z22q2u2a2t2i2o2n2.2D2S2M2T2422≥ 2202 2 2v2Û-i2(2x2 2;2 2z2
2D2¥-u2 2b2±-n2g2 2x2£-y2
2 2 2 2(2y2-2z2)222
222i2o2n2.2D2S2M2T2422≥ ;2 2y2 2D2¥-u22a22
22V22E2q2u2a2t2i2o2n2.2D2S2M2T2422E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422
2
22
2+2 2za2t
2
22 22E2M2Bn2.2D2S2M2T24z2 2+2 2z2x2.2 2D2¥-u2 2b2±-n2g2
2x2£-y2 2r2a2 2k2h
2b2)2T2a2 2x2é2t2 2h2i2Ç-u2:2 2x22E2M2B2E22D2222i2o2n2.2D2S2M2T2422
2
2 2+2 2z22E2M2B2E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422
2
22-2
2(22+22M2B2E2D2 2E2q2u42E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422
2
22 2+2
2z22E2M2B2E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422 2
x2z2 2 22yy2 2+E2qT2422
2
22t2i2 Í-i2 2x2;2y2;2z22E2M22n2.2Dy2 2x22E2M2B2E2D2 2E2q2u2a2t222 2+2
2y22E2M2B2E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422
2
22 2+2 2z22E2M2B2E2D2
2E2q2u422
2
222E2M2B2E2D2 S2M 2 2 ú2n22;22q22422 R∈ 22
£-y2 2r2a2 2k2h22T2a2 2x2E2M2B22.2Dy22E2M2B2E2D22S2M2T24222
Equat+
y +n.DSBED Equation.MBED E1
= (x-142
+n.DSMT42
+DSMT42
4≥
0
2222222222222222222222222222222222222222222222222222222222222222222222222
222222222222222222222222222222222222222222222222222222222222222222222222D
2¥-u2(2=2)2x2£-y2 2r22=21c2h2é-n2g2 2m2i2n)2 2 22E2M2B2E2D2
222
22





 +
≥
+ baba
2
2 2;2 2 2 2 2b2)2 22E2M2B2E2D2
2E2q2u2a2t2i2o2n2.2D2S2M2T2422 222E2M2Bn2.2D2S2M2T2422
2E2

+
33 3 3;3 3 3y3 3
2222
33





 ++
≥
++ cbacba
3
3 3 3 3c3)3 3 3H3ã3y3 3t3Õ-n3g3 3q3u3á3t3 3b3à3i3 3t3o3á3n3
3G3i3£-i3:3 3a3)3 3 3T3a3 3x3é3t3 3h3i3Ç-u3 3 3 3 3 33E3M3B3E3D3
33M3T3433
222
22





 +
−
+ baba
3
3 3 3 3 3E3q3u3a3t3i3o3n3.3D3S3M3T33D3
( )abbaba 222
4
1 2222
−−−+
3
3 3=33E3M3B3E3D3
( ) 0
4
1 2
≥− ba
3
3
3 3 3 3 3 3 3 3 3 3V33E3q3u3a3t3i3o3n3.3D3S3M3T3433
222
22





 +
≥
+ baba
3
3.3 3D3¥-u3
3b3±-n3g3 3x3£-y3 3r3a3 3k3h3i3 3 3a3=3b3
3b3)3T3a3 3x3é3t3 3h3i3Ç-u3
3 3 3 3 3 3 3 33E3M3B3E3D3 3E3q3u3433
2222
33





 ++
−
++ cbacba
3=33E3M3o3n3.3D3S3M3T3433
( ) ( ) ( )[ ] 0
9
1 222
≥−+−+− accbba
3
D3 3E3M3T3433
2222
33





 ++
≥
++ cbacba
3
3
3D3¥-u3 3b3±-n3g3 3x3£-y3 3r3a3 3k3h3i3 3 3a3 3=3 3g3
3q3u3á3t33E3M3B3n3.3D3S3M3T343i3 3c3 3c3h3é-n3g3 3m3i3n3h3
3A33E3M3B3E3D3 3E3q3u3a3t3i3o3n3.3D3S3M3T3 3 Ë-n3h3 3n3g3h3) a3
3B3° Û-c3 313:3 -u3 3 3H3 3=3 3A3 3-3 3B3
3B3° Û-c3 323:3B3i3¿-n3 3 Õ-i3 3H3=3(3C3+3D3)33E3M3B3E3D3 3EM3T3433
2
3h3o3·E3M3B3E3D3 3E3q3u3a3t3i3o3n3.3D3S3M3T3433)33E3M3B3E3D3
S3M3T3433
2
33
3B3° Û-c3 333:3K3¿-t3 3l3u3-n3 3A3 3(3 3B3
3V3í3 3d3å- 313:3 3(3m3,3n3,3p3,3q3 3t3a3 3 Á-u3 3cD3
2
333q3u3a3t3i3o33
3p333i3o3n3.3D3SE3M33n33(33
33G33 33M33D3 3E3q3u3a3t3i3o 3(3l3u3ô3n3 3 ú3n3g3)3
3D3¥-u3 3b3±-n3g3 3x3£-y3 3r3a3 3k3h3i3 3 33E3M3B3E3D3
3E3

+
44 4 4;4 4 4y4 4











=−
=−
=−
=−
01
2
0
2
0
2
0
2
m
q
m
p
m
n
m
4
44E4M4B4E4D4
4E4q4u4a4t4i4o4n4.4M4B4E4D4 4E4M4T4444









=
=
=
=
2
2
2
2
m
m
q
m
p
m
n
4
44E4M4B4E4D4
4E4q4u4a4t4i4o4n4.4D4S4M4T4444⇔
444E4M4B4E4D4 4S4M4T4444


===
=
1
2
qpn
m
4
4
4V4í4 4g4 4m4i4n4h4 4r4±-n4g4 4v4Û-i4 4m4Í-i4 4a4,4 4b4,4 4c4 4t4a4 4l4u4ô4n4 4c4ó4 4:44
4E4M4B4E4D4 4E4 44 )(444
cbaabccba ++≥++
44
4G4i4£-i4:4 4T4a4 4c4ó4 4:4 44 4E4M4B4E4D4 4E4q4u4a4t4i4o4n4 4E4M4B4E4D4
4E4q4u4a4t4i4o4n4.4M4B4E4D4 4E4q4u4a4t4i4o4n4.434 4 44
( ) ( ) ( )
( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) 0
0)2(
)2()2(
0222
222
0222222
0
222222222222
22222
2222222222222222222
222
222222222222222
222444
222444
≥−+−+−+−+−+−⇔
≥−++
−++−++−+−+−⇔
≥−−−
+−++−++−⇔
≥−−−++⇔
≥−−−++⇔
acabacbcbcabaccbba
abaacba
abcaccbacbcbbaaccbba
abcacbbca
caaccbcbbaba
abcacbbcacba
abcacbbcacba
4
4
4 ú4n4g4 4v4b4,4 4c4.4
4P4h4° ¡ n4g4D4ù4n4g4 4p4h4é4p4 4b4i4¿-n4 4 Õ-i4 4t4
4K4i4¿-n4 4t4h4é-c4:4
4 4 4 4 4 4 44T4a4 4b4i4¿-n4 4 Õ-i4 4b4¥-c4 4c4§-n4 4c4h4é-n n4g4 4 ° ¡ n4g4
4v4Û-i4 4b4¥-t4 4 ³-n4g4 4t4h4é-c4 4 ú4n4g4 ³-n4g4 4t4h4é-c4h4à4 4 4A4 4<4D4
4E4q4u4a4t4i4o4 4<4 4 4<4b4¥-t4 4 ³-n4n4 4n4h4i4ê4n4,4 4h4o4 4l4à4 4 ú4n4g4 4t4h4ì4 4 4c4ó4
4b4¥-t4 4 ³-n4g4 4 4.4
4C4h4ú4 4ý4 4c4á4c4 4h4±-n4g4 4 ³-n4g4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4B4E4D4
4E4q4u4a4t4i4o4n4.4D4S4M4T4444( ) 222
2 BABABA ++=+
44
4 4 4 4 4 4 4 4 4 4 4 4 4 44E4M4B4E4D4 4E4q4u4a4t4i4o4n4.4D4S4M4T444 4 4 4 4 4 4
4 4 4 4 4 4 4 4 4 4 4 4a4t4i4o4n4.4D4S4M4T4444( ) 32233
33 BABBAABA +++=+
44
4 4 4 4 4V4í4 4d4å- 414:4 4C4h4o4d4,4e4 4l4à4 4c4á4c4 4s4Ñ- 4t4h4ñ-c4
4c4h4é-n4g4 4m4i4n4h4 4r4±-n4g4
4E4

+
55 5 5;5 5 5y5 5
5 5 5 5 5 5 5 5 5 55E5M5B5E5D5 5E5q5u5455
ab
b
a ≥+
4
2
2
55E5M5o5n5.5D5S5M5T5455 55 5 5B5E5D5 5E5q5u5a5t5i5o5n5.5D5S5M5T5455 5 5
5 5 5 5 5 5c5)5 55E5M5B5E5D5 5E5q5u5a5t5i5o5n5.5D5S5M5T5455
( )edcbaedcba +++≥++++ 22222
55
5G5i5£-i5E5D5 5S5M5T5455
ab
b
a ≥+
4
2
2
5
55E5i5o5n5.5D5S5M5T5455
abba 44 22
≥+⇔
5u5a5t5i555E5M5B5E5D5 5E5q5u5a5t5i5o5n5.5D5S5M5T5455
( ) 02
2
≥−⇔ ba
55 5
5(5ô5n5 5 ú5n5g5)5.5 5V5-y55E5M5B5E5D5
5E5q5u5a5t5i5o5n5.5D5S5M5T545u5 5b5±-n5g5 5 525a5=5b5)5
5 5 5 5 55E5i5o5n5.5D5S5E5D5 5E5q5u5455 ) )(21(2 22
baabba ++>++⇔
55
5 5 5 5 5D5 5E5q5u5a5t5i5o5n5.5D5S5M5T545 5 5 55E5M5o5n5.5D5S5M5T5455B5¥-t5
5 ³-n5g5 5t5h5é-c5 5c5u5Ñ-i5 5 ú5n5 5 5 5 5V5-y5 55E5M5B5E5D5
5E5q5u5a5t5i5o5n5.5D5S-u5 5b5±-n5g5 5x5£-y5 5r5a5 5k5h5i5 5a5=5b5=515
5 5 5 5 5 5 5 5 5 5 5E5q5u5a5t5i5o5n5.5D5S5M5T5455
( )edcbaedcba +++≥++++ 22222
555E5M5B5E5D5 5E5q5u5a5t5i5o5n5.5D5S5M5T5455
⇔
555E5M5o5n5.5D5S5M5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
5 5 5 5 5 5 5 5 5 5 5D5 5E5q5u5455⇔
555E5M5B5E5.5D5S5M5T5455
( ) ( ) ( ) ( ) 044444444 22222222
≥+−++−++−++− cacadadacacababa
55
5 5 55E5M5B5n5.5D5S5M5D5 5S5M5T5455( ) ( ) ( ) ( ) 02222
2222
≥−+−+−+− cadacaba
55
5 5 5 5 5 5 5 5 5 5 5 5 5 5B5é-c5 5 ú5n5g5 5v5-y5 5t5a5 5c5ó5 5 i5Á-u5 5p5h5£-i5
5c5h5é-n5g5 5m5i5n5h5
5V5í5 5d5å- 525:h5 5r5±-n5g5:5 55E5M5B5E5D5
5E5q5u5a5t5i5o5n5.5D5S5M5T5455( )( ) ( )( )4488221010
babababa ++≥++
55
5G55D55S5M5T55D55S5MD5 S5M5T5455
128448121210221012
bbabaabbabaa +++≥+++
55
5 5 5 5 5 5 55E5Mo5n5.5D5S5M5T5455⇔
55 55E5M5B5E5D5
5E5q5u5a5t5i5o5n5.5D5S5M5T5455 ( ) ( ) 022822228
≥−+− abbababa
555E5M5B5E5D5
5E5q5u5a5t5i5 5a525
b525
(5565
)55t5i5o55055a5t5i5o5n5.5D5S5M5T5455⇔
5a525
b525
(5a525
-5b525
)525
(5a545
+5 5a525
b525
+5b545
)5 5 55E5M5B5E5D5
5E5q5u5a5t5i5o5n505
5B5¥-c5c5u5Ñ-i5 5 ú5n5g5 5v5-y5 5t5a5h5£-i5 5c5
5V5í5 5 5x5.5y5 5=515 5v5à5 5x55E5M5B5n5.5D5S5M5T5455〉 -n5g5 5m5i5n5h5
5 55E5M5B5E5D5
5E5q5u5a5t5i5o5n5.5D5S5M5T545E5q5u5a5t5i5o5n5.5D5S5M5T5455≥
55E5M5B5E5.5D5S5M5T5455 22
55
5E5

+
66 6 6;6 6 6y6 6
6G6i6£-i6:6 66E6M6B6E6D6 6E6q6u6a6t6i6o6n6.6D6B6E6D6 6E6q6T6466≥
66E6M6B6E6D6 6E6q6u6a6t6 22
66 6 6v6ì6 6:6x6a6t6i6o6n6.6D6n6ê6n6 6 6x6-6 6y6
66E6M6B6n6.6D6S6M6T6466〉
6E6D6 6E6q6u6a6t6i6o6n6.6D6S6M6T6466⇒
6x626
+6y626
6E6M6B6E6D6 6E6q6u6a6t6i6o 66E6M6B6E6D6
6E6q6u6a6t6i6o6n6.6D6S6MT4 22 ( x-y)
MT4⇒ x2
+y2
- EMBED Equation.DSMT4 22 x+EMBED Equation.DSMT4 22 y
EMBED Equation.DSMT4≥ 0EMBED Equation.DSMT4⇔ x2
+y2
+2- EMBED
Equation.DSMT4 22 SMT4 22 y -2 EMBED EqED Eq+(EMBED EquatD
Equation.DSMT4 22 x+ 22 y -2xy EMBED Equation.DSMT4≥ 0 vì x.y=1 nên 2.x.y=2
E(x-y-EMBED EquatioEquation.DSMT4≥ 66666666666666666666 6 66y6
ú6n6gc6ó6 c6h6é-n6g6 6m6i6n6h6
6V6í6 6d6å- 646:6 6C6h6én6g66 6 =66E6M66n6.6D6S6M6T6466
01269 222
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kkk
k
11
11
11a11t11i11o11n11.11311 11 1111
2 2 2 2
1 1 1 1 1 1 1 1 1 1 1 1 1 1 2
... ...
1 1 3 5 5 7 1 1 3 12 3 3
2 2 2 2 2 2 2 2 2 2
k nk k n n n
     
 ÷  ÷  ÷
⇒ < − ⇒ + + + < − + − + + − = − < ÷  ÷  ÷
 ÷  ÷  ÷− + − + +
     
11
11
11D11o11 11 ó11 11t11h11e11o11 11bé-c11 11B11u11n11h11
11E11M11Bn11.11311 11 1111
2
3
2
3
1
...
3
1
2
1
...
1
....
32 222
22
2
2
1
21
<<++++++≤
+
+++
n
aaa
n
aaa
n
n
11
11d11å-11411 11b1111
11m11i 11 11 11 11 11 11 11 1111E11M11B11E11D11
11E11q11u11a11t11i11o11n11.11D1111 11 11 1111D11ù11n 11t11h11é-c11 11
11B11u11n11h11i11a11c11o11p11s11k11i11:11 11T11a11c11ó11 11 11 11
11E11

a12t12i12o12n12.12D12S12M12T1241212 bca 22
+
1212 12 12;12 12 12y12 12
12a12c12+12b12d1212E12M12B12E12D12
12E12q12u12a12t12i12o12n12.12D12S12M12T1241212≤
u12a12t12i12o12n12.12D12S12M12T1241212
2222
. dcba ++
1212
E12q12u12a12t12i121212E12M12B12E12DD12S12M12T1241212
( ) 22222222
.2 dcdcbaba ++++++≤ q12u12a12t12i12o12n12.12D12S12MD12
12E12q12u12a12t12i12o12n12.12D12S12M12T1241212
222222
)()( dcbadbca +++≤+++
1212
12V12í12 12d12å- 12312:12 12 12 12C12h12é-n12g12
12m12i1212E12M1212n12.12D12S12-i12:12-t12 1212B12u1212 12 12 12 12
12C12á12c12h12 12112:12 12(12112,12112,12112)12 12v12à12 12(12ac12ó12
121212i12o12n12.12D12S12M1212 12 12 12
1212a12t12i12o12n12.12D1212M12B12E12DD12S12M1212M12B12E12D12
12E12q12u12a12t12i12121212i12o12n12.12D12S12M12T1241212
acbcabcba ++≥++ 222
1212 12 12 12 i12Á-u12 12p12h12£-i12 12c12h12é-n12g12
12m12i12n12h12 12D12¥-u12 12b12±-n12g12 12x12£-y12 12r12a12 12k1212° ¡ n12g12
12p12h12á12p12 12612:12 12 12 12B12¥-t12 12 ³-n12g12 12t12h12é-c12
12T12r12ê12-12 12b12° -12s12é12p12
12K12i12¿-n12 12t12h12 12E12M12B12E12D12
12E12q12u12a12t12i12o12n12.12312 12 1212


≤≤≤
≤≤≤
n
n
bbb
aaa
.....
.....
21
21
12
12 12t12h12ì12 1212
12E12M12B12E12D12 12E12q12u12a12t12i12o12n12.12312 12 1212
n
bababa
n
bbb
n
aaa nnnn +++
≤
++++++ ........
.
... 22112121
12
12.12
12D12¥-r12a12 - 12k12h12i1212 12E12M12B12E12D12
12E12q12u12a12t12i12o12n12.12312 12 1212 12E12M12B12E12D12
12E12q12u12a12t12i12o12n12.12312 12 1212


≥≥≥
≤≤≤
n
n
bbb
aaa
.....
.....
21
21
12
12D12
12E12q12u12a12t12i12o12n12.12312 12 1212
n
bababa
n
bbb
n
aaa nnnn +++
≥
++++++ ........
.
... 22112121
12
12
12D12¥-u12 12 =12h12i12 12v12à12 12c12h12É- 12k12h12i1212
12E12M12B12E12.12312 12 1212



===
===
n
n
bbb
aaa
....
....
21
21
12
12
12V12í12 12d12å- 121B12E12D12 12E12q12u12a12t12i12o12n12.12312 12 1212
12E12

a13t13i13o13n13.13D13S13M13T1341313 bca 22
+
1313 13 13;13 13 13y13 13
∆1313A1313 13n13-p13 13 ° Ý-n13g13 1313n13h
1313 13E13Mo13n13.13313 13 1313
.
3
2
sinsinsin
2sin.sin2sin.sin2sin.sin S
CBA
CCBBaA
=
++
++
13
13
13S13 13l13à13 13d13i13Ç-n13 13t13í1313.13 13c13h13é-n13g13
13m13i13n13h13 13r13±-n13g13 1313 13E13M13B13E13D13
13E13q13u13a13t13i13o13n 13l 13 Á-u13.13
13G1313g13i13£-m13 13t13í13nt13 13t13a13 M13B13E13D13
13E13q13u13a13t13i13o13n13.13313 13 1313
.
2
0
π
<≤≤< CBA
13
13S13u13y13
13r13a13:13
1313 13E13M13B13E13D13 13E13q13u13a13t13i13o13n13.13313 13 1313



≤≤
≤≤
CBa
CBA
2sin2sin2sin
sinsinsin
13
13
13Á13p13 13d13å-n13g13 13Bp13 13t13a13 13 ° ã-c13:13
1313 13E13M13B13E13D13 13E13q13u13a13t13i13o13n13.13313 13 1313
( )( )
( )
)2sin2sin2(sin
3
1
sinsinsin
2sin.sin2sin.sin2sin.sin
2sin.sin2sin.sin2sin.sin3
2sin2sin2sinsinsinsin
CBA
CBA
CCBBAA
CCBBAA
CBACBA
++≤
++
++
⇔
++≥
≥++++
13
13
13D13¥-13a1313 13E13M13B13E13D13 13E13q13u13a13t13i13o13n13.13313 13
1313
dêuABC
CBA
CBA
∆⇔


==
==
⇔
2sin2sin2sin
sinsinsin
13
13
13M13·-t13 13k13h13á13c13:13
1313 13E13M13B13E13D13 13E13q13u13a13t13i13o13n13.13313 13 1313
[ ] [ ]
)2(2sin..sin).sin2)(sin2(
sinsinsin4sin.sin2.sin2
)cos()cos(sin2cos)cos(sin2
2sin)cos().sin(22sin2sin2sin
SCbaCBRAR
CBABAC
BABACCBAC
CBABACBA
===
==
+−−=+−=
+−+=++
13
13
13T13h13a13y13 13(13213)13 13v13à13o13 13(13113)13 13t13a13 13c13ó13
13 1313 13E13M13B13E13D13 13E13q13u13a13t13i13o13n13.13313 13 1313
.
3
2
sinsinsin
2sin.sin2sin.sin2sin.sin S
CBA
CCBBaA
≤
++
++
13
13
13D13¥-u13 13 =13 13x13£-y13 13r13a13 1313 13E13M13B13E13D13
13E13q13u13a13t13i13o13n13. Á-u13.13
13
13 13 13 13 13 13 13 13 13V13íñ- 13g13i13£-i13)13:13 13
13a13+b+c=1 CMR: EMBED Equation.DSMT4 9
111
≥++
cba
>0 và x+y+z=1
CMR:x+2y+zEMBED Equation.DSMT4 )1)(1)(1(4 zyx −−−≥
13E13

a14t14i14o14n14.14D14S14M14T1441414 bca 22
+
1414 14 14;14 14 14y14 14
c/ Cho a>0 , b>0, c>0 D
Equation.DSMT4
2
3
≥
+
+
+
+
+ ba
c
ac
b
cb
a
d)Cho
xEMBEDEMB1
4141414141414141414141414141414141414141414141414141414141414141414141414
141414 14 14t14h14Ï-a14 14m14ã14n14 1414E14M14B14E14D14
14E14q14u14a14t14i14o14n14.14D14S14M14T1441414 12 =− yx
1414 14 14 14 1414
14 14x14+14y1414E14M14B14E14D14 14E14q14u14a1414 5
1
≥
14
14 14 1414 14 14 14 14
14 14 14
14V14í14 14d14å- 14314:14 14C14h14o14 14a14>14b14>14c14>14014 14 14
14v14à14 1414E14M14B14E14D14
14E14q14u14a14t14i14o14n14.14D14S14M14T1441414 1222
=++ cba
1414.14
14C14h14é-n14g14 14m14i14n14h14 14r14±-n14g1414E14M14B14E14D1414S14M:14
c14 14 14i1414D1414S1414E14D1414S14M14T144141414D14 14E14q14u14a14t14i14
14 a14t14i14o
14Á 14 1414 1414 14 14 14 14 1414E14M14B14E14D14
14E14q14u14a14t14i14o14n14.14D14S14M14T1441414






+
+
+
+
+
++
≥
+
+
+
+
+ ba
c
ca
b
cb
acba
ba
c
c
ca
b
b
cb
a
a .
3
...
222
222
14
1414a14t14i14o14n14
.14D14S14M14T1441414 2
3
.
3
1
14
14u14a14t14i14
14V14q14u141414 2
1333
≥
+
+
+
+
+ ba
c
ca
b
cb
a
14
14 14 14 14 14 14 14
14D14¥-14r14a=1414E14M1414n14.1414 14d14å- 14414:14 14 14 14 14C14h14o14
14a14,14b14,14c14,14d14>14014 14v14à14 1414C14h14-n14g 14EM14T1441414
( ) ( ) ( ) 102222
≥+++++++++ acddcbcbadcba
1414
14G14i14£-i14:14 14T14a14 14c14ó14 14 14 1414E14M14B14E14D14
14E14q14u14a14t14i1414
14 14 14 14 14 14 14 14 14 14D14
14E14q14u14a14t14i14o14n14.14D14S14M14T1441414 cddc 222
≥+
1414
14D14o14ê14n14 14c14d14 14=1414E14M14B14E14D14
14E14q14u14a14t14i1414 14 14(14d14ù14n14g14 1414E14M14B14E14D14
14S14M14T1441414
2
11
≥+
x
x 14E14M14o14n14.14D14S14M14T1441414
4)
1
(2)(2222
≥+=+≥++
ab
abcdabcba
14
14 14 14 14(14114)14 14 14 14 14
14E14

a15t15i15o15n15.15D15S15M15T1541515 bca 22
+
1515 15 15;15 15 15y15 15
15M15·-t15 15k15h15á15c15:15E15q15T1541515 ( ) ( ) ( )acddcbcba +++++
1515 15
15=15 15c15+15)15
15 15 15 15 1515E15M15B15E15D15 15E15q15u15a1515
222
111
++≥





++





++





+
bc
bc
ac
ac
ab
ab
15
15
15V15-y1515t15 ( ) ( ) ( ) 102222
≥+++++++++ acddcbcbadcba
1515
15P15h15° ¡ n15g15 15p15 15 ³-n15g15 15t15h15é-c15
15B15e15r15n15o15u15l15i15
15K15 15a15)15D15¡-n15g15 15n15g15u15y15ê15n15 15t1515
15E15M15B15n15.1515 15t15i15o15n15h15ì15q15u15a15t15i15o15n15.15315 15 1515
( ) naa
n
+≥+ 11
1515.15 15D15¥-u15 15 =15 15x15£-y15 15r15a15 - 15k15h15i1515
15E15M15o15n15.15 15D15¡-n15g15 15m15ß- 15r15Ù15o15 15a15 15>15 15-
15115,1515 15E15M15B15E15D15 15E15q15u15a1515 15t15h15ì15 1515
15E15M15B15E15D15 15E15q15u15a15t15i15 15D15¥-u15 15b15±-n15g15 15x15£-y15
15r15a15É- 15k15h15i15 15a15 15=1515
15t15i15t15h15E15q1515n15g15i15i1515a15t15i15o15n15.15315 15 1515



=
=
1
0
α
a
15
15.15
15 15V15í15 15d15å- 15m15i15E15M15B15E15D15 15E15q15u15a15t15.15
15G15i15£-i15
15N15¿-u15 1515 15E15M15B15E15D15 15E15q15u15 1≥a
1515 15h15a15y15
1515 15E15M15B15E15D15 15E15q15u15a15t15 15t15h15ì15 15B15 T15
15l15u15ô15n15 15 ú15 15a15,15b15 15<15 15B15i15:15q15u15
( )11 1
1 1 .
b b
bb aa a b a
a
a a a a a b
−− +   
= + < + < ⇒ > ÷  ÷
+   
15
15
15C15h15é-n15g15 15m15i15n15h15 15t15° ¡ n15g15 15t15ñ-:1515
15E15M15o15n15.15315 15 1515 1515 15t15i15o15n15 15 15 15(15
p15c15m15)15.15
15V15í15 15d15å- 15215:15 15>15 15015.15C15h15é-n15g15 15
15u15a1515.15 15 15
1515 15t15i15o15p15 15e15r15n15o15u15l15i15:15
1515 15E15M15B15E15D15 15E15q1515
( )
cba
acb
cba
acb
cba
a
++
−+
+≥





++
−+
+=





++
25
1
2
1
3
55
15 15 15 15(15215)15
15C15h15é-n15g15g15 15t15ñ- 15t15a15 15 u15ã-c15:15
1515 15E15M15B15E15D15 15E15q15u15a15t15i15o15n15.15315 15 1515
( )
cba
bac
cba
b
++
−+
+≥





++
25
1
3
5
15
15 15 15 15M15B15E15D15
15E15q15u15a15t15i15o15n15.15315 15(15415)15
15C15Ù15 15(15415)15¿- 15t15E15.15315)15 15c15ó15t15Õ-n15g15
15E15

a16t16i16o16n16.16D16S16M16T1641616 bca 22
+
1616 16 16;16 16 16y16 16
16q16u16á16t16 16s16a16u16 16 â16y16:16
16E16D16 16E16q16u16a16t16i16o16n16.16316 16 1616 .1;0,..., 21 ≥> raaa n
16C16h16é-n16g16 16m16i16n16h16 16r16±-n16g16 16
16 16 16 16 1616 16E16M16B16E16D16 16E16q16u16a16t16i16o16n16.16316 16
1616
r
n
r
n
rr
n
aaa
n
aaa







 +++
≥
+++ ........ 2121
16
16.16
16D16¥-u16 16 =16 1616 16E16M16B16E16D16
16E16q16u16a16t16i16o16n16.16316 16 1616 naaa ===⇔ ....21
16.16(16c16h16é-n16g16 16m16i16n16h16 16t16° ¡ n16g16 16t16ñ- 16b16à16i16
16t16r16ê16n16)16.16
16V16í16 16d16å- 16316:16 16C16h16o16 1616 16E16M16B16E16D16
16E16q16u16a16t16i16o16n16.16316 16 1616 1,,0 ≤≤ zyx
1616.16 16C16h16é-n16g16
16m16i16n16h16 16r16±-n16g16 16
1616 16E16M16B16E16D16 16E16q16u16a16t.16
16G16i16£-i16 ·-t16 1616 16t16i16o16n1616 16E16M16B16E16D16 16E16q1616
( ) ( ) 2 2
1 2 1 2 0 3 2 0 3 (1)a a a a a a
a
≤ ≤ ⇒ − − ≤ ⇒ − + ≤ ⇒ + ≤
16
16
16C16h16é-n16g16 16m16i16n16h16 16t16° ¡ n16g16 16t16ñ-:16
1616 16E16M16B16E16D16 16E16q16u16a16t16
16C16Ù-n16g16 16)16 16v16¿- 16t16h16e16o16 16v16¿- 16t16a16 16E16D16
16E16q16u16a16t16i16o16h16ú16 16ý16:16t16Õ-n16g16 16q16u16á16t16
16d16¡-n16g16 16n16à16y16
16 16C16h16o16 16n16 16s16Ñ- 1616 16E16M16B16E16D16
16E16q16u16a16t16i16o16n16.16316 16 1616 [ ] 1,,,....,, 21 >∈ cbaxxx n
161616
16E16M16B16E16D16 16E16q16u16a16t16i16o16n16.16316 16 1616 1616
16T16a16 16l16u16ô16n16 16c16ó16:16
1616 16E16M16B16E16D16 16E16q16u16a16t16iP16h161616° ¡ n16g16
16p16h16á16p16 16816:16 16 16 16 16 16 16 16 16 16S16í- 16d16å-n16g16
16t16í16n16h16 16c16h16¥-t16 16b16¯-c16 16c16§-u16
16K16i16¿-n16 16t16h16é-c16:16 16 16 16 16A16>16B16 16v16à16 16B16>16C16
16t16h16ì16 16 16A16>16C16
16V16í16 16d16å- 16116:16 16C16h16o16 16a16,16 16b16,16 16c16 16,16d16
16>16016 16t16h16Ï-a16 16m16ã16n16 16 16a16>16 16c16+16d16 16,16
16b16>16c16+16d16
16 16 16 16 16 16 16 16 16 16 16 16 16C16h16é-n16g16 16m16i16n16h16
16r16±-n16g16 16 16 16 16a16b16 16>16a16d16+16b 16G16i16£-i16:16T16a16c16ó16
a16t16i16 16 16 16 16 1616E16M16B16E16D16 16E16q16u16a16t16i16o
1616E16M16B16E16D16 16E16q16u16a16t16i16o16n16.16D1616B16E16D16
16E16q16u16a16t16i16o16n16.16D16S16M16T1641616⇒
1616 16 16 16 16 16 16(1616
16c16d16
16 16 16 16 16 16 16 16 16 16 1616E16M16B16E16D16 S16M16T1641616⇔
1616 16 16
16E16

a17t17i17o17n17.17D17S17M17T1741717 bca 22
+
1717 17 17;17 17 17y17 17
17a17b17-17a17d17-17b17c17+17c17d17 17>17c17d17 1717E17M17B17E17D17
17E17q17u17a17t17i17o17n17.17D17S17M17T1741717⇔
1717 17 17 17a17b17>17
17a17d17+17b17c17 17 17 17 17 17 17 17 17 17 17 17 17 17 1717£-i17
17c17h17é-n17g17 17m17i17n17h17)17
17V17í17 17d17å- 17217:17 17C17h17o17 17a17,17b17,17c17>17017 17 17
17t17h17Ï-a17 17m17ã17n17 1717E17M17B17E17DD17S17M17T1741717
3
5222
=++ cba i17n17h17 17 17 1717E17M17B17E17D17
17E17q17u17a17t17i17o17n17.17D17S17M17T174£-i17:17 1717+17b17-17
17c17)17217
=17 17 17a17217
+17b17217
+17c17217
+1721717b17c17)17
1717E17M17B17E17D17 17E17q17u17a17t17i17o17n17.17D17S17M17T17417 17 17
1717E17M17B17E17D17 17E17q17u17a17t17i17o17n17.1717+17bD17
17E17q17u171717〈
171717E17M17B17E17D17
17E17q17u17a17t17i17o17n17.17D17S17M17T1717
+17c17217
)B17E17D17
17E17q17u17a17t17i17o17n17.17D17S17M17T1741717⇒
1717 17a17c17+17b17c17-
17a17b17 17 1717E17M17B17E17D17
17E17q17u17a17t17i17o17n17.17D17S17M17T1741717 6
5
≤
1717E1717o17n17.17D17S17M17T1741717〈
1717 17117 17 17C17h17i17a17
17h17a17i17 17v17¿- 17c17h17o17 17a17b17c17 17>17 17017 17 17 17t17a17 17c17ó17
17 1717E17M17B17E17D17 17E17q17u17a17t17i17o17n1717M17B17E17D17
17E17q17u17a17t17i17o17n17.17D17S17M17T1741717〈
17q17u17a17t17i17o17n17.17D17S17M17T1741717 abc
1
17
17
17V17í17 17d17å- 17317:17 17C17h17o17 17017 17<17 17a17,17b17,17c17,17d17
17 17<17117 17 17C17h17é-n17g17 17m17i17n17h17 17r17±-n17g17 17(1717(17 17117-
17c17)17.17(17117-17d17)17 17>17 17 17117-17a17-17b17-17c17-17d17
17G17i17£-i17:17 17T17a17 17c17ó17 17(17117-17a17)17.17(171-b) = 1-a-b+ab
Do a>0 , b>0 nên ab>0EMBED Equation.DSMT4⇒ (1-a).(1-b) Do c <1 nên 1-
c >0 ta có EMB(1-a).(1-b) ( 1-c) > 1-a-b-c
EMBED Equation.DSMT4).(1- =1- ⇒ (1-a)> 1-a-
b171717171717171717171717171717171717171717171717171717171717(17 i17Á-u17
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17S17M17T17B17E17D17D17S17M17T1741717M17B17ESMT4 2
a T4 2
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Equation.DSMT4⇒ 1+EMBED Equation.DSMT4 2
a 2
b MT4 2
a + b
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a >
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17E17

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24 24 24 2424r24±-n24g24 24t24i24o24n24.24D24S24M24T2442424
)(2222
cabcabcbacabcab ++<++<++
2424
24 24 24 24 24 24 24 24224/24C24h 24c24h24i24Á-u24 24d24à24i24 24b24a24
24c24¡-n24h24 24c24ç-a24 24t 24c24h24u24 24v24i24 24b24±h24é-n24g24
24m24i24n24h24 E24M24B24E24D24 24E24q24u24a24t24i24o
24P24h24° ¡ n2424 24 24S24í-
24d24å-n24g2424à24 2424 24d24å- 24124:24
24C24h24é-n24g24 24m24i24n24h24 2424B24E24D24
24E24q24u24a24t24i24o24n24.24324 24 2424 0,)()( ≥≥∀≤−+− baabcbccac
24v24à24 2424 24E24M24B24E24D24 24E 2424 cb ≥
2424
24G24i24£-i24
24T24r24o24n24g24 24m24·-t24 24p24h24³-n24g24 24O24x24y2424B24E24D24
24E24q24u24a24t;24 24 24 24 24 2424 24E24M24B24E24D24
24E24q24u24a24t24i24o24n24.24324 24 2424 ),( ccav −=
2424
24T24h24ì24 2424 24E24M24B2424.24324 24 2424 bu =
24
24,24 24 2424
t24i24o24n24.24324 242424t2424H24¡ n24 24n24ï-a24:24 2424 24E24M24B24E24D24
24E24q24u24a24t24i24o24n24.24324 24 2424
⇒≤−+−⇒≤= abcbccacvuvuvuvu )()(.),cos(...
24
24(24 P24C24M24)24
24V24í24 24d24å- 24224:24
24C24h24o24 24224n24 24s24Ñ-:24 2424 24E24M24B24E24D24
24E24q24u24a24t24i24o24n24.24324 24 2424 niyx ii ,...,2,1,; =
2424 24 24t24h24Ï-a24
24m24ã24n24:24 2424 24E24M24B24E24D24 24E24q24u24a24t24i24o24n24.24324 24
2424
.1
11
=+ ∑∑ ==
n
i
i
n
i
i yx
24
24 24C24h24é-n24g24 24m24i24n24h24 24r24±-n24g24:2424
24E24M24B24E24D24 24E24q24u24a24t24i24o24n24.24324 24 24242424V24½-
24h24ì24n24h24
24
24
2424
24242424
242424
242424
2424
24242424
24
24E24

a25t25i25o25n25.25D25S25M25T2542525 bca 22
+
2525 25 25;25 25 25y25 25
25
25
25 25T25r25o25n25g25 25m25·-t25 25p25h25³-n25g25 25t25Í-a25 25 Ù-,25
25x25é25t25:25 25
25 2525 25E25M25B25E2525325 2525E2525325 25 2525 ),( 21212 yyxxM ++
25;25& ;2525 t25i25o25n25.25325 25 2525 ),( 11 nnn yyxxM ++++ 
2525
25G25i25£- 25t25h25i25¿-t25 25s25u25y25 25r25a25 2525 2525i25o25n25.25325
25 t25h=25 25125.25 25L25ú25c25 25 ó25:25
2525 25E25M25B25E25D25 25E25q25u25a25t25i25o25n25.25325 25 2525
2
1
2
11 yxOM +=
2525,25 25 25 2525 25E25M25B25E25D25
25E25q25u25a25t25i25o25n25.25325 25 2525
2
2
2
221 yxMM +=
2525,25 2525
25E25M25B25E25D25 25E25q25u25a25t25i25o25n25.25325 25 2525
2
3
2
332 yxMM +=
25,25& ,25 2525 25E25M25B25E25D25 25E25q25u25a25t2525V25à25 2525
25E25M25B25E25D25 25E25q25u25a25t25i25o25n25.25325 25 2525 1OM
252525
25E25M25B25n25.25325 25 2525 21MM+
252525 25E25M25B25E25D25
25E25q25u25a25t25i25o25n25.25325 25 2525 32 MM+
252525 25E25M25B25E25D25
25E25q25u25a25t25i25o25n25.25325 25 2525 2
2
1 =≥≥++ − OHOMMM nnn
25q25u25a25t25i25o25P25C25M25)25
25P25h252525° ¡ n25g25 25p25h25á25p25 25125i25 25b25i25¿-n25:25
25C25h25o25 25a25,25b25,25c25 25>25 25025 25 25C25h25é-n25g25
2525E25M25B25E25D25D25S25M25T254-i25:25 25 ·-t25 25x25=25b25+25c25 25;25
25y25=25c25+25a25 25t25a25 25c25ó25 25a25=2525E25M25B25E25D25
25E25q25u25a25t25i25o25n25.25D25S25M25T2542525 2
xzy −+
25
25
25E25M25B25E25D25
25E25q25u25a25t25i25o25n25.25D25S25M25T2542525M25B25E25D25
25E25q25u25a25t25i25o25n25.25D25S25M25T2542525 2
zyx −+
25
25
25t25a25 25c25ó25 25(25125)25 2525E25M25o25n25.25D25S25M25T25425D25
25E25q25u25a25t25i25o25 2525E25M25B25n25.25D25S25M25T25425
25E25q25u25a25t25i25o25n25 25 2525E25M25B25E25D25 25S25M25T2542525
3111 ≥−++−++−+
z
y
z
x
y
z
y
x
x
z
x
y
25
25 25u25a25t25i25o25n25.25D25S25M25T25E25D25
25E25q25u25a25t25
6)()() ≥+++++
z
y
y
z
z
x
x
z
y
x
x
y
25
25
25 25 25 25B25¥-t25 25 ³-n25g25 25t25h25é-c25 25c25u25Ñ-i25 25c25ù25 25
25E25

a26t26i26o26n26.26D26S26M26T2642626 bca 22
+
2626 26 26;26 26 26y26 26
26 26(2626E26M26B26n26.26D26S26M26T2642626
;2≥+
y
x
x
y
26
26
26E26q26u26a26t26i
26
26;26 26 2626E26M26B26E26D26 26S26M26T2642626
2≥+
z
y
y
z
26n26ê26n26 26 26p26h26£-i26 26c26 26 26V26í26 26d26å-226C26h26o26
26a26,26b26,26c26 26>26 26026 26 26v26à26 26a26+26b26+26c26 26<26126.26
26C26h26é-n26g26 26m26i26n26h26 26r26±-n26g26 26
26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 26 2626E26M26B26E26D26
26E26q26u26a26t26i26o26n26.26D26S26M26T2642626
9
2
1
2
1
2
1
222
≥
+
+
+
+
+ abcacbbca
26 26 26 26 26 26 26 26 26 26 26(26126)26
26G26i26£-i26:26 26 ·-t26 26x26 26=26 2626E26M26B26
26E26

19 phuong phap_chung_minh_bdt

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