36
a1t1i1o1n1.1D1S1M1T1411 bc a 2 2 1 1 1 1;1 1 1y1 1 1 1111111111111111111111111111111111111111111111111111111111111 1111111111111111111111111111111111111111111111111111111111111 1111111111111111111111111111111111111111111111111111111111111 1111111111111111111111111111111111111111111111111111111111111 111111111111111111111111111111111111111111111111 H1èÝ1 111/1 Ë-n1h1 11D1 11M1Tí1n1h1 11E1M1B1E1D1 1E1qT14111 1B1 1E1qT1411E1M1B11.1D1S1M1T1411 1 1A1+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.DSMT4 n >T4 1111111111111111111111111111111111111111Û-i1 1n1 M1B1E1DD1S1M1T1411 1 1 1>1 11E1M1B1E1D1 1EM1T1411 1 1 1 1 11E1M1B1E1D1 1E1q1u1a1t1i1o1n1.1D1S1M1T1411 1 1 1 11E1M1B1E1D1 1E1q1u1a1t1i1o1n1.1D1S1M1T1411 1 1A11E1M1B1E1D1 1E1q1u1a1t1i1o1n1.1D1S1M1T1411 n 1 1 1>1 1B11E1M1B1E1D1 1E1q1u1a1t1i1o1n1.1D1S1M1T1411 n 1 1 1 1v1Û 1 11 1v1à1 1A1 1>1 111 1 11E1M1B1E1D1 1E1q1u1a1t1i1o1n1.1D1S1M1T1411 11u1a1t1 1 1 1>11t1i1A11E1M1B1E1D1 1E1q1u1a1t1i1o1n1.11 1 1 11 1v1à11 11t1i1o1n1.1D1S1M1T1411 1 1 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 1 1 101 1v1Û-iu1a1t1i1o1n1.1D1S1M1T14 1=1 1x1£-y1 1r1a1 1k1h1i1 1A1 1=1 101 1)1 1+1 1A1 n1 11E1M1Bn1.1D1S1M1T1411 1 1 101 1 1v1Û-i11E1M1B1E1D1 1E1q1u1a1t1i1oA1 1(1 1d1¥-u1 1=1 1x1£-y1 1r1a1 1k1h1i1 1A1 1=1 101 1)1 1+u1a1t1i1 1v1Û 1E11T1411 1 1 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 1E1

19 phuong phap_chung_minh_bdt

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Page 1: 19 phuong phap_chung_minh_bdt

a1t1i1o1n1.1D1S1M1T1411 bca 22 11 1 1;1 1 1y1 1

11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111

H1èÝ1111/1 Ë-n1h1 11D1 11M1Tí1n1h1 11E1M1B1E1D1 1E1qT14111 1B1

1E1qT1411E1M1B11.1D1S1M1T1411 11A1+1C1 1>1B1 1+1 1C1

11 1 11D1 11M1T 1B1 1+1 1D11 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.DSMT4 n >T4

1111111111111111111111111111111111111111Û-i1 1n1 M1B1E1DD1S1M1T141111

1>1 11E1M1B1E1D1 1EM1T141111 1 1 11E1M1B1E1D1

1E1q1u1a1t1i1o1n1.1D1S1M1T1411 11 1 11E1M1B1E1D1

1E1q1u1a1t1i1o1n1.1D1S1M1T1411 11A11E1M1B1E1D1

1E1q1u1a1t1i1o1n1.1D1S1M1T1411 n11 1>1 1B11E1M1B1E1D1

1E1q1u1a1t1i1o1n1.1D1S1M1T1411 n11 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 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 1M11V1í1 1d1å- 111 1 1 1(1 1x1,1 1ym1i1n1h1 1r11 11 1x11E1M1B1E1D1 1E1q1u1a1t1i1o1n1.1D1S1M1T1411 11 1+1 11a1t1i1o1n1.1D1S1M1T14B1E1DD1S1ME1D1 1E1q1u1a1t1i11 1 1x1y1+1 1y1z1 1+1 1z1x11 1 1 1 1 1 1 1 1 1 1 11B1E.1D1S1ME1M1B1E1D1 1E1q1u1a1t1i1o1n1.1D1S1M1T1411

1 1E1q1u1a1t1i1o1n1.1D1S1M1T1411 111t1i1o1n1.1D1S1M1T1411 11 121x1y1 1

1E1

Page 2: 19 phuong phap_chung_minh_bdt

a2t2i2o2n2.2D2S2M2T2422 bca 22 22 2 2;2 2 2y2 2

222x2z2 2+2 222y2z22 2 2 2 2 2 2 22 2 2 2 2 22c22q2u2a2t2i2o 2+2 2y22E2M2B2E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422 22 2+2 2z22E22o2n2. 22E22o2n2. 2(2x2 2+2 2y2 2+2 2z2)2

2G2i2£-i2:h2i22D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422 22 22M22M22n22x2y2 2 2y2z2 2 2z2x2 2 2=22E2M2B2E2D2 2E2q2u2a2t2i22.222 2.2(2 2x22t2i2o2 2+2q2u2a2t2i2o2n2.2D2S2M2T2422 22 2+2 2z22E2M2B2E2D2

2E2q2u2a2t2i2o2n2.2D2S2M2T2422 222-2 2x2y2 2 2y2z2 2 2z2x2)2

2=22E2M2B2E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422 2

1

222E2M2B2E2D2

2E2q2u2a2t2i2o2n2.2D2S2M2T2422 0)()()( 222 zyzxyx22

ú2n2g2;2y2;2z22E2M2B2E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422 222 2 2 2 2 2 2 2 2 2V2ì2 2(2x2-2y2)222 22E2M2B2E2D2

2E2q2u2a2t2i2o2n2.2D2S2M2T24222202 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 22

2+2 2za2t 22 22E2M2Bn2.2D2S2M2T24z2 2+2 2z2x2.2 2D2¥-u2 2b2±-n2g2 2x2£-y2 2r2a2 2k2h

2b2)2T2a2 2x2é2t2 2h2i2Ç-u2:2 2x22E2M2B2E22D2222i2o2n2.2D2S2M2T24222 2+2 2z22E2M2B2E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422 2

22-2

2(22+22M2B2E2D2 2E2q2u42E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422 22 2+2

2z22E2M2B2E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422 2 x2z2 2 22yy2 2+E2qT242222t2i2 Í-i2 2x2;2y2;2z22E2M22n2.2Dy2 2x22E2M2B2E2D2 2E2q2u2a2t222 2+2

2y22E2M2B2E2D2 2E2q2u2a2t2i2o2n2.2D2S2M2T2422 22 2+2 2z22E2M2B2E2D2

2E2q2u422 222E2M2B2E2D2 S2M 2 2 ú2n22;22q22422 22£-y2 2r2a2 2k2h22T2a2 2x2E2M2B22.2Dy22E2M2B2E2D22S2M2T2422 Equat+

y +n.DSBED Equation.MBED E1= (x-14 +n.DSMT4 +DSMT4 4

02222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222222D2¥-u2(2=2)2x2£-y2 2r22=21c2h2é-n2g2 2m2i2n)2 2 22E2M2B2E2D2

2E2q2u2a2t2i2o2n2.2D2S2M2T242222 2;2 2 2 2 2b2)2 22E2M2B2E2D2

2E2q2u2a2t2i2o2n2.2D2S2M2T2422 222E2M2Bn2.2D2S2M2T2422

2E2

Page 3: 19 phuong phap_chung_minh_bdt

a3t3i3o3n3.3D3S3M3T3433 bca 22 33 3 3;3 3 3y3 3

33 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

33M3T343333 3 3 3 3E3q3u3a3t3i3o3n3.3D3S3M3T33D3

3E3q3u3a3t3i3o3n3.3D3S3M3T343333 3=33E3M3B3E3D3

3E3q3u3a3t3i3o3n3.3D3S3M3T3433 0

4

1 2 ba

33

3 3 3 3 3 3 3 3 3 3V33E3q3u3a3t3i3o3n3.3D3S3M3T343333.3 3D3¥-u3

3b3±-n3g3 3x3£-y3 3r3a3 3k3h3i3 3 3a3=3b33b3)3T3a3 3x3é3t3 3h3i3Ç-u3

3 3 3 3 3 3 3 33E3M3B3E3D3 3E3q3u3433

3=33E3M3o3n3.3D3S3M3T34333D3 3E3M3T3433

33

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 3B33B3° Û-c3 323:3B3i3¿-n3 3 Õ-i3 3H3=3(3C3+3D3)33E3M3B3E3D3 3EM3T3433

3h3o3·E3M3B3E3D3 3E3q3u3a3t3i3o3n3.3D3S3M3T3433)33E3M3B3E3D3 S3M3T3433 33

3B3° Û-c3 333:3K3¿-t3 3l3u3-n3 3A3 3(3 3B33V3í3 3d3å- 313:3 3(3m3,3n3,3p3,3q3 3t3a3 3 Á-u3 3cD3

3E3q3u3a3t3i3o3n3.3D3S3M3T3433 333q3u3a3t3i3o33 3p333i3o3n3.3D3SE3M33n33(33

33G33 33M33D3 3E3q3u3a3t3i3o 3(3l3u3ô3n3 3 ú3n3g3)33D3¥-u3 3b3±-n3g3 3x3£-y3 3r3a3 3k3h3i3 3 33E3M3B3E3D3

3E3

Page 4: 19 phuong phap_chung_minh_bdt

a4t4i4o4n4.4D4S4M4T4444 bca 22 44 4 4;4 4 4y4 4

4E4q4u4a4t4i4o4n4.4D4S4M4T4444

4

44E4M4B4E4D4

4E4q4u4a4t4i4o4n4.4M4B4E4D4 4E4M4T4444

4

44E4M4B4E4D4

4E4q4u4a4t4i4o4n4.4D4S4M4T4444 444E4M4B4E4D4 4S4M4T444444

4V4í4 4g4 4m4i4n4h4 4r4±-n4g4 4v4Û-i4 4m4Í-i4 4a4,4 4b4,4 4c4 4t4a4 4l4u4ô4n4 4c4ó4 4:44

4E4M4B4E4D4 4E4 44 44

4G4i4£-i4:4 4T4a4 4c4ó4 4:4 44 4E4M4B4E4D4 4E4q4u4a4t4i4o4n4 4E4M4B4E4D4 4E4q4u4a4t4i4o4n4.4M4B4E4D4 4E4q4u4a4t4i4o4n4.434 4 44

4

4

4 ú4n4g4 4v4b4,4 4c4.44P4h4° ¡ n4g4D4ù4n4g4 4p4h4é4p4 4b4i4¿-n4 4 Õ-i4 4t44K4i4¿-n4 4t4h4é-c4:44 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

4E4

Page 5: 19 phuong phap_chung_minh_bdt

a5t5i5o5n5.5D5S5M5T5455 bca 22 55 5 5;5 5 5y5 55E5q5u5a5t5i5o5 5<5 5 5<5b5¥-t5 5 ³-n5n5 5n5h5i5ê5n5,5 5h5o5 5l5à5 5 ú5n5g5 5t5h5ì5 5 5c5ó5 5b5¥-t5 5 ³-n5g5 5 5.5

5C5h5ú5 5ý5 5c5á5c5 5h5±-n5g5 5 ³-n5g5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5B5E5D5

5E5q5u5a5t5i5o5n5.5D5S5M5T5455 55

5 5 5 5 5 5 5 5 5 5 5 5 5 55E5M5B5E5D5 5E5q5u5a5t5i5o5n5.5D5S5M5T545 5 5 5 5 5 5

5 5 5 5 5 5 5 5 5 5 5 5a5t5i5o5n5.5D5S5M5T5455 55

5 5 5 5 5V5í5 5d5å- 515:5 5C5h5o5d5,5e5 5l5à5 5c5á5c5 5s5Ñ- 5t5h5ñ-c5 5c5h5é-n5g5 5m5i5n5h5 5r5±-n5g5

5 5 5 5 5 5 5 5 5 55E5M5B5E5D5 5E5q5u5455

55E5M5o5n5.5D5S5M5T5455 55 5 5B5E5D5 5E5q5u5a5t5i5o5n5.5D5S5M5T5455 5 5

5 5 5 5 5 5c5)5 55E5M5B5E5D5 5E5q5u5a5t5i5o5n5.5D5S5M5T5455

55

5G5i5£-i5E5D5 5S5M5T5455555E5i5o5n5.5D5S5M5T5455

5u5a5t5i555E5M5B5E5D5 5E5q5u5a5t5i5o5n5.5D5S5M5T5455

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 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=5155 5 5 5 5 5 5 5 5 5 5E5q5u5a5t5i5o5n5.5D5S5M5T5455

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

55

5 5 55E5M5B5n5.5D5S5M5D5 5S5M5T5455 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 55

5G55D55S5M5T55D55S5MD5 S5M5T5455

55

5 5 5 5 5 5 55E5Mo5n5.5D5S5M5T5455 55 55E5M5B5E5D5

5E5q5u5a5t5i5o5n5.5D5S5M5T5455 022822228 abbababa555E5M5B5E5D5

5E5

Page 6: 19 phuong phap_chung_minh_bdt

a6t6i6o6n6.6D6S6M6T6466 bca 22 66 6 6;6 6 6y6 6

6E6q6u6a6t6i6 6a626b626(6666)66t6i6o66066a6t6i6o6n6.6D6S6M6T64666a626b626(6a626-6b626)626(6a646+6 6a626b626+6b646)6 6 66E6M6B6E6D6 6E6q6u6a6t6i6o6n606

6B6¥-c6c6u6Ñ-i6 6 ú6n6g6 6v6-y6 6t6a6h6£-i6 6c66V6í6 6 6x6.6y6 6=616 6v6à6 6x66E6M6B6n6.6D6S6M6T6466 -n6g6 6m6i6n6h6

6 66E6M6B6E6D6 6E6q6u6a6t6i6o6n6.6D6S6M6T646E6q6u6a6t6i6o6n6.6D6S6M6T6466 666E6M6B6E6.

6D6S6M6T6466 66

6G6i6£-i6:6 66E6M6B6E6D6 6E6q6u6a6t6i6o6n6.6D6B6E6D6 6E6q6T6466

66E6M6B6E6D6 6E6q6u6a6t6 66 6 6v6ì6 6:6x6a6t6i6o6n6.6D6n6ê6n6 6 6x6-6 6y6

66E6M6B6n6.6D6S6M6T6466 6E6D6 6E6q6u6a6t6i6o6n6.6D6S6M6T6466

6x626+6y6266E6M6B6E6D6 6E6q6u6a6t6i6o 66E6M6B6E6D6 6E6q6u6a6t6i6o6n6.6D6S6MT4 ( x-y)

MT4 x2+y2- EMBED Equation.DSMT4 22 x+EMBED Equation.DSMT4 22 y EMBED Equation.DSMT40EMBED Equation.DSMT4 x2+y2+2- EMBED Equation.DSMT4 22 SMT4 y -2 EMBED EqED Eq+(EMBED EquatD Equation.DSMT4 x+ y -2xy EMBED Equation.DSMT40 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

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