Cyclic and noncyclic inequalities

391 Pages·2015·1.16 MB·English

by  Vasile Cîrtoaje

Vasile CCCCîrtoajje
MATHEMMATICAL
INEQQUAALITIES
Volume 3
CYCLIC AND NONCYCLIC
IINNEEQQUUAAAALLIITTIIEESS
UNIVERSITY OF PLOIESTI, ROMANIA
2015

Contents
1 Cyclic Inequalities 1
1.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2 Noncyclic Inequalities 241
2.1 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .241
2.2 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .263
3 Bibliography 387
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Chapter 1
Cyclic Inequalities
1.1 Applications
1.1. If a,b,c arepositiverealnumbers,then
1 1 1 3
+ + ≥ .
a(a+2b) b(b+2c) c(c+2a) ab+bc+ca
1.2. If a,b,c arenonnegativerealnumberssuchthat a+b+c=3,then
ab2+bc2+ca2≤4.
1.3. If a,b,c≥1,then
(a) ab2+bc2+ca2+6≥3(a+b+c);
(b) 2(ab2+bc2+ca2)+3≥3(ab+bc+ca).
1.4. If a,b,c arepositiverealnumberssuchthat a+b+c=3,then
a b c
+ + ≥1.
b2+2c c2+2a a2+2b
1.5. If a,b,c arepositiverealnumberssuchthat a+b+c≥3,then
a−1 b−1 c−1
+ + ≥0.
b+1 c+1 a+1
1
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1.6. If a,b,c arepositiverealnumberssuchthat a+b+c=3,then
1 1 1
+ + ≥1.
2ab2+1 2bc2+1 2ca2+1
1.7. If a,b,c arepositiverealnumberssuchthat a+b+c=3,then
ab bc ca 3
+ + ≤ .
9−4bc 9−4ca 9−4ab 5
1.8. If a,b,c arepositiverealnumberssuchthat a+b+c=3,then
a2 b2 c2
(a) + + ≥1;
2a+b2 2b+c2 2c+a2
a2 b2 c2
(b) + + ≥1.
a+2b2 b+2c2 c+2a2
1.9. Let a,b,c bepositiverealnumberssuchthat a+b+c=3. Then,
1 1 1
+ + ≤1.
a+b2+c3 b+c2+a3 c+a2+b3
1.10. If a,b,c arepositiverealnumbers,then
1+a2 1+b2 1+c2
+ + ≥2.
1+b+c2 1+c+a2 1+a+b2
1.11. If a,b,c arenonnegativerealnumbers,then
a b c 1
+ + ≤ .
4a+4b+c 4b+4c+a 4c+4a+b 3
1.12. If a,b,c arepositiverealnumbers,then
a+b b+c c+a 2
+ + ≥ .
a+7b+c b+7c+a c+7a+b 3
CyclicInequalities 3
1.13. If a,b,c arepositiverealnumbers,then
2a+b 2b+c 2c+a
+ + ≥3.
2a+c 2b+a 2c+b
1.14. If a,b,c arepositiverealnumbers,then
5a+b 5b+c 5c+a
+ + ≥9.
a+c b+a c+b
1.15. If a,b,c arepositiverealnumbers,then
a(a+b) b(b+c) c(c+a) 3(a2+b2+c2)
+ + ≤ .
a+c b+a c+b a+b+c
1.16. If a,b,c arerealnumbers,then
a2−bc b2−ca c2−ab
+ + ≥0.
4a2+b2+4c2 4b2+c2+4a2 4c2+a2+4b2
1.17. If a,b,c arerealnumbers,then
(a) a(a+b)3+b(b+c)3+c(c+a)3≥0;
(b) a(a+b)5+b(b+c)5+c(c+a)5≥0.
1.18. If a,b,c arerealnumbers,then
3(a4+b4+c4)+4(a3b+b3c+c3a)≥0.
1.19. If a,b,c arepositiverealnumbers,then
(a−b)(3a+b) (b−c)(3b+c) (c−a)(3c+a)
+ + ≥0.
a2+b2 b2+c2 c2+a2
1.20. Let a,b,c bepositiverealnumberssuchthat abc=1. Then,
1 1 1
+ + ≤1.
1+a+b2 1+b+c2 1+c+a2
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1.21. Let a,b,c bepositiverealnumberssuchthat abc=1. Then,
a b c 1
+ + ≥ .
(a+1)(b+2) (b+1)(c+2) (c+1)(a+2) 2
1.22. If a,b,c arepositiverealnumberssuchthat ab+bc+ca=3,then
(a+2b)(b+2c)(c+2a)≥27.
1.23. If a,b,c arepositiverealnumberssuchthat ab+bc+ca=3,then
a b c
+ + ≤1.
a+a3+b b+b3+c c+c3+a
1.24. Ifa,b,c arepositiverealnumberssuchthata≥ b≥c andab+bc+ca=3,then
1 1 1
+ + ≥1.
a+2b b+2c c+2a
1.25. If a,b,c∈[0,1],then
a b c 1
+ + ≥ .
4b2+5 4c2+5 4a2+5 3
(cid:149)1 (cid:152)
1.26. If a,b,c∈ ,3 ,then
3
a b c 7
+ + ≥ .
a+b b+c c+a 5
(cid:149) 1 (cid:112) (cid:152)
1.27. If a,b,c∈ (cid:112) , 2 ,then
2
3 3 3 2 2 2
+ + ≥ + + .
a+2b b+2c c+2a a+b b+c c+a
CyclicInequalities 5
1.28. If a,b,c arenonnegativerealnumberssuchthat a+b+c=3,then
(a) ab2+bc2+ca2+abc≤4;
a b c
(b) + + ≤1;
4−b 4−c 4−a
(c) ab3+bc3+ca3+(ab+bc+ca)2≤12;
ab2 bc2 ca2
(d) + + ≤1.
1+a+b 1+b+c 1+c+a
1.29. If a,b,c arenonnegativerealnumbers,notwoofwhicharezero,then
4abc a2+b2+c2
+ ≥2.
ab2+bc2+ca2+abc ab+bc+ca
1.30. If a,b,c arenonnegativerealnumberssuchthat a+b+c=3,then
1 1 1 1
+ + ≥ .
ab2+8 bc2+8 ca2+8 3
1.31. If a,b,c arenonnegativerealnumberssuchthat a+b+c=3,then
ab bc ca 3
+ + ≤ .
bc+3 ca+3 ab+3 4
1.32. If a,b,c arepositiverealnumberssuchthat a+b+c=3,then
(ab+bc+ca)(ab2+bc2+ca2)≤9.
1.33. If a,b,c arenonnegativerealnumberssuchthat a2+b2+c2=3,then
(a) ab2+bc2+ca2≤abc+2;
a b c
(b) + + ≤1.
b+2 c+2 a+2
1.34. If a,b,c arenonnegativerealnumberssuchthat a2+b2+c2=3,then
a2b3+b2c3+c2a3≤3.
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1.35. If a,b,c arenonnegativerealnumberssuchthat a+b+c=3,then
a4b2+b4c2+c4a2+4≥a3b3+b3c3+c3a3.
1.36. If a,b,c arenonnegativerealnumberssuchthat a+b+c=3,then
a b c 3
(a) + + ≥ ;
b2+3 c2+3 a2+3 4
a b c 3
(b) + + ≥ .
b3+1 c3+1 a3+1 2
1.37. Let a,b,c bepositiverealnumbers,andlet
1 1 1
x =a+ −1, y = b+ −1, z=c+ −1.
b c a
Provethat
xy+ yz+zx ≥3.
1.38. Let a,b,c bepositiverealnumberssuchthat abc=1. Provethat
(cid:129) 1 (cid:112) (cid:139)2 (cid:129) 1 (cid:112) (cid:139)2 (cid:129) 1 (cid:112) (cid:139)2
a− − 2 + b− − 2 + c− − 2 ≥6.
b c a
1.39. Let a,b,c bepositiverealnumberssuchthat abc=1. Provethat
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) 1(cid:12) (cid:12) 1(cid:12) (cid:12) 1(cid:12)
(cid:12)1+a− (cid:12)+(cid:12)1+b− (cid:12)+(cid:12)1+c− (cid:12)>2.
(cid:12) b(cid:12) (cid:12) c(cid:12) (cid:12) a(cid:12)
1.40. If a,b,c arepositiverealnumbers,notwoofwhichareequal,then
(cid:12) (cid:12)
(cid:12) a (cid:12) (cid:12) b (cid:12) (cid:12) c (cid:12)
(cid:12)1+ (cid:12)+(cid:12)1+ (cid:12)+(cid:12)1+ (cid:12)>2.
(cid:12) b−c(cid:12) (cid:12) c−a(cid:12) (cid:12) a−b(cid:12)
1.41. Let a,b,c bepositiverealnumberssuchthat abc=1. Provethat
(cid:129) 1 1(cid:139)2 (cid:129) 1 1(cid:139)2 (cid:129) 1 1(cid:139)2 3
2a− − + 2b− − + 2c− − ≥ .
b 2 c 2 a 2 4