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Inegalit at˘i simetrice ^ n 3 variabile · PDF fileInegalit at˘i simetrice ^ n 3 variabile...
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Inegalit˘at ¸i simetrice ˆ ın 3 variabile 1.Dac˘a a,b,c > 0 astfel ˆ ıncˆat a 2 + b 2 + c 2 = 2, determinat ¸i valoarea minim˘ a a expresiei E = ab c + bc a + ca b 2. Fie a, b, c ∈ R,n ∈ N * . Demonstrat ¸ic˘a: (a)Dac˘a a + b + c ≥ 0 atunci a 3 + b 3 + c 3 ≥ 3abc. (b) (a 2 - bc) 3 +(b 2 - ca) 3 +(c 2 - ab) 3 ≥ 3(a 2 - bc)(b 2 - ca)(c 2 - ab). (c) a 4 + b 4 + c 4 ≥ abc(a + b + c). (d) a 4 n + b 4 n + c 4 n ≥ (abc) 4 n -1 3 (a + b + c). (e) a 6n-2 + b 6n-2 + c 6n-2 ≥ (abc) 2n-1 (a + b + c). 3.Dac˘a a, b, c > 0 demonstrat ¸ic˘a: (a + b + c) 3 ≥ 9 4 [(a + b) 2 c +(b + c) 2 a +(c + a) 2 b] 4. Fie a, b, c ∈ (0, 1) astfel ˆ ıncˆ at abc = (1 - a)(1 - b)(1 - c). Demonstrat ¸i c˘ a abc ≤ 1/8, cu egalitate pentru a = b = c =1/2. 5. Fie a, b, c > 0 astfel ˆ ıncˆ at a + b + c = 1. Demonstrat ¸i c˘ a: a √ a + b √ b + c √ c ≥ q a 4 + b 4 + c 4 + 24abc 3 √ abc 6. Determinat ¸i numerele reale strict pozitive a, b, c ¸ stiind c˘aˆ ındeplinesc simultan condit ¸iile: a 2 + b 2 + c 2 =1 b 2 + c 2 a + c 2 + a 2 b + a 2 + b 2 c ≤ 2 √ 3 1
Transcript of Inegalit at˘i simetrice ^ n 3 variabile · PDF fileInegalit at˘i simetrice ^ n 3 variabile...
Inegalitati simetrice ın 3 variabile
1. Daca a, b, c > 0 astfel ıncat a2 + b2 + c2 = 2, determinati valoareaminima a expresiei
E =ab
c+
bc
a+
ca
b
2. Fie a, b, c ∈ R, n ∈ N∗. Demonstrati ca:
(a) Daca a + b + c ≥ 0 atunci a3 + b3 + c3 ≥ 3abc.
(b) (a2 − bc)3 + (b2 − ca)3 + (c2 − ab)3 ≥ 3(a2 − bc)(b2 − ca)(c2 − ab).
(c) a4 + b4 + c4 ≥ abc(a + b + c).
(d) a4n
+ b4n
+ c4n ≥ (abc)
4n−13 (a + b + c).
(e) a6n−2 + b6n−2 + c6n−2 ≥ (abc)2n−1(a + b + c).
3. Daca a, b, c > 0 demonstrati ca:
(a + b + c)3 ≥ 9
4[(a + b)2c + (b + c)2a + (c + a)2b]
4. Fie a, b, c ∈ (0, 1) astfel ıncat abc = (1− a)(1− b)(1− c). Demonstratica abc ≤ 1/8, cu egalitate pentru a = b = c = 1/2.
5. Fie a, b, c > 0 astfel ıncat a + b + c = 1. Demonstrati ca:
a√a + b
√b + c
√c ≥
√a4 + b4 + c4 + 24abc
3√abc
6. Determinati numerele reale strict pozitive a, b, c stiind ca ındeplinescsimultan conditiile:
a2 + b2 + c2 = 1
b2 + c2
a+
c2 + a2
b+
a2 + b2
c≤ 2√
3
1
7. Fie a, b, c > 0, distincte doua cate doua. Notam cu :
S(a, b, c) =(a2 − b2)3 + (b2 − c2)3 + (c2 − a2)3
(a− b)3 + (b− c)3 + (c− a)3
Demonstrati ca:
(a) Daca abc = 1, atunci S(a, b, c) ≥ 8.
(b) Daca a + b + c = 1, atunci S(a, b, c) ≤ 8/27.
(c) Daca ab + bc + ca = 1, atunci S(a, b, c) ≥ 8√
3/9.
8. Daca a, b, c sunt lungimile laturilor unui triunghi oarecare, demonstratica:
9
4(a + b + c)≤ 1
2a + b + c+
1
a + 2b + c+
1
a + b + 2c<
7
3(a + b + c)
9. Daca a, b, c ∈ [0, 1], atunci:
0 ≥ 3abc− (ab + bc + ca) ≥ abc− 1 ≥ a + b + c− 3
10. Sa se arate ca ın orice triunghi ABC are loc relatia:
(a3 + b3 + c3)(ab + bc + ca) ≥ abc(a + b + c)2
11. Fie a, b, c > 0 astfel ıncat 4abc + a + b + c = 18. Demonstrati ca2abc ≤ ab + bc + ca.
12. Daca a, b, c > 0, a + b + c = 1, atunci:
3(a2 + b2 + c2)− 2(a3 + b3 + c3) ≥ 7
9
13. Daca a, b, c > 0 si abc = 1, atunci(a− 1 +
1
b
)(b− 1 +
1
c
)(c− 1 +
1
a
)≤ 1
14. Fie a, b, c > 0 astfel ıncat a + b + c ≥ abc. Atunci
bc
a3(b + c)+
ca
b3(c + a)+
ab
c3(a + b)≥ 1
2
2
15. Fie a, b, c > 0 cu a2 + b2 + c2 = 3. Atunci
a
a + 3+
b
b + 3+
c
c + 3≤ 3
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16. Fie x, y, z > 0 cu 1/x + 1/y + 1/z = 3. Atunci (x + 1)(y + 1)(z + 1) ≤(x + y)(y + z)(z + x).
17. Fie a, b, c > 0. Atunci
ab
c(a + b)+
bc
a(b + c)+
ca
b(c + a)≥ 3
2
18. Fie a, b, c > 0 cu abc = 1. Atunci
a2
b + c + 1+
b2
c + a + 1+
c2
a + b + 1≥ 1
19. Fie a, b, c > 0. Atunci
a
a + 2b + 3c+
b
b + 2c + 3a+
c
c + 2a + 3b≥ 1
2
20. Daca a, b, c ≥ 0, atunci
(a + b + c)6 ≥ (2a + b)(2b + a)(2a + c)(2c + a)(2b + c)(2c + b)
21. Daca a, b, c > 0 si abc + a2 + b2 + c2 < 4, atunci a + b + c < 3.
22. Daca a, b, c ≥ 0 si a + b + c = 1, atunci
ab
c + 1+
bc
a + 1+
ca
b + 1≤ 1
4
23. Daca a, b, c > 0 si abc = 1, atunci
1
a(1 + b)+
1
b(1 + c)+
1
c(1 + a)≥ 3
2
24. Daca a, b, c ≥ 0 si a2 + b2 + c2 = 12, atunci (a + 1)(b + 1)(c + 1) ≤ 27.
3
25. Daca a, b, c > 0 si a + b + c = 1, atunci
8
(a
1− a2+
b
1− b2+
c
1− c2
)≤ a + 1√
bc+
b + 1√ca
+c + 1√
ab
26. Daca a, b, c > 0, atunci
a3
2a + 3b+
b3
2b + 3c+
c3
2c + 3a≥ a2 + b2 + c2
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27. Daca a, b, c > 0 si a2 + b2 + c2 = 3, atunci
ab
ab + a + b+
bc
bc + b + c+
ca
ca + c + a≤ 1
28. Daca a, b, c > 0 si abc ≥ 1, atunci
1
1 + b + c+
1
1 + c + a+
1
1 + a + b≤ 1
29. Daca a, b, c ≥ −3 si a3 + b3 + c3 = 0, atunci a + b + c ≤ 3.
30. Daca a, b, c ≥ 0, atunci a3 + b3 + c3 ≥ ab2 + bc2 + ca2. (rearanjamentesau medii!)
Nota: Inegalitatile 13, 19, 23, 26 si 30 nu sunt simetrice ci numai circularsimetrice.
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