Inequalities

1. Let a, b, c ∈ R_{+} so that \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1. Prove that:

∑ \frac{1}{a − 1} ≥ \frac{3}{2}.

2. Let a, b, c ∈ R_{+} so that a + b + c = 1. Prove that:

∑ \frac{1}{1 − a} ≥ \frac{9}{2}.

3. Prove the following inequality for a ∈ (1, ∞) and n ∈ N*:

∑_{k=2}^{n+1} k^{2a} > a · n(n + 1).

4. Prove that for every a, b, c ∈ R_{+} satisfying abc = 1 and for every n ∈ N* the following inequality holds:

∑ \frac{1}{a^{2^{n}+1}(b + c)} ≥ \frac{3}{2}.

5. Let a, b, c ∈ R_{+} so that a + b + c = 1. Prove that:

∏ \frac{1 + a}{1 − a} ≥ 8.

6. Let a, b, c ∈ R_{+} so that a + b + c = 1. Prove that:

∑ \frac{a^{3}}{1 + b − c} ≥ \frac{1}{9}.

7. Let a, b, c ∈ R_{+} so that ab + bc + ca = 1. Prove that:

∑ \frac{1 − a^{2}}{1 + a^{2}} ≤ \frac{3}{2}.

8. Let a, b, c ∈ R_{+} so that \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1. Prove that:

∑(a − 1)(b − 1) ≥ 12.

9. Let a, b, c ∈ R_{+} so that a + b + c = abc. Knowing that the numitors are positive prove

that:

∑ \frac{a^{7}}{a^{2} + 5bc − a^{2}bc} ≥ 9\sqrt{3}.

10. Let a, b, c ∈ R_{+}. Prove that:

∑ \frac{a^{3}}{(b + c)(a + 1)} ≥ \frac{27}{8}.

11. Let a, b, c ∈ R_{+} so that a + b + c = abc. Prove that:

∑ a ≥ 3 ∑ \frac{1}{a}.

12. Let a, b, c ∈ R_{+} so that a + b + c = abc. Prove that:

∑ \frac{a^{2}}{bc(ab + 3)} ≥ \frac{1}{2}.

13. Let a, b, c ∈ R_{+} so that a + b + c = 1. Prove that:

∑ \frac{a}{1 − a^{2}} ≥ \frac{27}{8}.

14. Let a, b, c ∈ R_{+} so that a + b + c = abc. Prove that: ∏(a^{2} + 1) ≥ 8 ∏(bc − 1).

15. Let a, b, c ∈ R_{+} so that a + b + c = abc. Prove that:

∏ (a − \frac{1}{b}) ≥ \frac{8}{∑ a}.

16. Let a, b, c ∈ R_{+} so that \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1. Prove that: ∏(a − 2) ≤ 1.

17. Let a, b, c ∈ R_{+} so that \frac{1}{a} + \frac{1}{b} + \frac{1}{c} = 1. Prove that: ∏(a − 1) ≥ 8 ∏(a − 2).

18. Let a, b, c ∈ R_{+} so that ab + bc + ca = 1. Prove that:

∑ (\frac{bc}{a})^{2} ≥ 1.

19. Let a, b, c ∈ R_{+} so that ab + bc + ca = 1. Prove that:

∑ ( \frac{b + c}{\frac{1}{bc} − 1})^{2} ≥ 1.

All problems are proposed by Călin Popa.