Problems using Cauchy-Schwarz不等式.doc

Problems using Cauchy-Schwarz inequality.Problem 1. (Jack Garfunkel)For non-negative numbers a,b,c , prove the inequality Solution.Using the Cauchy-Schwarz inequality, we haveThus, one only needs to proveThis is obviously true becauseThe proof is complete. Equality occurs if and only if Problem 2. (Vo Quoc Ba Can)For non-negative numbers a,b,c with sum equal to 1, prove thatSolution.Using the Cauchy-Schwarz inequality, we haveThus, one only needs to prove thatWe have We proveWe have Therefore the inequality is equivalent toLast inequality is obviously true so the result follows. Equality does not hold.Problem 3. (Vo Quoc Ba Can)For non-negative numbers a,b,c, prove thatSolution.Using the Cauchy Schwarz inequality, we haveThus, one only needs to prove thatWithout loss of generality, assume , the inequality is equivalent towhere Using the assumption , it is easy to prove that .The result follows, equality holds if and only if Problem 4.For positive numbers a,b,c, prove that Solution.Lemma.: Using the Cauchy Schwarz inequality, We have to prove We have Thus, one only needs to prove Without loss of generality, normalize the inequality by let Then we have The inequality is equivalent to It is clear that is an increasing function in r , we should have Inequality is completely proved. Equality occurs if and only if Problem 5. (Phan Thanh Nam)For the non-negative numbers a,b,c with sum equal to 1. Let prove thatSolution.Using the Cauchy Schwarz inequality,Thus, one only needs to prove Let , then we have The inequality is equivalent to So, we have Equality occurs if and only if or and its permutationsProblem 6. (Vo Quoc Ba Can)For positive numbers a,b,c, prove that Solution.Using the Cauchy Schwarz inequality, we haveSo, we have to prove Without l