4802Inequalities07

# 4802Inequalities07 - n-1. Prove that we can cut the...

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Math 4802 Fall 2007 Inequalities Problems (due Tuesday, November 6) 1. If a, b, c are positive real numbers, prove that a 4 + b 4 + c 4 abc ( a + b + c ) . 2. Given that a, b, c, d, e are real numbers for which a + b + c + d + e = 8 a 2 + b 2 + c 2 + d 2 + e 2 = 16 , ﬁnd the maximum possible value of e . 3. Prove that for all real numbers x , 2 x + 3 x - 4 x + 6 x - 9 x 1 . 4. Let P ( x ) be a polynomial with positive coeﬃcients. Prove that p P ( a ) P ( b ) P ( ab ) for all positive real numbers a and b . 5. Suppose we have a necklace of n beads. Each bead is labelled with an integer, and the sum of all these labels is
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Unformatted text preview: n-1. Prove that we can cut the necklace to form a string whose consecutive labels x 1 , x 2 , . . . , x n satisfy k X i =1 x i ≤ k-1 for all k = 1 , 2 , . . . , n . 6. If π is a permutation of { 1 , 2 , . . . , n } , prove that π (1) π (2)+ π (2) π (3)+ ··· + π ( n-1) π ( n )+ π ( n ) π (1) ≤ 2 n 3 + 3 n 2-11 n + 18 6 , with equality achieved for at least one permutation π for every n ≥ 2. 1...
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## This note was uploaded on 03/13/2010 for the course MATH 4801 taught by Professor Staff during the Spring '08 term at Georgia Tech.

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