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Homework 2 Solutions

# Homework 2 Solutions - HOMEWORK 2 COMMENTS 1 Non-book...

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HOMEWORK 2: COMMENTS 1. Non-book problems (2) (a) Since p | a and p | b there exist integers q 1 and q 2 such that a = pq 1 and b = pq 2 . Hence c = p ( q 1 - q 2 ) , and so p | c . (b) It does not follow. For example, let a = 13 b = 1 , c = 12 and p = 2. Then a = b + c and p | c , yet p does not divide a . (4) (b) We may apply the result of part (a) three times to conclude that x 2 + y 2 2 xy, y 2 + z 2 2 yz, and z 2 + x 2 2 zx. Adding these three inequalities we obtain 2( x 2 + y 2 + z 2 ) 2( xy + yz + zx ) . The result now follows after we divide the above inequality by 2. (5) Recall we proved the following lemma in class: Lemma 1.1. Let x, y, z and w be nonnegative real numbers. Suppose that x z and y w . Then xy zw . Since a, b and c are nonnegative real numbers, it follows from the AG-mean in- equality that ( a + b ) 2 ab, ( b + c ) 2 bc, and ( c + a ) 2 ca. We may multiply the above inequalities (thanks to Lemma 1.1, applied twice) to obtain ( a + b )( b + c )( c + a ) 8 ab bc ca = 8 abc.

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