Problem Set 3. Inequalities.

# Problem Set 3. Inequalities. - x 2-z 2 2 z 2 1 ≥ for all...

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Problem Seminar. Fall 2013. Problem Set 3. Inequalities. Classical results. 1. Jensen. For any convex function f , real x 1 , x 2 , . . . , x n and non-negative real α 1 , α 2 , . . . , α n such that n i =1 α i = 1 f ( α 1 x 1 + α 2 x 2 + . . . + α n x n ) α 1 f ( x 1 ) + α 2 f ( x 2 ) + . . . + α n f ( x n ) . 2. AM-GM. For any non-negative real x 1 , x 2 , . . . , x n n x 1 x 2 . . . x n x 1 + x 2 + . . . + x n n . 3. Cauchy-Schwarz. For any real x 1 , x 2 , . . . , x n , y 1 , y 2 , . . . , y n ( x 1 y 1 + x 2 y 2 + . . . + x n y n ) 2 ( x 2 1 + x 2 2 + . . . + x 2 n )( y 2 1 + y 2 2 + . . . + y 2 n ) . Problems. 1. Show that 1 2 · 3 4 · 5 6 · . . . · 2 n - 1 2 n < 1 2 n + 1 . 2. IMO 1994. Let m and n be positive integers. Let a 1 , a 2 , . . . , a m be distinct elements of { 1 , 2 , . . . n } such that whenever a i + a j n for some i, j (possibly the same) we have a i + a j = a k for some k . Prove that: a 1 + a 2 + . . . + a m m n + 1 2 . 3. GA 127. Prove that
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Unformatted text preview: x 2-z 2 2 z 2 + 1 ≥ . for all real numbers x,y,z . 4. GA 133. Let a,b,c ≥ ,a + b + c = 1 . Prove that ≤ ab + bc + ac-2 abc ≤ 7 27 . 5. Put 2003. A4. Let a,b,c,A,B,C be real, a,A non-zero such that | ax 2 + bx + c | ≤ | Ax 2 + Bx + c | for all real x . Show that | b 2-4 ac | ≤ | B 2-4 AC | . 6. Put 2003. B6. Show that Z 1 Z 1 | f ( x ) + f ( y ) | dx dy ≥ Z 1 | f ( x ) | dx for any continuous real-valued function f on [0 , 1] ....
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