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      Posted · Report

      Fun with real numbers

      For part I. 

      I get K=9.

      The inequality holds for K=0.

      Assume there is a largest K for which the inequality holds for all 0 <= a <= b <= c <= d. Then for some a=0, b=c=d > a= 0 we get (a + b + c + d)^2 >= K b c <=> (3d)^2 >=Kd^2. Since d>0, this means K<=9.

      For all 0 <= a <= b <= c <= d the following holds true:

      (a + b + c + d)^2 >= (0 + b + c + c)^2 = (b+2*c)^2 = b^2 + (2*c)^2 + 2*b*(2*c) = b^2 + c^2 + 3*c^2 + 2*b*c + 2*b*c =(b+c)^2 + 3*c^2 +2*b*c >= 4*b*c + 3*b*c + 2*b*c = 9*b*c.

      Last step uses at once:

      - (b+c)^2 >= 4bc which is equivalent to (b-c)^2 >=0

      - 3*c^2 = 3*c * c >= 3*b*c since c>=b, c,b>=0

      Equality holds for a=0, d=c, b=c.

       

       

       

      0

      Posted · Report

      Anagrams for Fun - I: Defintions?

      3. The Morse Code

      0

      Posted · Report

      Anagrams for Fun - I: Defintions?

      I'll get things started 

      1. Dormitory
      2. Desperation
      3. ??
      4. Slot Machines
      5. Animosity
      6.  ??
      7. Alec Guinness (only know that one from Simpsons!)
      8. Semolina
      9. ??
      10. ??
      11. ??
      12. Eleven plus two
      13. ??

      A couple more...

      10) Decimal Point

      11) The earthquakes

      13) Contradiction

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      Posted · Report

      Anagrams for Fun - II: Quotations
      1. In one of the Bard's best thought of tragedies, our insistent hero, Hamlet, queries on two fronts about how life turns rotten.
         
      2. A thin man ran; makes a large stride; left planet, pins flag on moon! On to Mars!
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      Posted (edited) · Report

      Anagrams for Fun - I: Defintions?

      An Anagram, as we all know, is a word or phrase made by transposing or rearranging the letters of another word or phrase. The following phrases can be anagrammed into their definition or description. Honor code is in force: :rolleyes:

      1. Dirty Room          (Clue: single word beginning with D.)
      2. A Rope Ends It
      3. Here Come Dots
      4. Cash Lost in 'em
      5. Is No Amity
      6. Alas! No More Z's
      7. Genuine Class
      8. Is No Meal
      9. Large Picture Halls, I bet
      10. I'm a Dot in Place
      11. That Queer Shake
      12. Twelve plus one
      13. Accord not in it

       

      Edited by bonanova
      0