Forums

New Brain Teasers (User Submitted)

 205349
 posts

 25861
 posts

 238
 posts

 159
 posts


Old Brain Teasers

 7040
 posts

 1938
 posts

 89
 posts


Miscellaneous

 85737
 posts

 139
 posts

 1858
 posts

 553
 posts

 3012
 posts

 295
 posts


Who's Online 2 Members, 0 Anonymous, 61 Guests (See full list)

Popular Contributors

Posts
 Dormitory
 Desperation
 ??
 Slot Machines
 Animosity
 ??
 Alec Guinness (only know that one from Simpsons!)
 Semolina
 ??
 ??
 ??
 Eleven plus two
 ??
 In one of the Bard's best thought of tragedies, our insistent hero, Hamlet, queries on two fronts about how life turns rotten.
 A thin man ran; makes a large stride; left planet, pins flag on moon! On to Mars!
 Dirty Room (Clue: single word beginning with D.)
 A Rope Ends It
 Here Come Dots
 Cash Lost in 'em
 Is No Amity
 Alas! No More Z's
 Genuine Class
 Is No Meal
 Large Picture Halls, I bet
 I'm a Dot in Place
 That Queer Shake
 Twelve plus one
 Accord not in it
araver 3
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 (bc)^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.
0Rainman 14
Pickett 11
Anagrams for Fun  I: Defintions?
I'll get things started
A couple more...
10) Decimal Point
11) The earthquakes
13) Contradiction
0bonanova 58
Anagrams for Fun  II: Quotations
0bonanova 58
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:
0