1971 AHSME Problems/Problem 6
Problem
Let 
be the symbol denoting the binary operation on the set 
of all non-zero real numbers as follows:
For any two numbers 
and 
of , 
. Then the one of the following statements which is not true, is
Solution
![\[a \ast b = b \ast a = 2ab\]](http://latex.artofproblemsolving.com/d/a/9/da98493b596a2b5f5e10d448fd4e2b3cbd2e3d0e.png)
Statement A is true.
![\[a \ast (b \ast c) = a \ast 2bc = 2a2bc = 4abc\]](http://latex.artofproblemsolving.com/4/7/a/47af5cba180e658900cb7d4484520f0144ee7be1.png)
![\[(a \ast b) \ast c = 2ab \ast c = 2(2ab)(c) = 4abc\]](http://latex.artofproblemsolving.com/1/1/b/11b920aa2324dbf366b81d71d884af14baf225d7.png)
Statement B is true.
![\[a \ast \frac{1}{2} = a\]](http://latex.artofproblemsolving.com/0/d/f/0dff967d0cda711f1bec225e5f939fe2b30d071c.png)
Statement C is true.
Statement D is true.
![\[\frac{1}{2a} \ast a = 1\]](http://latex.artofproblemsolving.com/a/c/4/ac497d0a34eb3576f69b01ee9f168d6372c9f381.png)
Since the identity is 
, not 1, we can see that statement E is false.
Hence, the answer is 
.
-edited by coolmath34 -fixed by DoctorSeventeen
See Also
| 1971 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 5 |
Followed by Problem 7 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 | ||
| All AHSME Problems and Solutions | ||
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