1956 AHSME Problems/Problem 43
Revision as of 22:38, 12 February 2021 by Coolmath34 (talk | contribs) (Created page with "== Problem 43== The number of scalene triangles having all sides of integral lengths, and perimeter less than <math>13</math> is: <math>\textbf{(A)}\ 1 \qquad\textbf{(B)}\...")
Problem 43
The number of scalene triangles having all sides of integral lengths, and perimeter less than 
is:
Solution
We can write all possible triangles starting with a minimum side length of 
![\[(1, 6, 6)\]](http://latex.artofproblemsolving.com/9/a/6/9a6cb018924af84a7af88cfb26a168896a4ab82a.png)
![\[(2, 5, 6)\]](http://latex.artofproblemsolving.com/8/1/7/81744e4c5d8f79985fe75aa7a3d01ac687afc7f1.png)
![\[(3, 4, 6), (3, 5, 7)\]](http://latex.artofproblemsolving.com/b/d/3/bd369b632328b75a1ca5bd53264b58447cae2fb8.png)
The first triple 
is not scalene, because two of the sides are equal. This leaves 
scalene triangles.
-coolmath34
(If you see any cases I missed out, edit them in.)
See Also
| 1956 AHSC (Problems • Answer Key • Resources) | ||
| Preceded by Problem 42 |
Followed by Problem 44 | |
| 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 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 | ||
| All AHSME Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
