The length of the three sides of a triangle are positive integers A, B and C, with A < B < C and satisfying:
{3^{A}/10^{4}}= {3^{B}/10^{4}}= {3^{C}/10^{4}}
where, {x}=x[x] and, [x] denotes the greatest integer ≤ x
Determine the minimum perimeter length of the triangle.
The instruction: {x}=x[x] and, [x] denotes the greatest integer less than or equal to x returns a fraction, less than 1, expressed in thousandths. Indeed, rather than do this, we might as well just inspect the last 3 digits of 3^n. It turns out that these are cyclic with period 100  see A216096 in Sloane (the caption is incorrect, and seems to have been switched with another entry).
To show that the values are distinct, the entries in the table at A216096 can conveniently be sorted after extracting the final 3 digits using the RIGHT function in Excel.
Now we are asked to find a triangle with suitable integer values, so that A+B>C: since the period is 100, A must be at least 101, giving 201 for B, and 301 for C. So the least perimeter is 603.
Note 1: 3^A,3^B,3^C are:
1546132562196033993109383389296863818106322566003,
7968419666276243080163439661073388804877003579601/
83487923724885217277472703906548983154097132003,
4106744371757651279739780821462649478993910868760/
12309414440570235106991532497229781400618467066824/
164751453321793982128440538198297087323698003,
respectively.
Note 2: Of the many ways in which this essentially mundane problem could be put, the current version could hardly be improved on for abstruseness.
Edited on September 20, 2013, 7:52 am

Posted by broll
on 20130920 07:26:26 