Originally Posted by
ebaines There are a lot more possibilities than these. Assuming that the total length of all sides may not exceed 1994, this does not mean that the total length has to equal 1994. For example a triangle with legs of 1, 1, and 2 satisfies the requirement.
Let y = length of the base and x = the lengths of the two equal sides. The requirement is that 2x + y <= 1994. Let's look at what happens for different values of y:
If y = 1, the x can be any value from 1 to (1994-1)/2 rounded down to an integer, or 996
if y= 2 then x can be any value from 1 to (1994-2)/2 = 996
if y = 3 then x can be any integer from 1 to 995
if y = 4 then x can be any integer from 1 to 995
etc.
...
if y = 1989 the x can be any integer from 1 to 2
if y = 1990 then x can be any value from 1 to 2
if y = 1991 then x = 1
if y = 1992 then x = 1.
So the total of all possibilities is $\displaystyle 2 \times \sum_{i=1} ^{996} i = 2 \times \frac {996 \times 997} 2 = 993,012$.
However, given the strange wording of the question perhaps OP means that no single leg may exceed length of 1994. If that's what he meant then there are 1994 choices for length of leg x and 1994 choices for leg y, so the total number of possibilities is 1994 x 1994 = 3,976,036.