It stands to reason that all men should finish at the same time. I can't explain why this is true, it just makes common sense.

Let me state up front that I made up this problem myself and I am not 100% positive there is no better answer. If anyone can think of a faster way to cross please e-mail me. That being said I strongly believe that the fastest way to get everyone across is for B and C to start out on foot and A to start out with the bicycle. At a point y A will get off the bicycle and walk the rest of the way. Eventually C will get to the bicycle abondoned by A, then ride backwards to a point x, leaving the bicycle there, then turning around and walk until he reaches the end. Person B will walk until he reaches the bicycle left by C and then ride the rest of the way.

Below are the times that each will take to cross, in terms of x and y:

A: 1*y + 10*(1-y)
B: 5*x + 1*(1-x)
C: 2*y + (y-x) + 2*(1-x)

Next equate these equations: 10 - 9y = -3x + 3y + 2 = 4x + 1.

To solve set up two linear equations:

10 - 9y = -3x + 3y + 2 -> 3x - 12y = -8
10 - 9y = 4x + 1 -> 4x + 9y = 9

Then solve for x and y:

x = 12/25, y=59/75.

Given these points it will take each person 73/25 = 2.92 minutes to cross.

Below are other crossing times, given various crossing times of the two fastest people, assuming the slowest still takes 10 minutes to cross.

 
                               Crossing time of fastest
        --------------------------------------------------------------------
Second
Fastest
Time	1	2	3	4	5	6	7	8	9
------- ----    ----    ----    ----    ----    ----    ----    ----    ----
1	1.00								
2	1.62	1.92							
3	1.90	2.41	2.74						
4	2.06	2.71	3.16	3.48					
5	2.16   (2.92)	3.45	3.85	4.16				
6	2.23	3.07	3.67	4.13	4.49	4.78			
7	2.29	3.18	3.84	4.35	4.75	5.08	5.34		
8	2.33	3.27	3.98	4.52	4.96	5.32	5.62	5.87	
9	2.36	3.34	4.09	4.67	5.14	5.52	5.85	6.12	6.36