The numbers
1/1, 1/2, ....., 1/2010
are written on a blackboard. A student chooses any two of the numbers, say
x, y, erases them, and then writes down xy+x+y.
He continues to do this until only one number is left on the blackboard. What is that number?
The process is commutative and also associative:
f(x,y) = xy+x+y = (x+1)(y+1) - 1
f(f(x,y),z)=f(x,f(y,z)) = (x+1)(y+1)(z+1) - 1
With reciprocals f(1/a,1/b)= (1+a+b)/(ab) = [(a+1)(b+1)-ab]/(ab)=(a+1)(b+1)/(ab) - 1
So with the given list
(2*3*...*2011)/(1*2*...*2010) - 1 = 2010
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Posted by Jer
on 2025-04-07 07:52:32 |
Solution
