3actually this was created for me,but i cant solve.
Q1 If a1,a2,..a101 is an arrangement of the numbers 1,2,...101 then prove that the product of (1-a1)(2-a2)...(101-a101) is an even integer
I wasnt in favour of starting aonther revision thread on P&C beocz already many exist.....but this is on request of "msp"
Answer will be confirmed tomorrow..and also new question will posted iff this is completely solved[1]
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10 Answers
62The hint here is read my post 2...
TOtal number of ai's is 101 which is odd..
some of these are themselves odd.. =51
Others are even = 50
Now is there a proof?
62Some one complete the proof.. This one is too simple to be hanging like this!
1there are 51 even numbers and and 50 odd
so we start placing ai in such a way that each bracket contains an odd and an even term
however by the last bracket, we would have exhausted all all even ai as well as numbers from the set {2,4,6,8,...,100}
so the last bracket would definately contain a difference of 2 odd numbers
so the product becomes even