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Suppose each of x and y is a real number > 0

Find the minimum value of:

 
2x2+2y2+3xy+1
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     x+y

Find the smallest whole number N such that N contains all of the digits from 0 through 9, and N² contains all of the digit pairs 00, 11, 22, ..., 99.

Determine all the numbers formed by three different and non-zero digits, such that the six numbers obtained by permuting these digits leaves the same remainder after the division by 4.

Arrange the integers from 1 to N in an order such that the sum of any two consecutive terms is a power of 2.

For what values of N do solutions exist?