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Problem of the Month (October 2017)

The pair of numbers 692 and 1118 have a very interesting property. Note that 6922 + 11182 = 1728788. And note that 6923 + 11183 = 1728788920 starts with exactly the same digits! What other pairs of positive integers x and y have the property that x3 + y3 starts with the digits of x2 + y2? More generally, what pairs of positive integers x and y have the property that xn + yn starts with the digits of xm + ym, where n>m>1? Even more generally, what collections of numbers {x, y, ... , z, m, n} have the property that xn + yn + . . . + zn starts with the digits of xm + ym + . . . + zm? What about other bases?


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