Putnam 1994

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Problem B1

For a positive integer n, let f(n) be the number of perfect squares d such that |n - d| ≤ 250. Find all n such that f(n) = 15. [The perfect squares are 0, 1, 4, 9, 16, ... .]

 

Solution

Answer: 315 - 325 and 332 - 350.

Straightforward.

We may readily compile the following table: 


0  -250	    ≥ 16      02 - 152

251-299     ≥ 16      72 - 222

300-314	      16       82 - 232

315-325       15       92 - 232

326-331       16       92 - 242

332-350       15      102 - 242

351-371       14      112 - 242

252 - 112 = 504 > 500 and hence the gap (n+15)2 - (n+1)2 > 500 for all n > 10. So no numbers n > 112 + 250 = 371 have f(n) > 14.

 


 

Putnam 1994

© John Scholes
jscholes@kalva.demon.co.uk
12 Dec 1998