John H. Lindsey

The Total Length of Short Cycles of a Random Permutation: 10732

Editorial comment. From f (a) = 2 , a(i)2 2 a (i)a(i + 1), it follows that the largest value of Ua(i)at(i + 1) is I3n1I i2 (2n 3) = (2n3 + 3n2 _ 11n + 18)/6. This is problem B-3 from the 1996 William Lowell Putnam Mathematical Competition (this MONTHLY 104 (1997) 744-754). John H. Lindsey II and Ricardo Garcia-Pelayo mentioned that the examples in (1) and (2) are also extremal for a large class of sums of the form iZ=I h(a (i) a (i + 1)), where h is even, nonnegative, and convex.

Maximum Compared to Average: 11056

11056 [2004, 64]. Proposed by John H. Lindsey II, Cambridge, MA. Let n be a positive integer, and let p be a polynomial of degree at most n. Show that on any nonempty finite real interval I, maxxe/ Ip(x)\ is at most 4n2 times the average on I of Ip(x)|. Solution by Omran Kouba, Higher Institute for Applied Sciences and Technology, Damascus, Syria. We establish a stronger bound: maxxE/ Ip(x)l is at most (n + 1)2 times the average on I of Ip(x) . By an affine change of variable, we may assume that I = [-1, 1]. Recall that the Legendre polynomials, defined for n E N by

Digit Frequencies in Powers of Random Numbers: 10590

10590 [1997, 362]. Proposed by Robb Muirhead, University of Michigan, Ann Arbor; MI, and Stephen Portnoy, University of Illinois, Urbana, IL. Let X have a uniform distribution on the interval [0, 1] and let Nm,k be the digit in the mth place to the right of the decimal point in Xk. (a) Find limm?0, P(Nm,m = i) for i = 0, 1, 2,... . 9. (b) Characterize those functions k(m) for which limm,00 P(Nm,k(m) = i) = 1/10 for i =0, 1, 2, ... ,9.