8/13/2019 Putnam 1995, Mathematics Competition

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The Fifty-Sixth William Lowell Putnam Mathematical CompetitionSaturday, December 2, 1995

A1 Let

be a set of real numbers which is closed undermultiplication (that is, if and are in

, then so is ).Let and be disjoint subsets of

whose union is

. Given that the product of any three (not necessarilydistinct) elements of is in and that the product of any three elements of is in , show that at least oneof the two subsets is closed under multiplication.

A2 For what pairs of positive real numbers does theimproper integral

converge?

A3 The number has nine (not necessarily dis-tinct) decimal digits. The number is suchthat each of the nine 9-digit numbers formed by replac-ing just one of the digits is by the corre-sponding digit ( ) is divisible by 7. Thenumber is related to is the sameway: that is, each of the nine numbers formed by replac-ing one of the by the corresponding is divisible by7. Show that, for each , is divisible by 7. [Forexample, if , then may be 2or 9, since and are multiples of 7.]

A4 Suppose we have a necklace of beads. Each bead islabeled with an integer and the sum of all these labelsis . Prove that we can cut the necklace to form astring whose consecutive labels satisfy

for

A5 Let be differentiable (real-valued) func-tions of a single variable which satisfy

..

.

..

.

for some constants . Suppose that for all ,as . Are the functions

necessarily linearly dependent?

A6 Suppose that each of people writes down the numbers1,2,3 in random order in one column of a matrix,with all orders equally likely and with the orders for

different columns independent of each other. Let therow sums of the resulting matrix be rearranged(if necessary) so that . Show that for some

, it is at least four times as likely that bothand as that .

B1 For a partition of , let bethe number of elements in the part containing . Provethat for any two partitions and , there are two dis-tinct numbers and in suchthat and . [A partition of a set

is a collection of disjoint subsets (parts) whoseunion is

.]

B2 An ellipse, whose semi-axes have lengths and , rollswithout slipping on the curve . How are

related, given that the ellipse completes one rev-

olution when it traverses one period of the curve?B3 To each positive integer with decimal digits, we as-

sociate the determinant of the matrix obtained by writ-ing the digits in order across the rows. For example, for

, to the integer 8617 we associate

. Find, as a function of , the sum of all the determi-nants associated with -digit integers. (Leading digitsare assumed to be nonzero; for example, for ,there are 9000 determinants.)

B4 Evaluate

Express your answer in the form , whereare integers.

B5 A game starts with four heaps of beans, containing 3,4,5and 6 beans. The two players move alternately. A moveconsists of taking either

a) one bean from a heap, provided at least two beansare left behind in that heap, or

b) a complete heap of two or three beans.

The player who takes the last heap wins. To win thegame, do you want to move rst or second? Give awinning strategy.

B6 For a positive real number , dene

Prove that cannot be expressed as the dis- joint union of three sets

and

. [Asusual, is the greatest integer .]