MathDB

1995 Putnam

Part of Putnam

Subcontests

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Putnam 1995 A5

Let x1,x2,⋯ ,xnx_1,x_2,\cdots, x_n be real valued differentiable functions of a variable tt which satisfy \begin{align*} & \frac{\mathrm{d}x_1}{\mathrm{d}t}=a_{11}x_1+a_{12}x_2+\cdots+a_{1n}x_n\\ & \frac{\mathrm{d}x_2}{\mathrm{d}t}=a_{21}x_1+a_{22}x_2+\cdots+a_{2n}x_n\\ & \;\qquad \vdots \\ & \frac{\mathrm{d}x_n}{\mathrm{d}t}=a_{n1}x_1+a_{n2}x_2+\cdots+a_{nn}x_n\\ \end{align*} For some constants aij>0a_{ij}>0. Suppose that lim⁡t→∞xi(t)=0\lim_{t \to \infty}x_i(t)=0 for all 1≤i≤n1\le i \le n. Are the functions xix_i necessarily linearly dependent?

Putnam 1995 B5

A game starts with four heaps of beans, containing 3, 4, 5 and 6 beans. The two players move alternately. A move consists of taking (a) either\text{either} one bean from a heap, provided at least two beans are left behind in that heap, (b) or\text{or} a complete heap of two or three beans. The player who takes the last heap wins. To win the game, do you want to move first or second? Give a winning strategy.
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Putnam 1995 A3

The number d1d2⋯d9d_1d_2\cdots d_9 has nine (not necessarily distinct) decimal digits. The number e1e2⋯e9e_1e_2\cdots e_9 is such that each of the nine 99-digit numbers formed by replacing just one of the digits did_i in d1d2⋯d9d_1d_2\cdots d_9 by the corresponding digit ei    (1≤i≤9)e_i \;\;(1 \le i \le 9) is divisible by 77. The number f1f2⋯f9f_1f_2\cdots f_9 is related to e1e2⋯e9e_1e_2\cdots e_9 is the same way: that is, each of the nine numbers formed by replacing one of the eie_i by the corresponding fif_i is divisible by 77. Show that, for each ii, di−fid_i-f_i is divisible by 77. [For example, if d1d2⋯d9=199501996d_1d_2\cdots d_9 = 199501996, then e6e_6 may be 22 or 99, since 199502996199502996 and 199509996199509996 are multiples of 77.]

Putnam 1995 B3

To each number with n2n^2 digits, we associate the n×nn\times n determinant of the matrix obtained by writing the digits of the number in order along the rows. For example : 8617↦det⁡(  86    17  )=508617\mapsto \det \left(\begin{matrix}{\;8}& 6\;\\ \;1 &{ 7\;}\end{matrix}\right)=50. Find, as a function of nn, the sum of all the determinants associated with n2n^2-digit integers. (Leading digits are assumed to be nonzero; for example, for n=2n = 2, there are 90009000 determinants.)
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Putnam 1995 A1

Let SS be a set of real numbers which is closed under multiplication (that is a,b∈S  ⟹  ab∈Sa,b\in S\implies ab\in S). Let T,U⊂ST,U\subset S such that T∩U=∅,T∪U=ST\cap U=\emptyset, T\cup U=S. Given that for any three elements a,b,ca,b,c in TT, not necessarily distinct, we have abc∈Tabc\in T and also if a,b,c∈Ua,b,c\in U, not necessarily distinct then abc∈Uabc\in U. Show at least one of TT and UU is closed under multiplication.

Putnam 1995 B1

For a partition π\pi of {1,2,3,4,5,6,7,8,9}\{1, 2, 3, 4, 5, 6, 7, 8, 9\}, let π(x)\pi(x) be the number of elements in the part containing xx. Prove that for any two partitions π\pi and π′\pi^{\prime}, there are two distinct numbers xx and yy in {1,2,3,4,5,6,7,8,9}\{1, 2, 3, 4, 5, 6, 7, 8, 9\} such that π(x)=π(y)\pi(x) = \pi(y) and \pi^{\prime}(x) = \pi^{\prime}(y).
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Putnam 1995 A6

Suppose that each of nn people writes down the numbers 1,2,31, 2, 3 in random order in one column of a 3×n3\times n matrix, with all orders equally likely and with the orders for different columns independent of each other. Let the row sums a,b,ca, b, c of the resulting matrix be rearranged (if necessary) so that a≤b≤ca \le b \le c. Show that for some n≥1995n \ge 1995 ,it is at least four times as likely that both b=a+1b = a+1 and c=a+2c = a+2 as that a=b=ca = b = c.

Putnam 1995 B6

For any a>0a>0,set S(a)={⌊na⌋∣n∈N}\mathcal{S}(a)=\{\lfloor{na}\rfloor|n\in \mathbb{N}\}. Show that there are no three positive reals a,b,ca,b,c such that S(a)∩S(b)=S(b)∩S(c)=S(c)∩S(a)=∅ \mathcal{S}(a)\cap \mathcal{S}(b)=\mathcal{S}(b)\cap \mathcal{S}(c)=\mathcal{S}(c)\cap \mathcal{S}(a)=\emptyset S(a)∪S(b)∪S(c)=N \mathcal{S}(a)\cup \mathcal{S}(b)\cup \mathcal{S}(c)=\mathbb{N}