1996 Putnam

Part of Putnam

Subcontests

(6)

2

Putnam 1996 A6

c\ge 0

be a real number. Give a complete description with proof of the set of all continuous functions

f: \mathbb{R}\to \mathbb{R}

f(x)=f(x^2+c)

x\in \mathbb{R}

Putnam 1996 B6

(a_1,b_1),(a_2,b_2),\ldots ,(a_n,b_n)

be the vertices of a convex polygon containing the origin in its interior. Prove that there are positive real numbers

x,y

(a_1,b_1)x^{a_1}y^{b_1}+(a_2,b_2)x^{a_2}y^{b_2}+\ldots +(a_n,b_n)x^{a_n}y^{b_n}=(0,0)

2

Putnam 1996 A5

p

be a prime greater than

3

p^2\Big| \sum_{i=1}^{\left\lfloor\frac{2p}{3}\right\rfloor}\dbinom{p}{i}.

Putnam 1996 B5

Given a finite binary string

S

X,O

\Delta(S)=n(X)-n(O)

n(X),n(O)

respectively denote number of

X

O

's in a string. For example

\Delta(XOOXOOX)=3-4=-1

. We call a string

S

\emph{balanced} if every substring

T

S

-2\le \Delta(T)\le 2

. Find number of balanced strings of length

n

2

Putnam 1996 A4

S

be a set of ordered triples

(a,b,c)

of distinct elements of a finite set

A

. Suppose that
[*]

(a,b,c)\in S\iff (b,c,a)\in S

(a,b,c)\in S\iff (c,b,a)\not\in S

(a,b,c),(c,d,a)\text{ both }\in S\iff (b,c,d),(d,a,b)\text{ both }\in S

Prove there exists

g: A\to \mathbb{R}

g

g(a)<g(b)<g(c)\implies (a,b,c)\in S

Putnam 1996 B4

For any square matrix

\mathcal{A}

\sin {\mathcal{A}}

by the usual power series.

\sin {\mathcal{A}}=\sum_{n=0}^{\infty}\frac{(-1)^n}{(2n+1)!}\mathcal{A}^{2n+1}

Prove or disprove :

\exists 2\times 2

A\in \mathcal{M}_2(\mathbb{R})

\sin{A}=\left(\begin{array}{cc}1 & 1996 \\0 & 1 \end{array}\right)

2

Putnam 1996 A3

Suppose that each of

20

students has made a choice of anywhere from

0

6

courses from a total of

6

courses offered. Prove or disprove : there are

5

2

courses such that all

5

have chosen both courses or all

5

have chosen neither course.

Putnam 1996 B3

S_n

be the set of all permutations of

(1,2,\ldots,n)

\max_{\sigma \in S_n} \left(\sum_{i=1}^{n} \sigma(i)\sigma(i+1)\right)

\sigma(n+1)=\sigma(1)

2

Putnam 1996 A2

\mathcal{C}_1

\mathcal{C}_2

be circles whose centers are

10

units apart, and whose radii are

1

3

. Find, with proof, the locus of all points

M

for which there exists points

X\in \mathcal{C}_1,Y\in \mathcal{C}_2

M

is the midpoint of

XY

Putnam 1996 B2

Prove the inequality for all positive integer

n

\left(\frac{2n-1}{e}\right)^{\frac{2n-1}{2}}<1\cdot 3\cdot 5\cdots (2n-1)<\left(\frac{2n+1}{e}\right)^{\frac{2n+1}{2}}

2

Putnam 1996 A1

Find the least number

A

such that for any two squares of combined area

1

, a rectangle of area

A

exists such that the two squares can be packed in the rectangle (without the interiors of the squares overlapping) . You may assume the sides of the squares will be parallel to the sides of the rectangle.

Putnam 1996 B1

Define a \emph{selfish} set to be a set which has its own cardinality as its element. And a set is a \emph{minimal }\text{ selfish} set if none of its proper subsets are \emph{selfish}. Find with proof the number of

\text{minimal selfish}

\{1,2,\cdots ,n\}