1974 Putnam

Part of Putnam

Subcontests

(12)

1

Putnam 1974 B6

n

elements, how many subsets are there whose cardinality is respectively

\equiv 0

3

\equiv 1

3

\equiv 2

3

)? In other words, calculate

s_{i,n}= \sum_{k\equiv i \;(\text{mod} \;3)} \binom{n}{k}

i=0,1,2

. Your result should be strong enough to permit direct evaluation of the numbers

s_{i,n}

and to show clearly the relationship of

s_{0,n}, s_{1,n}

s_{2,n}

to each other for all positive integers

n

. In particular, show the relationships among these three sums for

n = 1000

1

Putnam 1974 B5

1+\frac{n}{1!} + \frac{n^{2}}{2!} +\ldots+ \frac{n^{n}}{n!} > \frac{e^{n}}{2}

for every integer

n\geq 0.

1

Putnam 1974 B4

f: \mathbb{R}^{2} \rightarrow \mathbb{R}

is said to be continuous in each variable separately if, for each fixed value

y_0

y

f(x, y_0)

is contnuous in the usual sense as a function in

x,

f(x_0 , y)

is continuous as a function of

y

x_0

f: \mathbb{R}^{2} \rightarrow \mathbb{R}

be continuous in each variable separately. Show that there exists a sequence of continuous functions

g_n: \mathbb{R}^{2} \rightarrow \mathbb{R}

f(x,y) =\lim_{n\to \infty}g_{n}(x,y)

(x,y)\in \mathbb{R}^{2}.

1

Putnam 1974 B3

a

is a real number such that

\cos \pi a= \frac{1}{3},

a

1

Putnam 1974 B2

y(x)

be a continuously differentiable real-valued function of a real variable

x

y'(x)^2 +y(x)^3 \to 0

x\to \infty,

y(x)

y'(x) \to 0

x \to \infty.

1

Putnam 1974 B1

Which configurations of five (not necessarily distinct) points

p_1 ,\ldots, p_5

x^2 +y^2 =1

maximize the sum of the ten distances

\sum_{i<j} d(p_i, p_j)?

1

Putnam 1974 A6

n

k = k(n)

be the minimal degree of any monic integral polynomial

f(x)=x^k + a_{k-1}x^{k-1}+\ldots+a_0

such that the value of

f(x)

is exactly divisible by

n

for every integer

x.

Find the relationship between

n

k(n)

. In particular, find the value of

k(n)

corresponding to

n = 10^6.

1

Putnam 1974 A5

Consider the two mutually tangent parabolas

y=x^2

y=-x^2

. The upper parabola rolls without slipping around the fixed lower parabola. Find the locus of the focus of the moving parabola.

1

Putnam 1974 A4

An unbiased coin is tossed

n

times. What is the expected value of

|H-T|

H

is the number of heads and

T

is the number of tails?

1

Putnam 1974 A3

A well-known theorem asserts that a prime

p > 2

can be written as the sum of two perfect squares (

p = m^2 +n^2

m

n

integers) if and only if

p \equiv 1

4

). Assuming this result, find which primes

p > 2

can be written in each of the following forms, using integers

x

y

x^2 +16y^2,

4x^2 +4xy+ 5y^2.

1

Putnam 1974 A2

A circle stands in a plane perpendicular to the ground and a point

A

lies in this plane exterior to the circle and higher than its bottom. A particle starting from rest at

A

slides without friction down an inclined straight line until it reaches the circle. Which straight line allows descent in the shortest time?

1

Putnam 1974 A1

Call a set of positive integers "conspiratorial" if no three of them are pairwise relatively prime. What is the largest number of elements in any "conspiratorial" subset of the integers

1

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