1953 Putnam

Part of Putnam

Subcontests

(14)

1

Putnam 1953 B7

w\in (0,1)

be an irrational number. Prove that

w

has a unique convergent expansion of the form

w= \frac{1}{p_0} - \frac{1}{p_0 p_1 } + \frac{1}{ p_0 p_1 p_2 } - \frac{1}{p_0 p_1 p_2 p_3 } +\ldots,

1\leq p_0 < p_1 < p_2 <\ldots

are integers. If

w= \frac{1}{\sqrt{2}},

p_0 , p_1 , p_2.

1

Putnam 1953 B6

P

Q

be any points inside a circle

C

O

OP=OQ.

Determine the location of a point

Z

C

PZ+QZ

1

Putnam 1953 B5

Show that the roots of

x^4 +ax^3 +bx^2 +cx +d

, if suitably numbered, satisfy the relation

\frac{r_1 }{r_2 } = \frac{ r_3 }{r _4},

a^2 d=c^2 \ne 0.

1

Putnam 1953 B4

Determine the equations of a surface in three-dimensional cartesian space which has the following properties: (a) it passes through the point

(1,1,1)

and (b) if the tangent plane is drawn at any point

P

X,Y, Z

are the intersections of this plane with the

x, y

z-

axis respectively, then

P

is the orthocenter of the triangle

XYZ.

1

Putnam 1953 B3

Solve the equations

\frac{dy}{dx}=z(y+z)^n, \;\; \; \frac{dz}{dx} = y(y+z)^n,

given the initial conditions

y=1

z=0

x=0.

1

Putnam 1953 B2

a_0 ,a_1 , \ldots, a_n

be real numbers and let

f(x) =a_n x^n +\ldots +a_1 x +a_0.

f(i)

is an integer for all

i.

n! \cdot a_k

is an integer for each

k.

1

Putnam 1953 B1

Is the infinite series

\sum_{n=1}^{\infty} \frac{1}{n^{1+\frac{1}{n}}}

1

Putnam 1953 A7

Assuming that the roots of

x^3 +px^2 +qx +r=0

are all real and positive, find the relation between

p,q,r

which is a necessary and sufficient condition that the roots are the cosines of the angles of a triangle.

1

Putnam 1953 A6

Show that the sequence

\sqrt{7} , \sqrt{7-\sqrt{7}}, \sqrt{7-\sqrt{7-\sqrt{7}}}, \ldots

converges and evaluate the limit.

1

Putnam 1953 A4

From the identity \int_{0}^{\pi \slash 2} \log \sin 2x \, dx = \int_{0}^{\pi \slash 2} \log \sin x \, dx + \int_{0}^{\pi \slash 2} \log \cos x \, dx +\int_{0}^{\pi \slash 2} \log 2 \, dx, deduce the value of \int_{0}^{\pi \slash 2} \log \sin x \, dx.

1

Putnam 1953 A1

Prove that for every positive integer

n

\frac{2}{3} n \sqrt{n} < \sqrt{1} + \sqrt{2} +\ldots +\sqrt{n} < \frac{4n+3}{6} \sqrt{n}.

1

1953 Putnam A5

S is a parabola with focus F and axis L. Three distinct normals to S pass through P. Show that the sum of the angles which these make with L less the angle which PF makes with L is a multiple of π.

1

1953 Putnam A3

a, b, c

are real, and the sum of any two is greater than the third.
Show that

\frac{2(a + b + c)(a^2 + b^2 + c^2)}{3} > a^3 + b^3 + c^3 + abc

1

1953 Putnam A2

The complete graph with 6 points and 15 edges has each edge colored red or blue. Show that we can find 3 points such that the 3 edges joining them are the same color.