MathDB

1

sinh inequality

a

and

b

be positive numbers. Find the largest number

c

, in terms of

a

and

b

, such that for all

x

with

0<|x|\le c

and for all

\alpha

with

0<\alpha<1

, we have:

a^\alpha b^{1-\alpha}\le\frac{a\sinh\alpha x}{\sinh x}+\frac{b\sinh x(1-\alpha)}{\sinh x}.

B5

1

QR which is QR+1

Let

p>2

be a prime. How many residues

\pmod p

are both squares and squares plus one?

B4

1

binomial sum congruence

Let

p>2

be a prime. Prove that

\sum_{n=0}^p\binom pn\binom{p+n}n\equiv2p+1\pmod{p^2}

.

B3

1

tiling with 4x6 and 5x7 rectangles

Can we find

N

such that all

m\times n

rectangles with

m,n>N

can be tiled with

4\times6

and

5\times7

rectangles?

A6

1

sequences, n-sum of type 1/2

An

n

-sum of type

1

is a finite sequence of positive integers

a_1,a_2,\ldots,a_r

, such that:

(1)

a_1+a_2+\ldots+a_r=n

;

(2)

a_1>a_2+a_3,a_2>a_3+a_4,\ldots, a_{r-2}>a_{r-1}+a_r

, and

a_{r-1}>a_r

. For example, there are five

7

-sums of type

1

, namely:

7

;

6,1

;

5,2

;

4,3

;

4,2,1

. An

n

-sum of type

2

is a finite sequence of positive integers

b_1,b_2,\ldots,b_s

such that:

(1)

b_1+b_2+\ldots+b_s=n

;

(2)

b_1\ge b_2\ge\ldots\ge b_s

;

(3)

each

b_i

is in the sequence

1,2,4,\ldots,g_j,\ldots

defined by

g_1=1

,

g_2=2

,

g_j=g_{j-1}+g_{j-2}+1

; and

(4)

if

b_1=g_k

, then

1,2,4,\ldots,g_k

is a subsequence. For example, there are five

7

-sums of type

2

, namely:

4,2,1

;

2,2,2,1

;

2,2,1,1,1

;

2,1,1,1,1,1

;

1,1,1,1,1,1,1

. Prove that for

n\ge1

the number of type

1

and type

2

n

-sums is the same.

A4

1

existence of infinite disks

Can we find an (infinite) sequence of disks in the Euclidean plane such that:

(1)

their centers have no (finite) limit point in the plane;

(2)

the total area of the disks is finite; and

(3)

every line in the plane intersects at least one of the disks?

A2

1

M^2+N^2 invertible, M^3=N^3, M^2N=N^2M

M

and

N

are real unequal

n\times n

matrices satisfying

M^3=N^3

and

M^2N=N^2M

. Can we choose

M

and

N

so that

M^2+N^2

is invertible?

A1

1

area under curve after rotation

The rectangle with vertices

(0,0)

,

(0,3)

,

(2,0)

and

(2,3)

is rotated clockwise through a right angle about the point

(2,0)

, then about

(5,0)

, then about

(7,0

), and finally about

(10,0)

. The net effect is to translate it a distance

10

along the

x

-axis. The point initially at

(1,1)

traces out a curve. Find the area under this curve (in other words, the area of the region bounded by the curve, the

x

-axis and the lines parallel to the

y

-axis through

(1,0)

and

(11,0)

).

B2

1

Cute functional equations

Define functions

f

and

g

as nonconstant, differentiable, real-valued functions on

R

. If

f(x+y)=f(x)f(y)-g(x)g(y)

,

g(x+y)=f(x)g(y)+g(x)f(y)

, and

f'(0)=0

, prove that

\left(f(x)\right)^2+\left(g(x)\right)^2=1

for all

x

.

A3

1

Putnam 1991 A3 and follow-up

Find all real polynomials

p(x)

of degree

n \ge 2

for which there exist real numbers

r_1 < r_2 < ... < r_n

such that (i) p(r_i) \equal{} 0, 1 \le i \le n, and (ii) p' \left( \frac {r_i \plus{} r_{i \plus{} 1}}{2} \right) \equal{} 0, 1 \le i \le n \minus{} 1. Follow-up: In terms of

n

, what is the maximum value of

k

for which

k

consecutive real roots of a polynomial

p(x)

of degree

n

can have this property? (By "consecutive" I mean we order the real roots of

p(x)

and ignore the complex roots.) In particular, is k \equal{} n \minus{} 1 possible for

n \ge 3

?

A5

1

Putnam 1991 A5

A5) Find the maximum value of

\int_{0}^{y}\sqrt{x^{4}+(y-y^{2})^{2}}dx

for

0\leq y\leq 1

. I don't have a solution for this yet. I figure this may be useful: Let the integral be denoted

f(y)

, then according to the [url=http://mathworld.wolfram.com/LeibnizIntegralRule.html]Leibniz Integral Rule we have

\frac{df}{dy}=\int_{0}^{y}\frac{y(1-y)(1-2y)}{\sqrt{x^{4}+(y-y^{2})^{2}}}dx+\sqrt{y^{4}+(y-y^{2})^{2}}

Now what?

B1

1

Sequence

For each integer

n\geq0

, let

S(n)=n-m^2

, where

m

is the greatest integer with

m^2\leq n

. Define a sequence by

a_0=A

and

a_{k+1}=a_k+S(a_k)

for

k\geq0

. For what positive integers

A

is this sequence eventually constant?

1991 Putnam

Subcontests

sinh inequality

QR which is QR+1

binomial sum congruence

tiling with 4x6 and 5x7 rectangles

sequences, n-sum of type 1/2

existence of infinite disks

M^2+N^2 invertible, M^3=N^3, M^2N=N^2M

area under curve after rotation

Cute functional equations

Putnam 1991 A3 and follow-up

Putnam 1991 A5

Sequence