MathDB

p1. A matrix is a rectangular array of numbers. For example,

\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}

and

\begin{pmatrix} 1 & 3 \\ 2 & 4 \end{pmatrix}

are

2 \times 2

matrices. A saddle point in a matrix is an entry which is simultaneously the smallest number in its row and the largest number in its column. a. Write down a

2 \times 2

matrix which has a saddle point, and indicate which entry is the saddle point. b. Write down a

2 \times 2

matrix which has no saddle point c. Prove that a matrix of any size, all of whose entries are distinct, can have at most one saddle point.
p2. a. Find four different pairs of positive integers satisfying the equation

\frac{7}{m}+\frac{11}{n}=1

. b. Prove that the solutions you have found in part (a) are all possible pairs of positive integers satisfying the equation

\frac{7}{m}+\frac{11}{n}=1

.
p3. Let

ABCD

be a quadrilateral, and let points

M, N, O, P

be the respective midpoints of sides

AB

BC

CD

DA

. a. Show, by example, that it is possible that

ABCD

is not a parallelogram, but

MNOP

is a square. Be sure to prove that your construction satisfies all given conditions. b. Suppose that

MO

is perpendicular to

NP

. Prove that

AC = BD

.
p4. A Pythagorean triple is an ordered collection of three positive integers

(a, b, c)

satisfying the relation

a^2 + b^2 = c^2

. We say that

(a, b, c)

is a primitive Pythagorean triple if

1

is the only common factor of

a, b

, and

c

. a. Decide, with proof, if there are infinitely many Pythagorean triples. b. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form

(a, b, c)

where

c = b + 2

. c. Decide, with proof, if there are infinitely many primitive Pythagorean triples of the form

(a, b, c)

where

c = b + 3

.
p5. Let

x

and

y

be positive real numbers and let

s

be the smallest among the numbers

\frac{3x}{2}

\frac{y}{x}+\frac{1}{x}

and

\frac{3}{y}

. a. Find an example giving

s > 1

. b. Prove that for any positive

x

and

y,s <2

. c. Find, with proof, the largest possible value of

s

.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement