MathDB

p1. For what integers

x

is it possible to find an integer

y

such that

x(x + 1) (x + 2) (x + 3) + 1 = y^2 ?

p2. Two tangents to a circle are parallel and touch the circle at points

A

and

B

, respectively. A tangent to the circle at any point

X

, other than

A

B

, meets the first tangent at

Y

and the second tangent at

Z

. Prove

AY \cdot BZ

is independent of the position of

X

.
p3. If

a, b, c

are positive real numbers, prove that

8abc \le (b + c) (c + a) (a + b)

by first verifying the relation in the special case when

c = b

.
p4. Solve the equation

\frac{x^2}{3}+\frac{48}{x^2}=10 \left( \frac{x}{3}-\frac{4}{x}\right)

p5. Tom and Bill live on the same street. Each boy has a package to deliver to the other boy’s house. The two boys start simultaneously from their own homes and meet

600

yards from Bill's house. The boys continue on their errand and they meet again

700

yards from Tom's house. How far apart do the boy's live?
p6. A standard set of dominoes consists of

28

blocks of size

1

2

. Each block contains two numbers from the set

0,1,2,...,6

. We can denote the block containing

2

and

3

[2, 3]

, which is the same block as

[3, 2]

. The blocks

[0, 0]

[1, 1]

,...,

[6, 6]

are in the set but there are no duplicate blocks.
a) Show that it is possible to arrange the twenty-eight dominoes in a line, end-to-end, with adjacent ends matching, e. g.,

... [3, 1]

[1, 1]

[1, 0]

[0, 6] ...

.
b) Consider the set of dominoes which do not contain

0

. Show that it is impossible to arrange this set in such a line.
c) Generalize the problem and prove your generalization.
PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here.

Problem Statement