MathDB

p1. Determine all possible values of

k

so that there are no real number pairs

(x, y)

which satisfies the system of equations:

x^2-y^2 = 0

(x-k)^2 + y^2 = 1

p2. A number is said to be beautiful if it satisfies the following two conditions: (a) It is a perfect square, that is, the square of a natural number. (b) If the rightmost digit in the decimal writing is moved its position becomes the leftmost digit, then the number formed is still a perfect square .
For example,

441

is a beautiful number consisting of

3

digits, because

441 =21^2

and

144 = 12^2

. While

144

is not a pretty number because

144 = 12^2

but

414

is not a perfect square. Prove that there are beautiful numbers whose decimal representation consists of exactly

2011

digits..
p3. Let

A

be the set of all positive divisors of

10^9

. If two are chosen, any number

x

and

y

A

(may be the same), find the probability that

x

divides

y

.
[url=https://artofproblemsolving.com/community/c6h2371634p19389707]p4. Given a rectangle

ABCD

with

AB = a

and

BC = b

. Point

O

is the intersection of the two diagonals. Extend the side of the

BA

so that

AE = AO

, also extend the diagonal of

BD

so that

BZ = BO.

Assume that triangle

EZC

is equilateral. Prove that (i)

b = a\sqrt3

(ii)

EO

is perpendicular to

ZD

p5. Let

M

be a subset of

\{1, 2, 3, ..., 12, 13\}

and there are no three member of

M

whose product is a perfect square. Specify the maximum number of possible

M

members.

Problem Statement