MathDB

p1. Let

a

and

b

be different positive real numbers so that

a +\sqrt{ab}

and

b +\sqrt{ab}

are rational numbers. Prove that

a

and

b

are rational numbers.
p2. Find the number of ordered pairs of natural numbers

(a, b, c, d)

that satisfy

ab + bc + cd + da = 2016.

p3. For natural numbers

k

, we say a rectangle of size

1 \times k

k\times 1

as strips. A rectangle of size

2016 \times n

is cut into strips of all different sizes . Find the largest natural number

n\le 2016

so we can do that.
Note:

1\times k

and

k \times 1

strips are considered the same size.
[url=https://artofproblemsolving.com/community/c6h1549013p9410660]p4. Let

PA

and

PB

be the tangent of a circle

\omega

from a point

P

outside the circle. Let

M

be any point on

AP

and

N

is the midpoint of segment

AB

MN

cuts

\omega

C

such that

N

is between

M

and

C

. Suppose

PC

cuts

\omega

D

and

ND

cuts

PB

Q

. Prove

MQ

is parallel to

AB

.
p5. Given a triple of different natural numbers

(x_0, y_0, z_0)

that satisfy

x_0 + y_0 + z_0 = 2016

. Every

i

-hour, with

i\ge 1

, a new triple is formed

(x_i,y_i, z_i) = (y_{i-1} + z_{i-1} x_{i-1}, z_{i-1} + x_{i-1}-y_{i-1}, x_{i-1} + y_{i-1}- z_{i-1})

Find the smallest natural number

n

so that at the

n

-th hour at least one of the found

x_n

y_n

, or

z_n

are negative numbers.

Problem Statement