MathDB

p1. In the rectangle of the figure whose base is twice the height, we construct the two quarter-circles shown and the circles tangent to both quarter-circles and to the previous one (except the first one that is tangent to the upper side of the rectangle). Let

R

denote the height of the rectangle and list the tangent circles in order of decreasing size: https://cdn.artofproblemsolving.com/attachments/7/f/12dca5a35c34b14f963d78b0889f9eff328276.jpg a) Prove that

d_n =\frac{R}{n (n + 1)}

, where

d_n

denotes the diameter of the

n

-th circumference. b) Prove that

\lim_{n \to \infty} \left( \frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+...+\frac{1}{n(n + 1)}\right)=1

p2. Let

x

be a real number such that

x +\frac{1}{x}

is integer. Prove that

x^{2007} +\frac{1}{x^{2007}}

is integer.
p3. On the island of Camelot, there are

13

red,

15

green and

17

yellow chameleons. When two different colors are found, they change simultaneously to the third color. Can the situation occur in which all chameleons have the same color? Justify your answer.
p4. Let

n

be a natural number. It is known that we can write

n^3

as the sum of

n

consecutive odd natural numbers. For instance

1^3 = 1

2^3 = 3 + 5

3^3 = 7 + 9 + 11

4^3 = 13 + 15 + 17 + 19

. a) Given an arbitrary natural number

n

, explicitly describe one way to determine the

n

of consecutive odd numbers used to write

n^3

as above. b) Generalize the above for

n^k

, where

k

is a natural number greater than or equal to

2

.
p5. If

a, b

and

c

are any three positive reals, prove that it is true

\frac{a + b}{c}+ \frac{ab + c}{a}+ \frac{ac + a}{b} \ge 6

p6. Let two parabolas, from equations:

y = cx^2 + d

(with

c> 0

and

d <0

) and

x = ay^2 + b

(with

a> 0

and

b <0

), which intersect at four points. Show that these four points belong to the same circumference.
PS. Seniors P3 was also [url=https://artofproblemsolving.com/community/c4h2689976p23346234]Juniors P3.

Problem Statement