MathDB

p1. Determine for which real numbers

x

the identity

|x + 1| = |x| + 1

is satisfied.
p2. In the rectangle in the figure, whose base is twice the height, we construct the two quarter-circles with centers at the lower vertices as shown. And the circles tangent to both quarter-circles and to the previous one (except the first which is tangent to the top side of the rectangle). Let

R

denote the height of the rectangle and list the circles tangent in order of decreasing sizes. Prove that

d_1 =\frac{R}{2}

, where

d_1

denotes the diameter of the first circle. https://cdn.artofproblemsolving.com/attachments/9/0/cb88775ffeedea002b75bc760d7c2ccfe4c197.jpg
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

. Determine the first and last of the

72

consecutive odd numbers used to represent

72^3

as above.
p5. Let

a

be a digit between

1

and

9

. We will denote

\underbrace{aa...a}_{n \,\, times}

the number whose decimal expression is formed by

n

digits equal to

a

\bullet

Prove that the identity

\underbrace{aa...a}_{n \,\, times}= a^n

is not satisfied for any integer

n> 1

\bullet

For no

n> 1

can

\underbrace{aa...a}_{n \,\, times}

be a perfect square.
p6. Find all pairs of prime numbers such that their sum and difference are also primes.
PS. Juniors P3 was also [url=https://artofproblemsolving.com/community/c4h2692684p23377175]Seniors P3.

Problem Statement