MathDB

Problem 1. The longest side of a cyclic quadrilateral

ABCD

has length

a

, whereas the circumradius of

\triangle{ACD}

is of length 1. Determine the smallest of such

a

. For what quadrilateral

ABCD

results in

a

attaining its minimum?
Problem 2. In a box, there are 4 balls, each numbered 1, 2, 3 and 4. Anggi takes an arbitrary ball, takes note of the number, and puts it back into the box. She does the procedure 4 times. Suppose the sum of the four numbers she took note of, is 12. What's the probability that, while doing the mentioned procedure, that she always takes the ball numbered 3?
Problem 3. If

\alpha, \beta

and

\gamma

are the roots of

x^3 - x - 1 = 0

, determine the value of

\frac{1 + \alpha}{1 - \alpha} + \frac{1 + \beta}{1 - \beta} + \frac{1 + \gamma}{1 - \gamma}.

Problem 4. The lengths of three sides,

a, b, c

with

a \leq b \leq c

, of a right triangle, are integers. Determine all triples

(a, b, c)

so that the value of the perimeter and the area such triangle(s) are equal to each other.
Problem 5. Let

A

and

B

be two sets, each having consecutive natural numbers as their elements. The sum of the arithmetic mean of the elements of

A

and the arithmetic mean of the elements of

B

is 5002. If

A \cap B = \{2005\}

, then find the largest possible element of

A \cup B

Problem Statement