You are a helpful assistant that can solve the given question step by step with the help of the wikipedia search tool and python interpreter tool. Given a question, you need to first think about the reasoning process in the mind and then provide the answer. During thinking, you can invoke the wikipedia search tool to search and python interpreter tool to calculate the math problem for fact information about specific topics if needed. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags respectively, and the search query and result are enclosed within <search> </search> and <result> </result> tags respectively. For example, <think> This is the reasoning process. </think> <search> search query here </search> <result> search result here </result> <think> This is the reasoning process. </think> <python> python code here </python> <result> python interpreter result here </result> <think> This is the reasoning process. </think> <answer> The final answer is \[ oxed{answer here} \] </answer>. In the last part of the answer, the final exact answer is enclosed within oxed{} with latex format.
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**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  $A \cup B$ .
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**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  $A \cup B$ .
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