[b]p1.[/b] Prove that if $a, b, c, d$ are real numbers, then $$\max \{a + c, b + d\} \le \max \{a, b\} + \max \{c, d\}$$


[b]p2.[/b] Find the smallest positive integer whose digits are all ones which is divisible by $3333333$.


[b]p3.[/b] Find all integer solutions of the equation $\sqrt{x} +\sqrt{y} =\sqrt{2560}$.


[b]p4.[/b] Find the irrational number: $$A =\sqrt{ \frac12+\frac12 \sqrt{\frac12+\frac12 \sqrt{ \frac12 +...+ \frac12 \sqrt{ \frac12}}}}$$ ($n$ square roots).


[b]p5.[/b] The Math country has the shape of a regular polygon with $N$ vertexes. $N$ airports are located on the vertexes of that polygon, one airport on each vertex. The Math Airlines company decided to build $K$ additional new airports inside the polygon. However the company has the following policies: 
(i) it does not allow three airports to lie on a straight line, 
(ii) any new airport with any two old airports should form an isosceles triangle. 
How many airports can be added to the original $N$?


[b]p6.[/b] The area of the union of the $n$ circles is greater than $9$ m$^2$(some circles may have non-empty intersections). Is it possible to choose from these $n$ circles some number of non-intersecting circles with total area greater than $1$ m$^2$?


PS. You should use hide for answers.
To solve the problem, we need to find the value of \( A \) given the infinite nested radical expression:
\[ A = \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{ \frac{1}{2} + \frac{1}{2} \sqrt{ \frac{1}{2} + \cdots + \frac{1}{2} \sqrt{ \frac{1}{2}}}}} \]

1. **Assume the expression converges to a value \( A \):**
   \[ A = \sqrt{ \frac{1}{2} + \frac{A}{2} } \]

2. **Square both sides to eliminate the square root:**
   \[ A^2 = \frac{1}{2} + \frac{A}{2} \]

3. **Multiply through by 2 to clear the fraction:**
   \[ 2A^2 = 1 + A \]

4. **Rearrange the equation to standard quadratic form:**
   \[ 2A^2 - A - 1 = 0 \]

5. **Solve the quadratic equation using the quadratic formula \( A = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):**
   Here, \( a = 2 \), \( b = -1 \), and \( c = -1 \).
   \[ A = \frac{-(-1) \pm \sqrt{(-1)^2 - 4 \cdot 2 \cdot (-1)}}{2 \cdot 2} \]
   \[ A = \frac{1 \pm \sqrt{1 + 8}}{4} \]
   \[ A = \frac{1 \pm \sqrt{9}}{4} \]
   \[ A = \frac{1 \pm 3}{4} \]

6. **Evaluate the two possible solutions:**
   \[ A = \frac{1 + 3}{4} = 1 \]
   \[ A = \frac{1 - 3}{4} = -\frac{1}{2} \]

7. **Since \( A \) must be positive, we discard the negative solution:**
   \[ A = 1 \]

Thus, the value of \( A \) is \( \boxed{1} \).