[b]p1.[/b] A square is tiled by smaller squares as shown in the figure. Find the area of the black square in the middle if the perimeter of the square $ABCD$ is $14$ cm.
[img]https://cdn.artofproblemsolving.com/attachments/1/1/0f80fc5f0505fa9752b5c9e1c646c49091b4ca.png[/img]


[b]p2.[/b] If $a, b$, and $c$ are numbers so that $a + b + c = 0$ and $a^2 + b^2 + c^2 = 1$. Compute $a^4 + b^4 + c^4$.


[b]p3.[/b] A given fraction $\frac{a}{b}$ ($a, b$ are positive integers, $a \ne b$) is transformed by the following rule: first, $1$ is added to both the numerator and the denominator, and then the numerator and the denominator of the new fraction are each divided by their greatest common divisor (in other words, the new fraction is put in simplest form). Then the same transformation is applied again and again. Show that after some number of steps the denominator and the numerator differ exactly by $1$.


[b]p4.[/b] A goat uses horns to make the holes in a new $30\times 60$ cm large towel. Each time it makes two new holes. Show that after the goat repeats this $61$ times the towel will have at least two holes whose distance apart is less than $6$ cm.


[b]p5.[/b] You are given $555$ weights weighing $1$ g, $2$ g, $3$ g, $...$ , $555$ g. Divide these weights into three groups whose total weights are equal.


[b]p6.[/b] Draw on the regular $8\times 8$ chessboard a circle of the maximal possible radius that intersects only black squares (and does not cross white squares). Explain why no larger circle can satisfy the condition.


PS. You should use hide for answers. Collected [url=https://artofproblemsolving.com/community/c5h2760506p24143309]here[/url].
1. Given the conditions \(a + b + c = 0\) and \(a^2 + b^2 + c^2 = 1\), we need to find \(a^4 + b^4 + c^4\).
2. Consider the polynomial \(P(x) = x^3 + px^2 + qx + r\) with roots \(a, b, c\). By Vieta's formulas, we know:
   \[
   a + b + c = -p, \quad ab + bc + ca = q, \quad abc = -r
   \]
3. Given \(a + b + c = 0\), we have \(p = 0\).
4. Using the identity for the sum of squares:
   \[
   a^2 + b^2 + c^2 = (a + b + c)^2 - 2(ab + bc + ca)
   \]
   Substituting \(a + b + c = 0\) and \(a^2 + b^2 + c^2 = 1\), we get:
   \[
   1 = 0^2 - 2(ab + bc + ca) \implies ab + bc + ca = -\frac{1}{2}
   \]
5. To find \(a^4 + b^4 + c^4\), we use the identity:
   \[
   a^4 + b^4 + c^4 = (a^2 + b^2 + c^2)^2 - 2(a^2b^2 + b^2c^2 + c^2a^2)
   \]
6. We already know \(a^2 + b^2 + c^2 = 1\). Next, we need to find \(a^2b^2 + b^2c^2 + c^2a^2\). Using the square of the sum of products:
   \[
   (ab + bc + ca)^2 = a^2b^2 + b^2c^2 + c^2a^2 + 2abc(a + b + c)
   \]
   Since \(a + b + c = 0\), the term \(2abc(a + b + c)\) vanishes, leaving:
   \[
   \left(-\frac{1}{2}\right)^2 = a^2b^2 + b^2c^2 + c^2a^2 \implies \frac{1}{4} = a^2b^2 + b^2c^2 + c^2a^2
   \]
7. Substituting back into the identity for \(a^4 + b^4 + c^4\):
   \[
   a^4 + b^4 + c^4 = 1^2 - 2 \left(\frac{1}{4}\right) = 1 - \frac{1}{2} = \frac{1}{2}
   \]

The final answer is \( \boxed{ \frac{1}{2} } \)