Live Challenge 2

[This post originally appeared on the Brilliant blog on 9/28/2012.]

The following challenges will be discussed this coming week. Remember to keep discussion of a challenge to its own blog post.

Monday: What is the minimum number of divisors for the 8-digit number $\overline{abbaabba}$, where $a$ and $b$ are integers from 1 to 9?

Clarification: The number $\overline{1221} = 1221$, and is not equal to $4=1 \times 2 \times 2 \times 1$.

Tuesday: The vertices of a regular 10-gon are labeled $V_1, V_2, \ldots V_n$, which is a permutation of $\{ 1, 2, \ldots, 10\}$. Define a <strong>neighboring sum</strong> to be the sum of 3 consecutive vertices $V_i, V_{i+1}$ and $V_{i+2}$ [where $V_{11}=V_1, V_{12}=V_2$]. For each permutation $\sigma$, let $N_\sigma$ denote the maximum neighboring sum. As $\sigma$ ranges over all permutations, what is the minimum value of $N_\sigma$?

Clarification: If the integers are written as $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$ around the circle, then the neighboring sums are $6, 9, 12, 15, 18, 21, 24, 27, 20, 13$, and the maximum neighboring sum is 27.

Thursday: Let $A$ be a number with 2012 digits such that $A$ is a multiple of $10!$. Let $B$ be the digit sum of $A$, $C$ be the digit sum of $B$, and $D$ be the digit sum of $C$. What is the <span style="text-decoration:underline;">unit’s digit</span> of $D$?

Note: The <strong>digit sum</strong> of a number is the sum of all its digits. For example the digit sum of 1123 is $1 + 1 + 2 + 3 = 7$.

Note: $10! = 10 \times 9 \times 8 \times \ldots \times 1$.

Friday: Consider an infinite chessboard, where the squares have side length of 1. The squares are colored black and white alternately. The (finite) radius of the largest circle which can be drawn completely on the white squares (hence you can see the entire circle) has a radius of $\frac {a\sqrt{b}} {c}$, where $a, b$ and $c$ are integers, $a$ and $c$ are coprime, and $b$ is not divisible by the square of any prime. What is the value of $a + b + c$?

Clarification: $a, b$ and $c$ are all allowed to be 1. In particular, if you think the the largest radius is $1 = \frac {1 \sqrt{1}}{1}$, then your answer to this should be $1+1+1=3$.

Do not simply state your numerical answer. Provide a complete solution so that other students can learn from it.