Average and Variance

Denote the average of $k$ numbers $x_1, …, x_k$ by $\bar{x} = \frac{1}{k} \sum_i x_i$. Their variance is defined as $\frac{1}{k} \sum_i \left( x_i – \bar{x} \right) ^ 2$.
Let $S(n)$ be the sum of all quadruples of integers $(a,b,c,d)$ satisfying $1 \leq a \leq b \leq c \leq d \leq n$ such that their average is exactly twice their variance.
For $n=5$, there are $5$ such quadruples, namely: $(1, 1, 1, 3), (1, 1, 3, 3), (1, 2, 3, 4), (1, 3, 4, 4), (2, 2, 3, 5)$.
Hence $S(5)=48$. You are also given $S(10^3)=37048340$.
Find $S(10^8)$. Give your answer modulo $433494437$.

The problem involves both advanced combinatorics and number theory. The primary difficulty is finding an efficient way to count the number of relevant quadruples.

First, we need to reformulate the problem in a way that is easier to handle. Instead of working with the average and variance, let’s define a new function $T(n)$ to be the sum of all 4-tuples $(a,b,c,d)$ with $1 \le a \le b \le c \le d \le n$ such that 4 times their sum is equal to n squared. We want to figure out how many quadruples (a, b, c, d) satisfy these conditions, and then sum up their values.

Essentially, mathematically we’re looking for solutions to the equation $4(a + b + c + d) = n^2$. We can rewrite this equation as $a + b + c + d = \frac{n^2}{4}$.

Considering the problem’s symmetry, we only care about $n^2$ being a multiple of 16. Then we can analyze it in terms of partitions of $\frac{n^2}{16}$ into four integer parts, which provides us a more combinatorial angle.

The partition problem can be computationally expensive. To make it easier, we can compute the partitions of the number once, and then use those results to calculate $T(n)$ for all $n$.

Finally, we sum up $T(n)$ for all $n$ from 1 to $10^8$ which are multiples of 4, and that gives us the desired $S(10^8)$. Since the numbers involved are very large, we use the modulo operation to keep the numbers manageable and find the result modulo $433494437$.

This algorithm would require advanced programming techniques and knowledge in combinatorics, number theory and dynamic programming to be implemented and executed.

Please note that this problem seems to be from the Project Euler. You can check the solution for this problem specifically on the Project Euler forum which is only accessible when you solve it there.

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