Question 97
We consider sequences $a_1, a_2, \dots, a_n$ consisting of $n$ integers. For given $k \le n$, we can partition the numbers of the sequence into $k$ groups as follows: $a_1$ goes in the first group, $a_2$ in the second group, and so on until $a_k$ which goes in the $k$-th group. Then $a_{k+1}$ goes in the first group again, $a_{k+2}$ in the second group, and so on. The sequence is called $k$-composite if this partition has the property that the sums of the numbers in the $k$ groups are equal.
The sequence $1, 2, 3, 4, -2, 6, 13, 12, 17, 8$, for instance, is 4-composite as
$$
1 + (-2) + 17 = 2 + 6 + 8 = 3 + 13 = 4 + 12.
$$
However, this sequence is not 3-composite, as the sums $1 + 4 + 13 + 8$, $2 + (-2) + 12$, and $3 + 6 + 17$ do not give equal outcomes.
a. Give a sequence of 6 *distinct* integers that is both 2-composite and 3-composite.
b. Give a sequence of 7 *distinct* integers that is 2-composite, 3-composite, and 4-composite.
c. Find the largest $k \le 99$ for which there exists a sequence of 99 *distinct* integers that is $k$-composite. (Give an example of such a sequence and prove that such a sequence does not exist for greater values of $k$.)
The sequence $1, 2, 3, 4, -2, 6, 13, 12, 17, 8$, for instance, is 4-composite as
$$
1 + (-2) + 17 = 2 + 6 + 8 = 3 + 13 = 4 + 12.
$$
However, this sequence is not 3-composite, as the sums $1 + 4 + 13 + 8$, $2 + (-2) + 12$, and $3 + 6 + 17$ do not give equal outcomes.
a. Give a sequence of 6 *distinct* integers that is both 2-composite and 3-composite.
b. Give a sequence of 7 *distinct* integers that is 2-composite, 3-composite, and 4-composite.
c. Find the largest $k \le 99$ for which there exists a sequence of 99 *distinct* integers that is $k$-composite. (Give an example of such a sequence and prove that such a sequence does not exist for greater values of $k$.)