**Summer Vacation**

... in the House of Math. Eudoxus, even from afar in Europe where he is discussing set theory and other things I don't understand, is making sure the girls keep their mathematical engines in tune by sending questions, and I am tasked with the usual grunt-work of math lessons. If the opinionated readership cares to try, you may match your wits against the opinionated offspringen.

**15-year-old level:**

(summer math team questions)

Let {a

_{1}, a

_{2}, ..., a

_{n}} be a set of real numbers, such that a

_{1}< a

_{2}< ... < a

_{n}. We define the

*power sum*of {a

_{1}, a

_{2}, ..., a

_{n}} to be a

_{1}

*i*

^{1}+ a

_{2}

*i*

^{2}+ ... + a

_{n}

*i*

^{n}.

Given any

*n*, let S

_{n}be the sum of the power sums of all non-empty subsets of the set {1,2,...,n}.

S

_{8}is -176 - 64

*i*. What is S

_{9}?

**8-year-old level:**

(sent from Daddy)

To prepare yourself for fighting the dragon, you need a suit of armor and either a sword or an axe. You have three suits of armor, five swords, and two axes. How many different ways can you prepare yourself for fighting the dragon?

(lesson work from Mommy)

Evaluate [-(216/125)]^(-2/3)

**3-year-old level:**

(sent from Daddy)

You have a pizza and a cookie to eat, and you're trying to decide what order to eat them in. One order is to eat the cookie first and the pizza second. What is the other order?

(lesson work from Mommy)

Find the difference between 6 and 4. [using number line]

Personally, I'm going for the 3-year-old level math questions for the win, Alex. No rest for the non-mathy; I'm working my way through the Art of Problem Solving math books, starting at Introduction to Algebra and planning to move on to Geometry, Counting & Probability, and Intermediate Algebra, in the hope of at last comprehending the math I aced in high school by dint of short-term memorization.

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