Qualifying Quiz

### Problem 1

In Problem 1b, do the side lengths a, b, c, and d need to be integers?

No, the side lengths can be any positive real numbers.

In Problem 1b, what freedom do we have in cutting the second rectangle?

The second rectangle should be cut along a line segment that starts on the bottom edge and ends at the top edge; you can choose where it starts and ends as necessary to make your solution work.

### Problem 3

Does <example question> count as a paradox?

The universe explodes in paradox if, no matter which answer the mathematician gave, an observer would immediately be able to say, "That violated the rules that mathematician was supposed to follow." Some questions just don't have clear answers, or don't have known answers, and these will not cause the universe to explode. For example, nothing happens if you ask "Is the statement, 'This statement is false,' true?" or "Is the Riemann hypothesis true?" or "Will it rain tomorrow?"

Are compound questions allowed?

Questions can ask about several clauses (combined with, e.g., 'and', 'or', 'if'), but an answer will still only have a single truth value. For example, if you asked, "Is Mathcamp awesome, and is the sky green?", the true answer would be "no" because only one of the two parts connected by "and" is true.

### Problem 4

Do we have access to a random number generator? If so, what kind?

Yes; you should describe a method of giving out the slices of cake to Mathcampers which involves making some decisions at random at some point. You can use any tools you like to do this; all that needs to be true is that every time you make a random decision between several options, it's clear what the probability of each option is.

Are the numbers x1, x2, ..., xn real or rational?

We intended them to be real numbers. That being said, if you have a solution that only works for rational numbers, we'd also be very interested in seeing it! (It's possible that your solution would be equivalent to a fully general solution, if only you had used a slightly different tool to make random decisions.)

### Problem 5

In Problem 5a, are you sure that the minimum length is correct?

We're sure, but it's easy to get a smaller minimum length if you don't follow all the rules of stitching. If you're getting a smaller answer, check that every front stitch is the diagonal of a single 1 by 1 square; check that front and back stitches alternate; check that every back stitch has a positive length.

### Problem 6

What are the "hops" allowed in unchanging rule III?

Imagine starting on one circle, moving to a second circle that shares a marked point with the first, then maybe moving to a third circle that shares a marked point with the second, and repeating this any number of times you like. You should be able to get from any circle to any other this way. The purpose of this condition is connectedness: otherwise, you'd be able to solve Problem 6a by drawing two copies of our example diagram side-by-side!

In Problem 6a, the circles are required to be equal in size; is this true for the rest of the problem?

Very much no, because in the rest of the problem, they might no longer be geometrical objects! They might be drawn with circles of the same size, or with circles of different sizes, but they might not be representable by circles at all. All that matters is which marked points are part of which circles, and of course that the unchanging rules are satisfied.

Do all intersection points between circles have to be marked?

No, you can choose which points on the circles to mark to satisfy the "unchanging rules". (The rules don't apply to intersection points that are not marked points.)