The Qualifying Quiz is the centerpiece of the Mathcamp application: it's a set of 5-10 problems, designed to be fun, challenging, and thought-provoking for you, and to give us a good read on how you think about math. We assume no background outside of the standard high-school curriculum, but the problems are anything but standard: you'll have to think hard to solve them. If the problems stick with you, nag you in the shower, and make you want to learn more, then that's a good indication that Mathcamp's classes will be fun for you.
We call it a Quiz, but it's really a challenge: a chance for you to show us how you approach new problems and new concepts in mathematics. What matters to us are not just your final results, but your reasoning. Correct answers on their own will count for very little: you have to justify all your assertions and prove to us that your solution is correct. (For some general tips on writing proofs, see proof tips, or take a look at examples of solutions to past QQ problems.) Sometimes it may take a while to find the right way of approaching a problem. Be patient: there is no time limit on this quiz.
The problems are roughly in increasing order of difficulty, but even the later problems often have some easier parts. We don't expect every applicant to solve every problem: in the past, we have sometimes admitted people who could do only half of them, occasionally even fewer. However, don't just solve three or four problems and declare yourself done! The more problems you attempt, the better your chances. We strongly recommend that you try all the problems and send us the results of your efforts: partial solutions, conjectures, methods – everything counts. Also, if you come up with a solution that is messy and ugly, see if you can find a better way of thinking about the problem: elegance and clarity count too! None of the problems require a computer; you are welcome to use one if you'd like, but first read a word of warning about computers.
If you need clarification on any problem, please contact us. You may not consult or get help from anyone else. You can use books or the Web to look up definitions, formulas, or standard techniques, but any information obtained in this way must be clearly referenced in your solution. Please do not try to look for the problems themselves: we want to see how well you can do math, not how well you can use Google! For further clarification, read more about our policy on getting help. Any deviation from these rules will be considered plagiarism and may permanently disqualify you from attending Mathcamp.
You may handwrite or type your Quiz. For those interested in typing mathematics beautifully, we also offer you the LaTeX source file for the Qualifying Quiz. Please don't let it distract you from solving the problems, though! For more information on LaTeX, take a look at our LaTeX tutorial for The QQ. No matter how you write up your solutions, we require that you upload them with your application as PDF files; visit our PDF tutorial if you need help.
Finally: don't write your name on your Quiz. (Really!) Instead, write your applicant ID number on your Quiz. (You will receive an applicant ID number when you start your online application, on the welcome page.) Why not your name? When we grade Quizzes, we want to be thinking just about the math. So we're doing an experiment in which we evaluate all applicants just by ID number and then uncover the names later. We appreciate your help with the experiment.
Good luck and have fun!
Lotta is a consulting mathematician who specializes in very large numbers. She runs a business with 100 clients ranked 1 through 100 in order of importance. (The most important client is ranked 1.) Each day, Lotta has time to visit only one of her clients.
A client feels mistreated if Lotta has never visited them, or if Lotta has visited someone less important since the last time she visited them. Every day, Lotta visits the most important client that feels mistreated. On the first day, she visits client 1; on the second day, she visits client 2; on the third day, she goes back to client 1, and so on.
When none of Lotta's clients feel mistreated, she can finally retire.
This problem is about some curious relations between the sums of certain entries of Pascal's triangle. You may find the following background reading useful: Pascal's Triangle and Binomial Coefficients.
In this problem, we define
|k mod m|
|k + m|
|k + 2m|
where the sum includes every mth element between 0 and n inclusive, starting at k. For example,
|2 mod 5|
|0 mod 2|
|1 mod 2|
|0 mod 3|
|0 mod 5|
|1 mod 5|
|2 mod 5|
|3 mod 5|
|4 mod 5|
It's the week before Mathcamp, and the N mentors are frantically preparing their classes. The Mathcamp library is open around the clock, and each mentor visits it once in every 24 hour period. They follow a strict schedule: each mentor has a set time when they enter the library and a set time when they leave. (Some mentors work through the night, arriving in the evening and leaving in the morning.) No mentor arrives or departs at the exact same time as another: all 2N times are different.
It so happens that:
Let k be a positive integer, and let Ek be the equation
A solution to Ek is a pair of positive integers m and n that satisfy the equation. For example, solutions to E2017 include m = 3459, n = 2595 and m = 4484, n = 3588. (There are others as well.)
Wizards live in towers that have infinitely many floors, numbered 1, 2, 3, ... . The floors are not connected by staircases or any other mundane means. Instead, for every positive integer N, there is a red magic portal connecting floor N to floor N + 10, and a blue magic portal connecting floor N to floor 3N + 1. The portals go both ways; for example, starting at floor 26, you could descend to floor 16 by a red portal, descend to floor 5 by a blue portal, and ascend to floor 15 by a red portal.
Of course, an infinitely tall tower would have enough room for multiple wizards. But wizards refuse to share: two wizards refuse to both live in the tower if it's possible to get from one wizard's floor to the other wizard's floor using the magic portals.
(This problem first appeared on Mathcamp 2015's weekly Team Problem Solving competition.)
A triangle function assigns a nonnegative real number to all nondegenerate triangles. We call a triangle function f consistent if any two congruent triangles are assigned the same value: f(△ABC) = f(△DEF) whenever △ABC ≅ △DEF.
Suppose that for some △ABC, points X, Y, and Z are chosen in the interior of sides AB, AC, and BC, respectively. If a triangle function f satisfies
for all choices of △ABC, X, Y, and Z, we say that f has the triforce property. (In the diagram below, the function's values for each of the shaded triangles must add up to the function's value for the large triangle.)
Is there a consistent triangle function with the triforce property, other than the trivial function assigning the value 0 to every triangle?
Waldo and Carmen play a guessing game played in N cities located on a large ring around the earth. Two cities that are next to each other in the ring are chosen uniformly at random. Waldo is sent to one of them and Carmen is sent to the other. Neither player knows which of the adjacent cities the other is in (but each knows her or his own location).
Starting with Waldo, the players take turns guessing where the other is. More precisely:
Problem #1 is due to Bill Kuszmaul, MC '12–'13; problem #2 is due to Austin Shapiro, who teaches mathematics at Proof School in San Francisco; problem #5 is due to Drake Thomas, MC '14–'16; all other problems were written by the Mathcamp staff.