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
n |
k mod m |
n |
k |
n |
k + m |
n |
k + 2m |
where the sum includes every m^{th} element between 0 and n inclusive, starting at k. For example,
20 |
2 mod 5 |
20 |
2 |
20 |
7 |
20 |
12 |
20 |
17 |
n |
0 mod 2 |
n |
1 mod 2 |
n |
0 mod 3 |
n |
0 mod 5 |
n |
1 mod 5 |
n |
2 mod 5 |
n |
3 mod 5 |
n |
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 E_{k} be the equation
A solution to E_{k} is a pair of positive integers m and n that satisfy the equation. For example, solutions to E_{2017} 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.