MUSA

Problem of the Week Archive

At MUSA, we love to talk about tricky math problems of all levels! Here's some of our past problems of the week. If you can't figure them out yourself, try asking at office hours and see if anyone else can make heads or tails of them.

Don't forget to check out this week's problem on the homepage!

March 10, 2025 - Points on an Ellipse

Let \(\varepsilon\) be any ellipse. Choose three points on \(\varepsilon\), which we call \(A, B, C\). Through those three points, draw two congruent ellipses \(\varepsilon_1\) and \(\varepsilon_2\) with axes of length 2 and 4 such that their major axes are orthogonal to each other.

Further, let \(\varepsilon\) intersect \(\varepsilon_1\) (but not \(\varepsilon_2\)) at \(D\) and \(\varepsilon_2\) (but not \(\varepsilon_1\)) at \(E\). Take \(X \neq D\) on \(\varepsilon_1\) and let \(l_1 = DX\). Also, let \(Y \neq E\) on \(\varepsilon_2\) such that \(l_2 = EY\) intersects \(l_1\) at a point \(F\) on \(\varepsilon\). If \(XY\) intersects \(\varepsilon_1\) at \(P \neq X\) and \(\varepsilon_2\) at \(Q \neq Y\), what is \(AP + AQ\)? (Problem credit: Atharv Sampath)

March 3, 2025 - Recursive sequence

Define the sequence \(a_n\) recursively as \(a_{n+2} = c(a_{n+1} + a_{n}) - a_{n-1}\), where \(a_1 = 1, a_2 = 1, a_3 = 4\) and \(c\) is some integer greater than \(1\). What values of \(c\) satisfy that the sequence contains only perfect squares?

February 25, 2025 - Rational Approximation

Let \(R\) denote a non-negative rational number. Determine a fixed set of integers \(a,b,c,d,e,f\), such that for every choice of \(R\), \[\left|\frac{aR^2+bR+c}{dR^2+eR+f}-\sqrt[3]{2}\right|<|R-\sqrt[3]{2}|\]

Integer Pairs

Compute the number of integer pairs \((x, y)\) such that \(0 \leq \left|x\right|, \left|y\right| \leq 10^{10}\) , and \(x^2-24y^2 = 1\).

Highlight/copy to show answer: 42

Topological invariants and the open mapping theorem

Prerequisite: Math 185; Math 142 or Math 202A will help

Say that a metric space \(X\) is locally connected if for each point \(x\) of \(X\) there is a connected open set \(U\) containing \(x\). Let \(K\) be a locally connected compact subset of the complex plane \(\mathbb{C}\) and \(V(K)\) denote the vector space of all holomorphic functions \(V(K)\to\mathbb{C}\). Give a formula for the dimension of \(V(K)\) in terms of topological invariants of \(K\).

If that was pretty easy, what happens if \(K\) is not locally connected? In particular, what is the dimension of \(V(K)\), for \(K\) the Cantor set?

Fractal topology

Prereqs: Math 104

If \(X\) is a metric space, say that \(X\) has a fractal topology if, for each nonempty open set \(U\) in \(X\), there is a homeomorphism from \(U\) to \(X\). What's an example of \(X\)?

Cuboids on the sphere

Prison break!

Prereqs: None

There are \(N\) prisoners in a circular prison; each isolated in their own soundproof prison cell. Each prisoner is given a button and a switch; the button controls the lightbulb in the prisoner one cell clockwise from the button-holder. When a prisoner presses their button, the corresponding lightbulb will flash on and then off at the coming noontime, whenever that is. The goal of the prisoners is to find N; the total number of prisoners. To make things challenging; the prison warden shuffles all the prisoners every night.

Can this be done; given that you are allowed to give an algorithm \(A\) for all the other prisoners to follow; and you yourself can follow an algorithm \(B\)?

Towers of boxes

Prereqs: None

How many different ways can a person stack n cubical boxes if (a) the person must stack all the boxes in the same two dimensional plane, (b) on the bottom layer of boxes, the boxes must be placed side by side with no gaps in between, and (c) on all higher levels, the boxes must be stacked directly on top of an existing box?

Homeomorphism classes in \(\mathbb{R}^n\)

Prereqs: Math 104

We say that two subsets, \(X\) and \(Y\), of \(\mathbb{R}^n\) are homeomorphic if there is a continuous bijection from \(X\) to \(Y\) whose inverse is also continuous.

Show that there exist two subsets of \(\mathbb{R}^n\) which are not homeomorphic, but do have continuous bijections between them.