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A billiard rack is a rack, usually a triangle, that can hold a certain number of equal size billiard balls, such that the balls' centres cannot move within the rack. Can the rack be a square, for ever …

A regular $n$-gon contains a regular $m$-gon, where $n$ and $m$ are coprime, with no sides coinciding. What is the maximum number of contact points between the $n$-gon and the $m$-gon? (I'm not aski …

A triangle is drawn in a region bounded by the $x$-axis and a polynomial curve with all real roots, with one side of the triangle on the $x$-axis, as shown in this example: What is the supremum of t …

Let $u_1,u_2,u_3,\dots$ be a permutation of the integers greater than $2$. A unit circle is in a regular $u_1$-gon, which is a regular $u_2$-gon, which is in a regular $u_3$-gon, ad infinitum. Each po …

Consider a regular $n$-gon of side length $1$ with diagonals. Here is an example with $n=11$ (from geogebra applet). I've been trying to find, in terms of $n$, bounds on the area of the largest cell, …

Originally posted at MSE. A circle with integer radius $R$ is inscribed in a triangle. Three other circles with integer radii $a,b,c$ are each tangent to the large circle and two sides of the triangl …

Draw $n$ random chords in a circle, where each chord connects two independent uniformly random points on the circle. As $n\to\infty$, which kind of polygon (triangle, quadrilateral, pentagon, etc.) o …

On a "bottom" disk of area $\pi$, we place "top" disks of area $\pi,\frac{\pi}{2},\frac{\pi}{3},\dots$ such that the centre of each top disk is an independent uniformly random point on the bottom disk …

The question has been answered at the original MSE question. (A simulation with a larger sample size shows that a $99.9$% confidence interval for the probability is $0.49994\pm0.00002$.)

This question was posted at MSE but was not answered. The vertices of a pentagram are five uniformly random points on a circle. The areas of three consecutive triangular "petals" are $a,b,c$. The pet …

The following question resisted attacks at Math SE, so I thought I would try posting it here. Is the following conjecture true or false: Given any five coplanar points, we can always draw at least on …

Suppose we choose $n$ uniformly random points in a disk, then draw the smallest circle that encloses all of those points. There is evidence suggesting that the probability that the enclosing circle is …