Problem Bank

Let $\theta$ be a constant, and define a sequence of functions by

In a regular tetrahedron $D\text{-}ABC$ of edge length $1$, let $O$ be the center of $\triangle ABC$. A line through $O$ meets segments $AC$ and $BC$ at $M$ and $N$ (endpoints allo…

In a rectangle $ABCD$, a point $P$ is chosen at random on the side $CD$. The probability that the largest side of $\triangle APB$ is $AB$ equals $\dfrac 12$. Find $\dfrac{AD}{AB}$.

As shown in the figure, a “cross-gable” roof is the solid obtained as the union of two right triangular prisms which overlap …

A line $BC$ is perpendicular to the plane of a unit circle $O$ and meets the circle at $A$, with $AB=BC=1$. Let $P$ be any point of the circle other than $A$, and let $l$ be the ta…

An engineer uses a detector, with effective range $\dfrac{\sqrt 3}2$ km, to sweep for mines in an equilateral triangular region of side length $2$ km. Starting from one vertex of t…

Determine all rational numbers $r$ such that the equation

has roots which are all integers.

Call a sequence $\{a_n\}$ ratio-arithmetic with ratio-difference $t$ if

Deter…

Given an ellipse, a hyperbola, and a parabola (each drawn as a curve), describe a straightedge-and-compass construction of the focus (or foci) of each.

In $\triangle ABC$, $AB=2AC$ and $\angle BAC=60^\circ$. An interior point $P$ satisfies $PA=\sqrt 3$, $PC=2$, $PB=5$. Find the area of $\triangle ABC$.

The Riemann zeta function $\zeta(s)$ is closely tied to the distribution of primes. Writing $\operatorname{Re}(s)$ for the real part of a complex number $s$, define

Let $a,b,c\in[0,1]$. Find the range of