Problem Bank

Consider the curve

Find the minimum distance from a point of $C$ to the origin.

Prove the following.

1. For every integer $k\geqslant 2$,

2. The squares with side lengths $1,\dfrac 12,…

A right triangle $DEF$ has its three vertices on the sides $AB$, $BC$, $CA$ of an equilateral triangle $ABC$ respectively, with $\angle DEF=90^\circ$ and $\angle EDF=30^\circ$. Fin…

Unit vectors $\vec e_1, \vec e_2$ make an angle of $120^\circ$, and reals $x, y$ satisfy

Find the range of $\left|x\vec e_1 - y\v…

The lateral surface of a cone unrolls into a circular sector of fixed area with central angle $\alpha$. When the volume of the sphere inscribed in the cone is largest, find $\alpha…

In the coordinate plane, let $A(-1, 0)$, and let $B, C$ move along the unit circle centered at the origin $O$ so that the line $BC$ has inclination $30^\circ$. Find the range of po…

Real numbers $x, y$ satisfy the constraints

If the objective $z = y - ax$…

A sequence satisfies $a_0 = 0$, $a_1 = \dfrac13$, and

Prove that $|a_n| < \dfrac12$ for every positive integer $n$.

Sequences $\{x_n\}$ and $\{y_n\}$ are defined by $x_1 = 1$, $y_1 = 39$, and

Prove that …

A function $f(x)$ and its derivative $f'(x)$ are defined on $\mathbb{R}$, with $f(x)$ increasing. Write $g(x) = f'(x)$. Suppose

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…

Let $p = (a_1, a_2, \dots, a_9)$ be a permutation of $1, 2, \dots, 9$, and define

the sum of …