Problem Bank

Let $f(x) = \dfrac13 x^3 - a\left(x^2 + x + 1\right)$.

1. For $a = 3$, find the intervals of monotonicity of $f$. 2. Prove that $f$ has exactly one real zero (for every real $a$).

An oil pot has a body shaped as a cylinder (wall thickness negligible) with base diameter $6.4\text{ cm}$ and height…

Let $A(1,-1)$, $B(3,0)$, $C(2,1)$. The region $D$ consists of all points $P$ satisfying

Consider the line $\ell: y = x + t$ and the circle $C: x^2 + (y-2)^2 = 8$.

Suppose there is a fixed point $M$ such that for every point $P$ on $\ell$, the tangent length from …

The inequality

holds for all $x > 1$. Find the range of possible values of $a$.

The four vertices of a (not necessarily planar) quadrilateral $ABCD$ lie on a sphere $O$. Let $E, F$ be the midpoints of $AB$ and $CD$, with $EF \perp AB$ and $EF \perp CD$. If $AB…

Define a sequence $\{b_n\}$ of continued fractions by

A sequence $\{a_n\}$ satisfies $a_1=\dfrac 12$ and

1. Prove that $1\leqslant\dfrac{a_n}{a_{n+1}}\leqslant 2$ for all $n\in\mathbb N…

In an infinite city, all streets run either east-west or north-south along straight lines; the intersections are called lattice points. A person starts at some lattice point, passe…

Let $f(x)=x+\sin x$. If the inequality $f(x)\geqslant ax\cos x$ holds for all $x\in\left[0,\dfrac{\pi}2\right]$, find the range of the real number $a$.

Through the left focus $F_1$ of the ellipse $W:\dfrac{x^2}{2}+y^2=1$, a line $l_1$ meets the ellipse at points $A$ and $B$, where $A(0,1)$. A second line $l_2$ through $F_1$ meets …

A sequence $\{a_n\}$ has $a_1 = 1$ and

Determine, with proof, which of the following hold for all applicable $n$:

1. $\dfrac{1}{a_{n+…