Problem Bank

Two distinct points $A$ and $B$ on the ellipse $\dfrac{x^2}2+y^2=1$ are symmetric about the line $y=mx+\dfrac 12$.

1. Find the range of $m$. 2. Find the maximum area of $\triangle…

Let

where $a > 0$ and $a \neq 1$. If the graph of $f(x)$ contains exactly one pair of poi…

Let $A,B$ be two points on the ellipse $\dfrac{x^2}8+\dfrac{y^2}2=1$ with chord length $|AB|=\dfrac 83$. Find the range of the area of $\triangle AOB$, where $O$ is the origin.

Let

and suppose the line $y = a$ meets the graph of $f$ at consecutive …

Sequences $\{a_n\}$ and $\{b_n\}$ satisfy $a_1 = 1$, $b_1 = 2$, and

1. If additionally $b_n = 2a_n$ for all $n$, prove that for $n \geqslant …

Let $\mathrm{e}$ be the base of the natural logarithm, let $f(x) = \mathrm{e}^x + \sin x - 2x$, and let $g(x)$ be the derivative of $f(x)$.

1. Find the equation of the tangent lin…

Find the number of permutations of $1, 2, 3, 4, 5, 6$ such that for each $k = 1, 2, 3, 4, 5$, at least one of the first $k$ terms of the permutation is greater than $k$.

In triangle $ABC$, the sides opposite angles $A, B, C$ are $a, b, c$, the altitude to side $BC$ has length $h$, and

1. If $h = \dfrac{2}{3}a$ and $k\sin A - \…

In the plane, $\boldsymbol e_1$ and $\boldsymbol e_2$ are perpendicular unit vectors. A vector $\boldsymbol a$ satisfies $|\boldsymbol a-4\boldsymbol e_1|=2$, and a vector $\boldsy…

Evaluate

Let $a_0 = \dfrac{1}{2}$ and

Prove that

Suppose the function

has exactly three zeros in $(0,3]$. Find the range of the real number $r$.