Problem Bank

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

1. Prove that $\dfrac{a_{n+2}}{n} = \dfrac{a_n}{n+1}$. 2. Prove that

The function

has exactly $3$ zeros. Find the range of possible values of the real number $a$.

Let $A(0, 6)$, let $B$ move along the $x$-axis, and let $C$ be a point of the plane with

Find the minimum possible length of segment $…

Determine whether the statement "$k\sin x\cos x<x$ for all $x\in\left(0,\dfrac{\pi}2\right)$" is a sufficient condition, a necessary condition, both, or neither, for "$k<1$". Justi…

In a sequence $\{a_n\}$, $a_1=3$ and $a_{n+1}a_n+\lambda a_{n+1}+\mu a_n^2=0$ for $n\in\mathbb N^{*}$.

1. If $\lambda=0$ and $\mu=-2$, find the general term of $\{a_n\}$. 2. If $\…

A chord $AB$ of the parabola $x^2 = 2py$ ($p > 0$) has midpoint $M$ and length $l$. Find, in terms of $p$ and $l$, the minimum possible distance $h$ from $M$ to the $x$-axis.

Let $f(x) = ax^3 + bx^2 + cx + d$ with $a \neq 0$, and suppose $|f(x)| \leqslant 1$ for all $x \in [-1, 1]$. Find the maximum possible value of

Let $f(x) = \dfrac{x^2}{2} - x\ln x - (a-1)x + a$.

1. If $f$ has two extreme points $x_1, x_2$, find the range of $a$. 2. Under that condition, if $m > f(x_1) + f(x_2)$ always hol…

As shown in the figure, the cube $ABCD\text{-}EFGH$ has edge length $2$, where $E$, $F$, $G$, $H$ lie directly abov…

In the coordinate plane, the graph of the parabola $y = ax^2 - 3x + 3$ (with $a \neq 0$) is the reflection of the graph of the parabola $y^2 = 2px$ (with $p > 0$) across the line $…

Let $f(x) = \dfrac{1}{2}x^2 - (a+1)x + a\ln x$ with $a \in \mathbb{R}$.

1. If $x = 1$ is a local minimum point of $f(x)$, find the range of possible values of $a$. 2. Suppose $f(x…

Vectors $\boldsymbol{a}$ and $\boldsymbol{b}$ satisfy $|\boldsymbol{a}| + |\boldsymbol{b}| = 2$. Find the maximum and minimum values of