Problem Bank

Let $f(x) = \ln(x+1) + \sqrt{x+1} + ax + b$, where $a, b \in \mathbb{R}$ are constants, and suppose the curve $y = f(x)$ is tangent to the line $y = \dfrac{3}{2}x$ at the point $(0…

The function $f(x) = ax(\ln x - 1) - x^2$ (with $a \in \mathbb{R}$) has exactly two extreme points $x_1 < x_2$.

1. Find the range of possible values of $a$. 2. If the inequality $…

A sequence $\{a_n\}$ satisfies $a_1=1$, $a_2=6$, and for $n\geqslant 3$,

A random variable $X$ takes values $0, 1, 2$ with probabilities $a, b, c$ respectively. If $4a, b, c$ form a geometric sequence, find the maximum value of the variance $D(X)$.

Real numbers $x, y$ satisfy

Find the maximum possible value of $x$.

Consider the ellipse $\dfrac{x^2}4+y^2=1$ and the hyperbola $\dfrac{x^2}4-y^2=1$. From the left vertex $P(-2,0)$, two lines are drawn meeting the ellipse at $A$, $C$ and the hyperb…

Let $|a|, |b|, |c| \leqslant 1$. Prove that

The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a,b>0$) has right vertex $A$. The circle centered at $A$ with radius $b$ meets one asymptote of $C$ at two points $M$ and $N…

A sequence $\{a_n\}$ satisfies $a_{n+1}=\dfrac 14(a_n-6)^3+6$ ($n\in\mathbb N^{*}$). Determine, with proof, which of the following claims are true.

1. When $a_1=3$, $\{a_n\}$ is d…

The graph of $f(x) = \mathrm{e}^x - ax\sin x - bx + c$ is tangent to the $x$-axis at the origin.

1. Find $b$ and $c$. 2. If $f(x)$ has a unique zero in $(0, \pi)$, find the range …

Let $a, b, c$ be real numbers and

1. When $b = 1$ and $c = 0$, find the minimum value of $f(x)$. 2. If $f(x) \geqslant -1$ for all $x$,…

For a set $A = \{a_1, a_2, \dots, a_n\}$ of positive reals ($n \geqslant 2$), define

If $A + A$ has exactly $\dfrac{n(n-…