Problem Bank

Sequences $\{x_n\}, \{y_n\}$ satisfy $x_0 = 5$, $y_0 = 2$, and

Find $\lim_{n\to\infty} x_n$ and $\lim_{n\to\i…

Given $\tan 4x = \dfrac{\sqrt3}{3}$, evaluate

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) is given. A line $l$ passes through the origin, and a line parallel to $l$ meets the ellipse $C$ at two points $A$ and…

Real numbers $a, b$ satisfy

Find the range of possible values of $a^2 + b^2$.

In quadrilateral $ABCD$, the diagonals satisfy $AC = a$ and $BD = b$. Find

Find the number of positive integer solutions $(x, y, u, v)$ of

Let $A, B, C$ be three points on the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$), and let $O$ be the origin. Can the quadrilateral $OABC$ be a rectangle? Justif…

The equation

has exactly two real solutions. Find the range of $m$.

Differentiable functions $f(x), g(x)$ on $\mathbb{R}$ satisfy

and $g(2x + 1)$ is an even function. Determine, with proof, …

As shown in the figure, an aquaculture center is located on an island $A$. Two towns $B$ and $C$ lie on a straight coas…

Let $a_1 = 1$, $a_2 = 8$, and

1. Prove that there exists $c > 0$ with $a_n \leqslant cn^2$ for all $n$. 2. Prov…

Real numbers $x, y$ satisfy

Find the maximum value of $2x + y$.