Problem Bank

Let $0 < x < \dfrac{\pi}{2}$.

1. Prove that $\tan x + \sin x > 2x$. 2. Find the largest positive integer $n$ such that

holds for all such $x$.

Let

Prove that the sequence $\{a_n\}$ has a limit, and find that limit.

A sequence $\{a_n\}$ has $a_n > 0$ and

1. Prove that $a_{n+2} < a_{n+1} < 2$. 2. Prove that $a_n > 1$ …

Let $f(x) = x^2 + ax + b$ and $g(x) = \mathrm{e}^x(cx + d)$. The curves $y = f(x)$ and $y = g(x)$ both pass through the point $P(0, 2)$ and have the same tangent line $y = 4x + 2$ …

A plane quadrilateral $ABCD$ has $AB = 1$, $BC = 4$, $CD = 2$, $DA = 3$. Find

In an acute triangle $ABC$,

Find the minimum value of

A function $f$ on $\mathbb{R}$ satisfies $f(0) = 0$,

and $f$ is nondecreasing on $[0, 1]$. Find

The ellipse $\dfrac{x^2}{2} + y^2 = 1$ has right focus $F(1, 0)$. A line through the fixed point $P(2, 0)$ meets the ellipse at $A$ and $B$, and $k_1, k_2$ denote the slopes of $FA…

For a positive integer $n$, let $f(n)$ be the digit sum of $n$ written in base $4$, and $g(n)$ the digit sum of $f(n)$ written in base $8$. (Example: $f(2020) = 10 = 12_{(8)}$, so …

Let $f(x) = \dfrac{a}{x} + \dfrac12\ln x - 1$.

1. Discuss the monotonic intervals of $f$. 2. If $f$ has two zeros $x_1, x_2$, prove that

Consider the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{3} = 1$, points $A, B$ on it with $A$ in the first quadrant, and $O$ the origin; let $S$ be the area of $\triangle OAB$.

1. If $…

The point $P(x, y)$ satisfies

Find the range of possible values of $\dfrac{x + y}{\sqrt{x^2 + y^2}}$.