Problem Bank

The function $f(x)=(x+b)\left(\mathrm e^x-a\right)$ ($b>0$) has tangent line

at the point $(-1,f(-1))$.

1. Find $a$ and $b$. 2. Suppos…

Points $A$ and $B$ lie on skew lines $a$ and $b$ respectively, and the segment $AB$ is perpendicular to both. Moving points $P\in a$ and $Q\in b$ satisfy that $PA+QB$ is constant. …

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

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

1. Discuss the monotonicity of $f(x)$. 2. If $f(x)$ has two zeros, find the range of possible values of the real number $a$.

Let $f(x) = a\mathrm{e}^x - \mathrm{e}x - a$ with $a < \mathrm{e}$.

1. If the local minimum value of $f(x)$ is $-1$, find $a$. 2. If $a = 1$, prove that for $x \geqslant 0$,

Let $A, B, C$ be the interior angles of a triangle and $x, y, z$ real numbers. For each of the following, determine (with proof or counterexample) whether it holds for all such $A,…

Let $n \in \mathbb{N}^*$, and let $a_1, a_2, \ldots, a_n$ be positive reals with partial sums $S_k = a_1 + \cdots + a_k$ and $S_n = 1$. Prove that

Let $m$ be a positive integer and

Suppose that for every integer $n$,

A positive integer $a$ has $n$ digits in base ten, and $a^3$ has $m$ digits. Determine which of the values

are possible, with…

Prove that a ray of light emitted from one focus of an ellipse, after reflecting off the ellipse, passes through the other focus.

Find the number of real solutions of

Let $x, y > 0$. Find the range of possible values of