Problem Bank

For a positive integer $n$, let $s(n)$ be the sum and $p(n)$ the product of its digits. Call $n$ a coincidence number if

Find the sum of all coincidence nu…

The sequence $\{a_n\}$ has partial sums $S_n$ with

and

Suppose $a, b \in \mathbb{R}$ satisfy

When $a + b$ attains its minimum value, what is $a$?

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

1. If $a = 2$, prove that $f(x) > \dfrac{1}{6}$. 2. If $a = \dfrac{9}{4}$, prove that $f(x) > 0$.

Suppose that for every $x \in [0, 1]$,

Let $\mu$ be the maximum possible value of the real number $a$. Prove that

Let $f(x) = ax - \ln x$ with $a \in \mathbb{R}$.

1. Find the intervals on which $f(x)$ is monotonic, discussing all cases. 2. If $f(x)$ has two zeros $x_1, x_2$, prove that

If the constant term in the expansion of

equals $\dfrac{35}{2}$, find the positive integer $n$.

Plane vectors $\vec{a}, \vec{b}, \vec{c}$ satisfy

Find the range of p…

For a triangle with sides $a \leqslant b \leqslant c$, define its tilt

Let $f(x) = \mathrm{e}^x - \ln(x + m)$.

1. Suppose $x = 0$ is an extreme point of $f(x)$. Find $m$, and discuss the monotonicity of $f(x)$. 2. Prove that if $m \leqslant 2$, then …

Three lines $l_1,l_2,l_3$ are pairwise parallel; the distance between $l_1$ and $l_2$ is $1$, between $l_2$ and $l_3$ is $\dfrac 12$, and between $l_1$ and $l_3$ is $\dfrac 32$ (so…

A cube $ABCD\text{-}A_1B_1C_1D_1$ of edge length $2$ moves in a rectangular coordinate system $O\text{-}xyz$ so that the vertex $A$ stays on the $z$-axis and the vertex $B_1$ stays…