Problem Bank

Let

1. Discuss the monotonicity of $f(x)$ on the interval $[0, \pi]$. 2. Determine, with proof, the number of zeros of $y = …

Let

Suppose $f(x)$ has no zero in the interval $(\pi,2\pi)$. Find the range of $\…

Consider the parabola $y=x^2$ and the circle $P:x^2+(y-2)^2=1$. From a point $A$ on the parabola, the two tangent lines to the circle meet the parabola again at points $B$ and $C$;…

For what value of $t$ is the ellipse

tangent to the hyperbola $xy = 1$?

Consider the line $\ell: y = x + t$ and the circle $C: x^2 + (y-2)^2 = 8$.

Suppose there is a fixed point $M$ such that for every point $P$ on $\ell$, the tangent length from …

In a cube $ABCD\text{-}A_1B_1C_1D_1$ with edge length $4$, $P$ is the center of the square face $BCC_1B_1$, a…

Sequences $\{a_n\}$, $\{b_n\}$, $\{c_n\}$ satisfy

If $\{c_n\}$ is an arithmetic sequence and $b_n \leqslant b_{n+…

Consider the circle $O \colon x^2 + y^2 = 4$ and the line $l \colon y = kx + 5$.

1. If there exist a point $A$ on $l$ and a point $B$ on the circle $O$ with $\angle OAB = \dfrac{\…

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 …

Let $a_n = 1 + 2 + \cdots + n$ and $S_m = a_1 + a_2 + \cdots + a_m$. Among $S_1, S_2, \dots, S_{2017}$, how many are divisible by $2$ but not by $4$?

If the inequality

holds for all $x \in (2, +\infty)$, find the range of possible values of the real number $a$.

Let $C$ be a hyperbola with center $O$. Suppose there is exactly one pair of lines $A_1B_1$ and $A_2B_2$ through $O$, making an angle of $60^\circ$ with each other, such that $|A_1…