Problem Bank

Let $x_1, \dots, x_n$ ($n \ge 2$) be positive reals with $x_1x_2\cdots x_n > 1$. Prove that

As shown in the figure, $F$ is the focus of the parabola $y^2=2px$ ($p>0$), and $M$ is the intersection point of its di…

Define the operation

Evaluate

The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a,b>0$) has eccentricity $2$ and left and right foci $F_1$, $F_2$. A line $l$ through $F_2$ meets the right branch of $C$ at…

Let

If the equation $f(f(x)) = m$ has exactly two real solutions $x_1, x_2$, find the mi…

Let $f(x) = e^x + x^2 - x$ and $g(x) = x^2 + ax + b$, with $a, b$ real.

1. When $a = 1$, find the intervals of monotonicity of $F(x) = f(x) - g(x)$. 2. Suppose the tangent line $l…

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

1. If $f(x)$ has two zeros, find the range of possible values of $a$. 2. If for every $x \in [0, +\infty)$,

A sequence $\{a_n\}$ satisfies $a_{n+1}=\dfrac 14(a_n-6)^3+6$ ($n\in\mathbb N^{*}$). Determine, with proof, which of the following claims are true.

1. When $a_1=3$, $\{a_n\}$ is d…

A positive sequence $\{a_n\}$ satisfies $a_1 = \dfrac{3}{2}$ and

Let $S_n$ be the sum of the first $n$ terms. Find

As shown in the figure, in the cube $ABCD\text{-}A_1B_1C_1D_1$ with edge length $2$, the point $E$ is the midpoint o…

Let $A \colon a_1, a_2, \ldots, a_{2m}$ be a permutation of $1, 2, \ldots, 2m$, where $m \in \mathbb{N}^*$ and $m \geqslant 3$. Say $A$ has property $P$ if there is at least one …

Consider the circles

A moving circle $P$ is externally tangent to $M$ and internally tangent to $N$; the locus of its cen…