Problem Bank

A plane $\alpha$ makes angles $\theta_i$ ($i = 1, 2, \dots, 6$) with the six faces of a rectangular box. Find the value of

Suppose there exists $a \in \mathbb{R}$ such that the inequality

holds for all $x \in [0, 1]$. Find the range of possible values of the real number $m$.

The function $g(x) = x^2 - 2ax + 1$ has range $[0, 4]$ on the interval $[1, 3]$.

1. Find $a$. 2. If $g\left(2^x\right) - k\cdot 4^x \geqslant 0$ for all $x \in [1, +\infty)$, find…

Let $H,P$ be two points in the plane of $\triangle ABC$, distinct from $A,B,C$, and write $\vec a=\overrightarrow{PA}$, $\vec b=\overrightarrow{PB}$, $\vec c=\overrightarrow{PC}$, …

Let $a > 0$ and $f(x) = ax - x\mathrm{e}^x$.

1. Find the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$. 2. Prove that $f(x)$ has a unique extreme p…

Let

Find the greatest integer not exceeding $M$.

Let

(the exponents need not be distinct). A subset $B \subseteq A$ consists of $n$ consecutive natural num…

A closed container in the shape of a right square prism stands on a level surface. Embedded in its base is a solid decorative righ…

Let $ABCDEF$ be a hexagon whose interior angles are all equal, with $AB=6$, $BC=8$, $CD=10$, $DE=12$. Let $d$ be the diameter of the largest circle contained in the hexagon. Find $…

Nonempty sets $A$ and $B$ satisfy $A \cup B = \{1, 2, 3, \dots, 10\}$ and $A \cap B = \varnothing$. The number of elements of $A$ is not an element of $A$, and the number of elemen…

Let $n$ be a positive integer. If

holds for every positive integer $m$, find $n$.

Let $f(x) = x$ and $g(x) = x^2 - x + 3$. Suppose there exist $x_1, x_2, \dots, x_n \in \left[0, \dfrac{9}{2}\right]$ such that