Problem Bank

In a regular $n$-gonal pyramid $P\text{-}A_1A_2\cdots A_n$ (where $n \geqslant 3$), let $O$ be the center of the regular $n$-gon base $A_1A_2\cdots A_n$, and let $B$ be the midpoin…

Let

1. If the equation $f(x) = 0$ has exactly $3$ real solutions, find the range of possibl…

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 $…

A sequence $\{a_n\}$ has $a_1 = 1$ and

Find

Sequences $\{a_n\}$ and $\{b_n\}$ satisfy

with $a_1 = 2$ and $a_2 = 4$.

1. Find $a_3, a_4, a_5$. 2.…

Call $f$ triangle-preserving if, whenever $a, b, c$ lie in its domain and are the side lengths of some triangle, the values $f(a), f(b), f(c)$ are also the side lengths of some t…

A regular octahedron has faces numbered $1$–$8$; one throw gives sample space $\Omega = \{1, \dots, 8\}$ (uniform). Construct events $A, B, C$ with

but…

Let $F_1,F_2$ be the left and right foci of the ellipse $\dfrac{x^2}{25}+\dfrac{y^2}{16}=1$. If $M$ is a point on the ellipse such that the incircle of $\triangle MF_1F_2$ has circ…

Suppose the function $f(x)=(x^2+a)\mathrm e^x$ has a minimum value. Find the number of zeros of $g(x)=x^2+2x+a$.

Find the units digit of $\left(5 + \sqrt{22}\right)^{2016}$, i.e., of $\left\lfloor\left(5 + \sqrt{22}\right)^{2016}\right\rfloor$.

The ellipse $C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) passes through $P(-2, 1)$ with eccentricity $\dfrac{\sqrt2}{2}$. Two perpendicular lines through $P$ meet $C$…

Let $PA$, $PB$, $PC$ be three rays emanating from a common point $P$ in space. Denote