Problem Bank

The $8$ edges of a regular quadrilateral pyramid $P\text{-}ABCD$ are colored using $6$ different colors, so that the edges meeting at each vertex all have different colors. Find th…

A fair six-sided die with faces numbered $1$ through $6$ is rolled three times, the rolls being independent, giving results $a_1, a_2, a_3$ in order. Find the probability of the ev…

Define $\max\{a,b\}=\begin{cases}a,&a\geqslant b,\\ b,&a<b.\end{cases}$ Real numbers $x,y$ satisfy $|x|\leqslant 2$ and $|y|\leqslant 2$. Find the range of

1. The ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has right focus $F$; a line through $F$ meets the ellipse at $A, B$, and $C$ is the reflection of $A$ ac…

The ellipse $C:\dfrac{x^2}9+\dfrac{y^2}3=1$ has left vertex $A$. A line $l$ through the point $B(1,0)$ meets $C$ at two points $P$ and $Q$. Let $N$ be the circumscribed circle of $…

Let $M$ be a point of the midsphere (the sphere tangent to all edges) of a cube $ABCD\text{-}A_1B_1C_1D_1$ of edge length $2$, and let $N$ be a point of the circumcircle of the tri…

A cube $ABCD\text{-}A_1B_1C_1D_1$ has edge length $2$. The vertices $A_1, B_1, C_1, D_1$ lie on the lateral surface of a cone, and the vertices $A, B, C, D$ lie in the plane of the…

Let $H$ be the orthocenter of $\triangle ABC$, and

Find $\cos\angle AHB$.

Let

If $f(x)$ has at least $3$ zeros, find the range of possible values of the real number $a$.

You may use the fact that for $x \in \left(0, \dfrac{\pi}{2}\right)$,

1. Prove that $\dfrac{\sin x}{x} > \dfrac{1}{2}$ for $x \in \left(0, \d…

Let

Prove that there is a closed interval $D$ of length greater than $1$ (the length of $[m, n]$ being $…

The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a>0$, $b>0$) has left and right foci $F_1$, $F_2$, eccentricity $3$, and the two intersection points of the line $y=2$ with …