Problem Bank

Let $P$ be the bottom vertex of the ellipse $E \colon \dfrac{x^2}{4} + y^2 = 1$. Two mutually perpendicular lines $l_1, l_2$ pass through $P$: the line $l_1$ meets the circle $x^2 …

The hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a,b>0$) has right focus $F$ and right vertex $A$. The line through $F$ perpendicular to $AF$ meets the hyperbola at points $B$…

A five-term sequence $a_1,a_2,a_3,a_4,a_5$ satisfies $0\leqslant a_1<a_2<a_3<a_4<a_5$, and for all $i,j$ with $1\leqslant i\leqslant j\leqslant 5$, the difference $a_j-a_i$ is a te…

Define

1. If $f(x)$ has exactly two zeros, find the range of the real number $a$. 2. If $a\leqslant -2$, find all …

The ellipse $G \colon \dfrac{x^2}{6} + \dfrac{y^2}{b^2} = 1$ (with $0 < b < \sqrt{6}$) has foci $F_1, F_2$ and minor-axis endpoints $B_1, B_2$. A point $P$ on $G$ satisfies

A quadrilateral pyramid has one lateral edge perpendicular to its base, its base is a parallelogram, and the set of lengths of its $8$ edges is $\{1,\sqrt 2,\sqrt 3\}$. Find the nu…

Let $x, y \ge 0$, and set

Given that $a$ and $b$ are integers that are not consecutive, find $a$ and $b$.

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

Find $\cos\angle AHB$.

The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ $(a>b>0)$ shown has left and right foci $F_1$ and $F_2$. A…

The three lines

bound a triangle of area $4$, and $k_1<0<k_2<k_3$. Find the maximum value of $\dfrac{k_2}{k_3}$.

Joey, Chloe, and their daughter Zoe share a birthday. Joey is one year older than Chloe, and Zoe turns exactly $1$ today. Today Chloe's age is an integer multiple of Zoe's age, and…

The ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has eccentricity $\dfrac{1}{2}$. Two lines through the interior point $P(1, 1)$ meet the ellipse a…