Problem Bank

A line $\ell$ meets the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{2} = 1$ at $A$ and $B$, the negative $x$-axis at $C$, and the positive $y$-axis at $D$, with $A, C, D, B$ in order alo…

In the plane, $\overrightarrow{AB_1} \cdot \overrightarrow{AB_2} = 0$, $\left|\overrightarrow{OB_1}\right| = \left|\overrightarrow{OB_2}\right| = 1$, and $\overrightarrow{AP} = \ov…

The figure shows part of the graph of a function $f(x)$. Determine, with proof, which of the following four expression…

The ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) has eccentricity $\dfrac{\sqrt2}{2}$; two non-vertex points $M, N$ on it satisfy $\angle MON = 90^\circ$ and

Consider the curve $C$ with equation

Determine, with proof, which of the following are true:

1. $C$ is both axially symmetric and centra…

A Reuleaux tetrahedron is the solid obtained by intersecting the four balls whose centers are the vertices of a regular te…

For a positive integer $n$, the expansion of

after combining like terms in $x^i y^j$ ($i, j = 0, 1, \dots, n$), has at least $2021$ terms. Find the min…

The sequence $\{a_n\}$ satisfies $a_1 = a_2 = 1$ and $a_{n+2} = a_{n+1} + a_n$. Determine, with proof, which of the following are true:

1. only finitely many positive integers $m$…

In a regular triangular frustum $ABC\text{-}A_1B_1C_1$, $AB = 2A_1B_1 = 2\sqrt{3}$ and $AA_1 = 2$. Let $D$ be the midpoint of edge $AB$, and let $l$ be a moving line in the plane $…

In an acute triangle $ABC$ with orthocenter $H$, the feet of the three altitudes are $D,E,F$. Determine which notable…

A function $f(x)$ on $\mathbb{R}$ is odd with period $6$, and for $x \in (0, 3)$,

If $f(x)$ has exactly $5$ zeros in $[-3, 3]$, find the…

The line $x - 3y + m = 0$ ($m \ne 0$) meets the asymptotes of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a, b > 0$) at points $A$ and $B$. If $P(m, 0)$ satisfies $|P…