Problem Bank

Determine, with proof, which of the following objects can be placed entirely inside a cube-shaped container of edge length $1$ m (wall thickness negligible).

1. A ball of diameter…

Let $0<x<y<z<\dfrac{\pi}2$. Prove that

The ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ has foci $F_1, F_2$ and center $O$. For a point $P$ on the ellipse, the line through $F_2$ perpendicular to $OP$ meet…

Let $O$ be the circumcenter of $\triangle ABC$, and suppose $x\overrightarrow{OA}+y\overrightarrow{OB}+z\overrightarrow{OC}=\vec 0$ with $xyz\neq 0$, where $C$ is an interior angle…

Simplify and evaluate:

Let $f(x) = \mathrm{e}^x - x - 1$.

1. Prove $f(x) \ge 0$ for all real $x$. 2. Prove that for every positive integer $n$,

In the rectangle $ABCD$, $|AB|=4$ and $|BC|=2\sqrt{3}$. Let $E,F,G,H$ be the midpoints of $AB,BC,CD,DA$ respectively, and…

In $\triangle ABC$, $AB = AC$, and $D$ is the midpoint of $AC$ with $BD = 3$. Find the maximum possible area of $\triangle ABC$.

Let $m$ be real, and set $f(x) = e^{x+1} - ma$ and $g(x) = ae^x - x$. If there exists a real $a$ such that $f(x) \leqslant g(x)$ for all $x \in \mathbb{R}$, find the range of possi…

Consider the ellipse $\dfrac{x^2}{4} + \dfrac{y^2}{3} = 1$ and the point $P(4, 0)$. A secant line through $P$ meets the ellipse at $A$ and $B$; let $C$ be the reflection of $B$ acr…

Consider the ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{3y^2}{a^2} = 1$ (with $a > 0$), and points $P, Q, R$ on it such that the distances from $R$ to the lines $OP$ and $OQ$ both…

For $n \in \mathbb{N}^*$, write $|A|$ for the number of elements of a set $A$ and $\min(A)$ for its smallest element. A nonempty set $A \subseteq \{1, 2, \ldots, n\}$ is called an …