Problem Bank

The ellipse $\dfrac{x^2}{2} + y^2 = 1$ has right focus $F(1, 0)$. A line through the fixed point $P(2, 0)$ meets the ellipse at $A$ and $B$, and $k_1, k_2$ denote the slopes of $FA…

Define the diameter of a plane region as the largest distance between two of its points. Find the diameter of the region enclosed by

Let

1. Find the intervals of monotonicity of $f(x)$. 2. For $a \in \left[\dfrac{5}{2}, \dfrac{17}{4}\right]$, let $…

For a closed interval $I$, let $M_I$ denote the maximum value of the function $y=\sin x$ on $I$. Suppose a positive number $a$ satisfies

Find $a$.

The graphs of $f(x) = m\ln x$ and $h(x) = \dfrac{x-1}{2x}$ (for $x > 0$) have exactly one common tangent line. Find the real number $m$.

Let $a_1, a_2, \ldots, a_n$ be a strictly increasing sequence of positive integers. The set $\{a_1, a_2, \ldots, a_n\}$ has $2^n$ subsets, whose element sums (with the empty set co…

Let $f(x)=2+3\sin x$ for $x\in\left[0,\dfrac{\pi}2\right]$. Suppose $\theta$ is such that for every $x_1\in\left[0,\dfrac{\pi}2\right]$ there exists $x_2\in\left[0,\dfrac{\pi}2\rig…

Dissolve $a$ grams of sugar in $b$ grams of sugar water ($b > a > 0$). Adding $m > 0$ grams of sugar makes the water sweeter — which is the everyday content of the inequality

A line $m$ is perpendicular to a plane $\alpha$, with foot $O$. A regular tetrahedron $ABCD$ has edge length $4$; the vertex $C$ moves in the plane $\alpha$ while the vertex $B$ mo…

Let $f(x) = \dfrac{2}{3}x^3 - mx^2 + m^2 x$ with $m \in \mathbb{R}$, and let $f'(x)$ be its derivative.

1. If $g(x) = f(x) - f'(x)$ has extreme values, find the range of possible …

Find the area of the region defined by

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…