Problem Bank

Let $a, b$ be positive. If

find the range of possible values of $a + b$.

A square sheet of paper $ABCD$ with side length $1$ is folded so that vertex $B$ lands at a point $B'$ on side $AD$, as show…

Complex numbers $z_1, z_2$ satisfy

Find the minimum value of $|z_1^3 + z_2^3|$.

In an acute triangle $ABC$, the sides opposite $A, B, C$ are $a, b, c$, and

Find the range of possible values of $B$.

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

1. If $a = 0$, find the minimum value of $f(x)$. 2. If $f(x)$ has a local maximum at $x = 0$, find the range of possible values of …

Let $I = \{1, 2, 3, 4, 5\}$. Choose two nonempty subsets $A, B$ of $I$ such that the smallest element of $B$ is greater than the largest element of $A$. How many such choices are t…

Given $2\tan(\alpha + \beta) = 3\tan\alpha$, find the maximum value of

The ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has eccentricity $\dfrac{\sqrt{3}}{2}$, and the quadrilateral formed by its foci and the endpoints…

In $\triangle ABC$, the altitudes to sides $a, b, c$ have lengths $h_a, h_b, h_c$, and

1. If $S$ is the area o…

Consider the sequence $1,1,2,1,2,4,1,2,4,8,1,2,4,8,16,\dots$, where the first term is $2^0$, the next two terms are $2^0,2^1$, the next three are $2^0,2^1,2^2$, and so on. Find the…

In a (not necessarily planar) quadrilateral $ABCD$ in space, $AB = 2$, $BC = 8$, $CD = 10$, $DA = 4$. Find $\overrightarrow{AC}\cdot\overrightarrow{BD}$.

The line $y=k(x+2)$ ($k>0$) meets the parabola $C:y^2=8x$ at two points $A$ and $B$, and $F$ is the focus of $C$. If $FA=2FB$, find $k$.