Problem Bank

In quadrilateral $ABCD$, the diagonals satisfy $AC = a$ and $BD = b$. Find

Let $\omega$ be the circumcircle of an acute scalene triangle $ABC$. The tangents to $\omega$ at $B$ and $C$ meet at a point $T$, and $X,Y$ are the feet of the perpendiculars from …

The function $f(x) = \ln x + \dfrac{1}{2}x^2 + ax$ has two extreme points $x_1, x_2$, and

Find the range of possible values of the real number $…

Let $a>b>c>0$. Find the minimum value of

An Android-style unlock pattern connects some of $9$ points arranged in a $3\times 3$ grid, subject to these rules: choose a starting point, then move the finger through a sequence…

Let

Find the value of

Vectors $\vec a, \vec b$ satisfy

Find the maximum and minimum values of $|\vec b|$.

Let $M$ be the midpoint of edge $AD$ of a cube $ABCD\text{-}A_1B_1C_1D_1$ with edge length $2$, and let $P$ lie in the plane of face $BCC_1B_1$. If the plane $D_1PM$ makes equal ac…

Let $a, b, m, n$ be positive numbers with $a + b = 1$ and $mn = 2$. Find the minimum value of

For a function $f(x)$ and a real number $a$, say $f(x)$ has property $D(a)$ if there exists $b \in \mathbb{R}$ such that

Let $a, b > 0$ and $f(x) = ax^2 + b$. Suppose

Find all conditions on $a$ and $b$ for this to h…

Let $f(x) = a^2\ln x - \dfrac{1}{2}ax^2 - \left(a^2 - a\right)x$ with $a \neq 0$.

1. When $a = 1$, find the intervals on which $f(x)$ is monotonic. 2. If $f(x)$ attains a local ma…