Problem Bank

Let $f(x) = a\ln x + \dfrac{a-1}{x} - x$ with $a < 2$.

1. Find the extreme values of $f(x)$, discussing all cases. 2. Let $m \in \mathbb{Z}$ and $a = 1$. If the inequality

The base of the tetrahedron $P\text{-}ABC$ is an equilateral triangle of side $2$. Given $PA = PB = \sqrt{2}$ and that the dihedral angle $P\text{-}BA\text{-}C$ measures $60^\circ$…

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

If the bisector $AD$ of angle $A$ has length $2$, fin…

The equation

has exactly one real solution in $(0, +\infty)$. Find the range of $a$.

Of $5$ table-tennis balls, $3$ are new and $2$ are old. For each match, $2$ balls are drawn, used, and returned (used balls become old). Let $\xi$ be the number of new balls drawn …

Positive $x, y$ satisfy $xy(x+y) = 4$. Find the maximum of $xy$ and the minimum of $2x + y$.

A sequence $\{a_n\}$ of positive integers satisfies

Find $a_n$.

Consider the hyperbola $\dfrac{x^2}3-y^2=1$ with left and right foci $F_1$, $F_2$. Let $P$ be a moving point on the hyperbola. The lines $PF_1$ and $PF_2$ meet the hyperbola again …

In a regular triangular prism $ABC\text{-}A_1B_1C_1$ with all edges of length $1$, let $D$ and $E$ be the midpoints of $BB_1$ and $B_1C_1$. The cross-section through $A$, $D$, $E$ …

Let $n$ be a positive integer and $x \in \mathbb{R}$. Prove that

Let $f(x) = (x - 1)^2(x - 4)$. Determine, with proof, which of the following are true:

1. $x = 1$ is a local minimum point of $f(x)$; 2. $f(2 + x) + f(2 - x) = -4$ for all $x$; 3.…

In $\triangle ABC$ with the usual side notation, suppose

and $b = 2$. Find the minimum possible area of the circumcircle of $…