Problem Bank

1. Find the maximum of $f(x) = x\ln x - (1-x)\ln(1-x)$ on $0 < x \leqslant \dfrac{1}{2}$. 2. For $0 < x < 1$, prove that

Three vertices of a regular $n$-gon are chosen at random. The probability that they form an obtuse triangle is $\dfrac{93}{125}$. Find all possible values of $n$.

Let $\alpha, \beta, \gamma \in [0, 2\pi)$ be pairwise distinct. Determine, with proof, which of the values $0$, $1$, $2$, $4$ can be the number of solutions $(x, y)$ of the system …

Let $f(x) = \dfrac{1}{x-a} + \dfrac{1}{x-b}$ with $a, b \in \mathbb{R}$.

1. If $a = b = 1$, solve the inequality $f(x) > 1$. 2. Define the length of each of the intervals $(m, n)$…

As shown in the figure, the triangle $ABC$ has area $2$. Points $D$ and $E$ lie on the sides $AB$ and $AC$ respectively, a…

Take the $4$ vertices and $6$ edge-midpoints of a tetrahedron — $10$ points in all. Among all lines determined by pairs of these points, how many pairs of lines are skew?

Among the following curves, determine which one has the greatest number of intersection points with the curve $y=x^3-3x$:

Ju…

Solve

Positive reals $a, b$ satisfy $a^2 + b^2 = 1$. Find the range of

Consider the ellipse $\dfrac{x^2}4+\dfrac{y^2}3=1$ with foci $F_1$, $F_2$. Perpendicular lines $l_1$ and $l_2$ through $F_1$ and $F_2$ meet the ellipse at $A,B$ and $C,D$ respectiv…

Let $f(x) = \mathrm{e}^x - \dfrac{1}{2}ax^2 - x$.

1. Let $f'(x)$ be the derivative of $f(x)$. Discuss the monotonicity of $y = f'(x)$. 2. When $a \leqslant 1 - \dfrac{1}{\mathrm{e…

In the coordinate plane $xOy$, consider the circles

1. A line $l$ through the point $A(-4,0)$ meets the circ…