Problem Bank

A disease has overall prevalence $5\%$ in a population. A machine test for it has accuracy $90\%$ (it correctly reports the true status $90\%$ of the time). Given that a person is …

Let $f(x) = \mathrm{e}^{mx} + x - x\ln x$ with $m \geqslant 0$.

1. When $m = 1$, find the range of $f(x)$ on $[1, \mathrm{e}]$. 2. Let $f'(x)$ be the derivative of $f(x)$. Determi…

A circular wooden board is measured with a carpenter's square, an L-shaped tool formed by two perpen…

The numbers $a, b, c$ form a geometric sequence, and $a, \dfrac{b(b-1)}{2}, c$ form an arithmetic sequence. If $1 < a < 3 < c < 7$, find the range of possible values of $b$.

Let $a \in \mathbb{R}$, and let $m$ be the number of integer solutions in $(0, 2025)$ of the inequality

Let $P$ be a moving point on the ellipse $C:\dfrac{x^2}3+\dfrac{y^2}2=1$. The two tangent lines from $P$ to the circle $x^2+y^2=1$ meet the ellipse again at points $A$ and $B$; fro…

Let $a, b, c > 0$ with $a^2 + b^2 = c^2$. Find the largest real number $k$ such that

holds for all s…

Two nonzero complex numbers $x, y$ satisfy $x^3 + y^3 = 0$ with $x \neq y$. Find the value of

In a parallelogram $ABCD$ of area $2$, a point $P$ moves along the line $AD$. Find the minimum value of

The four vertices of tetrahedron $P\text{-}ABC$ lie on a sphere $O$. We have $PA = PB = PC$, triangle $ABC$ is equilateral with side length $2$, and $E, F$ are the midpoints of $AC…

Let $a_n = n(n+1)\cdot 2^n$ for $n \in \mathbb{N}^*$. Prove:

1. $\dfrac{3}{a_1} + \dfrac{4}{a_2} + \cdots + \dfrac{n+2}{a_n} < 1$; 2. $\dfrac{4}{a_1} + \dfrac{5}{a_2} + \cdots + \…

Let $f(x) = \ln x - a\sqrt x$.

1. Discuss the monotonicity of $f$. 2. If some positive $k$ makes $g(x) = kx - 1 + f(x)$ have three zeros, find the range of $a$.