Problem Bank

Let $a$ and $b$ be positive numbers with $\dfrac{1}{a} + \dfrac{1}{b} = 1$, and let $n \in \mathbb{N}^*$. Prove that

Find all primes $x$ and $y$ with

Let $f(x)=2|x+1|-|x-2|$.

1. Solve the inequality $f(x)\leqslant 0$. 2. Let $g(x)=|3x-a|$. If $g(x)\geqslant f(x)$ for all $x\in\mathbb R$, find the range of $a$.

A moving point $P(x, y)$ is equidistant from the point $F(1, 0)$ and the line $x = -1$.

1. Find the equation of the locus $L$ of $P$. 2. Let $M(x_0, y_0)$ with $y_0 \geqslant 0$ b…

As shown in the figure, from a point $P$ outside an ellipse, the two tangent lines $PA$ and $PB$ are drawn, touching …

Let $z_1, z_2, \dots, z_{673}$ be distinct complex numbers such that

where $g(x)$ is a p…

In $\triangle ABC$, the sides opposite $A, B, C$ are $a, b, c$. The height on side $AC$ equals $c - a$.

Find the value of

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 $k$.

A tetrahedron $P\text{-}ABC$ is inscribed in a sphere $O$, with $PA : PB : PC = 1 : 2 : 3$. When the lateral surface area of the tetrahedron (the total area of the three faces meet…

Let $f(x) = |x + a| + |x^2 + b|$ for $x \in [0, 1]$, and let $M(a, b)$ denote the maximum of $f(x)$ on $[0, 1]$. If the minimum possible value of $M(a, b)$ is $1$, find all possibl…

Let

Suppose the equation $f(x) = m$ has $3$ real solutions $x_1 < x_2 < x_3$.

1. Find the ra…

Let $f(x) = \dfrac{\arcsin x}{\sqrt{1 - x^2}}$.

1. Prove that for $n \in \mathbb{N}^*$,

2. Fin…