Problem Bank

Let

Call a positive integer $m$ ($m \geqslant 4$) a related number of $A_{2n}$ if every $m$-el…

Fix the parabola $y^2 = 2px$ and points $A(a, b)$, $B(-a, 0)$ with $ab \ne 0$ and $b^2 \ne 2pa$. For a variable point $M$ on the parabola, the lines $AM$ and $BM$ meet the parabola…

For $0 < a < b < \dfrac{\pi}{2}$, arrange in increasing order:

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

A sequence $\{a_n\}$ satisfies $a_1=a$ and

Determine, with proof, which of the following claims are true.

1. If $a…

Let $m$ be a positive integer and

Suppose that for every integer $n$,

Suppose that for $x\in[0,1]$, the graphs of $y=(mx-1)^2$ and $y=\sqrt x+m$ have exactly one common point. Find the range of the positive real number $m$.

Let $f(x) = \sin^n x + \cos^n x$, where $n \in \mathbb{N}^*$.

1. When $n = 4$, determine whether $f(x)$ is even, odd, or neither. 2. For even $n$, the equation $f(x) = \dfrac{1}{2…

Given

find the maximum value of $\tan\gamma$.

Let $f(x) = \dfrac{x+1}{x^2+1}$. Find the number of real solutions of

The graph of

contains three points $M, N, P$ with $x$-coordinates $x_1, x_2, 1$ respectively.

1. Find the equation of the tangent li…

Find the least real $a$ such that

holds for all $x, y$ with $|x| + |y| \le 1$.