Problem Bank

Find the minimum value of

As shown in the figure (which illustrates the case $n = 4$), an equilateral triangle $ABC$ is divided into $n$ rows…

The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a>0$, $b>0$) has left and right foci $F_1$, $F_2$. A point $A$ lies on $C$ and a point $B$ lies on the $y$-axis, with

A class has two math study groups: group $A$ with $10$ students and group $B$ with $30$ students. On a test, group $A$ has mean score $130$ and variance $115$, while group $B$ has …

Given

find $\sin!\left(\alpha + \dfrac{\pi}{6…

Find the least positive period of

Let $a>0$ and

1. For $a=1$, find the maximum value of $f(x)$. 2. Determine the number of zeros of $f(x)$, and justify your answer.

Let $P(a, b)$ be any point on the ellipse $\dfrac{x^2}{3} + \dfrac{y^2}{2} = 1$. From the origin $O$, two tangent lines to the circle

Consider the circle $M:(x+\cos\theta)^2+(y-\sin\theta)^2=1$ and the line $l:y=kx$. Determine, with proof, which of the following claims are true.

1. For all real $k$ and $\theta$,…

A quadratic function $f(x)=ax^2+bx+c$ ($a,b,c\in\mathbb R$, $a\neq 0$) satisfies the following conditions:

1. $f(x-4)=f(2-x)$ for all $x\in\mathbb R$, and $f(x)\geqslant x$ for al…

Let $f(x) = \dfrac{a}{x} - \ln x$ with $a \in \mathbb{R}$.

1. Discuss the monotonicity of $f(x)$. 2. If $x_1, x_2$ are two real solutions of the equation $f(x) = 2$, prove that

A sequence satisfies $a_1 = 1$ and

Prove that $a_{2015} > 63$.