Problem Bank

As shown in the figure, $ABCD$ is a quadrilateral inscribed in the base circle of a cylinder, $AC$ is a diameter of tha…

A sequence $\{a_n\}$ has general term

Find all real numbers $x$ for which

Each cell of a $3 \times 3$ grid is colored red or blue. How many colorings have both three collinear blue cells and three collinear red cells (rows, columns, or diagonals)?

Find the number of $7$-tuples of positive integers $(a, b, c, d, e, f, g)$ that satisfy the system

Consider $a_n = \tan(11n)$ (radians, $n \in \mathbb{N}^*$).

1. Prove that $a_1 < a_3 < a_5 < \cdots < a_{709}$. 2. Prove that the sequence $\{a_{2k-1}\}$ (all odd indices) is **no…

Let $f(x)=\sqrt{2x^4-18x^2+12x+68}+x^2-x+1$. Determine, with proof, the minimum value of $f(x)$ and the number of points at which it is attained.

Let $f(x) = e^x - e^{-x} - ax$ with $a > 0$, and suppose $f(x)$ has two critical points $x_1, x_2$. Let $k$ be the slope of the line through $A(x_1, f(x_1))$ and $B(x_2, f(x_2))$. …

Suppose the graphs of $f(x)$ and $g(x)$ meet the line $x = m$ at points $A, B$ respectively, and meet the line $x = n$ at points $C, D$ respectively, where $m < n$. If the slopes o…

The ellipse $\Gamma:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has foci $F_1(-c,0)$ and $F_2(c,0)$. A line $l$ is tangent to $\Gamma$, and $d_1$, $d_2$ denote the distances fro…

In a sequence $\{a_n\}$, $a_1=3$ and $a_{n+1}a_n+\lambda a_{n+1}+\mu a_n^2=0$ for $n\in\mathbb N^{*}$.

1. If $\lambda=0$ and $\mu=-2$, find the general term of $\{a_n\}$. 2. If $\…

In $\triangle ABC$, $A+B=3C$ and $2\sin(A-C)=\sin B$.

1. Find $\sin A$. 2. If $AB=5$, find the altitude to the side $AB$.

Let $\alpha, \beta \in \mathbb{R}$ satisfy

Find the minimum value of $\sin\alpha + \cos\beta - 2$.