Problem Bank

In sector $AOB$ the central angle is $120^\circ$. $P$ is a point on arc $AB$, and $\overrightarrow{OP} = x,\overrightarr…

Given $AB = 10$, point $M$ is a moving point on segment $AB$. On the same side of $AB$, equilateral triangles…

In the coordinate plane $xOy$, let $B$ be the reflection of $A(-1,1)$ across the origin, and let $P$ be a moving point such that the product of the slopes of $AP$ and $BP$ is $-\df…

Let $\omega \in \mathbb{R}$ and $f(x) = (x - 6)^2 \sin(\omega x)$. Suppose there exists a constant $a \in \mathbb{R}$ such that $f(x + a)$ is an even function. Determine which of t…

Let $a \in \mathbb{R}$ and

1. When $a = 1$, prove that $f(x) > \mathrm{e}$. 2. If $f(x) > \m…

Find all complex numbers $\alpha$ such that for any distinct $z_1, z_2$ with $|z_1|, |z_2| < 1$,

Sequences $\{a_n\}$ and $\{b_n\}$ satisfy

with $a_1 = 2$ and $a_2 = 4$.

1. Find $a_3, a_4, a_5$. 2.…

A sequence $\{a_n\}$ satisfies $a_1 = 3$ and

1. Prove that $2 < a_{n+1} < a_n$ for all $n$. 2. Let $S_n = a_1…

A complex number $z$ satisfies $|z| = 1$ and

Find the least positive integer $n$ for which such a $z$ exists.

If the inequality

holds for all $x$ in the domain, find the range of possible values of the real number $m$.

Let $(1+x)^{50}=a_0+a_1x+a_2x^2+\cdots+a_{50}x^{50}$. Find the value of

Let $F$ be the right focus of the ellipse $\dfrac{x^2}{5} + y^2 = 1$, and let $M$ be the point of the ellipse in the first quadrant with $MF$ perpendicular to the $x$-axis. The lin…