Problem Bank

Let $a \in \mathbb{R}$ and $f(x) = |x^2 - 1| - x^2 + ax$.

1. If $f(x)$ is an even function, find $a$. 2. Suppose the graph of $f(x)$ meets the line $y = 2x$ at exactly $2$ points …

As shown in the figure, in the plane quadrilateral $ABCD$ we have $AB\perp BC$, $AD\perp DC$, $AB=AD=1$, and $\angle BAD=\dfrac{2…

The ellipse $E \colon \dfrac{x^2}{4} + y^2 = 1$ has left and right foci $F_1, F_2$. A moving point $P$ on $E$ is such that lines $PF_1$ and $PF_2$ meet $E$ again at points $Q$ and …

A sequence $\{a_n\}$ satisfies $a_1 = \dfrac{4}{3}$ and

Let $S_n$ be the sum of the first $n$ terms of $\left{\dfrac{1}{…

A sequence $\{a_n\}$ satisfies $a_1 = 1$ and

Prove that infinitely many terms of $\{a_n\}$ are divisible by $7$.

In the equilateral triangle $ABC$ with side length $2$ shown in the figure, $D$ is the midpoint of $BC$, and $E$, $F…

A regular tetrahedron can contain $10$ balls of radius $1$. Find the minimum possible edge length of the tetrahedron.

In an acute triangle $ABC$,

Find the maximum value of $\cos C$.

Vectors $\vec a, \vec b, \vec c$ satisfy $|\vec a| = |\vec b| = 2$, $|\vec c| = 1$, and

Find the range of $\vec a \cdot \vec b$…

Let $f(x)$ be a fourth-degree polynomial with $f(k)=\dfrac 1k$ for $k=1,2,3,4,5$. Find $f(6)$.

Let $H,P$ be two points in the plane of $\triangle ABC$, distinct from $A,B,C$, and write $\vec a=\overrightarrow{PA}$, $\vec b=\overrightarrow{PB}$, $\vec c=\overrightarrow{PC}$, …

Let $f(x) = a\ln x - \dfrac{x - b}{x + 1}$, and suppose the tangent line to $y = f(x)$ at $(1, f(1))$ is $x + 4y - 1 = 0$.

1. Find $a$ and $b$. 2. Find the intervals of monotonici…