Problem Bank

Plane vectors $\boldsymbol a,\boldsymbol b,\boldsymbol c$ satisfy

Find the range o…

Let $E, F$ be the trisection points of the hypotenuse $AB$ of an isosceles right triangle $ABC$. Find $\tan\angle ECF$.

Let $A$ and $B$ be two points on the ellipse $\dfrac{x^2}4+y^2=m$ ($m>1$), with $P(0,1)$ and $\overrightarrow{AP}=2\overrightarrow{PB}$. Find the maximum value of the $x$-coordinat…

Let $A(x_1, y_1)$ and $B(x_2, y_2)$ be two distinct intersection points of the graphs of

Find the range …

Let $f(x) = ax - m$ (with $m \in \mathbb{N}^*$) and $g(x) = \ln\dfrac{x}{a}$. If

find the range of pos…

Let $f(x) = \dfrac{1}{x} - x + 2a\ln x$, where $a$ is a real number.

1. If $a = \dfrac{1}{2}$, find the equation of the tangent line to $y = f(x)$ at $(1, f(1))$. 2. Find the inte…

The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ $(a>b>0)$ shown has left and right foci $F_1$ and $F_2$. Points $A$…

Say an ellipse or hyperbola has a $K$-point if there is a point $P$ on it whose distances to the two foci are in ratio $2 : 1$. Determine, with proof, which of the following curv…

As shown in the figure, an ellipse is centered at the origin $O$ with its major axis on the $x$-axis and eccentri…

A function $f$ on $\mathbb{R}$ satisfies $f(1) = \dfrac14$ and

Find $f(2016)$.

A square sheet of paper $ABCD$ with side length $1$ is folded so that vertex $B$ lands at a point $B'$ on side $AD$, as show…

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

1. Prove that $\cos B = 2…