Problem Bank

The graph of

is symmetric about the line $x = 3$. Find the maximum value of $f(x)$.

The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has foci $F_1(-c,0)$ and $F_2(c,0)$ ($c>0$). A line $l$ through the right focus $F_2$ is tangent to the circle $x^2+y^2=…

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has focal distance $4$ and passes through the point $P\left(\sqrt 2,\sqrt 3\right)$.

1. Find the equation of the elli…

Consider the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). Let $PQ$ be a focal chord through the left focus $F$, not perpendicular to the $x$-axis. Suppose …

A sequence has $a_1 = \dfrac13$ and

1. Prove $a_n < a_{n+1} < 1$ for all $n$. 2. Prove $a_n > \dfrac12 - \dfrac{1}{4n}$ for all $n$.

Let

If $y = f(x) - g(x)$ has exactly $2$ distinct …

The figure shows part of the graph of a function $f(x)$. Determine, with proof, which of the following four expression…

In a circular sector $AOB$, $OA = OB = 1$ and $\angle AOB = \dfrac{\pi}{3}$. A point $C$ on the arc $AB$ (excluding the endpoints) is written as

Let $x, y \geqslant 0$ with $(1 + x)(1 + y) = 2$. Prove that

From $\{1, 2, \dots, 2018\}$, at most how many numbers can be chosen so that no two of them have ratio exactly $\dfrac32$?

Let $n$ be a positive integer and $x \in \mathbb{R}$. Prove that

In a right triangle $ABC$ with the right angle at $B$, $AB=1$ and $BC=2$. A point $D$ moves along the hypotenuse $AC$. The triangle $ABD$ is folded along $BD$ to a triangle $A_1BD$…