Problem Bank

The sequence $2, 3, 4, a, b, 20, 30, 40$ consists of positive integers and is strictly increasing. Suppose every strictly increasing sequence formed by choosing $4$ of its terms is…

In equilateral triangle $ABC$, $D$ and $E$ are the midpoints of $AB$ and $AC$ respectively. $M$ is a moving …

Suppose $C, L \in \mathbb{R}$ with $L \neq 0$ satisfy

A function $f(x)$ defined on all of $\mathbb{R}$ satisfies $f(x) = 2^x - x$ for $x \in [0, 1)$, and

Let $a = \log_2 3$. Find…

In the coordinate plane, the equation

describes an ellipse. Find the range of possible values of $m$.

Let $A$ be the right vertex of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$), and let $PQ$ be a chord not through $A$ whose slope is a fixed constant $k$…

Let $a \neq 0$, and suppose $x = a$ is a local maximum point of

Determine the relationship between $ab$ and $a^2$, with proof.

Prove that $x^x>4\ln x$ for all $x>0$.

Let $f(x)=\mathrm e^x+x$. For three points $A,B,C$ on the curve $y=f(x)$ whose $x$-coordinates form an arithmetic progression, determine, with proof, which of the following are tru…

Order from smallest to largest:

A point $(x,y)$ is rational if both $x$ and $y$ are rational. For a given positive real $r$, determine the maximum possible number of rational points on the circle

As shown in the figure, $OABC$ is a square in the coordinate plane, where $O$ is the origin, $A = (10,0)$, and $C = (…