Problem Bank

The point $P$ lies on the right branch of $x^2 - \dfrac{y^2}{3} = 1$ with foci $F_1, F_2$, and $|PF_2| = 2$. Let $I$ be the incenter of $\triangle PF_1F_2$, $O$ the origin, and $M$…

In right triangle $ABC$, $\angle C = 90^\circ$, $AC = 4$, and $BC = 2$. Points $A$ and $C$ lie on the $x$-axis a…

Let $G$ be the centroid of $\triangle ABC$ with $AG\perp BG$, and

Find $\lambda$.

The real-coefficient equations

have four distinct roots whose points in the complex plane are concyclic. Find the ra…

Let $m, n \geqslant 0$, and suppose

is decreasing on the interval $\left[\dfrac{1}{2}, 2\right]$. Find the maximum value of $mn$.

In $\triangle ABC$,

Find $\sin\left(A + \dfrac{\pi}{4}\right)$.

A strictly increasing sequence of positive reals $\{x_n\}$ has the property: $x \in \{x_n\}$ iff $x + \{x\}^2$ is an integer, where $\{x\}$ is the fractional part of $x$. Find …

The curve $C$ (a lemniscate) is the locus of points whose distances to $F_1(-a, 0)$ and $F_2(a, 0)$ have product $a^2$. Determine, with proof, which of the following are true:

1…

Simplify and evaluate:

Real numbers $x, y, t$ satisfy

Find the minimum value of $x^2 + (y - 2t)^2$.

Let

where the real number $a < 3$.

1. Is $x = 1$ a critical (extremum) point of $f(x)$? Explain. 2. If $f(x) \leqslant 0$ on the interval …

The ellipse $\Gamma:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) passes through the point $P(0,\sqrt 3)$; its left and right vertices are $A_1$, $A_2$, and the product of the slo…