Problem Bank

Solve over the complex numbers:

The quadratic $f(x) = ax^2 + bx + c$ ($a \ne 0$) satisfies

Find the maximum possible value of $|f(x)|$ over $x \in [-…

Find the minimum and maximum of

A real function $f$ satisfies

Do there exist integers $n$ with $f(n)$ also an integer? Find all such $n$ or prove none exist.

Let $\omega$ be the circumcircle of an acute triangle $ABC$ with orthocenter $H$. The tangent line at $H$ to the circumcircle of $\triangle HBC$ meets $\omega$ at points $X$ and $Y…

The hyperbola $C:x^2-\dfrac{y^2}{24}=1$ has left and right foci $F_1$, $F_2$. Let $P$ be a point of $C$ in the first quadrant, and let $l$ be the bisector of $\angle F_1PF_2$. The …

Suppose the function

has exactly three zeros in $(0,3]$. Find the range of the real number $r$.

Let $f(x) = (x^2 + ax + b)e^x$ with $b < 1$, and suppose $f(x)$ is increasing on both $(-\infty, -2)$ and $(1, +\infty)$. Find the range of possible values of $\dfrac{a+b}{a-2}$.

The ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has eccentricity $\dfrac{\sqrt{2}}{2}$, and the circle whose diameter is the minor axis of $C$ is …

A convex pentagon $ABCDE$ has $AB=5$, $BC=CD=DE=6$, $EA=7$, and has an inscribed circle. Find the area of the pentagon $ABCDE$.

Let

If the equation $f(x) = g(x)$ has exactly two real solutions on $(0, +\infty)$, find the ra…

In $\triangle ABC$, the centroid is $G$ and the circumcenter is $O$, and $OG\parallel BC$. Find $\tan B\tan C$.