Problem Bank

Let $f(x) = (x-a)^2(x+b)e^x$, where $a, b \in \mathbb{R}$.

1. When $a = 0$ and $b = -3$, find the intervals of monotonicity of $f(x)$. 2. When $a = 0$: if $x = 0$ is a local maxim…

Let $f(x) = x - 1 - \ln x$. If distinct positive reals $x_1, x_2$ satisfy $f(x_1) = f(x_2)$, prove that

Positive real numbers $x, y, z$ satisfy

where $\lg$ denotes the base-$10$ logarithm…

A sequence $\{a_n\}$ satisfies $a_1 = 1$ and

1. Prove that $a_n < 5$ for every positive integer $n$. 2. Pro…

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 $f(x) = \sin \pi x$ and $g(x) = -x^2 + 2ax - a^2 + 1$. Writing $\max\{a, b\}$ for the larger of $a$ and $b$, define

If $M(x)$ has exactly two z…

Suppose that for all $m, n \in \mathbb{R}$, the inequality

holds for every real $x$. Find the maximum possible value of the r…

As shown in the figure, $F_1$ and $F_2$ are the two foci of the hyperbola

and $O$ is the…

A line through the fixed point $A(-1, 0)$ meets the parabola $C \colon y^2 = 4x$ at $M$ and $N$, and $Q$ is a point of the parabola different from $M, N$. If the line $QM$ always p…

A right triangle $DEF$ has its three vertices on the sides $AB$, $BC$, $CA$ of an equilateral triangle $ABC$ respectively, with $\angle DEF=90^\circ$ and $\angle EDF=30^\circ$. Fin…

The function $f(x) = ax^3 + bx^2 + cx + d$ has local extrema at the two points $O$ and $A$, where $O$ is the origin and $A$ lies on the curve

Let $f(x) = a^x + b^x - c^x$, where $c > a > 0$ and $c > b > 0$.

1. Let $M$ be the set of triples $(a, b, c)$ such that $a, b, c$ cannot be the side lengths of a triangle and $a =…