Problem Bank

In a sequence $\{a_n\}$, $a_1 = 2$ and

1. Find a formula for $a_n$. 2. A sequence $\{b_n\}$ has $b_1…

A differentiable function $f$ on $\mathbb{R}$ satisfies

Prove that $f(x) < 0$ for all $x \in \mathbb{R}$.

Consider the hyperbola $C_1:x^2-y^2=1$ and the curve

1. Find the number of intersection points of $C_1$ and $C_2$. 2. Find the minimum distance …

The function

is monotonic on its domain. Find the range of possible values of the real number $a$.

Points $C$ and $D$ lie on the same side of the line $AB$, with $\triangle ABC\cong\triangle BAD$, $AB=9$, $BC=AD=10$, and $CA=DB=17$. The area of the intersection of the two triang…

Let $A = \{1, 2, m\}$ where $m$ is a real number. Let $B = \{a^2 \mid a \in A\}$ and $C = A \cup B$. If the sum of all elements of $C$ is $6$, find the product of all elements of $…

The graphs of

have exactly one common point. Find the range of $a$.

Let $x, y, z > 1$ with

Prove that

Let $A$ be the right vertex of the ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). Two mutually perpendicular lines $AP$ and $AQ$ through $A$ meet th…

Call a set $P$ discrete if it has at least $2$ elements and the absolute difference of any two of its elements is at least $3$. How many subsets of $M = \{1, 2, 3, \dots, 10\}$ a…

The inequality

holds for all $x > 0$. Find the minimum value of $m^2 + n^2$.

Prove that there are infinitely many odd positive integers $m$ for which $8^m + 9m^2$ is composite.