Problem Bank

Let $x, y, z$ be positive reals with $xyz = 1$. Prove that

Let $f(x)=|\lg x|-kx-2$. Determine, with proof, which of the following claims are true.

1. If $k=0$, then $f(x)$ has two zeros. 2. There exists $k<0$ such that $f(x)$ has exactly …

For $n \in \mathbb{N}^*$, let $f(n)$ be the number of positive divisors of $n$; for example $f(1) = 1$ and $f(4) = 3$. Determine, with proof, which of the following are true:

1. $…

A sequence satisfies

where $\lfloor x \rfloor$ denotes th…

1. Prove that for $x \in [0, 1]$,

2. If the inequality

holds for a…

Let $a > 0$ and

1. Let $g(a)$ denote the maximum value of $f(x)$ on the interval $[0, 4]$. Find an explicit expression for $g(a)$. …

Given $\left(\dfrac{12}{11}\right)^{10} \approx 2.3872$, prove that for all $n \ge 11$,

A sequence has $a_1 = \dfrac{1}{2}$ and

and $b_n = na_n$.

1. Prove that $\{a_n\}$ is decreasing. 2. Prove that $b_n \leqslant \dfrac{3}…

In the plane, $\overrightarrow{AB_1} \cdot \overrightarrow{AB_2} = 0$, $\left|\overrightarrow{OB_1}\right| = \left|\overrightarrow{OB_2}\right| = 1$, and $\overrightarrow{AP} = \ov…

Let $F_1, F_2$ be the foci of the hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), and let $P$ be a point on $C$. Drop perpendiculars from $P$ to th…

Let $a, b, c > 0$ with $ab + bc + ca = 1$. Prove that

Let $f(x) = \sin\omega x + \sin 2x$, where $\omega$ is a positive integer with $\omega \leqslant 2023$. If $f(x) < 2$ for all $x$, how many values of $\omega$ are possible?