Think Deeper.
Do Better Math.

Positive reals $a_1, \dots, a_{100}$ satisfy $a_i \ge a_{101-i}$ for $i = 1, \dots, 50$. Define

P…

Let $a$ be a constant and

1. Discuss the monotonicity of $f(x)$. 2. Suppose $f(x)$ has two zeros $x_1 < x_2$.

1. Find the ran…

A finite sequence $\{a_n\}$ with $k \geqslant 2$ terms is called conservative if

Let $T_m$ be …

Let $f(x) = ax^3 + bx^2 + cx + d$ with $a \neq 0$, and suppose $|f(x)| \leqslant 1$ for all $x \in [-1, 1]$. Find the maximum possible value of

A function $f(x)$ on $(0, +\infty)$ is defined piecewise: for $n - 1 < x \leqslant n$ (with $n \in \mathbb{N}^*$),

Determine, with proof, which o…

A finite geometric sequence $\{a_n\}$ has $N$ terms ($N \geqslant 3$), first term $a_1 > 0$, and common ratio $q \in (0, 1)$. Its terms are rearranged into a sequence $\{b_n\}$ suc…