Problem Bank

Let $O$ be the circumcenter of $\triangle ABC$, with $|AB| = 2$ and $|AC| = 4$. Find the minimum value of

Call a function a Taiji function of a circle if its graph divides both the area and the perimeter (circumference) of the circle into two equal parts. Determine, with proof, which…

Let $f(x) = \mathrm{e}^x - \dfrac{1}{2}ax^2 - x$.

1. If $f(x)$ is increasing on all of $\mathbb{R}$, find the value of $a$. 2. For $a > 1$:

1. Prove that $f(x)$ has two extreme…

Define the positive logarithm by

Consider the following four propositions, where $a > 0$ and …

An ellipse $E$ centered at the origin $O$ with foci on the $x$-axis passes through $A\left(\sqrt{2}, -\dfrac{\sqrt{2}}{2}\right)$ and $B(0, 1)$.

1. Find the standard equation of $…

As shown in the figure, in square $ABCD$, $E$ is the midpoint of side $AB$, and $P$ is a point on the circular arc from $B$ to…

Let $f(x) = a^x + \mathrm{e}^{-x}$ with $a > 1$.

1. Prove that $f(x)$ has an extreme value. 2. Suppose $f(x)$ attains its extreme value at $x = x_0$. If there exist integers $m, n…

For an $n$-dimensional vector $A=(a_1,\dots,a_n)$ with each $a_i\in\{0,1\}$, call $A$ an $n$-dimensional T-vector. For two such vectors $A,B$, define $d(A,B)=\sum_{i=1}^n|a_i-b_i…

The coordinate axes of a rectangular system $O\text{-}xyz$ are rotated about a line $l$ through $O$ so that the $Ox$-axis maps to the $Oy$-axis and the $Oy$-axis maps to the $Oz$-a…

Given $2\tan(\alpha + \beta) = 3\tan\alpha$, find the maximum value of

Let $a, b, m > 0$. The inequality

holds for all $a, b > 0$. Find the maximum value of $m$.

A function $f \colon \mathbb{R} \to \mathbb{R}$ satisfies, for all real $x, y, z$,

Find