Problem Bank

Let $f(u) = u^2 + au + b - 2$, where $u = x + \dfrac{1}{x}$ for real $x \neq 0$. If the composite function of $x$ has a zero, find the minimum value of $a^2 + b^2$.

Let $B$ be the top vertex of the ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$). If every point $P$ on $C$ satisfies $|PB| \leqslant 2b$, find the ra…

Let $f(x) = \mathrm{e}^x - \dfrac{1}{2}x^2 - ax - 1$ with $a \in \mathbb{R}$.

1. If $f(x) \geqslant 0$ for all $x \in [0, +\infty)$, find the range of possible values of $a$. 2. P…

Let $C$ be the locus of points in the plane whose distances to the fixed point $F(0, -2)$ and to the fixed line $l \colon y = 2$ sum to $6$. For a point $P$ on $C$ and a given poin…

Let $f(x)=x^3+ax^2-x+1-a$. Suppose that

Find the range of the real number $a$.

In $\triangle ABC$, the sides opposite $A, B, C$ are $a, b, c$, and

Let $f(x) = mx\ln x$ with $m \in \mathbb{R}$.

1. Find the intervals on which $f(x)$ is monotonic, discussing all cases. 2. When $0 < m \leqslant \dfrac{\mathrm{e}^2}{2}$, prove th…

Real numbers $x, y, z$ satisfy

Fin…

The set $A = \{1011, 1012, \dots, 2022\}$ is partitioned arbitrarily into two disjoint nonempty subsets. Prove that at least one of the subsets contains two numbers whose sum is a …

An ellipse centered at the origin $O$ has foci on the $x$-axis and right vertex $A$. If some point $P$ on the ellipse (other than $A$) satisfies $\angle OPA = 90^\circ$, find the r…

Compute

Vectors $\vec{a}, \vec{b}$ make an angle of $\dfrac{\pi}{3}$ and $\left|\vec{a} - \vec{b}\right| = 5$; the vectors $\vec{c} - \vec{a}$ and $\vec{c} - \vec{b}$ make an angle of $\df…