Problem Bank

Let $C$ be the curve $x^2 + y^2 = 2x - 4y$. Points $P(x_1, y_1)$, $Q(x_2, y_2)$, $R(x_3, y_3)$ lie on $C$. Find the minimum value of

Let $O$ be the center of an ellipse, and $A$ a point on the ellipse other than a vertex. Drop the perpendicular from $A$ to the major axis, with foot $M$. Extend $AO$ to meet the e…

The parabola $y^2=2px$ ($p>0$) meets the hyperbola $y=-\dfrac 1x$ at a point $R$. A common tangent line of the two curves touches the parabola at $S$ and the hyperbola at $T$. Prov…

Let $f(x) = x^2 + (a-2)x - a\ln x$ with $a \in \mathbb{R}$.

1. If $a = 1$, find the extreme values of $f(x)$. 2. Discuss the monotonicity of $f(x)$. 3. For $n \in \mathbb{N}^*$, p…

Arrange $1, 2, \dots, n^2$ clockwise in a spiral to fill an $n \times n$ table $T_n$, first row $1, 2, \dots, n$; for example

The ellipse $C \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has eccentricity $\dfrac{2\sqrt{2}}{3}$, bottom vertex $A$, right vertex $B$, and $|AB| = \sqrt{10…

Let

Arrange $a, b, c$ in increasing order, with proof.

The line $ax+by+c=0$ meets the circle $O:x^2+y^2=16$ at two points $M$ and $N$, where the coefficients satisfy $c^2=a^2+b^2$. Let $P$ be an arbitrary point on the circle $O$. Find …

Let $0 < a < b$. Lines $\ell$ and $m$ pass through $A(a, 0)$ and $B(b, 0)$ respectively, and meet the parabola $y^2 = x$ in four distinct points. When those four points are concycl…

A point $O$ in the plane of $\triangle ABC$ satisfies

with $|\overrightarrow{OA}| = 2$, …

Prove that $\ln 2<\lg 5$.

A sequence $\{a_n\}$ satisfies

Find