Problem Bank

Let $F$ be the focus of the parabola $C \colon y = x^2$, and let $P$ move along the line $l \colon x - y - 2 = 0$. From $P$, the two tangent lines $PA, PB$ touch the parabola at $A…

In a planar quadrilateral $ABCD$, $AD=1$, $BD=\sqrt 5$, $AB\perp AC$, and $AC=2AB$. Find the minimum value of $CD$.

In the isosceles right triangle $OPQ$, $\angle POQ = 90^\circ$ and $OP = 2\sqrt{2}$. A point $M$ lies on the segm…

Let $O$ be the origin, and let $A, B, C$ be three points on the ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) satisfying

The quadratic $f(x) = ax^2 + 2bx + c$ with $c > b > a$ passes through $(1, 0)$ and meets the line $y = -a$. Prove that

Let $x, y \in \mathbb{R}$ with

Find the minimum value of $x - |y|$.

Let $A, B$ be the left and right vertices of the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$), and let $P_1, P_2, \ldots, P_n$ be distinct points on the h…

Let $ABCDEF$ be an equiangular hexagon. Prove that

A function $f$ has domain and range $[0, 1]$. Define $f_1 = f$ and $f_n = f \circ f_{n-1}$; solutions of $f_n(x) = x$ are called $n$-th order periodic points of $f$.

For

Two walls meet at a right angle. A fence of total length $60$ meters, consisting of two straight segments, is placed so that the two fence segments and the two walls bound a quadri…

Nonnegative reals $a, b, c$ satisfy $a + b + c = 1$. Find the maximum value of

A player enters a best-of-three match (first to win two games wins the match). Each game is won with probability $p$ ($0 < p < 1$) and lost with probability $1 - p$, independently …