Problem Bank

Find all positive integer pairs $(x, y)$ with

Prove:

1. The equation $x^3-x-1=0$ has exactly one real root $\omega$, and $\omega$ is irrational. 2. $\omega$ is not a root of any quadratic equation $ax^2+bx+c=0$ with integer c…

Let $f(x) = ax^2 + |x - a| + b$. Suppose that for every $b \in [0, 1]$ and every $x \in [-3, 3]$,

Find the range of possible values of $a$.

In a sequence $\{a_n\}$, $a_1 = \dfrac{2}{3}$, $a_2 = \dfrac{8}{9}$, and for $n \geqslant 2$,

1. Prove that $\{a_{n+1} - a_n\}$ is geometric. 2. F…

Three parallel lines $a, b, c$ are given, with $b$ between $a$ and $c$, and a point $A$ lies on $a$. Construct (with justification) an equilateral triangle $ABC$ with $B$ on $b$ an…

Let $O$ be the origin and let $F_1, F_2$ be the left and right foci of the hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) with eccentricity $\dfrac…

A sequence has $a_1 = \dfrac52$ and $2a_{n+1} = a_n^2 + 1$. Find the largest integer $m$ such that

Let $f(x) = a\ln x + (x+1)^2$ with $a \neq 0$ and $x > 0$.

1. Find the intervals on which $f(x)$ is monotonic. 2. If

holds for every $x \in …

Find the equation of the conic that passes through all intersection points of

and whose chord cut by the $…

Let $f(x) = \mathrm{e}^x$ for $x \in \mathbb{R}$.

1. Find the tangent line to the graph of the inverse function of $f(x)$ at the point $(1, 0)$. 2. Prove that the curves $y = f(x)…

In a tetrahedron $P\text{-}ABC$, $PB=PC$, $AB=AC=\sqrt 2$, $AB\perp AC$, and the plane $PBC$ is perpendicular to the plane $ABC$. Let $V$ be the volume of $P\text{-}ABC$ and $r$ th…

Prove that for all reals $a, b, c$,

and determine exactly when equality holds.