Problem Bank

A sequence $\{a_n\}$ has $a_n > 1$, $a_1 = 2$, and

1. Prove that $\dfrac{n+7}{4} \leqslant a_n < a_{n+1} \leqslant n + 2$. 2. Prove that…

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has a vertex $A(2,0)$ and eccentricity $\dfrac{\sqrt 2}2$. The line $y=k(x-1)$ meets $C$ at two distinct points $M$ an…

Find the range of the function

The hyperbola $C:\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a>0$, $b>0$) has right focus $F$. Let $A$ be a point on the right branch of $C$, and let $D$, $E$ be the reflections of $A$ …

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…

Let $a,b,c\in\mathbb R$, and consider $f(x)=ax^2+bx+c$ and $g(x)=ax+b$, both on $-1\leqslant x\leqslant 1$. Prove that if $|f(x)|\leqslant 1$ for all $x\in[-1,1]$, then $|g(x)|\leq…

Call a positive integer $n$ good if for every arithmetic progression $a_1, a_2, \ldots, a_n$: whenever

is rational, at least one of $a_1, …

Let

where $a\leqslant b\leqslant c$. Determine, with proof, which of the following claims are true.

1. $f(b)\leqslant 0$. 2. If $b=\dfra…

Consider the ellipse $\dfrac{x^2}4+\dfrac{y^2}3=1$ with foci $F_1$, $F_2$. Perpendicular lines $l_1$ and $l_2$ through $F_1$ and $F_2$ meet the ellipse at $A,B$ and $C,D$ respectiv…

Let $a$ and $b$ be real numbers, and let $f(x)=|a^2x^2-1|+ax$. Suppose that

Find the minimum value of

Let $f(x) = x^2 + 2x + 4$ and $g(x) = 2\ln x + 2x + 5$.

1. Determine the number of zeros of $g(x)$, with justification. 2. Find the equations of all common tangent lines of the cu…

Real numbers $a, b, c$ satisfy $a^2 + b^2 + c^2 = 1$. Find the maximum and minimum of