Problem Bank

Positive reals $a, b$ satisfy $a^2 + b^2 = 1$. Find the range of

A regular quadrilateral pyramid $O\text{-}ABCD$ has base edge length $\sqrt 6$ and height $3$. The sphere centered at $O$ with radius $\sqrt 2$ intersects the sphere $O_1$ passing …

A function $f(x)$ is defined on $\mathbb{R}$, $f(x+1)$ is odd, $f(x+2)$ is even, and for $x \in [1, 2]$, $f(x) = ax^2 + b$. If $f(0) + f(3) = 6$, find $f!\left(\dfrac{9}{2}\right)…

Let $f(x)$ be an odd function on $\mathbb R$ with $f(x)=x^3-3x$ for $0\leqslant x\leqslant 2$, and suppose the graph of $f$ is centrally symmetric about the point $(2,2)$. Determin…

As shown in the figure, $ABCD$ is a rhombus with side length $\sqrt 3$ and $\angle DAB=\dfrac{\pi}3$. Point $E$ lies on…

Let $\{a_n\}$ be an arithmetic sequence with all terms positive. Prove that

Let $a > b > 0$. Find the maximum value of

In a cube $ABCD\text{-}A_1B_1C_1D_1$ with edge length $3$, let $E$ and $F$ be the centroids of $\triangle A_1DB$ and $\triangle A_1DC_1$ respectively. Find the length of the chord …

The ellipse $E \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has left and right vertices $A, B$ and top vertex $C$, eccentricity $\dfrac{\sqrt{3}}{2}$, and the…

The line $y = kx$ meets the hyperbola $\dfrac{x^2}{4} - \dfrac{y^2}{3} = 1$ at points $P$ and $Q$, with $P$ in the first quadrant. Let $N$ be the foot of the perpendicular from $P$…

Let $f(x) = x - x^3\mathrm{e}^{ax+b}$, and suppose the tangent line to $y = f(x)$ at the point $(1, f(1))$ is $y = -x + 1$.

1. Find $a$ and $b$. 2. Let $g(x) = f'(x)$. Find the in…

Let $a \in \mathbb{R}$ and $\theta \in [0, 2\pi)$, and set

Find the number of pa…