Problem Bank

Let $S_n$ be the partial sums of a sequence $\{a_n\}$ with $a_1 = 1$, $a_2 = 6$, $a_3 = 11$, and

where $A, B$…

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 $A_n=\{(x_1,\dots,x_n)\mid x_i\in\{-1,1\}\}$. For $\vec x,\vec y\in A_n$ define $\vec x\cdot\vec y=x_1y_1+\cdots+x_ny_n$; if $\vec x\cdot\vec y=0$, call $\vec x$ and $\vec y$ *…

Evaluate

Let $f(x)=|x+1|+|ax-2|$ ($a>0$) with domain $\mathbb R$, and let $M(a)$ be its minimum value. Determine, with proof, which of the following claims are true.

1. The minimum value o…

The figure shows part of the graph of the function $f(x)=2\cos(\omega x+\varphi)$, where $\omega>0$; the graph att…

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

Find the minimum value of

The hyperbola $E \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) has a vertex on the line $l \colon y = x + 1$ and eccentricity $\sqrt{5}$.

1. Find the standard…

The point $P(x, y)$ satisfies

Find the range of possible values of $\dfrac{x + y}{\sqrt{x^2 + y^2}}$.

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…

A robot plays a shooting game: a made shot scores $+1$, a miss scores $-1$. The game ends when the cumulative score $X$ reaches $2$ (a win) or $-2$ (a loss); otherwise play continu…

In the tetrahedron $P\text{-}ABC$ shown in the figure,

A poi…