Problem Bank

Prove that for all real numbers $a, b, c$,

A sequence $\{a_n\}$ satisfies $a_1 = 1$ and

Prove that for every $n \geqslant 2$,

is a positive in…

Let $f(x) = x^2 - \pi x$, and let

The hyperbola $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ ($a,b>0$) has right focus $F$ and right vertex $A$. The line through $F$ perpendicular to $AF$ meets the hyperbola at points $B$…

A shooting contest has two stages, and each team consists of two players. In stage one, one team member shoots $3$ times; if all $3$ shots miss, the team is eliminated with score $…

In $\triangle ABC$,

Find $\sin\left(A + \dfrac{\pi}{4}\right)$.

Let

1. Prove that on $(0, +\infty)$, $f(x)$ has a unique critical point and a unique zero. 2. Let $x…

A sequence has $a_0 = 1$ and

1. Prove that every $a_n$ is a positive integer. 2. Prove that $a_na_{n+1} - 1$ is always a perf…

Let $A$ be the right vertex of the ellipse $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$), and let $PQ$ be a chord not through $A$ whose slope is a fixed constant $k$…

In right triangle $ABC$, $\angle B=\dfrac{\pi}{2}$ and $AC=2BC=4$, and $D$ is the midpoint of segment $AC$. Triangle…

Points $A$ and $B$ lie on skew lines $a$ and $b$ respectively, and the segment $AB$ is perpendicular to both. Moving points $P\in a$ and $Q\in b$ satisfy that $PA+QB$ is constant. …

A sample consists of the data $0, 1, 2, 3, 4$. A random value $X$ with

is appended to form a new sampl…