Problem Bank

Real numbers $a_1, a_2, \ldots, a_{2017}$ satisfy $a_1 + a_2 + \cdots + a_{2017} = 0$ and

P…

Let $\max\{a, b, c\}$ denote the largest of the real numbers $a, b, c$. For $x, y, z > 0$, find the minimum value of

Let $f(x) = e^x - \dfrac{a}{x}$, where $a$ is a real constant.

1. For $a > 0$, find the intervals of monotonicity of $f(x)$. 2. If $f(x)$ has a critical point in $(0, +\infty)$ an…

Let

Arrange $a, b, c$ in increasing order, with proof.

Let $A(2, 3)$ and $B(2, 1)$. The hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a > 0$, $b > 0$) has an asymptote $y = x$, and the chord cut by line $AB$ on $C$ has…

The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has eccentricity $e=\dfrac 12$, left vertex $A$, and lower vertex $B$; let $C$ be the midpoint of the segment $OB$ ($O$ …

Nonnegative reals $x, y$ satisfy $x + y = 4$. Find the minimum of

A sequence satisfies $a_1 = 1$ and

Prove that $a_{2015} > 63$.

The angles of $\triangle ABC$ satisfy

Prove that $A = B$.

As shown in the figure, in the coordinate plane $xOy$, let $M(x_0, y_0)$ be a point on the ellipse

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…

Solve the inequality