Problem Bank

The two foci of the ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) together with one endpoint of the minor axis are the vertices of a right triangle. The line $l:y=-x+3$…

An arithmetic sequence $\{a_n\}$ with nonzero common difference satisfies $a_1 = 2$, and $a_1, a_3, a_7$ form a geometric sequence.

1. Find a closed formula for $a_n$. 2. A sequen…

Let $O$ be the circumcenter of $\triangle ABC$, whose sides $a, b, c$ (opposite $A, B, C$) satisfy

Find the range of possible values of $\overrightarrow{B…

Let $x, y > 0$. Find the range of possible values of

Let $f(x) = x - 1 - \ln x$. If distinct positive reals $x_1, x_2$ satisfy $f(x_1) = f(x_2)$, prove that

Let $F_1, F_2$ be the left and right foci of the hyperbola $x^2 - \dfrac{y^2}{3} = 1$. A line through $F_2$ with inclination angle $\theta$ meets the right branch at two points $A$…

The function

has no zeros. Find the range of the positive real $a$.

Define the operation

Evaluate

Consider the hyperbola $\Gamma \colon x^2 - \dfrac{y^2}{b^2} = 1$ (with $b > 0$) with left and right vertices $A_1, A_2$. A line $l$ through the point $M(-2, 0)$ meets $\Gamma$ at …

The inequality

holds for all $x \in (0, 3e]$. Find the range of possible values of $a$.

Suppose $|2x-a|\geqslant x-1$ for all real $x\in[0,2]$. Find the range of $a$.

Let $f(x) = \left|x^2 - a\right|$ with $a > 0$. Suppose there are exactly two pairs $(m, n)$ with $m < n$ such that $f$, restricted to the domain $[m, n]$, has range exactly $[…