Problem Bank

Let $f(x) = x|x - a| - 2a^2$. If $f(x) > 0$ for all $x > 2$, find the range of possible values of $a$.

Let $a \geqslant b \geqslant c > 0$, and suppose the function

has range $\mathbb{R}$. Find the minimum value of

The parabola $y = x^2 + bx + c$ meets the coordinate axes at three points $A, B, C$. Prove that the circumcircle of $\triangle ABC$ always passes through a fixed point $P$, indepen…

Let $P$ be a point on an ellipse $E$ with foci $F_1, F_2$, and let $d$ be the distance from the center of $E$ to the tangent line of $E$ at $P$. Prove that

Triangle $DEF$ has vertex $E$ on the $x$-axis and $F\left(\dfrac{1}{4}, 0\right)$, with $|DF| = |EF|$, and the midpoint $M$ of side $DE$ lies on the $y$-axis. Let $\Gamma$ be the l…

Three people each independently choose one of $5$ destinations uniformly at random. Find the probability that exactly $1$ of them chooses a particular given destination.

The function $f(x) = ax^3 - \dfrac{12x}{a}$ (with $a \neq 0$) attains its local minimum at $x = 2$, that is, the local minimum value is $f(2)$. Find $a$.

Let $f(x) = \dfrac{a\ln x + 1}{x}$ with $a \in \mathbb{R}$, and $g(x) = \mathrm{e}^x - 1$.

1. Discuss the monotonicity of $f(x)$. 2. If $f(\mathrm{e}) = \dfrac{2}{\mathrm{e}}$, pr…

Let $f(x) = \dfrac{x}{1+x} - \ln(1+x)$ and $g(x) = \ln(1+x) - bx$.

1. Find the equation of the tangent line to $y = f(x)$ at $(1, f(1))$. 2. Is there a real number $b$ such that $…

A line $l$ meets the parabola $y^2 = 4x$ at two points $A$ and $B$. If the quadrilateral $OAMB$ (where $O$ is the origin) is a rectangle, find the maximum value of the slope of the…

Positive real numbers $a, b$ satisfy

Compare $a$ and $2b$, with proof.

Let $A = \{0, 1, 2, 3, 4, 5, 6\}$, and let $f$ be a function from $A$ to the integers such that $f(0) = 0$, $f(6) = 12$, and