Problem Bank

Let $f(x) = ax^2 + 4x - 2$. Suppose the real number $a$ can be chosen so that

Find the minimum possible value of $m$, …

Find all $k$ such that

holds for all $x > 0$ with $x \ne 1$.

Let $f(x) = \log_{x+1}(x+2)$ for $x \geqslant 1$, and

Let $F(n)$ be the maximum of $g(x)$ on $[1, +\infty)$.…

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 …

Let $f(x) = |x - a| - \dfrac4x + a$ with $a \in \mathbb{R}$. Determine whether there exists $a$ such that the equation $f(x) = 3$ has exactly three distinct real roots forming an a…

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

is appended to form a new sampl…

Let $A, B$ be two points on the hyperbola $x^2 - \dfrac{y^2}{9} = 1$. For each of the points

determine whether it can …

Let $f(x) = |x + a| + |x^2 + b|$ for $x \in [0, 1]$, and let $M(a, b)$ denote the maximum of $f(x)$ on $[0, 1]$. If the minimum possible value of $M(a, b)$ is $1$, find all possibl…

Consider the ellipse $E \colon \dfrac{x^2}{4} + \dfrac{y^2}{3} = 1$. A line through $P(2, 1)$ meets the ellipse at $M$ and $N$; through $N$, the line of slope $-\dfrac{3}{2}$ meets…

In the coordinate plane, $B$ moves along the curve $y = \sqrt{1 - x^2}$, and $A(2, 0)$. A point $C$ in the first quadrant makes $\triangle ABC$ an isosceles right triangle with the…

As shown in the figure, in a regular tetrahedron $ABCD$, points $P$, $Q$, $R$ lie on edges $AB$, $AD$, $AC$, respectively, …

As shown in the figure, the equilateral triangle $A'BC$ has side length $1$. Its vertex $A'$ lies in a plane $\alpha$…