Problem Bank

A positive real number $a$ satisfies

Find the range of possible values of $a$.

Let $f(x) = \ln x + x^2 - kx + 1$ with $k \in \mathbb{R}$, and $g(x) = x^2 - 3x + x\mathrm{e}^x$.

1. Find the intervals on which $f(x)$ is monotonic, discussing all cases. 2. If t…

In a tetrahedron $P\text{-}ABC$, $PB=PC$, $AB=AC=\sqrt 2$, $AB\perp AC$, and the plane $PBC$ is perpendicular to the plane $ABC$. Let $V$ be the volume of $P\text{-}ABC$ and $r$ th…

Let $f(x) = a\ln x - x$.

1. Discuss the monotonicity of $f(x)$. 2. Let $0 < a < 1$. If the function

has a zero on $[1, +\infty)$, find the …

The equation

has exactly two real solutions. Find the range of $m$.

Triangle $ABC$ has integer side lengths with $AB=AC$. A circle $\omega$ is centered at the incenter of $\triangle ABC$. The excircle opposite to the side $BC$ is internally tangent…

Let $a, b, c > 0$. Prove that

In quadrilateral $ABCD$ the diagonals satisfy $AC\perp BD$. Prove that

Let $f(x) = x^3 - 3x^2 - x + 3$.

1. Compute $\displaystyle\sum_{i=1}^{7} f\!\left(\frac{i}{4}\right)$. 2. If the curve $y = f(x) + a\ln x$ (for $x \in (3, 4)$) has two mutually pe…

Six students sit in a row of connected desks and can leave only from the two ends of the row. They finish their exams in a uniformly random order, and a student who wants to leave …

In parallelepiped $ABCD\text{-}A_1B_1C_1D_1$ the base is a rhombus, and the three edges meeting at vertex $C$ ma…

Climbing a staircase, each step is $1$ stair with probability $\dfrac{2}{3}$ or $2$ stairs with probability $\dfrac{1}{3}$, independently. Given that the climber lands exactly on s…