Problem Bank

A closed cylindrical container (wall thickness negligible) has base radius $4\text{ cm}$ and height $9\text{ cm}$, and contains two iron balls of equal radius. Find the maximum pos…

A regular hexagon of side length $126$ is divided into unit equilateral triangles whose sides are parallel to the sides of the hexagon and have length $1$. Let $n$ be the number of…

Let $f(x) = \dfrac13 x^3 - a\left(x^2 + x + 1\right)$.

1. For $a = 3$, find the intervals of monotonicity of $f$. 2. Prove that $f$ has exactly one real zero (for every real $a$).

Prove that for every nonzero real number $x$,

Consider the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{6} = 1$ with foci $F_1, F_2$ and center $O$. A point $P$ on the ellipse satisfies

Find $|…

Let $f(x) = e^x - ax$ and $g(x) = \ln x - ax$. Suppose that for every $x_1 \in (0, +\infty)$ there exists $x_2 \in (0, +\infty)$ such that

Find the …

The function

attains a maximum value. Find the range of possible values of $a$.

A geometric sequence $\{a_n\}$ with first term $\dfrac{3}{2}$ is not non-increasing, and its partial sums $S_n$ satisfy: $S_3 + a_3$, $S_5 + a_5$, $S_4 + a_4$ form an arithmetic se…

The function

has exactly $4$ zeros. Find the range of possible values of the real number $a$.

As shown in the figure, the triangle $ABC$ is inscribed in the parabola $E: x^2 = y$, and the lines containi…

Complex numbers $x, y, z$ satisfy

Find $|x + y + z|^2$.

Let $a \in (0, 1)$, and define

1. Prove that $f(x)$ has exactly $2$ zeros. 2. For $x \in (0,…