Problem Bank

Let $f(x) = |\sin x|$.

1. Prove that

2. Prove that for every positive integer $n$,

Prove that for every $n \in \mathbb{N}^*$,

The graph of a cubic function $f(x)$ passes through the points $A(2, 4)$, $B(3, 9)$, $C(4, 16)$. Lines $AB$, $AC$, $BC$ meet the graph of $f(x)$ again at points $D$, $E$, $F$ respe…

As shown in the figure, in a regular tetrahedron $ABCD$ with edge length $1$, points $M$, $N$, $K$ lie on edges $AB$,…

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

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 $\{a_n\}$ be a geometric sequence with positive common ratio, partial sums $S_n$, $a_1 = 1$, and $S_2 = a_3 - 1$.

1. Find $S_n$. 2. Define a sequence $\{b_n\}$ by

Let $O$ be the circumcenter of $\triangle ABC$ with $A = \dfrac{\pi}{3}$, and suppose

Let

1. When $a = \sqrt{3}$, find the intervals of monotonicity of $f(x)$. 2. Suppose $f(x)$ has two zeros.

1. Find the range of possible…

Let $f(x) = \left(2x^2 - x^3\right)\mathrm{e}^{1-x}$ for $x > 0$.

1. Find the maximum value of $f(x)$. 2. If the inequality

holds …

Find the maximum value of

where $t \in \…

Let $f(x) = \mathrm{e}^x$ and $g(x) = \ln x$.

1. If $h(x) = f(x) + a\,g(x)$ has a local minimum, find the range of possible values of the real number $a$. 2. If $m > 0$ and