Problem Bank

For functions on $[-1, 1]$, consider the claims:

1. if $f(\cos x) = \cos nx$ with $n$ odd, then $f$ is odd; 2. if $f(\cos x) = \cos nx$ with $n$ even, then $f$ is even; 3. some od…

In $\triangle ABC$, the centroid $G$ satisfies $AG \perp BG$, and $AB = 2$.

1. Find the minimum value of $\dfrac{1}{\tan A} + \dfrac{1}{\tan B}$. 2. Prove that $\dfrac{\tan C}{\ta…

A sequence $\{x_n\}$ ($n \in \mathbb{N}^*$) satisfies $x_n > 1$. Let $\mathrm{e}$ be the base of the natural logarithm, and define

Positive reals $x, y, z, w$ satisfy $x \ge y \ge w$ and $x + y \le 2(z + w)$. Find the minimum of

In $\triangle ABC$, the sides opposite the angles $A,B,C$ are $a,b,c$. The area of $\triangle ABC$ is $\sqrt 3$, $D$ is the midpoint of $BC$, and $AD=1$.

1. If $\angle ADC=\dfrac{…

Given

find the minimum value of $(a - b)^2 + (c - d)^2$.

Let $f(x) = x - 1 + \dfrac{a}{\mathrm{e}^x}$, where $a \in \mathbb{R}$ and $\mathrm{e}$ is the base of the natural logarithm.

1. If the tangent line to $y = f(x)$ at the point $(1…

The sequence $\{a_n\}$ has partial sums $S_n$, with $a_1 = \dfrac{1}{2}$ and

where $\lambda$ is a co…

In $\triangle ABC$, $\angle C=90^\circ$, $AC=1$, $BC=\sqrt{3}$, and $D$ is the midpoint of side $AB$. A point $M$ lies o…

Each square in a row of $10$ squares is colored blue or red, with at most $2$ consecutive squares blue (no run of $3$ or more blue squares). How many colorings are there?

Let $f(x) = \mathrm{e}^x - \dfrac{1}{2}ax^2 - x$.

1. If $f(x)$ is increasing on all of $\mathbb{R}$, find the value of $a$. 2. For $a > 1$:

1. Prove that $f(x)$ has two extreme…

A square sheet of paper with side length $1$ is folded along a line through its center. Find the maximum possible area covered by the folded sheet.