Problem Bank

Let $O$ be the circumcenter of $\triangle ABC$, and suppose $x\overrightarrow{OA}+y\overrightarrow{OB}+z\overrightarrow{OC}=\vec 0$ with $xyz\neq 0$, where $C$ is an interior angle…

Nonnegative reals $a, b$ satisfy $a + b = \dfrac{3}{2}$. Find the maximum value of

Let $f(x)=x+\sin x$. If the inequality $f(x)\geqslant ax\cos x$ holds for all $x\in\left[0,\dfrac{\pi}2\right]$, find the range of the real number $a$.

If the constant term in the expansion of

equals $\dfrac{35}{2}$, find the positive integer $n$.

$F_1, F_2$ are the foci of $C: \dfrac{x^2}{4} + \dfrac{y^2}{3} = 1$. Points $A(x_1, y_1)$, $B(x_2, y_2)$ on $C$ satisfy $x_1 + x_2 = \dfrac12$ and

Consider the sequence

Is every term the product of two consecuti…

Let $f(x) = \mathrm{e}^x\left(x\ln x + \dfrac{2}{\mathrm{e}}\right)$ and $g(x) = ax$ with $a \in \mathbb{Z}$, where $\mathrm{e}$ is the base of the natural logarithm.

1. Find the …

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

1. Discuss the monotonicity of $f(\sin x)$ on $\left(-\dfrac{\pi}2,\dfrac{\pi}2\right)$, and determine whether it has extreme values (finding them when they e…

Let $l$ be a space diagonal of a cube, and let $S$ be the set consisting of the cube's vertices and the midpoints of its edges. How many segments joining two points of $S$ are perp…

In the coordinate plane, let $\Gamma$ be the graph of $y = \dfrac{1}{|x|}$. Points $P$ and $Q$ on $\Gamma$ are such that $P$ lies in the first quadrant, $Q$ lies in the second quad…

The function $f(x) = \ln x + \dfrac{m}{x} - 3$ has two zeros.

1. Find the range of possible values of $m$. 2. Let $a, b$ be the two zeros of $f(x)$. Prove that

How many pairs of integers $(m, n)$ satisfy