Problem Bank

Let $f(x) = -x^2 + bx + |x - a|$ with $a, b \in \mathbb{R}$. Suppose that for every real $a$, the equation $f(x) = a + 1$ has at most two distinct solutions. Find the range of poss…

Complex numbers $a, b, c$ satisfy $|a|^2 + |b|^2 + |c|^2 = 1$. Find the maximum value of

A sequence $\{a_n\}$ satisfies

Find

In the prism $ABCD\text{-}A_1B_1C_1D_1$, the edge $AA_1$ is perpendicular to the base plane $ABCD$, and the …

Let $f(x) = 2x - a\ln x + 4a$ with $a \in \mathbb{R}$.

1. Discuss the monotonicity of $f(x)$. 2. Let $g(x) = f(x) - \sin x$. If there exist $x_1, x_2 \in (0, +\infty)$ with $x_1 \…

The ellipse $C:\dfrac{x^2}{25}+\dfrac{y^2}{m^2}=1$ ($0<m<5$) has eccentricity $\dfrac{\sqrt{15}}4$. Let $A$ and $B$ be the left and right vertices of $C$.

1. Find the equation of …

Fix a positive integer $m$. A sequence of positive integers has $a_1 = 1$ and

Functions $y=f(x)$, $y=g(x)$, and $h(x)=kx+b$ ($k,b\in\mathbb R$) satisfy

1. If $f(x)=x^2+2x$, $g(x)=-x^2+2x$, and …

In $\triangle ABC$, $AC = 2AB = 2$ and $BC = \sqrt{3}$. A point $P$ inside the triangle satisfies

For points $P_1,\dots,P_n$ in a plane $\alpha$, call a point $P$ (in $\alpha$) a median point if the sum of distances from $P$ to $P_1,\dots,P_n$ is minimal. (For example, every …

A function $f(x)$ on $(0, +\infty)$ is defined piecewise: for $n - 1 < x \leqslant n$ (with $n \in \mathbb{N}^*$),

Determine, with proof, which o…

The circumcenter of $\triangle ABC$ is $O$, and $AB=AC$. $D$ is the midpoint of $AB$, and $E$ is the centroid…