Problem Bank

In $\triangle ABC$, $AB=AC$, and a point $P$ satisfies

Find $\angle APC$.

Let $f(x) = x^3 + (a+2)x^2 + bx + c$ with $a, b, c \in \mathbb{R}$. Suppose there exist real numbers $m$ and $n$, distinct from each other and both different from $a$, such that

Consider the hyperbola $\dfrac{x^2}2-y^2=1$ with fixed points $A(-2,1)$, $B(2,1)$ on it. A line through the fixed point $Q(0,-3)$ meets the hyperbola at points $C$ and $D$; the lin…

The inequality

holds for all $x \in (0, +\infty)$. Find the range of possible values of the real number $a$.

Let $S \subseteq \mathbb{N}^*$ have at least two elements. A set $T$ is called a coupled set of $S$ if:

1. $T \subseteq \mathbb{N}^*$ and $T$ has at least two elements; 2. for a…

An arithmetic sequence $\{a_n\}$ has partial sums $S_n$, and

Find $\dfrac{S_3}{S_4}$.

Let

1. If $a = -\dfrac{1}{2}$, find the intervals of monotonicity of $f(x)$. 2. If $a > 1$, prove that

Let $f(x) = x|x - 4|$ for $x \in \mathbb{R}$. Suppose a positive real $k$ makes the equation $f(x) = k$ have two solutions $a < b$ in $(2, +\infty)$. Find the range of possible val…

The function

has a minimum value. Find the range of $a$.

Let $x, y \in \mathbb{R}$ and $\theta \in \left(\dfrac{\pi}{4}, \dfrac{\pi}{2}\right)$ satisfy

Complex numbers $z_1, z_2, z_3, z_4$ satisfy

Find $\left|z_1 z_4 - z_2 z_3\right|…

Given $2n$ points in the plane ($n > 1$), no three collinear, all segments between pairs are drawn and some $n^2 + 1$ of them are colored red. Prove that there are at least $n$ tri…