Problem Bank

Let $a$ be a constant and $f(x) = x\ln x + ax^2$.

1. When $a = 1$, find the equation of the tangent line to $f(x)$ at $x = 1$. 2. Determine the number of zeros of $f(x)$, discussi…

In a right triangle $ABC$ with $\angle ACB=90^\circ$, $BC=a$, and $AC=b$, points $D,E,F$ lie on the sides $BC$, $CA$, $AB$ respectively so that $\triangle DEF$ is equilateral. Find…

In $\triangle ABC$, $AC = 5$ and

Find $BC + AB$.

1. Prove that for $0 < x < 1$,

2. Let $f(x) = \cos(ax) - \ln\left(1 - x^2\right)$. If $x = 0$ is a local maximum point of $f(x)$, find the range of poss…

Let $A,B,C,D$ be four points on the ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$, with the lines $AB$ and $CD$ meeting at a fixed point $P(x_0,y_0)$. Let $M$, $N$ be the midpoints…

The function

attains a maximum value. Find the range of possible values of $a$.

The ellipse $C:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has upper vertex $A(0,1)$ and eccentricity $\dfrac{\sqrt 3}2$.

1. Find the equation of the ellipse $C$. 2. Let $B$ be…

Let $\vec{e_1},\vec{e_2}$ be unit vectors with angle $\dfrac{\pi}6$ between them, and let $\vec b=x\vec{e_1}+y\vec{e_2}$ be a nonzero vector with $x,y\in\mathbb R$. Find the maximu…

In the coordinate plane, the ellipse $C_1 \colon \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ (with $a > b > 0$) has foci $F_1, F_2$, where $F_2$ is also the focus of the parabola $C_2…

For $M \subseteq \mathbb{N}^*$ with $|M| = n \ge 5$: if $M$ contains four distinct elements $a, b, c, d$ with $a + b = c + d$, call $M$ linked and $\{a,b,c,d\}$ a linked subset…

Let $F_1, F_2$ be the foci of the hyperbola $C \colon \dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ ($a, b > 0$), and let $P$ be an intersection point of $C$ with the circle $x^2 + y^2 …

For $x \in \mathbb{R}$, let

How many positive integers not exceeding $2024$ can occu…