Problem Bank

The system

has two solutions $(x_1, y_1)$ and $(x_2, y_2)$ with $\dfrac{x_1}{y_1} \ne \dfrac{x_2}{y_2}$. Find …

Let

and suppose $f(x)$ is increasing on its domain. Discuss the number of solutions of the equation

Let $f(x) = 2\ln x - ax^2 + 1$. Suppose there is a real $m$ such that the equation $f(x) = m$ has two real roots $\alpha, \beta$, both in $[1, 4]$, with $\beta - \alpha = 1$. Find …

Let $f(x) = \mathrm{e}^x - \dfrac{x^2}{2}$, and let $l$ be the tangent line to $y = f(x)$ at the point $P(x_0, f(x_0))$ with $x_0 > 0$.

1. Prove that $l$ meets the curve $y = f(x)…

Let $a > 0$, let $s_a$ be the minimum of $y = \sin x$ on $[a, 2a]$, and $t_a$ its minimum on $[2a, 3a]$. As $a$ varies, determine which of the following sign patterns cannot occur,…

Let $x_1, x_2, \dots, x_n > 0$. Prove that

In $\triangle ABC$, $AD$ is the altitude to side $BC$. If $BC+AD=AB+AC$, find the range of $\angle BAC$.

1. Let $P$ be any point in the plane of rectangle $ABCD$. Prove that

2. Generalize to a point $P$ anywhere in space, and prove your statement.
3. G…

The triangle $ABC$ has perimeter $6$, and its sides $a, b, c$ (opposite $A, B, C$) form a geometric sequence. Find the range of possible values of $\overrightarrow{BA}\cdot\overrig…

In a regular $n$-gonal pyramid $P\text{-}A_1A_2\cdots A_n$ (where $n \geqslant 3$), let $O$ be the center of the regular $n$-gon base $A_1A_2\cdots A_n$, and let $B$ be the midpoin…

The ellipse $\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has right focus $F$ and upper vertex $B$, with eccentricity $\dfrac{2\sqrt 5}5$ and $|BF|=\sqrt 5$.

1. Find the equatio…

Let $f(x) = x - \dfrac12\sin x - \ln x$.

1. Find the maximum of $f$ on $\left[1, \dfrac{\pi}{3}\right]$. 2. Prove that $f'$ has a unique zero. 3. If $x_1 \ne x_2$ and $f(x_1) = f(…