Problem Bank

Positive reals $a, b, c$ satisfy $a + b + c = 1$. Find the integer part of

Positive reals $a, b$ satisfy

Find the range of $3a + 4b$.

Let $f(x) = \mathrm{e}^x(a - x) + x + 1$.

1. If the graph of $f(x)$ has a tangent line of slope zero at some point with $x$-coordinate in $[0, 1]$, find the range of possible valu…

Find all primes $x$ and $y$ with

Let $f(x) = \dfrac{\mathrm{e}^x}{x^2 + 1}$ be defined on $(0, +\infty)$.

1. Prove that $f(x) > 1$. 2. Let $g(x) = \dfrac{\mathrm{e}^x - 1}{x}$. Find the intervals on which $g(x)$ …

The ellipse $E:\dfrac{x^2}{a^2}+\dfrac{y^2}{b^2}=1$ ($a>b>0$) has upper vertex $D(0,3)$, and the quadrilateral formed by its four vertices has area $18\sqrt 2$.

1. Find the equati…

For $a, b \geqslant 2$ and $n \in \mathbb{N}^*$, let $S_n$ be the area enclosed by the closed curve

Prove that $4 < …

In the coordinate plane, let $A(-1, 0)$, and let $B, C$ move along the unit circle centered at the origin $O$ so that the line $BC$ has inclination $30^\circ$. Find the range of po…

Prove that for all real numbers $a, b, c$,

Let $f(x)=ax^2-2016x+2017$ ($a>0$). On the interval $[t-1,t+1]$, let $M(t)$ and $m(t)$ be the maximum and minimum of $f(x)$, and let $h(t)=M(t)-m(t)$. If the minimum value of $h(t)…

In equilateral triangle $ABC$, $D$ and $E$ are the midpoints of $AB$ and $AC$ respectively. $M$ is a moving …

Let $A = \{0, 1, 2, 3, 4, 5, 6\}$, and let $f$ be a function from $A$ to the integers such that $f(0) = 0$, $f(6) = 12$, and