Problem Bank

A set $A$ consists of $40$ elements of $\{1, 2, \dots, 50\}$, and $S$ is the sum of its elements. How many distinct values can $S$ take?

With $[\,\cdot\,]$ the floor function, find the range of

Let $f(x)=\sin(x-\varphi)$, and suppose $\displaystyle\int_0^{2\pi/3}f(x)\,\mathrm dx=0$. Find an axis of symmetry of the graph of $f(x)$.

As shown in the figure, the quadrilaterals $ABCD$ and $ADPQ$ are both squares, and the planes containing them are per…

In rectangle $ABCD$ with $AD<CD$, triangle $ACD$ is folded up along $AC$ to triangle $ACD_1$, so that the dihedral …

The inequality

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

Let $f(x) = x\ln x - \dfrac12 m x^2 - x$.

1. For $m = 4$, find the tangent line to $y = f(x)$ at $(1, f(1))$. 2. Suppose $f$ has two extreme points $x_1 < x_2$, and

Three vertices of a regular $n$-gon are chosen at random. The probability that they form an obtuse triangle is $\dfrac{93}{125}$. Find all possible values of $n$.

As shown in the figure, let $A$ be the top vertex of the ellipse $\dfrac{x^2}{4} + y^2 = 1$. The two lines through $A$ tange…

A function $f$ on $\mathbb{R}$ satisfies $f(1) = \dfrac14$ and

Find $f(2016)$.

A function $f(x)$ satisfies

Find $f(84)$.

Real numbers $a,b$ satisfy

Determine the sign of $a$ and the sign of $b$ (each $>0$ or $<0$).