Problem Bank

Call a positive integer $n$ $k$-pretty if $n$ is divisible by $k$ and $n$ has exactly $k$ positive divisors. For example, $18$ is $6$-pretty: it is divisible by $6$, and its divi…

Let $f(x) = |\sin x|$.

1. Prove that

2. Prove that for every positive integer $n$,

A drawing tool is built as follows. $O$ is the midpoint of a straight slot $AB$. A short rod $ON$ can rotate about $O$, and …

As shown in the figure, points $A$ and $B$ lie on the ellipse $\dfrac{x^2}{m} + y^2 = 1$ (where $m > 1$), and $P$ and $Q$ a…

How many lattice points lie on the circle centered at $(199, 0)$ with radius $199$?

Let $g(x) = |x^2 - ax - a|$. Suppose that for every real $a$ there exists $x_0 \in [0, 1]$ with $g(x_0) \geqslant k$. Find the range of possible values of $k$.

Triangle $ABC$ has side lengths $AB = 9$, $BC = 5\sqrt{3}$, and $AC = 12$. Points $P_0 = A, P_1, P_2, \dots, P_{2449}, P_{2450} = B$ lie in this order on segment $AB$, and points $…

Let $f(x) = (x + a)\ln(x + b)$. If $f(x) \geqslant 0$ for all $x$ in the domain, find the minimum value of $a^2 + b^2$.

Suppose $|ax^2 + bx + c| \le 1$ for all $x \in [-1, 1]$. Prove that

A segment $AB$ of fixed length $4$ moves with its endpoints on the two intersecting lines $x + 2y = 0$ and $x - 2y = 0$ (one endpoint on each line). Let $G$ be the midpoint of $AB$…

A segment $AB$ of length $3$ has $A$ on the $x$-axis and $B$ on the $y$-axis, and a point $P$ on the line $AB$ satisfies $\overrightarrow{BP} = 2\overrightarrow{PA}$.

1. Find the …

As shown in the figure, in the pyramid $P\text{-}ABCD$ the base $ABCD$ is a parallelogram with $\angle ABC=120^\circ…