id int64 1 6 | problem stringclasses 6
values |
|---|---|
1 | There are $2026$ integers greater than $1$ written on a blackboard, not necessarily different. In a move, Confucius chooses two integers $m>1$ and $n>1$ from different places on the blackboard and replaces these two integers with
\[
\gcd(m,n)\qquad\text{and}\qquad
\frac{\operatorname{lcm}(m,n)}{\gcd(m,n)}.
\]
He co... |
2 | Let $ABC$ be a triangle and let points $M$ and $N$ be the midpoints of sides $AB$ and $AC$, respectively. Let points $K$ and $L$ be chosen strictly inside triangles $BMC$ and $BNC$, respectively, such that $K$ lies strictly inside triangle $ABL$ and $L$ lies strictly inside triangle $AKC$. Suppose that
\[
\angle KBA=... |
3 | Let $n$ be a positive integer. Liu Bang and Xiang Yu have a stick of length $1$ and want to divide it between themselves. Liu marks at most $n$ points on the stick, and then Xiang marks at most $n$ points on the stick. The marked points are distinct. Then, the stick is cut at all marked points, creating a number of pie... |
4 | Shan-Yu and Mulan are playing a game. Let $\theta$ be an angle with $0^\circ<\theta<180^\circ$ known to both players. Initially, Shan-Yu makes a paper triangle $T$ with measurements of his choice. Then, they repeatedly perform the following steps:
\begin{itemize}
\item If $T$ has at least one angle measuring exactly $\... |
5 | Let $\mathbb{R}_{>0}$ be the set of positive real numbers. Determine all functions $f:\mathbb{R}_{>0}\to\mathbb{R}_{>0}$ such that
\[
\sqrt{\frac{x^2+f(y)^2}{2}}
\ge \frac{f(x)+y}{2}
\ge \sqrt{x f(y)}
\]
for every $x,y\in\mathbb{R}_{>0}$. |
6 | Let $a_1,a_2,a_3,\dots$ be an infinite sequence of positive integers greater than $1$. Suppose that for all positive integers $n$, the number $a_{n+1}$ is the smallest positive integer greater than $a_n$ such that
\[
\gcd(a_{n+1},a_i)>1
\qquad\text{for every }i=1,2,\dots,n.
\]
Prove that there exist positive intege... |
No dataset card yet
- Downloads last month
- 38