Found problems: 329
1998 AMC 12/AHSME, 7
If $ N > 1$, then ${ \sqrt [3] {N \sqrt [3] {N \sqrt [3] {N}}}} =$
$ \textbf{(A)}\ N^{\frac {1}{27}}\qquad
\textbf{(B)}\ N^{\frac {1}{9}}\qquad
\textbf{(C)}\ N^{\frac {1}{3}}\qquad
\textbf{(D)}\ N^{\frac {13}{27}}\qquad
\textbf{(E)}\ N$
2013 NIMO Summer Contest, 8
A pair of positive integers $(m,n)$ is called [i]compatible[/i] if $m \ge \tfrac{1}{2} n + 7$ and $n \ge \tfrac{1}{2} m + 7$. A positive integer $k \ge 1$ is called [i]lonely[/i] if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$. Find the sum of all lonely integers.
[i]Proposed by Evan Chen[/i]
2000 APMO, 5
Given a permutation ($a_0, a_1, \ldots, a_n$) of the sequence $0, 1,\ldots, n$. A transportation of $a_i$ with $a_j$ is called legal if $a_i=0$ for $i>0$, and $a_{i-1}+1=a_j$. The permutation ($a_0, a_1, \ldots, a_n$) is called regular if after a number of legal transportations it becomes ($1,2, \ldots, n,0$).
For which numbers $n$ is the permutation ($1, n, n-1, \ldots, 3, 2, 0$) regular?
2014 Tajikistan Team Selection Test, 4
In a convex hexagon $ABCDEF$ the diagonals $AD,BE,CF$ intersect at a point $M$. It is known that the triangles $ABM,BCM,CDM,DEM,EFM,FAM$ are acute. It is also known that the quadrilaterals $ABDE,BCEF,CDFA$ have the same area. Prove that the circumcenters of triangles $ABM,BCM,CDM,DEM,EFM,FAM$ are concyclic.
[i]Proposed by Nairy Sedrakyan[/i]