by giangtruong13, Apr 15, 2025, 3:56 PM 
to satisfy that the equation 
has at least 1 roots 
by CHESSR1DER, Apr 14, 2025, 7:07 PM 
such for any 
:
.by Lukaluce, Apr 14, 2025, 11:01 AM 
be an integer. In a configuration of an 
board, each of the 
cells contains an arrow, either pointing up, down, left, or right. Given a starting configuration, Turbo the snail starts in one of the cells of the board and travels from cell to cell. In each move, Turbo moves one square unit in the direction indicated by the arrow in her cell (possibly leaving the board). After each move, the arrows in all of the cells rotate 
counterclockwise. We call a cell good if, starting from that cell, Turbo visits each cell of the board exactly once, without leaving the board, and returns to her initial cell at the end. Determine, in terms of 
, the maximum number of good cells over all possible starting configurations.by InterLoop, Apr 13, 2025, 12:38 PM 
of positive integers is called central if for every positive integer , the arithmetic mean of the first 
terms of the sequence is equal to .
, 
, 
, 
of positive integers such that for every central sequence 
, 
, 
, , there are infinitely many positive integers with 
.by CrazyInMath, Apr 13, 2025, 12:38 PM 
be an acute triangle. Points 
, and 
lie on a line in this order and satisfy 
. Let 
and 
be the midpoints of 
and 
, respectively. Suppose triangle 
is acute, and let 
be its orthocentre. Points 
and 
lie on lines 
and 
, respectively, such that 
and are concyclic and pairwise different, and 
and are concyclic and pairwise different. Prove that 
and are concyclic.by Rohit-2006, Apr 9, 2025, 5:22 AM 
satisfy,
Where {
} is a permutation of . Prove that 
by JetFire008, Apr 7, 2025, 7:14 AM by Sayan, Feb 14, 2012, 3:56 PM 
of the equation![\[n^3=p^2-p-1\]](http://latex.artofproblemsolving.com/f/f/8/ff88093bb0aa93afcfdff3ae25149ec5fbacb61f.png)
is a prime number and is an integerby Amir Hossein, Jul 19, 2011, 11:54 AM 
be an integer. We are given a balance and weights of weight 
. We are to place each of the weights on the balance, one after another, in such a way that the right pan is never heavier than the left pan. At each step we choose one of the weights that has not yet been placed on the balance, and place it on either the left pan or the right pan, until all of the weights have been placed.The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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