Inspired by old results
by sqing, Apr 15, 2025, 1:43 PM
Let 
and 
Prove that
and 
Prove that
an unsolved number theory problem
by GhostVN, Apr 15, 2025, 1:36 PM
Find all x, y, m and n are integers greater than 0, p is a prime that satisfy:
be reals such that 
Prove that
Turbo's en route to visit each cell of the board
by Lukaluce, Apr 14, 2025, 11:01 AM
Let 
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.
Proposed by Melek Güngör, Turkey
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.Proposed by Melek Güngör, Turkey
This post has been edited 1 time. Last edited by Lukaluce, Yesterday at 11:54 AM
pairwise coprime sum gcd
by InterLoop, Apr 13, 2025, 12:34 PM
For a positive integer 
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.
, let 
be all the positive integers smaller than that are coprime to . Find all 
such that
for all 
.This post has been edited 2 times. Last edited by InterLoop, Apr 13, 2025, 12:52 PM
Easy perpendicularity
by a_507_bc, Mar 15, 2024, 12:21 PM
The rhombuses 
and 
are arranged so that 
lies on the segment 
and 
lies on the segment 
. Point 
is the midpoint of 
. Prove that 
.
and 
are arranged so that 
lies on the segment 
and 
lies on the segment 
. Point 
is the midpoint of 
. Prove that 
.This post has been edited 1 time. Last edited by a_507_bc, Mar 15, 2024, 12:21 PM
Unlimited candy in PAGMO
by JuanDelPan, Oct 6, 2021, 10:28 PM
Celeste has an unlimited amount of each type of types of candy, numerated type 1, type 2, ... type n. Initially she takes 
candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:
She eats a candy of type 
, and in its position in the row she places one candy type 
followed by one candy type 
(we consider type 
to be type 1, and type 0 to be type ).
She chooses two consecutive candies which are the same type, and eats them.
Find all positive integers for which Celeste can leave the table empty for any value of 
and any configuration of candies on the table.
types of candy, numerated type 1, type 2, ... type n. Initially she takes 
candy pieces and places them in a row on a table. Then, she chooses one of the following operations (if available) and executes it:
She eats a candy of type 
, and in its position in the row she places one candy type 
followed by one candy type 
(we consider type 
to be type 1, and type 0 to be type ).
She chooses two consecutive candies which are the same type, and eats them.Find all positive integers
for which Celeste can leave the table empty for any value of 
and any configuration of candies on the table.
This post has been edited 4 times. Last edited by JuanDelPan, Oct 7, 2021, 12:35 AM
Constructing triangles holding many similarities
by WakeUp, Nov 6, 2011, 1:10 PM
Let be an interior point of the convex quadrilateral 
. Construct triangles 
and 
on the outside of the quadrilateral such that the similarities 
and 
hold. Let 
and 
be the projections of on the lines 
and 
, respectively. Prove that if the quadrilateral 
is cyclic, then
![\[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]](//latex.artofproblemsolving.com/0/8/e/08e13ecbf07fb1e8950fa34e40cb7f08a8c5a7e1.png)
be an interior point of the convex quadrilateral 
. Construct triangles 
and 
on the outside of the quadrilateral such that the similarities 
and 
hold. Let 
and 
be the projections of on the lines 
and 
, respectively. Prove that if the quadrilateral 
is cyclic, then![\[EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.\]](http://latex.artofproblemsolving.com/0/8/e/08e13ecbf07fb1e8950fa34e40cb7f08a8c5a7e1.png)
IMO 2008, Question 3
by delegat, Jul 16, 2008, 1:13 PM
Prove that there are infinitely many positive integers 
such that 
has a prime divisor greater than 
.
Author: Kestutis Cesnavicius, Lithuania
such that 
has a prime divisor greater than 
.Author: Kestutis Cesnavicius, Lithuania
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