Divisibility NT FE
by CHESSR1DER, Apr 14, 2025, 7:07 PM
Find all functions 
such for any 
:
where a natural number n is given.
such for any 
:
where a natural number n is given.This post has been edited 1 time. Last edited by CHESSR1DER, 2 hours ago
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, Today at 11:54 AM
I found this question really easy, but it is a P4...
by Sadigly, Apr 13, 2025, 7:17 PM
Take a sequence 
such that
a) Prove that there exists infitely many primes that divides at least 1 term of the sequence.
b) Prove that there exists infitely many primes that doesn't divide any term of the sequence.
such that
a) Prove that there exists infitely many primes that divides at least 1 term of the sequence.
b) Prove that there exists infitely many primes that doesn't divide any term of the sequence.
Inequality while on a trip
by giangtruong13, Apr 12, 2025, 12:24 PM
I find this inequality while i was on a trip, it was pretty fun and i have some new experience:
Let
such that: 
. Find the maximum: 
Let
such that: 
. Find the maximum: 
This post has been edited 1 time. Last edited by giangtruong13, Apr 12, 2025, 12:25 PM
NEPAL TST 2025 DAY 2
by Tony_stark0094, Apr 12, 2025, 8:40 AM
Consider an acute triangle 
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.
Define
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .
If
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
. Let 
and 
be the feet of the altitudes from 
to 
and from 
to 
respectively.Define
and 
as the reflections of across lines 
and , respectively. Let 
be the circumcircle of 
. Denote by 
the second intersection of line 
with , and by 
the intersection of ray 
with .If
is the circumcenter of , prove that , , and are collinear if and only if quadrilateral 
can be inscribed within a circle.
This post has been edited 1 time. Last edited by Tony_stark0094, Yesterday at 12:37 AM
Reason: typo
Reason: typo
Rotating segment by 45 degrees and interchanging endpoints.
by Goutham, Feb 9, 2011, 9:58 AM
A needle (a segment) lies on a plane. One can rotate it 
round any of its endpoints. Is it possible that after several rotations the needle returns to initial position with the endpoints interchanged?
round any of its endpoints. Is it possible that after several rotations the needle returns to initial position with the endpoints interchanged?Equation in naturals
by Ahiles, Apr 30, 2009, 12:17 PM
Solve the equation
![\[ 3^x - 5^y = z^2.\]](//latex.artofproblemsolving.com/6/4/d/64da66e4dcb59114cc7a3b967c84f82211e6f6de.png)
in positive integers.
Greece
![\[ 3^x - 5^y = z^2.\]](http://latex.artofproblemsolving.com/6/4/d/64da66e4dcb59114cc7a3b967c84f82211e6f6de.png)
in positive integers.
Greece
This post has been edited 3 times. Last edited by Amir Hossein, Aug 16, 2012, 11:25 AM
Reason: Edited.
Reason: Edited.
A nice collinearity problem
by April, Jul 13, 2008, 2:00 AM
Point 
lies on side 
of a convex quadrilateral 
. Let 
be the incircle of triangle 
, and let 
be its incenter. Suppose that is tangent to the incircles of triangles 
and 
at points 
and 
, respectively. Let lines 
and 
meet at 
, and let lines 
and 
meet at 
. Prove that points , , and are collinear.
Author: Waldemar Pompe, Poland
lies on side 
of a convex quadrilateral 
. Let 
be the incircle of triangle 
, and let 
be its incenter. Suppose that is tangent to the incircles of triangles 
and 
at points 
and 
, respectively. Let lines 
and 
meet at 
, and let lines 
and 
meet at 
. Prove that points , , and are collinear.Author: Waldemar Pompe, Poland
Italian WinterCamps test07 Problem5
by mattilgale, Jan 29, 2007, 8:40 AM
A sequence of real numbers 
is defined by the formula
![\[ a_{i + 1} = \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]](//latex.artofproblemsolving.com/5/c/2/5c2ac5789507cdeaabc16284447f28294dd4fbdf.png)
here 
is an arbitrary real number, 
denotes the greatest integer not exceeding 
, and 
. Prove that 
for 
sufficiently large.
Proposed by Harmel Nestra, Estionia
is defined by the formula![\[ a_{i + 1} = \left\lfloor a_{i}\right\rfloor\cdot \left\langle a_{i}\right\rangle\qquad\text{for}\quad i\geq 0;
\]](http://latex.artofproblemsolving.com/5/c/2/5c2ac5789507cdeaabc16284447f28294dd4fbdf.png)
here 
is an arbitrary real number, 
denotes the greatest integer not exceeding 
, and 
. Prove that 
for 
sufficiently large.Proposed by Harmel Nestra, Estionia
This post has been edited 2 times. Last edited by djmathman, Jun 27, 2015, 12:02 AM
Reason: changed wording and formatting to match that of english version of ISL2006
Reason: changed wording and formatting to match that of english version of ISL2006
Stay insane,Coz it's your will, labour and pain,which takes you to the top of the mountain.
September 2017
February 2017
September 2016
July 2016
April 2016
March 2016
December 2015
November 2015
August 2015
April 2015
March 2015
February 2015
January 2015
October 2014
September 2014
August 2014
July 2014