An inequality problem
by Arithmetic_fighter, Apr 3, 2025, 3:28 AM
Given 
such that 
. Prove that
such that 
. Prove that
Game About Passing Pencils
by WilliamSChen, Apr 3, 2025, 3:21 AM
A group of 
children sit in a circle facing inward with 
, and each child starts with an arbitrary even number of pencils. Each minute, each child simultaneously passes exactly half of all of their pencils to the child to their right. Then, all children that have an odd number of pencils receive one more pencil.
Prove that after a finite amount of time, the children will all have the same number of pencils.
I do not know the source.
children sit in a circle facing inward with 
, and each child starts with an arbitrary even number of pencils. Each minute, each child simultaneously passes exactly half of all of their pencils to the child to their right. Then, all children that have an odd number of pencils receive one more pencil.Prove that after a finite amount of time, the children will all have the same number of pencils.
I do not know the source.
Coaxial circles related to Gergon point
by Headhunter, Apr 3, 2025, 2:48 AM
Hi, everyone.
In
, 
is the Gergon point and the incircle 
touch 
, 
, 
at 
, 
, 
respectively.
Let the circumcircles of
, 
, 
be 
, 
, 
respectively.
Reflect in 
and then we get the circle 
Reflect in 
and then the circle 
Reflect in 
and then the circle 
Prove that , , are coaxial.
In
, 
is the Gergon point and the incircle 
touch 
, 
, 
at 
, 
, 
respectively.Let the circumcircles of
, 
, 
be 
, 
, 
respectively.Reflect
in 
and then we get the circle 
Reflect
in 
and then the circle 
Reflect
in 
and then the circle 
Prove that
, , are coaxial.Geometry :3c
by popop614, Apr 3, 2025, 12:19 AM
Quadrilateral 
has an incenter 
Suppose 
. Let 
be the midpoint of 
. Suppose that 
. 
meets 
again at point 
. Let points 
and 
be such that is the midpoint of 
and is the midpoint of 
. Point 
lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.
The angle bisectors of
and 
meet 
at 
and 
. Prove that 
.
has an incenter 
Suppose 
. Let 
be the midpoint of 
. Suppose that 
. 
meets 
again at point 
. Let points 
and 
be such that is the midpoint of 
and is the midpoint of 
. Point 
lies on the plane such that 
is a parallelogram, and suppose the angle bisectors of 
and 
concur on 
.The angle bisectors of
and 
meet 
at 
and 
. Prove that 
.This post has been edited 1 time. Last edited by popop614, 5 hours ago
Reason: asfdasdf
Reason: asfdasdf
and 
: Prove that :
Finding pairs of complex numbers with a certain property
by Ciobi_, Apr 2, 2025, 1:22 PM
Find all pairs of complex numbers 
such that the relation ![\[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]](//latex.artofproblemsolving.com/1/2/2/1229a609403904449c447225094b3d97e063efd4.png)
holds for all positive integers .
such that the relation ![\[|z^{2n}+z^nw^n+w^{2n} | = 2^{2n}+2^n+1 \]](http://latex.artofproblemsolving.com/1/2/2/1229a609403904449c447225094b3d97e063efd4.png)
holds for all positive integers .
such that ![\[f(2x+y) + f(x+f(2y)) = f(x)f(y) - xy\]](http://latex.artofproblemsolving.com/0/6/2/0623b7c328af6d088cfc5dd288afe3e17a94ff29.png)
for all reals 
and 
.(Original version) Same number of divisors
by MNJ2357, Aug 12, 2024, 9:46 AM
For a positive integer 
, let 
denote the number of positive divisors of . Determine whether there exists a positive integer triple 
such that there are exactly 
positive integers 
not greater than 
that satisfies the following: the equation
![\[ \tau(x) = \tau(y) = \tau(z) = \tau(ax + by + cz) = K \]](//latex.artofproblemsolving.com/a/8/6/a86ab0dd1aacf681d5d6761a4da73face1647f48.png)
holds for some positive integers 
.
, let 
denote the number of positive divisors of . Determine whether there exists a positive integer triple 
such that there are exactly 
positive integers 
not greater than 
that satisfies the following: the equation![\[ \tau(x) = \tau(y) = \tau(z) = \tau(ax + by + cz) = K \]](http://latex.artofproblemsolving.com/a/8/6/a86ab0dd1aacf681d5d6761a4da73face1647f48.png)
holds for some positive integers 
.This post has been edited 2 times. Last edited by MNJ2357, Aug 12, 2024, 10:04 AM
Reason: Added version
Reason: Added version
such that 
for reals 
.An nxn Checkboard
by MithsApprentice, Oct 3, 2005, 10:41 PM
Some checkers placed on an 
checkerboard satisfy the following conditions:
(a) every square that does not contain a checker shares a side with one that does;
(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.
Prove that at least
checkers have been placed on the board.
checkerboard satisfy the following conditions:(a) every square that does not contain a checker shares a side with one that does;
(b) given any pair of squares that contain checkers, there is a sequence of squares containing checkers, starting and ending with the given squares, such that every two consecutive squares of the sequence share a side.
Prove that at least
checkers have been placed on the board.The oldest, shortest words — "yes" and "no" — are those which require the most thought.
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