Nice FE from Canada Winter Camp
by AshAuktober, Apr 12, 2025, 10:47 AM
Find all functions 
such that 
and 
for all reals 
.
such that 
and 
for all reals 
.Unusual Hexagon Geo
by oVlad, Apr 12, 2025, 9:47 AM
Let 
be a convex hexagon, such that the triangles 
and 
are equilateral and the diagonals 
and 
are concurrent. Prove that 
or 
be a convex hexagon, such that the triangles 
and 
are equilateral and the diagonals 
and 
are concurrent. Prove that 
or 
Determining Integers From Sums
by oVlad, Apr 12, 2025, 9:45 AM
Let 
be a positiv integer. Ana chooses the positive integers 
and for any non-empty subset 
she computes the sum ![\[s_A=\sum_{k
\in A}a_k.\]](//latex.artofproblemsolving.com/7/8/a/78afe6d1036a006b2252023d8f9619285fe0c6d5.png)
She orders these sums 
Prove that there exists a subset 
with 
elements such that, regardless of the integers chosen by Ana, these can be determined by only knowing the sums 
with 
be a positiv integer. Ana chooses the positive integers 
and for any non-empty subset 
she computes the sum ![\[s_A=\sum_{k
\in A}a_k.\]](http://latex.artofproblemsolving.com/7/8/a/78afe6d1036a006b2252023d8f9619285fe0c6d5.png)
She orders these sums 
Prove that there exists a subset 
with 
elements such that, regardless of the integers chosen by Ana, these can be determined by only knowing the sums 
with 
This post has been edited 1 time. Last edited by oVlad, an hour ago
Almost Squarefree Integers
by oVlad, Apr 12, 2025, 9:35 AM
A positive integer is almost squarefree if there exists a prime number 
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
the ratio 
is an integer.
is almost squarefree if there exists a prime number 
such that 
and 
is squarefree. Prove that for any almost squarefree positive integer 
the ratio 
is an integer.NEPAL TST DAY 2 PROBLEM 2
by Tony_stark0094, Apr 12, 2025, 8:37 AM
Kritesh manages traffic on a 
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the grid.
Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove
cars from the grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.
What is the minimum value of that guarantees that Kritesh's job is possible?
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the grid.Kritesh realizes that it may not always be possible for all the cars to leave the grid. Therefore, before the process begins, he will remove
cars from the grid in such a way that it becomes possible for all the remaining cars to eventually exit the grid.What is the minimum value of
that guarantees that Kritesh's job is possible?Nepal TST DAY 1 Problem 1
by Bata325, Apr 11, 2025, 1:21 PM
Consider a triangle 
and some point 
on 
. The perpendicular from to 
intersects the circumcircle of 
at 
and the perpendicular from to 
intersects the circumcircle of 
at 
. Show that the line 
does not depend on the choice of .(Shining Sun, USA)
and some point 
on 
. The perpendicular from to 
intersects the circumcircle of 
at 
and the perpendicular from to 
intersects the circumcircle of 
at 
. Show that the line 
does not depend on the choice of .(Shining Sun, USA)This post has been edited 2 times. Last edited by Bata325, Yesterday at 1:23 PM
Reason: title
Reason: title
Navid FE on R+
by Assassino9931, Apr 9, 2025, 10:31 PM
Determine all functions 
such that
![\[ f(x)f\left(x + 4f(y)\right) = xf\left(x + 3y\right) + f(x)f(y) \]](//latex.artofproblemsolving.com/d/b/1/db14b64aa71cb3c8dbf53707f719f5d9a43bd7e4.png)
for any positive real numbers .
such that![\[ f(x)f\left(x + 4f(y)\right) = xf\left(x + 3y\right) + f(x)f(y) \]](http://latex.artofproblemsolving.com/d/b/1/db14b64aa71cb3c8dbf53707f719f5d9a43bd7e4.png)
for any positive real numbers .Number Theory Chain!
by JetFire008, Apr 7, 2025, 7:14 AM
I will post a question and someone has to answer it. Then they have to post a question and someone else will answer it and so on. We can only post questions related to Number Theory and each problem should be more difficult than the previous. Let's start!
?This post has been edited 1 time. Last edited by JetFire008, Apr 7, 2025, 7:14 AM
A geometry about a parallelogram ABCD
by nAalniaOMliO, Mar 28, 2025, 8:22 PM
On the side 
of parallelogram 
a point 
is chosen. The perpendicular from 
to 
and the perpendicular from 
to 
intersect at . Point 
is the midpoint of 
.
Prove that the perpendicular from to passes through the center of parallelogram .
Matsvei Zorka
of parallelogram 
a point 
is chosen. The perpendicular from 
to 
and the perpendicular from 
to 
intersect at . Point 
is the midpoint of 
.Prove that the perpendicular from
to passes through the center of parallelogram .Matsvei Zorka
This post has been edited 2 times. Last edited by nAalniaOMliO, Apr 9, 2025, 8:25 AM
JBMO Shortlist 2020 N1
by Lukaluce, Jul 4, 2021, 2:47 PM
Determine whether there is a natural number for which 
is prime.
for which 
is prime.Stay insane,Coz it's your will, labour and pain,which takes you to the top of the mountain.
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