[/quote]
,
and 
. Prove that
Where 
such that![\[
x+f(xy)=f(1+f(y))x
\]](http://latex.artofproblemsolving.com/4/3/3/433f95836dad1224754d480aff8432ef033a37cb.png)
for all 
.
vertices is given. To each vertex, a natural number from the set 
is assigned. Prove that there are 
vertices 
such that if the numbers 
are assigned to them respectively, then 
and 
is a parallelogram.
to 
cards, and she is told a number that is written on one of the chosen cards, but not which specific card it is on. After several moves, Madina must determine the written numbers on as many cards as possible. What is the maximum number of cards Madina can ensure to determine?
is such that number 
has exactly 
positive divisors. Prove that GCD 
be a triangle inscribed in a circle 
with orthocenter 
and altitude 
.
be the midpoint of 
.
meets 
.
meets 
.
be the orthogonal projection of 
on the line 
.
meets 
.
.
satisfying 
,
on 
and 
respectively in 
such that 
then draw line 
through 
such that 
let 
and 
reflection of 
and 
and 
be a set of 
distinct integers all of whose prime factors are at most 
Prove that 
such that
for all 
.
Y by internationalnick123456
Is it possible to color all lattice points in plane into 3 colors such that
1. every line passing through lattice points and parallel to x axis has these three colors infinitely many(that is, every color appears infinitely many times in those lines).
2. every line passing through lattice points and not parallel to x axis cannot have three different color lattice points on it.
I think the answer is yes, but I couldn't find an example...
1. every line passing through lattice points and parallel to x axis has these three colors infinitely many(that is, every color appears infinitely many times in those lines).
2. every line passing through lattice points and not parallel to x axis cannot have three different color lattice points on it.
I think the answer is yes, but I couldn't find an example...
This post has been edited 5 times. Last edited by CBMaster, Apr 11, 2025, 6:58 PM
Reason: .
Reason: .
