be a triangle and let 
be a point in its interior. Lines 
,
,
intersect sides 
,
,
at 
,
,
,![\[ [PAF]+[PBD]+[PCE]=\frac{1}{2}[ABC] \]](http://latex.artofproblemsolving.com/b/6/f/b6f492b82d7b43d9a5a0a7f9e81205cb8e732bc9.png)
![$[XYZ]$](http://latex.artofproblemsolving.com/c/7/6/c7608a3e97e44fc5bcf0cd62a1f7dfc0a0a2e7d4.png)
denotes the area of triangle 
.
,
.
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
be non-negative real numbers such that 
and![\[
\frac{25a + 36b + 49c}{5a + 6b + 7c} + \frac{25b + 36c + 49a}{5b + 6c + 7a} + \frac{25c + 36a + 49b}{5c + 6a + 7b} = 18.
\]](http://latex.artofproblemsolving.com/a/8/a/a8a56fe414b713138e1adc48cdcedcc23200843c.png)
Prove that exactly two of the numbers 
has side length 
.
and the circle centered at 
occurs when 
for the function 
.
Configuration 
:
,
:
.
,
,
,
.
islands. One day, people decide to create island-states as follows. Each island randomly chooses one of the other five islands and builds a bridge between the two islands (it is possible for two bridges to be built between islands 
and 
be the roots of the polynomial 
.
.
,
for prime 
and 
for 
.
such that 
is a perfect power of 
.
be a triangle whose 
-
,
,
,
and the perimeter of 
,
of functions 
satisfying:
,
,
.
can be written as 
where 
are distinct primes, compute 
.
and that the first (leftmost) two digits of 
are 10. Compute the number of integers 
such that 
starts with either the digit 
or 
(in base 
)
be the circumcenter of 
be the midpoint of 
intersects 
(on the major arc 
,
lies on the line 
such that 
is perpendicular to 
.
and 
,
.
be the isosceles triangle with side lengths 
.
within this triangle, and compute 
,
denote the regular polygon with 
sides such that the vertices of 
denote the polygon that results after applying f a total of k times. The area of 
where 
for some positive integers 
where 
is not divisible by the square of any prime and 
and 
.
.
. This function can be expressed in the form 
for sequences of integers 
,
.
.
be the centroid and let the circumcenters of 
,
,
be 
,
and the midpoint of 
at 
.
,
is 
,
?
. Prove that
Prove that
Let 
Prove that
Prove that
satisfy![\[
(\lvert x^2-12x+20 \rvert^{\log x^2})^{-1+\log x}=\lvert x^2-12x+20 \rvert^{1+\log(1/x)}?
\]](http://latex.artofproblemsolving.com/3/9/2/39233a1044197f10403587b51443c39b8568f4a9.png)
Prove that
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: .
