[/quote]
,
'is a perfect square.
and 
be positive real numbers with 
.
For which 
such that 
for all 
,
.
be a positive integer such that 
.
.
be lines in the plane such that no two of them are parallel and no three of them are concurrent. Let 
be the intersection point of lines 
.
on 
and 
on 
such that 
Interior points.
)
.
. Prove that: 
,
be the orthocenter of it and 
be any point on the side 
.
are on the segments 
,
and 
are cyclic. The segments 
and 
intersect at 
is a point on 
such that 
is tangent to the circumcircle of triangle 
at 
and 
intersect at 
.
and 
and 
then prove 
be the set of positive real numbers.
such that there exists a function 
with the property:![\[
f(y^2f(x) + y + c) = xf(x+y^2)
\]](http://latex.artofproblemsolving.com/e/3/0/e30ee6d9f69cc43422a5c2b9813192a632d359cb.png)
for all 
.
be non-negative real numbers such that 
Prove that
and 
.
,![\[ a^{b+c} b^{c+a} c^{a+b} \leq 1 . \]](http://latex.artofproblemsolving.com/0/c/e/0ce097b0b507403c959c2992cae1217ecdd7e581.png)
be a finite sequence of positive integers. Prove that there exist nonnegative integers 
and 
such that 
holds for all integers 
Y by
The set of 18 numbers is given
Here the sign
indicates the color red and the sign 
indicates the color blue.
Prove that no matter how we choose 10 elements from the set, there are always two of different colors with a sum of 10.
Here the sign
indicates the color red and the sign 
indicates the color blue.Prove that no matter how we choose 10 elements from the set, there are always two of different colors with a sum of 10.
This post has been edited 3 times. Last edited by cip1703, Oct 2, 2024, 2:35 PM
Reason: correction
Reason: correction
.
,
