denote the set of positive rational numbers. Find, with proof, all functions 
such that, for all positive rational numbers 
and 
we have ![\[f(x)=f(x+y)+f(x+x^2f(y)).\]](http://latex.artofproblemsolving.com/9/e/a/9ea71d2ee277dc3c689ba2d791b451e29353edb3.png)
be a triangle with circumcenter 
.
be a point on the perpendicular bisector of 
(see figure) and 
,
be the intersections of the perpendicular bisectors of 
and 
,
and 
.
and 
and the point 
is given. Let 
segments of real line of type ![$[i, j]$](http://latex.artofproblemsolving.com/a/6/4/a644edd67caa59eaded765c417b6d9822f782ec2.png)
,
and 
are integers, 
.
is said to be valuable if the intersection of any two segments from 
are there such that 
and 
?
and 
.
be real numbers such that 
Prove that
Let 
Prove that
and 
, then we can deduce 
\le 2 $ ?
satisfying 
.![\[
0 \le (x^2 + 1)(y^2 + 1) + 4(x - 1)(y - 1) \le 164.
\]](http://latex.artofproblemsolving.com/4/2/d/42d0b88388462f16dd99d928b94024e689ba7c2e.png)
be a sequence defined by 
.
if 
.
Y by Leo.Euler
At an informatics competition each student earns a score in 
on each of six problems, and their total score is the sum of the six scores (out of 
). Given two students 
and 
, we write 
if there are at least five problems on which scored strictly higher than .
Compute the smallest integer
such that the following statement is true: for every integer 
, given students 
, \dots, 
satisfying 
, the total score of is always at most points more than the total score of .
Jiahe Liu
on each of six problems, and their total score is the sum of the six scores (out of 
). Given two students 
and 
, we write 
if there are at least five problems on which scored strictly higher than .Compute the smallest integer
such that the following statement is true: for every integer 
, given students 
, \dots, 
satisfying 
, the total score of is always at most points more than the total score of .Jiahe Liu
This post has been edited 1 time. Last edited by YaoAOPS, Jan 22, 2025, 2:10 AM
is obviously the max, and is obtainable starting from 
and can increment it by a maximum of 
.
moves until we arrive at a net increase of 
,
.
.
,
.
