[/quote]
,
?
Prove that
vertices, each edge has one of the colors 
,
,
.
,
groups such that any two vertices connected by an edge of color 
are in different groups, find the minimum possible value of 
.
such that, for all 
,
such that for all integer 
and a subset 
of ![$[n],$](http://latex.artofproblemsolving.com/d/2/7/d27d2715f4f95c24a28b7c6371a006d4c803b2cf.png)
satisfy 
for all 
then 
be the set of positive integers. Find all functions 
such that for all 
,![\[m^2f(m) + n^2f(n) + 3mn(m + n)\]](http://latex.artofproblemsolving.com/0/f/f/0ff783fa217b7be4bda5006e417f1471ea3a6fca.png)
is a perfect cube.
and 
.
![$\mathbb{R}[x]$](http://latex.artofproblemsolving.com/a/8/8/a88b9f4858016b1771635c611d44b56161a08009.png)
be the set of all polynomials with real coefficients. Find all functions ![$f: \mathbb{R}[x] \rightarrow \mathbb{R}[x]$](http://latex.artofproblemsolving.com/0/3/3/033be3ab043bd47e537f01791c813739ad5c7da5.png)
satisfying the following conditions:
maps the zero polynomial to itself,![$P \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/f/7/c/f7c4c970908aa5570a9afcc45fc16c1d36cc3d5d.png)
,
,![$P, Q \in \mathbb{R}[x]$](http://latex.artofproblemsolving.com/c/3/0/c301578bade5b93056d77454921cba0a9812212c.png)
,
and 
have the same set of real roots.
be a square of center 
and let 
be the symmetric of the point 
be the intersection of 
and 
,
and 
.
is the angle bisector of 
.
be reals such that 
Show that
Let 
and 
be a polygon formed by the edges of an infinite chessboard, which does not intersect itself. Let the numbers 
represent the number of unit squares that have exactly 
edges on the boundary of 
such that the inequality 
holds for each polygon constructed with these conditions.
and 
Find the value range of 
and 
with real coefficients satisfying
Y by
At a chess tournament, every pair of contestants played each other at most once. If any two con-
testants,
and 
, failed to play each other, then exactly two other contestants, 
and 
, played
against both and during the tournament. Moreover, no 
contestants played exactly 
games
between them. Prove that every contestant played the same number of games.
Authored by Mirko Petrushevski
testants,
and 
, failed to play each other, then exactly two other contestants, 
and 
, playedagainst both
and during the tournament. Moreover, no 
contestants played exactly 
gamesbetween them. Prove that every contestant played the same number of games.
Authored by Mirko Petrushevski
This post has been edited 2 times. Last edited by steppewolf, Feb 9, 2025, 10:40 PM
play each other. Now we have that whenever there is an edge 
in the graph, there is also an edge 
such that 
,
,
and 
.
.
which is an endpoint of the conjugate of an edge 
.
of 
is an edge, or 
.
has maximal degree, and now we can prove every vertex has maximal degree using breadth first search, since the graph is connected.
