[/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 
with respect to 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 Mango247, Mango247, Mango247
Let 
be an odd prime number. Around a circular table, students sit. We give pieces of candy to those students in the following manner. The first candy we give to an arbitrary student. Then, going around clockwise, we skip two students and give the next student a piece of candy, then we skip 4 students and give another piece of candy to the next student... In general in the 
th turn we skip 
students and give the next student a piece of candy. We do this until we don't give out all pieces of candy.
How many students won't get any pieces of candy?
How many pairs of neighboring students (those students who sit next to each other on the table) both got at least a piece of candy?
be an odd prime number. Around a circular table, students sit. We give pieces of candy to those students in the following manner. The first candy we give to an arbitrary student. Then, going around clockwise, we skip two students and give the next student a piece of candy, then we skip 4 students and give another piece of candy to the next student... In general in the 
th turn we skip 
students and give the next student a piece of candy. We do this until we don't give out all pieces of candy.
How many students won't get any pieces of candy?
How many pairs of neighboring students (those students who sit next to each other on the table) both got at least a piece of candy?This post has been edited 1 time. Last edited by Steve12345, May 22, 2022, 7:12 AM
