[/quote]
,
and 
with sum 
.![\[\sum_{i=1}^n \min\{{x_i^2, x_{i+1}}\}\leq \frac{1}{2}.\]](http://latex.artofproblemsolving.com/7/f/0/7f01a8d065c8e03233fe2254946522a5762f80e2.png)
where 
.
be non-negative real numbers such that 
Prove that
and 
Prove that
![\[ a^2+b^2+c^2+abc=4\]](http://latex.artofproblemsolving.com/e/3/0/e30d8ebeabd9ea1c2892d3de685e518a75f62da3.png)
![\[ab+bc+ca-\frac{a+b+c-1}{2}\cdot abc\le 2\]](http://latex.artofproblemsolving.com/8/4/6/8464472df19ce2853344510353965107392c7243.png)
and 
Prove that
satisfying the following conditions:
for all 
Define the set ![\[S = \left\{ \left( \sum_{i=1}^{10} a_i x_i, \sum_{i=1}^{10} a_i y_i \right) : a_1, a_2, \cdots, a_{10} \in \{0, 1\} \right\},\]](http://latex.artofproblemsolving.com/d/e/b/deb50830b9fe10c86ea169c044c1b26f515b2514.png)
then 
，
with 
,
,
,![\[|az^3+bz^2+cz+d|\le M.\]](http://latex.artofproblemsolving.com/a/9/b/a9b4f124bf07c5a5aaa2d2e46c249ac44b7aaf57.png)
and 
,
[list=a]
and 
,
satisfying this equation.
denotes the greatest common divisor of positive integers 
and 
.
be the set of all 
such that each 
is a subset of 
.
denote the number of elements of the set 
.![\[ \sum_{(A_1, \ldots, A_n)\in F} |A_1\cup A_2\cup \cdots \cup A_n| \]](http://latex.artofproblemsolving.com/a/0/8/a0802b66ee9ec20913bbbf2c4a2ba546ff0e0dd3.png)
Y by clarkculus, tanxin002
A group of 
people will be partitioned into 
indistinguishable -person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as 
where 
and 
are positive integers and is not divisible by 
What is 
people will be partitioned into 
indistinguishable -person committees. Each committee will have one chairperson and one secretary. The number of different ways to make these assignments can be written as 
where 
and 
are positive integers and is not divisible by 
What is 
This post has been edited 1 time. Last edited by zhoujef000, Nov 13, 2024, 7:07 PM
and find the coefficient of the 
term and then multiply by6 4 times, then divide by 4! to get 5 as the answer since there is 5 factors of 3 left
gives 5, i forgot to divide by 
on the test
on 10a :skull:
instead of 
so I got it right
no way bro
