,
and 
,
.
is assigned to a positive integer in 
,
,
.
of 
,
'
colors, and for an arbitrary color 
,
that are continuous on 
satisfying
be a positive integer. Define 
as the number of integer pairs 
satisfying 
with 
odd. Define 
as the number of integer pairs 
of degree at most 
and 
distinct integers 
such that ![\[P(x_1)+P(x_2)+\cdots +P(x_k)=P(x_{k+1})\]](http://latex.artofproblemsolving.com/e/e/8/ee8adc79dc258b021529245fec16e081da2d1b0a.png)
.
. Prove that
have real roots and real coefficients with 
.
?
,
.
.
.
(I won't be proving this here). This new information allows us to utilize the mean inequality chain as follows:
Similarly, applying AM-HM yields 
.
.
and 
Show that
Let 
Show that
is a perfect cube.
digits are distinct?
Solve for 
,
,
,
.
be the point on 
such that 
,
be the midpoint of 
.
is a point such that 
is a rectangle, compute the area of 
.
has a sidelength of 
,
,
are chosen on 
,
,
,
,
,
are length 
.
.
satisfies for all integers 
If 
and 
,
.
children have black hair, all but 
have red hair, and all but 
have blond hair. How many children are there in the class?
can be partitioned into n Akash subsets.
be a prime number. How many different pairs 
satisfy the equation:
:
.
Let 
.
,![$ x \in [-1 / 10,0] $](http://latex.artofproblemsolving.com/1/e/f/1ef43e8c80ca68c6262a0f96555efb0a00e531d1.png)
.
is the inverse function of 
]
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I often request that posters use the search function before posting a [not so] new problem (look at the little magnifying glass symbol just at the right of the forum name on the top of forum window, or also here).
I received in PM some remarks about the fact that search function seems very often to give tons of results without any link with the searched problem. Here are some tips I frequently use :
1) use quotes for strings : search "P(x^2-1)" instead of P(x^2-1)
2) dont hesitate to use the character "+" which means "mandatory"
3) Split long strings in order to find problems where strings are not in the same order :
Instead of +"f(2x+f(y))=f(2x)+f(f(y))+xf(2y)", prefer +"f(2x+f(y))" +"f(2x)" +"f(f(y))" +"xf(2y)"
So you'll find
4) If the problem you are looking for contains strings as "2015" or "2016", try to exclude these strings from search term since this is likely an "annual problem" and it may have been posted wu=ith older years?
For example +"f(x-f(y)" +"f(x+y^" +"f(f(y)+y^" will find
as well as 
5) if the problem you are looking for uses function
or 
, dont hesitate to search also for function name 
which is generally the name of the "unknown" function in a FE problem.
6) Dont hesitate also to search for similar problem, just swapping variables
(for example search also for FE "
" when you want to solve "
")
For example :
Searching for problem
:
+"P(x^2-1)" +"P(-x)"
(I avoid using +"P(x)P(-x)" which would miss
or 
)
Searching for problem
:
+"f(x+yf(x))" +"f(f(x))"
(I avoid including the +f(y) in the second item in order to find both f(y)+f(f(x)) and f(f(x))+f(y))
Searching for problem
:
+"f(x+y)+f(x)f(y)" +"f(xy)+f(x)+f(y)"
(I prefered using
instead of $g())
I received in PM some remarks about the fact that search function seems very often to give tons of results without any link with the searched problem. Here are some tips I frequently use :
1) use quotes for strings : search "P(x^2-1)" instead of P(x^2-1)
2) dont hesitate to use the character "+" which means "mandatory"
3) Split long strings in order to find problems where strings are not in the same order :
Instead of +"f(2x+f(y))=f(2x)+f(f(y))+xf(2y)", prefer +"f(2x+f(y))" +"f(2x)" +"f(f(y))" +"xf(2y)"
So you'll find
4) If the problem you are looking for contains strings as "2015" or "2016", try to exclude these strings from search term since this is likely an "annual problem" and it may have been posted wu=ith older years?
For example +"f(x-f(y)" +"f(x+y^" +"f(f(y)+y^" will find
as well as 
5) if the problem you are looking for uses function
or 
, dont hesitate to search also for function name 
which is generally the name of the "unknown" function in a FE problem.6) Dont hesitate also to search for similar problem, just swapping variables
(for example search also for FE "
" when you want to solve "
")For example :
Searching for problem
:+"P(x^2-1)" +"P(-x)"
(I avoid using +"P(x)P(-x)" which would miss
or 
)Searching for problem
:+"f(x+yf(x))" +"f(f(x))"
(I avoid including the +f(y) in the second item in order to find both f(y)+f(f(x)) and f(f(x))+f(y))
Searching for problem
:+"f(x+y)+f(x)f(y)" +"f(xy)+f(x)+f(y)"
(I prefered using
instead of $g())This post has been edited 7 times. Last edited by pco, Jan 26, 2021, 9:36 AM
Reason: Added tip 6
Reason: Added tip 6
