[/quote]
,
,
,
,
.
such that for every 
![\[
f_n(x) = \frac{x}{1 + nx}, \quad \text{if and only if } f(x) = \frac{x}{1 + x}, \quad \forall x \geq 0.
\]](http://latex.artofproblemsolving.com/0/f/c/0fc36d9264eb7e103128c489aeae521a859c1fd4.png)
(For 
represents 
.
be a triangle with circumcircle 
.
on 
respectively. Circumcircle 
of triangle 
intersects 
.
intersects 
at 
.
cuts 
respectively. Construct parallelogram 
.
are collinear.
that are continuous on 
satisfying
of integer numbers, we define 
as the sum of the elements of 
and 
whose intersection is empty, whose union is the set 
and such that the product 
is a perfect square.
,
and 
,
.
is assigned to a positive integer in 
,
,
.
,
'
colors, and for an arbitrary color 
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)
.
.
is the midpoint of 
.
cut 
at 
meets 
at 
and 
is the projection of 
onto 
.
lie on a common circle.
with integer coefficient that satisfy the following
for each 
be the midpoint of side 
,
is such that the quadrilateral 
is cyclic and the circumcircle of 
is tangent to the line 
is the second common point of the line 
and the circumcircle of 
.
and 
.
and 
meet at points 
as 
and 
all pass through a single point on the circumcircle of 
.
.![$(a+b+c)^4(ab+bc+ca) - 9\sum{\frac{a}{c}} \ge 54[(a+b)(b+c)(c+a)+abc-1]$](http://latex.artofproblemsolving.com/d/3/c/d3cf439c79cb84ca23083e51b6190ab5118f34e8.png)
Y by Derive_Foiler, Tintarn, lmht, rafayaashary1, nicky-glass, sansae, 62861, bobthesmartypants, bel.jad5, talkon, yassinelbk007, Python54, shinichiman, rkm0959, randomusername, Ankoganit, fattypiggy123, Med_Sqrt, pi_Plus_45x23, adityaguharoy, Filipjack, lminsl, artsolver, TheOneYouWant, Ghoshadi, Akatsuki1010, Lsway, Churent, v_Enhance, muradmurad, Gluncho, ydr202020, sheripqr, TomMarvoloRiddle, equinox145111, Vimath, Snakes, Shaddoll1234, Wizard_32, QWERTYphysics, AnArtist, saiii456, Supercali, Amir Hossein, gmail.com, MNJ2357, lkarhat, enthusiast101, NiltonCesar, opptoinfinity, AlastorMoody, harry1234, mathisreal, Centralorbit, Jerry37284, MathPassionForever, Pluto1708, Plasma_Vortex, BEHZOD_UZ, Durga01, Aryan-23, amar_04, Feridimo, AmirKhusrau, Mathmick51, OliverA, Antara_Dey_temporaryacc, Abidabi, p_square, Bumblebee60, FatherOfIngenuity, aa1024, Kar-98k, IAmTheHazard, fjm30, Fedor Petrov, AFSA, Aritra12, test20, matinyousefi, hsiangshen, TheBarioBario, iman007, lneis1, mathingbingo, Kanep, Functional_equation, arqady, Supernova283, GCA, teomihai, Pluto04, animath_314159, OlympusHero, tenebrine, ZHEKSHEN, mango5, megarnie, tigerzhang, Darkztar, EmilXM, RubixMaster21, centslordm, mrdriller, rayfish, laikhanhhoang_3011, pi_quadrat_sechstel, JUSTemom, qwertyboyfromalotoftime, myh2910, ghu2024, Quidditch, CyclicISLscelesTrapezoid, David-Vieta, IMUKAT, thedodecagon, Seungjun_Lee, math_comb01, cos111, FreDER, Adventure10, Fatemeh06, Mango247, Tellocan, Arslan, ehuseyinyigit
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
