[/quote]
,
,
be all positive integers smaller than 
such that 
for all 
is the largest positive integer that divides both 
and 
.
.
,![\[
A = \left\{ r \, \middle| \, r = \frac{a}{3}\left(\frac{1}{x} + \frac{1}{y} + \frac{1}{z}\right) + b\sqrt[3]{xyz}, \quad x, y, z \in \left[1, \frac{a}{b}\right] \right\},
\]](http://latex.artofproblemsolving.com/e/9/2/e92becb6264b8e5a9753f2288436a48523a25f29.png)
and![\[
B = \left[2\sqrt{ab}, a + b\right].
\]](http://latex.artofproblemsolving.com/2/d/2/2d25aa634389b768dc26408480bb71b316c0bf17.png)
.
be a set with seven elements, where each element is a positive integer not exceeding 
.
)![\[
x_1 + x_2 + \cdots + x_t = x_{t+1} + x_{t+2} + \cdots + x_m
\]](http://latex.artofproblemsolving.com/7/0/7/707e7e9a0ce45adb307545bbc88c1f0bf90ca0c9.png)
has a solution in the set 
,
are all distinct.
such that: 
.
satisfies the equation![\[
(AB + CD)^2 + (AD + BC)^2 = (AC + BD)^2,
\]](http://latex.artofproblemsolving.com/7/d/a/7daf2d6ac00609b5252b2b8b05d96a5ca0952a94.png)
then 
,
touch sides 
at 
,
lie on the line through 
perpendicular to 
.
be the intersections of 
with 
,
be the projections of 
onto line 
be the second intersections of 
with the incircle 
be the intersection of 
and 
.
bisects segment 
.
can we find three positive integers 
,
and 
such that 
?
satisfy both of the following conditions.
,
,
.
that satisfies the equation 
and 
.
,
,
-
,
.
at 
.
parallel to 
,
meets 
.
is constant.
.
be the feet of the altitudes from 
to 
respectively.
and 
as the reflections of 
and 
be the circumcircle of 
.
with 
the intersection of ray 
with 
is the circumcenter of 
can be inscribed within a circle.
and 
. Prove that
Where 
grid consisting of 2025 unit squares. Within each unit square is a car, facing either up, down, left, or right. If the square in front of a car in the direction it is facing is empty, it can choose to move forward. Each car wishes to exit the 
cars from the 
be defined by![\[
a_1 = 1, \quad a_{n+1} = a_n + \frac{1}{\sqrt[2024]{a_n}} \quad \text{for } n \geq 1, \, n \in \mathbb{N}
\]](http://latex.artofproblemsolving.com/8/8/f/88f423d87c08b20dc552703fcf30f2a8e9585902.png)
Prove that![\[
a_n^{2025} >n^{2024}
\]](http://latex.artofproblemsolving.com/a/6/8/a6806c817edb7ce29a399981137c1a3e93f63c63.png)
for all positive integers 
.
and 
Prove that
Where 
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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
