[/quote]
,
be a random point on the smaller arc 
of the circumcircle of square 
,
be the intersection point of segments 
and 
.
to the circumcircle of the triangle 
are 
and 
, where 
is the center of the square. Prove that points 
,
,
and 
such that
such that: 
such that for every 
is satisfied? Explain the answer.
be an integer, and let 
be positive real numbers such that 
.![\[(1 + a_2)^2 (1 + a_3)^3 \dotsm (1 + a_n)^n > n^n.\]](http://latex.artofproblemsolving.com/7/7/f/77fb719e975225b59c64f1f3aa102c98a6c25f50.png)
colors, there exists a monochromatic regular hexagon.
is the smallest number such that if 
,
with two colors there exists a monochromatic arithmetic progression of length 
.
.
and 
.
points in plane. Prove that the number of isosceles triangles having their vertices among these 
.
.
and 
,
denotes the sum of all positive divisors of 
denotes the largest prime divisor of 
be a polynomial with real coefficients.
.
cannot have 2024 real roots.
and 
.
for all real numbers 
grid, one can find 
points such that no four of them are vertices of a square with sides parallel to lines of the grid. Imagine yourself as Erdos (!) and guess what is the best exponent instead of 
!
denote the set of positive integers. Find all functions 
satisfying the condition
for all 
be a scalene triangle, 
its medial triangle, and 
)
).
be the circumcevian triangle of 
be the pedal triangle of 
be the reflection in 
of 
.
,
cyclically.
be the reflection in 
of 
,
cyclically.
be the reflection in 
of 
,
cyclically.
,
,
and the orthocentroidal circle of 
Y by
A positive integer 
is given, find the largest real number 
such that there exists a geometric sequence 
with common ratio satisfying
for all positive integers 
. Here, 
denotes the distance from the real number 
to the nearest integer.
is given, find the largest real number 
such that there exists a geometric sequence 
with common ratio satisfying
for all positive integers 
. Here, 
denotes the distance from the real number 
to the nearest integer.This post has been edited 1 time. Last edited by David-Vieta, Sep 8, 2024, 11:53 PM
,
is equivalent to 
,
.![$\{x\}=x-[x]$](http://latex.artofproblemsolving.com/6/1/0/61036a03415081af535bd57c408b7be79aae9458.png)
)
is odd, them 
is feasible for 
(the decimal part of all terms is 
)
and 
.
and 
,
is feasible. For the sake of contradiction, assume that 
.
,
(otherwise 
would be an integer).
If 
,
.
,![$[2t\{a_1\}]=t-1$](http://latex.artofproblemsolving.com/1/8/5/185b64ae286ac87edd91c21635ec80a86e3550d7.png)
.
,
.
,
.
,
,
.
when 
or 
where 
(which is maximal by the definition of 
)
.
,
is the most we can achieve.
,
and 
(when even it's 
on numerator and when odd it's 
,
so if we had that 
then we have that 
so 
notice if for some 
we had 
then 
contradicting that 
,
and notice that 
so we get that 
which implies that 
but we have that then 
,
,
.
and 
