[/quote]
,
be an equilateral triangle, 
be a point inside it, and 
be the intersections of 
with the sides of 
are the midpoints of 
,
,
,
,
and 
.
such that 
![\[f(x)+f(\lfloor x\rfloor)f(\{x\})=x,\]](http://latex.artofproblemsolving.com/6/4/1/6411452c9f27ac7a8a25141927ed2f34e810e6dd.png)
and draw its graph. Find all 
for which the equation 
has solutions for any 
.
be a fixed integer and 
.
and 
,![\[\sum\limits_{k=1}^n \dfrac{|a-r\omega^k|^2}{|a|^2+r^2}\]](http://latex.artofproblemsolving.com/e/e/3/ee3b0568b9a758cf22c23f96dbb7e7afe3601964.png)
is an integer. Interpet this result geometrically.
be two integers and 
be a positive integer such that 
.![\[(n-t)(d-t)>\dbinom{n-t+r}{r+1},\]](http://latex.artofproblemsolving.com/e/5/f/e5f9486f980870ff72858b57f7b50cd2aa9d142a.png)
for 
is the midpoint of the arc 
,
,
,
be the 
intersects 
.
is perpendicular to 
,
is the incenter of 
consisting of 
consecutive non-negative integers can be partitioned in two disjoint sets 
and 
each containing 
elements so that 
for 
such that 
and let 
.![\[\sum_{s\in A} M(s)=(n+1)^{k+1}-(1^k+2^k+\ldots +(n+1)^k).\]](http://latex.artofproblemsolving.com/7/6/e/76e7fce0da6bb9537b73d8699f7a21a6b8cb61fa.png)
with pairwise disjoint interiors. Determine the locus of the centre of the sphere intersecting each 
along a great circle of 
be a positive integer not divisible by 3. Prove that there are infinitely many positive integers 
such that 
and 
are divisible by 
is the sum of digits of 
.
be a prime number, and 
be a sequence defined by 
and 
,
.
terms of the sequence is not prime.
![\[\max(|\sin x|,|\sin (x+1)|)>\frac13.\]](http://latex.artofproblemsolving.com/e/d/e/ede9b524370c457fe41038cca5663e3c9bdaf320.png)
of lattice points with positive coordinates in the plane. For each point 
from ![\[S(P)=\{(a+b,c)\mid c\in\mathbb{Z}, \, c>a+b\}.\]](http://latex.artofproblemsolving.com/3/d/0/3d091c92ad9bf8644c59b77a1ad0209c41765093.png)
Show that there is no colouring of the points in 
in the plane of a square 
for which ![\[\max (PA, PC)=\frac1{\sqrt2}(PB+PD).\]](http://latex.artofproblemsolving.com/3/9/8/3984b1970837a85af0bac93fd711843630a1fd99.png)
,![\[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]](http://latex.artofproblemsolving.com/5/d/f/5df80f885e63852e4f53a6145dafd18d63dc3526.png)
is divisible by 
.
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)
.
,
.
,
.
,
,
.
is 
when 
or 
where 
is a natural number. 
(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 
