AoPS Academy
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, do not answer with "
is a solution" only. Either you post any kind of proof or at least something unexpected (like "
is the smallest solution). Someone that does not see that is a solution of the above without your post is completely wrong here, this is an IMO-level forum.
we have![\[\max(|\sin x|,|\sin (x+1)|)>\frac13.\]](http://latex.artofproblemsolving.com/e/d/e/ede9b524370c457fe41038cca5663e3c9bdaf320.png)
of lattice points with positive coordinates in the plane. For each point 
from , we draw a segment between it and each of the points in the set ![\[S(P)=\{(a+b,c)\mid c\in\mathbb{Z}, \, c>a+b\}.\]](http://latex.artofproblemsolving.com/3/d/0/3d091c92ad9bf8644c59b77a1ad0209c41765093.png)
Show that there is no colouring of the points in with a finite number of colours such that every two points joined by a segment are coloured with different colours.
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 
.
in the plane of a square 
for which ![\[\max (PA, PC)=\frac1{\sqrt2}(PB+PD).\]](http://latex.artofproblemsolving.com/3/9/8/3984b1970837a85af0bac93fd711843630a1fd99.png)
, where 
is a prime number. Show that if 
is an even number, then the polynomial ![\[-1+\prod_{k=0}^{n-1} P\left(X^{p^k}\right)\]](http://latex.artofproblemsolving.com/5/d/f/5df80f885e63852e4f53a6145dafd18d63dc3526.png)
is divisible by 
.
then![\[\sum |\pm z_1\pm z_2\pm z_3|^2=2^3\sum_{i=1}^3|z_k|^2.\]](http://latex.artofproblemsolving.com/6/d/1/6d1cd28a47704aa0cac2a6c6b572816fc1db790a.png)
Generalize the problem.
and 
lying in parallel planes. Describe the locus of the midpoint of 
when 
varies along 
for 
.
the set of all positive real numbers.
which have the property:![\[f(x)f(y)=2f(x+yf(x))\]](http://latex.artofproblemsolving.com/4/f/f/4ffbde35ff082585a188c1a4368d39338d1b07a4.png)
and 
.
be the set of positive integers. Find all functions 
such that, for all 
:
, the following inequality holds:![\[
\sqrt{a + b} + \sqrt{b + c} + \sqrt{c + a} \geq \frac{4(ab + bc + ca)}{\sqrt{(a + b)(b + c)(c + a)}}
\]](http://latex.artofproblemsolving.com/e/7/2/e72fd0f134289c359bc8766440babc52ae131961.png)
be real numbers such that 
Prove that
Let be real numbers such that 
Prove that
satisfying the equation![\[
(f(x+y))^2= f(x^2) + f(2xf(y) + y^2), \quad \forall x, y \in \mathbb{R}.
\]](http://latex.artofproblemsolving.com/2/5/e/25eed911f303ed7cc4e03765a2f943ad4641c451.png)
.
as the midpoint of 
,
be a point on segment 
such that 
.
.
.
,
.
and 
.
.
.