AoPS Academy
[/quote]
, 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.
, if any permutation of the first 
natural numbers are taken, there will be at least 
positions 
such that ![$\gcd(p[i],p[i+1])=1$](http://latex.artofproblemsolving.com/8/1/2/812f410bd64e18c74379c56468b9dea260548a0f.png)
.(Here, ![$p[i]$](http://latex.artofproblemsolving.com/0/b/6/0b6cc7e1108ab6b64f3a644a3b28891f826b3bea.png)
denotes the number at the 
position in the permutation).
is a factor of 
, find the sum of 
be the set of all rational numbers 
such that 
and , when written as a decimal, is in the form 
, where 
are (not necessarilly) distinct digits. Find the sum of all elements in . are all the same, this is not counted in the set .
Prove that
Let 
Prove that
Let 
Prove that![$$ a+b^2+c\geq \frac{135 \sqrt[3]{90}-2}{2}$$](http://latex.artofproblemsolving.com/8/6/6/866623c65f54d6147a43b1b241c2cba96863b624.png)
such that 
for all 
such that 
.
be a positive integer such that the sum of its positive divisors is at least 
. Prove that has at least 
distinct prime factors.
is known as an interesting number if satisfies![\[{\ \{\frac{n}{10^k}} \} > \frac{n}{10^{10}} \]](http://latex.artofproblemsolving.com/0/c/a/0ca11fa76313b65fd76a3cceee90955960e44ef6.png)
.
is given. Find all functions 
such that for all 
we have
be an acute triangle, 
its orthocenter and 
the center of its nine point circle. Let 
be a point on the parallel through 
to 
such that 
and and 
are on different sides of 
and 
a point on the parallel through 
to 
such that 
and and are on different sides of . If 
and 
are the reflections of over and respectively, and 
are the intersections of 
and 
respectively with the circumcircle of , prove that the intersection of lines 
and 
lies on 
. be an acute triangle and its orthocenter. Let be a point on the parallel through to such that 
. Define and 
as points on the parallels through to and through to similarly. If 
are positioned around the sides of as in the given configuration, prove that are collinear.
points on the plane, all pairwise distances between which are different. Is there always a polyline with vertices at these points, passing through each point once, in which the link lengths increase monotonously?
is defined by 
. Find all terms which are perfect squares.
of the pyramid 
are equal to 
, and , respectively the base of the ABC is a regular triangle. It is known that the area of the lateral faces of the pyramid are equal to each other. Find the volume of the pyramid .
of the pyramid are equal to , and , respectively the base of the ABC is a regular triangle. It is known that the area of the lateral faces of the pyramid are equal to each other. Find the volume of the pyramid .
Prove that
Prove that
Let 
Prove that![$$ (a+b)(a+1)(b+1) \leq \frac{3}{2}+\sqrt[3]{6}+\sqrt[3]{36}$$](http://latex.artofproblemsolving.com/f/5/e/f5e68204507aa9c3e71c28844bbee3a804546b94.png)
.
has sides 
,
has sides 
,
for convenience), we have 
,
,
.
,
.
times the square of the volume) of this purported tetrahedron to check if it positive. However, 
)