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.
satisfies the conditions![\[\begin{cases}f(x+24)\le f(x)+24\\ f(x+77)\ge f(x)+77\end{cases}\quad\text{for all}\ x\in\mathbb{R}\]](http://latex.artofproblemsolving.com/5/5/6/556f545540d4941d058a1c4099b4b86eb35caf0d.png)
for all real 
.
such that
for all 
.
, there exists an arithmetic sequence 
of rational numbers, where 
are relatively prime positive integers for each 
such that the positive integers 
are all distinct.
![\[A = \left\{1+\frac{1}{k} : k=1,2,3,4,\cdots \right\}.\]](http://latex.artofproblemsolving.com/3/d/f/3dfb1d52835627a2977f0074f8df7e7ed677f2a5.png)
can be written as the product of one or more elements of 
, which are not necessarily different. let 
denote the minimum integer such that can be written as the elements of , which are not necessarily different.
of integers with 
, 
, and ![\[f(xy)<f(x)+f(y).\]](http://latex.artofproblemsolving.com/c/f/f/cff5554b8e72d99f8989448c2621794242d784d8.png)
(Pairs 
and 
are different if 
or 
).
denote the set of natural numbers, and let 
, 
, be nine distinct tuples in 
. Show that there are three distinct elements in the set 
whose product is a perfect cube.
satisfy the equality, 
prove that the triangle must have a right angle.
, let the diagonals 
and 
intersect at 
. Let the circumcircles of triangles 
and 
intersect again at 
. If is the incentre of triangle 
, show that 
.
be a positive integer. A graph on 
vertices is given such that the size of the largest clique in the graph is . Prove that there exists a vertex that is present in every clique of size 
, suppose 
. Let 
be the midpoint of 
, and let 
be a point inside the triangle such that 
and 
.
, Find 
. be a set of 
point (
is integer number, 
). On a plane take 3 points randomly which can form a triangle that has the area 
. Prove that: All points in set lie inside a triangle that has the area 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Let 
Prove that
Prove that
Where 
Prove that
Let 
Prove that
Let 
Prove that
be the unit circle in the complex plane. Let 
be the map given by 
. We define 
and 
for 
. The smallest positive integer such that 
is called the period of 
. Determine the total number of points in 
of period 
.
)
lie on side 
, 
on side 
, and 
on side 
of a triangle so that the following applies: 
, 
and 
. Construct the triangle if only the points 
and are given.
lie on side , on side , and on side of a triangle so that the following applies: , and . Construct the triangle if only the points and are given.
