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 
.
)
on 
s.t. 
. Now let 
. The line 
passes through (because of well-known projective properties of the complete quadrilateral). Let 
. We have 
, and since is the midpoint of 
, we find 
, so is the midpoint of . Furthermore, we have 
, which is fixed, so 
is constant, meaning that 
lies on a circle obtained from 
by a homothety of centers 
and 
(one of them is the external center, the other one is the internal center).
, with directed segments. Let p be a number such that 
, and consider the circle P(D, p).
. On the other hand, let the parallel to the diamater MN through the point D meet the line BM at V. Then, Thales yields 
. Since , we have 
, or, equivalently, 
. Since MA = AN, this implies DU = DV. This implies that the points U and V coincide, i. e. the parallel to the diameter MN through the point D meets the lines CN and BM at the same point. This point, of course, must then coincide with the point of intersection K of the lines CN and BM. Thus, K = U = V. Therefore, the equation becomes 
. On the other hand, we know that 
; thus, 
. Now, AN = r; hence, DK = p, and it follows that the point K lies on the circle P(D, p), completing our proof., it follows that the points B and C are the two centers of similitude of the circles T and P.]
